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((js. ()5) == (., t1§) -
273
r:
method uses the underlying statistical structure of the problem to replace cumbersome or expensive numerical maximizations with analytical or simpler maximizations.
e, -
X
f(y, z: 0) = f(y I x; (}s )f(x; 0)
(2)
i.e., the conditional distribution f{y I x; Os) must be independent of () s- In other words, X s must be a complete-data space (in the sense of [1]) for (}s given that (}s is known. A few remarks may clarify this definition's relationship to related methods. • The complete-data space for the classical EM algorithm et al. [1] is contained as a special case of Definition 2 by choosing S = {1, ... ~ p} and requiring Y to be a deterministic function of -J'Y S [4]. • Under the decomposition (2), one can think of Y as the output of a noisy channel that may depend on () 5 but not on () s- as illustrated in Fig. 1. • We use the term "hidden" rather than "complete" to describe X s, since in general ..X" 5 will not be complete for () in the original sense of Dempster et ale [1]. Even the aggregate of _X" S over all of S will not in general be an admissible complete-data space for (). • The most significant generalization over the EM complete-data that is embodied by (2) is that the conditional distribution of Y on ..X" 5 is allowed to depend on all of the other parameters () 5 (Fig. 1). In the superimposed signal application described in Section IV, it is precisely this dependency that leads to improved convergencerates. It also allows significantly more flexibility in the design of the distribution of X 5 . • The cascade EM algorithm [28] is an alternative generalization based on a hierarchy of nested complete-data spaces. In principle, one could further generalize the SAGE method by allowing hierarchies for each X S .
c.
Fig. 1. Representing the observed data Yas the output of a possibly noisy channel C whose input is the hidden-data X 5 .
For 1) 2) 3) 4)
i
= 0, 1, ...
SAGE Algorithm
{
Choose an index set S = s'. Choose an admissible hidden-data space X S\ for () St. E-step: Compute ljJsl(()St;()'i) using (4). M-step: (5)
(6) 5) Optional'? Repeat steps 3 and 4.
},
where the maximization in (5) is over the set
(7) If one chooses the index sets and hidden data spaces appropriately, then typically one can combine the E -step and i\tl-step via an analytical maximization into a recursion of the form 1 = g5 (()i). The examples in later sections illustrate this important aspect of the SAGE method. Note that if for some index set S one chooses _Y 5 == Y, then for that S one sees from (3) and (4) that cb s (8s : 1)1) == <1>( ()5 ~ (}i_). Thus, grouped coordinate-ascent [271 is a special case of rhe SAGE method, which one can use with index sets S for which 4>(f)s~ O§) is easily maximized. Rather than requiring a strict maximization in (5), one could settle simply for local maxima [4], or for mere increases in ¢s, in analogy with GEM algorithms [1]. These generalizations provide the opportunity to further refine the tradeoff between convergence rate and computation per iteration.
r:
1
D. Choosing Index Sets
Algorithm
An essential ingredient of any SAGE algorithm is the following conditional expectation of the log-likelihood of X S :
QS(Os; 0)
.D-y
1/
B. Hidden-Data Space
To generate the functions ljJs for each index set S of interest, we must identify an admissible hidden-data space defined in the following sense: Definition 2: A random vector X 5 with probability density function f(x; 0) is an admissible hidden-data space with respect to Os for f(y; 0) if the joint density of X S and Y satisfies
S
= QS(Os;Os, »»
=6 E { log f(~J;.rS ; Os, -(}s) I Y == y; -} (}
=
J
(3)
f(x I Y = y;O) log f(x; (}s,Bs)dx.
To implement a SAGE algorithm, one must choose a sequence of index sets s', i == 0, 1~ .... This choice is as much art as science, and will depend on the structure and relative complexities of the E- and lvI-steps for the problem. To illustrate the tradeoffs, we focus on imaging problems, for which there are at least four natural choices for the index sets: 1) the entire image, 2) individual pixels, i.e.
Si
We combine this expectation with the penalty function:
¢s(Bs;O) £ QS(Os;O) - P(()s,7J s).
(4)
Let ()o E e be an initial parameter estimate. A generic SAGE algorithm produces a sequence of estimates {()i}~o via the following recursion: 274
= {I + (i modulo p)}
(8)
2 Including the optional subiterations of the E- and M -steps yields a "greedier" algorithm. In the few ex.amples we have tried in image reconstruction, the additional greediness was not beneficial. (This is consistent with the benefits of under-relaxation for coordinate-ascent analyzed in [29].) In other applications however, such subiterations may improve the convergence rate, and may be computationally advantageous over line-search methods that require analogous subiterations applied directly to ep.
(this was used in the IeM-EM algorithm of [18]), 3) grouping by rows or by columns, and 4) "red-black" type orderings. These four choices lead to different tradeoffs between convergence rate and ability to parallelize. A "red-black" grouping was used in a modified EM algorithm in [15] to address the M -step coupling introduced by the smoothness penalties. However, those authors recently concluded [16] that a new simultaneous-update algorithm by De Pierro [17] is preferable. Those methods use the same complete-data space as in the conventional EM algorithm for image reconstruction [3], so the convergence rate is still slow. Since the E-step for image reconstruction naturally decomposes into p separate calculations (one for each element of (}), it is natural to update individual pixels (8). By using the less informative hidden-data spaces described in Section III, we show in [8] and [30] that the SAGE algorithm converges faster than the GEM algorithm of Hebert and Leahy [10], which in tum is faster than the new method of De Pierro [17]. Thus, for image reconstruction, it appears that (8) is best for serial computers. As noted by the reviewers, for image restoration problems with spatially-invariant systems, one can compute the Estep of the conventional EM algorithm using fast Fourier transforms (Ff'T's). A SAGE algorithm with single-element index sets (8) would require direct convolutions. Depending on the width and spectrum of the point-spread function, the improved convergence rate of SAGE using (8) may be offset by the use of direct convolution. A compromise would be to group the pixels alternately by rows and by columns. This would allow the use of 1-0 Ff'T's for the E -step, yet could still retain some of the improved convergence rate. Nevertheless. the SAGE method may be most beneficial in applications with spatially-variant responses. Regardless of how one chooses the index sets, we have constructed cps to ensure that increases in s lead to monotone increases in
inequality [1], one can easily show that
HS(Bs;B) ~ H S(Bs;8),
(lr+ 1 )
==
J
j(x I Y
(Os , 7} 5)
WS(B)
(9)
where
- ~(B)
(cPS
(os; 8)
- H S (71 5 ; 0)).
F. Convergence For a well-behaved objective <1>, the monotonicity property ensures that the sequence {f)i} will not diverge, but it does not guarantee convergence even to a local maximum of <1>. (Some EM algorithms have fixed points that are not local maxima [ 1J, [31].) Therefore, in the appendices we provide additional theorems that give sufficient conditions for convergence in norm, and that characterize the asymptotic convergence rate. To summarize briefly, these theorems show under suitable regularity conditions that:
• If a SAGE algorithm is initialized in a region suitably close to a local maximum in the interior of 8, then
the sequence of estimates will converge monotonically in norm to it. (This may not apply when the local maximum lies on the boundary of e, as often happens in the example in Section III.) • For suitably regular objectives, the region of monotone convergence in norm is guaranteed to be nonempty [43~. • The asymptotic convergence rate of a SAGE algorithm will be improved if one chooses a less informative hiddendata space.
This last point is subtle, but is perhaps one of the most important conclusions of our analyses since it emphasizes the need for careful design of the hidden-data spaces. Less informative hidden-data spaces yield faster convergence, but more informative hidden-data spaces may yield easier M -steps
[5], [8], [30].
(10)
W S (7J) £
((P) ~ ¢S((}~+l;(P) _ ¢S(()~;fP).
Thus, if ¢s (f) S; 0) ~ ¢s (lJ 5; 0), then ( ()5, (js) ~ <1>(0) using (11). The results then follow from the definition of the SAGE algorithm. 0 Standard numerical methods require evaluation of (B i + 1 ) - ((}i) to ensure monotonicity. That requirement is obviated for SAGE methods by the monotonicity theorem above.
H 5 ( Os; 0) ~ E {log f (-,y slY == y: f)5 , (j 5) I }T == y; (j} and due to (2)
-
== cPs (() 5; (1) - H S (f) 5; lJ) -
= y;7J) logj(x; 6s ,7J s )dx
ue., 8s ) + HS({}s; 8) -
(11)
Proof' From (4) and (9) it follows that
Let Sand X S respectively denote an index set and hidden data space used in a SAGE algorithm. Under mild regularity conditions [1]. [4], one can apply Bayes' theorem to (3) to see that
=
V8
from which the following theorem follows directly. Theorem 1: Let ()t denote the sequence of estimates generated by a SAGE algorithm (5). Then 1) <1>( ()i) is monotonically nondecreasing, 2) if {) maximizes , then {) is a fixed point of the SAGE algorithm, and 3)
E. Monotonicity
QS(Bs;B)
'riBs,
J
III. EXAMPLE 1 LINEAR POISSON MEASUREMENTS
j(x I Y = y;B)logj(y I x; Bs)dx.
Note that W S is independent of Os, so it does not affect the maximization (5). Using these definitions and Jensen's
275
The EM method has been used for over a decade to compute ML estimates of radionuclide distributions from tomographic data, such as that measured by positron emission tomography (PET) [3], [32]. In this section we present a brief review of the
classical EM algorithm for this problem, and then introduce two SAGE algorithms. The second SAGE algorithm is based on a new hidden-data space, and converges faster than even an accelerated EM algorithm. For simplicity we focus in this paper on ML estimation; the penalized version is described in [8] and [30]. Assume that a radionuclide distribution can be discretized into p pixels with emission rates A [AI, ... , Ap ]' . Assume that the emission source is viewed by N detectors, and let N nk denote the number of emissions from the kth pixel that are detected by the nth detector. Assume the variates N n k have independent Poisson distributions:
=
(12)
For this complete-data space, the Q function (3) becomes (see (4) of [3]) N
Ql(A; Ai)
=L
L p
Maximizing Q1(.; Ai) analytically leads to the algorithm which follows. ML-EM Algorithm for Poisson Data
for i = 0,1, ... {
L ankA~ + p
Yn :=
Yn = LNnk+Rn
for
k=l
k
=
Poisson{r n }
pOisson{tankAk+rn}.
ek
=L
(17) (14)
Given realizations {Yn} of {Yn } , the log-likelihood- for this problem is given by [3]:
=L lV
(-Yn(A)
n=l
+ Yn logYn(A))
where p
Yn(A) =
L
ankAk
k=l
+ Tn·
We would like to compute the ML estimate ~ from y. To apply coordinate ascent directly to this likelihood, one might try to update Ak by equating the derivative of the likelihood to zero
O=
-a.k
+
N ~ ~ank n=l ank(Ak -
Yn.
"
Ak) + Yn(A 1 )
ankYn/Yn
n=l
k=l
log f(y; A)
= 1, ... ~ N
IV
(13)
with means {Tn} assumed known for simplicity. Thus I"V
n
I. .... p {
where {Rn} are independent Poisson variates
Yn
r«.
k=l
p
I"V
+ N nk log(ankAk))
where [3]
where the ank are nonnegative constants that characterize the system [3]. The detectors record emissions from several source locations, so at best one would observe only the sums 2:~=1 N.",k, rather than each N n k . Background emissions, random coincidences, and scatter contaminate the measurements, so we observe
u;
(-ankAk
n=lk=l
}.
}
In words. the previous parameter estimate is used to compute predicted measurements, those predictions are divided into the measurements and backprojected to form multiplicative correction factors, and the estimates are simultaneously updated using those correction factors. This EM algorithm converges globally (3], [5], but slowly. The root-convergence factor is very close to 1 (even if p = 1 [5]), since the complete-data space is considerably more informative than the measurements [5], [8], [30]. We now derive two SAGE algorithms for this problem, both of which use individual pixels for the index sets: Si = {k}, where k = 1 + (i modulo p). The most obvious hidden-data for Ak is just
(15)
2::=1
where a.k = ank. Unfortunately, this equation has no analytical solution. A line-search method would require multiple evaluations of (15), which would be expensive-hence the popularity of EM-type algorithms [3] that require no line searches. The complete-data space for the classical EM algorithm [3] for this problem is the set of unobservable random variates (16)
276
which is a subset of the classical complete-data space (16). The QSk function for the kth parameter is:
QSk (Ak; Ai) =
L (-ankAk + N nk log(ankAk))' N
n=l
Maximizing QSk (.; Ai) analytically yields the following algorithm:
ML-SAGE-l Algorithm for Poisson Data
+ k Tn,
. l lze ' ,,",P ,0 Ln i. t La : -Yn=L.k=lank/l
f or i = 0,1 , . . . { fo r k = 1, . . . , p {
Maximum Likelihood
n= 1 , . . . , N .
6
5.8 5.6 N
ek =
L UnkYn/Yn
"8 5 .4
:5
n=1 ,/lki+1 = /lkek , i / a.k .\i+1 ) = N) ' J' r~ k
-Yn := -Yn + (,/\ki+1 - /\k 'i) ank ,
}.
~5.2
::i
(18)
8'
....I
4.8
vn :ank r
~o
\.J
5
lIE
ML-5AGE·2
o
ML-LINU
X
ML-EM
4.6
}
4.4
This SAGE algorithm updates the parameters sequentially, and immediately updates the predicted measurements Yn within the inner loop, whereas the ML-EM algorithm waits until all parameters have been updated . ML-SAGE-I is the unregularized special case of the ICM-EM algorithm of [18] ; a local convergence result for ICM-EM was mentioned in [ 18]. We found that ML -SAGE-I converges somewhat faster than \1L -EM for well-conditioned problems. but the difference is nin imal for poorly-conditioned problems . The reason is that X S • is still overly informative since the background events are isolated from the parameter being updated (cf (12) and (13)) [8], [30] . Therefore. we now introduce a new. less informative hidden-data space that associates some of the uncertainty of the background events R n with the particular parameter .\k as it is updated [8], [30]. Whereas the ordinary complete-data space has some intuitive relationship with the underlying image formation physics. this new hidden-data space was developed from a statistical perspective on the problem and its Fisher information. First. define
and define unob servable independent Poisson variates
Znk
"oJ
Poisson{an k(Ak + Zk)}
e., '" pOisson{r
n
-
I.L nkZk
+L
Iln} .\j }
(19)
o
10
15
20 iteration
25
30
3S
40
Fig. 2. Comparison of log-likelihood increase log f ( y; 8') - log f (y : 1J0 ) versus iteration i for ML-EM . ML-LINU . and ML-SAGE-2 algorithms. for image reconstruction from PET measurements with 9% random coincidences. ML-SAGE-2 clearly reaches the asymptote sooner.
derivation as in [3] (see [8] and [30] for details), one can show
where
Znk
= E {Znk I Y = y ; Ai } = (Ak + zk )ankYn /Yn (Ai ).
Maximizing Q S' (-: Ai ) analytically (subject to the nonnegativity constraint) yields the ML-SAGE-2 algorithm, which has the same sequential structure as ML-SAGE-I , except that (18) is replaced by: A~T l := max { (Ai, + zk)e k/ a.k - Zk, o].
Provided Zk i= 0, which is always the case in PET since random coincidences are pervasive, this remarkably small modification yields significant improvements in convergence rate . The Fisher information for the classical complete-data space with respect to A is diagonal with entries
a' k /~k
r#
and let the hidden-data space for Ak only be
provided the ML estimate ~ is positive . In contrast, the Fisher information for the new hidden-data space is diagonal with entries
Then, clearly
has the appropriate distribution (14) for any particular k. We have absorbed all of the background events into the terms Z"k and B"k which are associated with Ak. Thus . the aggregate of all p of the hidden -data spaces is not an admissible hiddendata space for the entire parameter vector A. Using a similar
which is clearly smaller since Zk > 0. The improved convergence rate of ML-SAGE-2 is closely related to this difference. To illustrate, Fig. 2 displays the likelihood 4>( Bi ) versus iteration for the ML-EM algorithm and for ML-SAGE-2 applied to a simulation of PET data . The image was an 80 x 110 discretization of a central slice of the digital 3-D Hoffman brain phantom (2 mm pixel size) . The sinogram size was 70
277
radial bins (3 nun wide) by 100 angles. A 900000-count noisy projection was generated using (6-mm-wide) strip-integrals for {ank} [29], including the effects of nonuniform head attenuation and nonuniform detector efficiency. A uniform field of random coincidences was added, reflectin~ a scan with 9% of the total counts due to randoms (i.e., ~'n=1 Tn ~ 0.12::=1 Yn(A)), a typical fraction for a PET study. Further details can be found in [8] and [30], including comparisons over a large range of Tn'S. Also shown in Fig. 2 is the LINU unbounded line-search acceleration algorithm described by Kaufman [23]. The ML-SAGE-2 likelihood clearly increases faster and reaches its asymptote sooner than both the ML-EM and ML-LINU algorithms.' (ML-SAGE-2 was also considerably easier to implement than the bent-line LIND method.) The convergence in norm given by Theorem 3 of Appendix A is inapplicable to this Poisson example when the ML estimate has components that are zero, Le., when the ML estimate lies on the boundary of the nonnegative orthant [33]. See [30] for a global convergence proof for ML-SAGE-I and ML-SAGE-2 similar to the proofs in [3] and [17]. The reader may wonder whether one can also find a better complete-data space for the classical EM algorithm. Because the EM update is simultaneous, one must distribute the background events among all pixels; therefore, the terms Zk are reduced by a factor of roughly [8], [30]. Since Vp is in the hundreds, the change in convergence rate is insignificant, which is consistent with the small reduction in Fisher information [8], [30]. Other simultaneous updates (17] similarly do not improve much [30]. Apparently one benefits most from this less informative hidden-data space by using a SAGE method with the parameters grouped into many small index sets. An alternative to SAGE is the coordinate-wise sequential Newton-Raphson updates recently proposed by Bouman and Sauer [19]. That method is not guaranteed to be monotonic, but when it converges it might do so somewhat faster than SAGE since it is even greedier. One can obtain similar (but monotonic) greediness by using multiple subiterations of the E- and M-steps in the SAGE algorithm, as indicated by Step 5 of the generic SAGE algorithm. However, for the few cases we have tested, we have not observed any improvement in convergence rates using multiple subiterations. Although further investigation of the tradeoffs available is needed, including comparisons with possibly superlinear methods such as preconditioned conjugate gradient [23], [34], it appears that the statistical perspective inherent to the SAGE method is a useful addition to conventional numerical tools.
vp
analysis of the convergence rates. In this section, we analyze the problem of estimating superimposed linear signals in Gaussian noise [2], [9]
where A = fa1 ... ap ] , and f. is additive zero-mean Gaussian noise with covariance II, Le., f. N(O, II). For simplicity we consider a quadratic penalty P(O) = ~O'PO, so the penalizedlikelihood objective function is f'"V
- cll(B)
= ~(y -
AB)'JI-l(y - AB)
+ ~B'PB.
Such objective functions arise in many inverse problems [9]. We assume A has full column rank, P is symmetric nonnegative definite, and the intersection of the null spaces of P and A is empty, in which case the (unique) penalizedlikelihood estimate is
fJ
= (A'rr- 1A + p)-l A'rr-1y.
(21)
If A is large, or if positivity constraints on () are appropriate, then (21) is impractical and iterative methods may be useful. (One can also think of (20) as a linearization of the more interesting nonlinear problem [2].) We present the linear version here since we can derive exact expressions for the convergence rates. We first present admissible hidden-data spaces for this problem, derive EM and SAGE algorithms, and then prove that the SAGE algorithm converges faster. Since the mean of Y is linear in 0, the conventional complete-data [2], [9] for the EM algorithm for this problem is also linear in f}. Here. we restrict our attention to hidden-data spaces ~yS whose means are also linear in f}, and for which the conditional mean of Y given X S is linear in ~Y 5 and OS' Considering a general index set 5, the natural hidden-data space for () 5 is
. ~ ~ ./V(BO s + BO s' C) S
YIX
5
-
=X~jV(Gx+G(}s~W)
which is admissible provided the two normal distributions are independent and consistent with (20), i.e., As = GB, As = GB + G, and II = W + GCG'. The log-likelihood for ~~s is given by
IV. EXAMPLE 2 LINEAR GAUSSIAN MEASUREMENTS
The Poisson problem has important practical applications, but the nonlinearity of the algorithms complicates a formal 3 Fast convergence is clearly desirable for regularized objective functions, but we advise caution when using "stopping rules" in conjunction with coordinate-based algorithms for the unregularized case, since for such algorithms the high spatial frequencies converge faster than the low frequencies [26].
where Cl and C2 are independent of Os. By standard properties of joint normal distributions
278
X
S
= E{X s I y" == u: Oi} = B(}~
+ B(}§ + CG'II-1(y -
A()i).
The ¢s function of (4) is thus i
c/>s(8 s; 8
)
SAGE Algorithm for Superimposed Signals
= (B 8s + B8§)'C- 1 ( X 1
- 2
[0 s ] 8§
I
[pP'2P3
1P 2 ]
S
-
Initialize: for i == 0,1, ...
~(B8s + B8§)),
k == 1 + (i modulo p),
[B s ]
S == {k}, O~+l :== (a~rr-lak + Pkk)-lpkO i + ( ak'IT-l ak + P kk )-1 ak'IT"E " . . (oi+l (}ik ) o,k, E :== E + k -
01 -
8§ + C2
which, maximized over Os, yields the generic combined Eand M -step
85+ 1
= (Fxs + Pd-1[B'C-1(X S -
fJ ki +
B8§) - P 28§]
== fJ~ + (F};s + Pl)-lA~rr-l[y - AtJl] - (F.ys + Pl)-1[P 1 P2]Oi
(22)
A. EM Algorithm
The ordinary EM algorithm [2], [9] is based on the following choices for the complete-data space: S == {1..... P}, ~ == diag{ak}, C == ~Ip I~ IT, G == (l/p.~ I p), and B ==
+
(ll~rr-lak
+ Pkk/p)-l(l~rr-l£LJ,J)~
ABl)
(23)
/ H==LH+DH+L H
B. SAGE Algorithm Because of the additive form of (20)~ it is natural for the SAGE algonthm to update each parameter Ok individually. i.e., S' == {k} where k == 1 + Ci modulo p). In light of the discussion in Appendix C. we would like the Fisher information of the hidden-data space for () k to be small, so we associate all of the noise covariance with the signal vector (Lk f'
r-.J
}v
(24)
where D H is a diagonal matrix with the diagonal entries of H, and L H is a strictly lower triangular matrix. Similarly. let
F == A'Il-1A == L F + D F + L~. DH==DF+D p
where D p == diag{ P k k } and F is the Fisher information for y. with respect to f}. Let II x II denote the standard Euclidian norm of a vector r, and for a nonsingular matrix T define !I~rjlT == IIT.rll, which induces the matrix nonn
IIAIIT =
Sk
+ 1~ ... ~p.
To establish convergence of the EM and SAGE algorithms, we use Definition 3 and Theorem 3 of Appendix A. A few definitions are needed. Let H == A'rr- 1 A + P be the Hessian for this problem, and decompose it by
for k == 1..... p.
.Y
1, k
C. Convergence
G == W == 0, where diag l} denotes a diagonal or blockdiagonal matrix appropriately formed. Ip denotes the p vector of ones. and :~ is the Kronecker matrix product. Note that these choices distribute a fraction 1.p of the noise covariance II to .ach signal vector ak. Thus, F.\ == p diag{a~I1-1(Lk}. which being a diagonal matrix is easily inverted. However. since S == {1..... p}, the penalized EM algorithm (22) requires inversion of F.\ + P. which could be just as difficult as inverting i\'TI- 1 A + P for a general P.Therefore. we consider the case where P == diag{ Pk k } . for which the EM algorithm simultaneouslv updates all parameters via + p kk / p)·-1 akIIJ-l(· .l/ -
e.j' J"-- 1, ... , k -
where P kk is the kth diagonal entry of P, and P k is the kth row of P. Note that unlike the EM algorithm, the SAGE algorithm circumvents the need to invert P by performing a sequential update, so a nondiagonal smoothness penalty P is entirely feasible.
where F.\s == B/C-1B is the Fisher information of ..~s for
iJl+l == -1.,\ akIrr- 1ak P
l . '-
}
Os·
ok
t==y-AOO
rn;x II~~~T
=
IITAT-ill.
In addition. let p( A) denote matrix spectral radius. the maximum magnitude eigenvalue of A. SAGE Algorithm: From the SAGE algorithm given above. one can show (cf. proof of Theorem 3) that g(i+l)p _ {)
== M p
.....
M
1
.
(Oi p
-
(25)
(})
where
M, == I - ekH;;klek/H~ = H- 1 / 2 Hl/2ek(Hkk)-lek'Hl/2)Hi/2,
(I -
(a k () k:» IT)
Y ==.oX" s' + ~ ~aj()j. J=I=k
Thus, F.\_.,.k == a~II-lak, which is p times le~s informative than the EM case. which associates only a fraction 1/p of the noise covariance with each signal. (This provides a statistical interpretation of the modified EM algorithms in [35] and [36].) The above choice for the hidden-data space corresp,?nds to B == o,k, C == IT, B == W == 0, G == I. and G == [al ... ak-l ak+l ... a p ], which, substituted into (22), yields the following algorithm. 279
== T- 1 (I - tk(t~tk)-lt~)T, == T-1ptT
pt
and where T = H 1j 2 , the kth column of T is tk, is the orthogonal projection onto tk, and ek is the kth unit vector of length p. Since an orthogonal projection is nonexpansive, IIM k liT ~ 1, which confirms condition 2 of Definition 3. To confirm condition 3, rewrite the SAGE algorithm using (24) as fjC'i+l)p _
0 == [1 - (D H + L H )-1 HJ((}ip
-
0)
which is the Gauss-Siedel iteration (see p. 72 of [37]). Condition 3 follows from p. 109 of [37] since
III-
(DH
+ L H )-lHIIT
= liMp'"
MIII T
< 1.
where x = H 1/ 2 v. Rearranging and multiplying both sides by x'
EM Algorithm: One can use (21) and (23) to show that (Ji+ 1 - {)
=M
. ((Ji
-
IIxl1 2 =
0)
+ P)-lH.
X'DHX
Thus, the EM algorithm is closely related to the simultaneous overrelaxation (JOR) iteration (p. 72 of [37]). To establish that IIMIIT < 1 for T = H 1 / 2 using Theorem 4, we must show that S + S' > H, where in this case S == pDF + P. Since H == LF + D F + L' r + P and P ~ 0, it suffices to show that pD r > LF + D r + L' F, or equivalently that pI > L + I + where L = Dpl/2LpDpl/2. Since A'rr- 1 A
is positive definite by assumption, x' (L + I + [')x > 0 for any nonzero z; therefore, using x == ej ± ek, we see that I i j E (-1,1). Thus, for any nonzero x, x'(E + 1+ r')x < (Lk I Xk 1)2 ~ pllxl1 2 , where the second inequality is a special case of Holder's inequality. The result then follows. We have thus established that both the EM and SAGE algorithm converge globally. The convergence is globally monotonic in norm with respect to the norm T = H 1 / 2 , i.e., R+ is all of lRP •
= p(I -
PEM
= p(I -
(pDp + P)-lH) == p(1 - ((p - I)D F + DH)-lH) == p(I - (DH + LH)-lH)
(26)
(27)
PEM
~
(Ilxll~
-
x'D Hx)/2.
l)D F + D H )- 1/ 2z 11 2 IIz11 2
1 - 1I((p - l)DF
= 1_
IIxll 2
+ DH)1/2H-l/2xII2 2 IIxII
x'[(p - 1)H-l/2DpH-1/2
> 1- IIxll
+ DH]x
2
X'DHX
= 1 - (1 +
>u =
V)~(l _ v)
= (1 ~ v )
u
PSAGE
where the last inequality follows from v E [O~ 1). 0 The inequalities in this proof are rather loose. and often the difference in convergence rate between EM and SAGE is more dramatic than the proof might suggest. To illustrate. consider the case where P = O. Then returning to (25), for the EM algorithm we have
< PEM < 1.
= x'L~x =
+ DH )-lH) + DH )-1/2H
for any z (by definition of spectral radius). In particular, for z = ((p - I)D F + DH )1/2H- 1 / 2 x :
Proof' The right inequality follows from PE:\t1 ~ IIMIIT < 1. From (24), 1 = LH + D H + L~ where L H = H- 1/ 2L HH - 1/ 2 and D H = H-l/2D HH --1/ 2. Thus, for any vector x
x'LHx
((p - l)DF
IIH 1/ 2 ((p -
~ 1-
since P = D p for diagonal P. Theorem 2: For linear superimposed signals in Gaussian noise with a diagonal penalty matrix, the SAGE algorithm asymptotically converges faster than the EM algorithm, i.e. PSAGE
I-v
x ((p-1)Dp+D H )- 1/2)
-
To compare the root-convergence factors of EM and SAGE. we focus on the case where P is diagonal, since otherwise the EM algorithm is in general impractical. Therefore, from the results above
= 1 + v llxlI 2 .
= p(1 - ((p - l)D F
D. Convergence Rates
PSAGE
+ DH)X.
By the invariance of eigenvalues under similarity transforms
r,
PE~1
(1 - v)x'(L H
Combining with (28)
for the EM algorithm, where (cf. (37))
M == I - (pDp
thus
(28)
Since I - G -1 H is similar to the real symmetric matrix 1 - G- 1 / 2 H G - l / 2, the eigenvalues of 1- (D H + L H )-lH are real. For u = PSAGE E [0,1) there exists v :I 0 such that
[I - (D H + LH)-lH]v = vu 280
M =
~
tM =
P k=1
k
T-
1
(~tPt)T. P k=l
Since eigenvalues are invariant to similarity transforms, it follows that root-convergence factors for the two algorithms are given by the spectral radii PEM
PSAGE
= p(~ p
=P
t Pt), k=l
(Ii Pt) k=l
i.e., for the EM algorithm we have a convex combination of orthogonal projections and for the SAGE algorithm we have the product of those projections. Thus, this SAGE algorithm is closely related to the method of alternating projections [38], [39]. In particular, if P = 0 and the columns of A are orthogonal, then PSAGE == 0 whereas PEM 2: 1 - lip, i.e.,
1,....--....---.,...--....-----.,,----
demonstrated that SAGE algorithms yield faster convergence than EM algorithms in two signal processing applications. The particular SAGE algorithms that we presented in this paper sacrifice one important characteristic of the EM algorithm: they are less amenable to a parallel implementation since they are coordinate-wise methods. However, the general SAGE method is very flexible, and work is in progress on more parallelizable algorithms using index sets S consisting of several elements of () [30]. The benefits of parallelization must be weighed against the convergence rates for each application. It is probably no coincidence that the applications we put forth are ones in which the terminology "incomplete-data" and "complete-data" as introduced in [1] are somewhat unnatural. In most of the statistical applications discussed in [I], there is a clearly identifiable portion of the data that is "missing," and hence one natural complete-data space. In contrast, there is nothing really "incomplete" about tomographic measurements; the problem is simply that the log-likelihood is difficult to maximize. The EM algorithm is thus just a computational tool. (To further illustrate this point, note that in classical missing data problems the estimates of the missing data may be of some intrinsic interest, whereas the "complete-data" for tomography is never explicitly computed and would be of little use anyway.) SAGE algorithms may be most useful in such contexts. We have emphasized that the SAGE algorithm improves the asymptotic convergence rate. The actual convergence rate will certainly depend on how close the initial estimate is to a fixed-point. In tomography and image restoration, fast linear algorithms can provide good initializers for penalized likelihood estimation. A greedy algorithm like SAGE is likely to be most beneficial in applications where such initializers are available.
0.9
•
Conventional EM
0.8
1 ...
0•7
~0.6
c
4D
~0.5
a0.4 4D
> u
~
80•3
a:
0.2
SAGE Algorithm
0.1
0'--.......---......- ----------.......- -......- - - - ' o 0.2 0.4 0.& 0.8 1 cos(Complementary Angle 8etw"n Sub_paces) Fig. 3. Comparison of root-convergence factors for conventional EM algorithm and proposed SAGE algorithm versus complementary angle between subspaces of superimposed signals. The SAGE algorithm has a significantly improved convergence rate.
the SAGE algorithm converges in one iteration, while EM converges very slowly. When p = 2, using a Gram-Schmidt argument one can show that t 1 = [1 0]' and t2 == [Q~)' where a = la~II-la21/(llallllla211) is the cosine of the complementary angle between al and a2. Thus PSAGE
= P(
[
~ ~])
=0 2
PE~I = p(~ [-~;l ~
(12
-a~]) 1+a2
_~ + ::
-
2
2'
Fig. 3 illustrates that the root-convergence factor of SAGE is significantly smaller than that of EM, which substantially reduces the number of iterations required. Not only is PSAGE < PEM, but also PSAGE < P~M' so one SAGE iteration is better than two EM iterations, at least when p = 2. Thus, even though the EM algorithm appears to have the advantage that one can parallelize the M -step using p processors that simultaneously update all parameters, in this case the convergence rate of the parallel algorithm is so much slower that a sequential update may be preferable. This depends, of course, on how difficult the M -step is; in the nonlinear case discussed in [2], the Mstep is presumably fairly difficult, so parallelization may be advantageous. Equations (26) and (27) help one examine these types of tradeoffs. V.
DISCUSSION
We have described a generalization of the classical EM algorithm in which one alternates between several hidden-data spaces rather than using just one, and updates only a subset of the elements of the parameter vector each iteration. By updating the parameters sequentially, rather than simultaneously, we
ApPENDIX A
MONOTONE CONVERGENCE IN NORM
Because the SAGE "algorithm" is so general, a single convergence theorem statement/proof cannot possibly cover all cases of interest (see, for example, the variety of special cases considered for the classical EM algorithm in [40].) Here we adopt the Taylor expansion approach of [4] since it directly illuminates the convergence rate properties and prescribes a region of monotone convergence in norm, However, this general approach has the drawback that it assumes the fixed point lies in the interior of e. This restriction is often not a necessary condition, and at least for some applications one can often find specific convergence results without the restriction, e.g., [3] and [30]. Readers who are satisfied with the assurance of monotonicity of the objective 4?((Ji), as provided by Theorem 1, may wish to simply skim this Appendix. For simplicity, we discuss only the case where the index sets s' are chosen cyclically with period K, i.e., Si = Sk where k == 1 + (i modulo K). We also assume that U~=l Sk == {1, ... , p} so that each parameter is updated at least once per cycle. Before stating the convergence theorem, several definitions are needed. Consider an index set S, and let m denote its
281
cardinality. Bearing in mind our notational convention that cPs (() S; 7J) cPs (() s: (j s, 7Js), we define the m x m matrices
=
Theorem 3: Assume i) Si = Sk where k = I+(i modulo K) and U:=l Sk = {I, ... ,p}, ii) iJ is a fixed point of the SAGE algorithm (5) in the interior of 8, iii) for all 7J E R+ the Sk maximum over ()Sk of cjJSk ((}SIc; 8) is in the interior of 8 (0), iv) s» ; 6) is twice differentiable in both arguments V8 E e Sk and V(}Sk E 8 (8), and v) the region of monotone convergence R+ for a nann II . liT is nonempty. 1. If E R+ then
«: (()
and
eo
and the m x (p - m) matrix
110 + 1 1
-
OIlT ~ IIIP - OIlT
Vi
and where \7 denotes the (row) gradient operator and \7' its transpose. Let {} be a fixed point of the SAGE algorithm, and define
1I()(l+l)K -
-1 1 1
('V 200¢S)(tf)s
+ (1 -
t)Os; to
V ((}s; 0) 1 = ('VllO¢S) (tf)s
+ (1 -
t)Os; to + (1 - t)O)dt
S
+ (1 -
t)O)dt
(29)
(30)
WS(f)s; 0) = 1\'V 101¢S)(tf)s
+ (1- t)Os; to + (1- t)O)dt. (31)
Let R S denote the p x p permutation matrix that reorders the elements of {S, S} into {I, .... p}. Then define the p x p composite matrix
MS(Os; 8) == R S [U S ( 0s: 0)-1 [Y S (
eS: (j) W S ( ()S; 0)] ] (R S )' Ip-
O(p-m)xm 11,
x
11,
(32)
m
(JIlT
(34)
p(MSK(OSK;B) ... MSI(Bsl;B)) which is bounded above by
and
where In denotes the In addition, define
~ QIIOll< -
where a < 1 is defined by (33). Therefore, IIBll< - 0IIT converges monotonically to zero with at least linear rate. 2. The root-convergence factor [41] of the subsequence {e,K } ~o is given by the spectral radius
US((}S; 0)
=
BIIT
Q
(35)
< 1.
Note that by the equivalence of matrix noons (p. 29 of [37]), monotone convergence with respect to the norm II . liT implies convergence with respect to any other norm, although probably nonmonotonically. Since the index sets are chosen cyclically, a "full iteration" consists of K updates; therefore, (34) bounds the convergence rate of the subsequence {8 i K }~o. Proof: Consider the ith iteration and let S == Sk where k == 1 + Ci modulo K). Define
and let ¢s ( z )
= cPs (f)S ~ 7]).
Let
d(z) == d(()s; 8) == (v~s(ps)(z)
identity matrix.
then by assumption iv) we can apply the Taylor formula with remainder [42] to expand d(z) about z
With the above definitions, we can define the following region of monotone convergence in norm to B. Definition 3: R+ c e is a region of monotone convergence in norm if there exists a nonsingular p x p matrix T such that R+ is an open ball with respect to the norm II . liT and Sk 1. For k == 1, ... ,K, U (OSk; B) is invertible for all () E R+ sk and for all f)Sk E (see J.7)). _ 2. For k == 1, ... ,K,JjM (Osk;(I)IIT ~ 1 for all (I E R+ sk and for all ()s» E (0),
e (!L e
3. There exists Q: < 1 such that for any () , ... , e -- k -k and ()Sk E s (0), k == 1, ... , K
~I<
-1
e
IIMSK(esK;8K) ... MSl(Bsl;81)!IT ~
Q.
E
d(z) = d(i)
+
i
1
('Vd)(tz
+ (1 -
t)i)dt (z - i).
Since Bis a fixed point of the SAGE algorithm, by assumption iii) and iv) d(z) O. Observe that by the definitions (29)-(31):
=
(\7d)(z) = [-Us(Os; 0) Y S(8 s: 0) WS((}s; 0)].
d(O~11; (}i) = 0 for the SAGE
By assumptions iii) and iv) algorithm (5), so
R+
U
Sk
i+ 1.
i
i+ 1
(OSk ~O )(()Sk - OSk) A
_
V
Sk
i+ 1.
(()Sk ,()
i
i
)(8S k
i+l. i i + W Sk (OSk ,8 )((}Sk
282
- fJ S k )' A
_
(36)
(33)
In general, T may depend on {), but we do not allow T to vary with iteration. The hard work is to verify condition 3 (see Appendix B), but if one can do so, the reward is the following theorem.
,.
- OSk)
By property 1 (invertibility) of Definition 3
(}i+ sl.: 1
(Oi+ 1. ()i) -1 VSk (Oi+ 1. ()i) (Oi _ {) sA: ) Sk , 51.: , Sk Sk (f)~t1; f)i)-1 W (f)~t1; f)i)( f)lk - 0sd.
_ {) _ USk Sk -
+U
Sk
From (6) the components of permuting using R S (32)
Oi+l _ {} == M
Sk
O§k
(O~"tl; tri ) ( Oi - {})
Sk
where M was defined by (32). Therefore, since by property 2 of Definition 3
IIO i + 1 - BIIT
~
to Os, and HS's is the coupling between Os and Os induced by 4l. Combining the above definitions with (29)-(31)
are just copied, so after
(37) (}o E
R+
IIf)i - eli T
Sk
... M
MS
(f)~~ +1;
f)'iK)( (pK -
0IIT
== Fxs 1y
W
S
== _Hs,s.
= R S [(US)~lVS
(U
= [(HS + Fxsciy)-lFXSIY aliT
~ rtllf)iK -
(38)
S);l WS]
(Rs )' .
(Rs)/MsIl s
0).
Thus, by property 3 of Definition 3
IIO(i+l)K -
v
S
Substituting in (38)
(g~:l)K; o(i+l)K -1) X S1
== H S
If H is positive definite, then US will be invertible; therefore, by (32)
and therefore, it follows by induction that f)t E R+. A full cycle consists of one update over each of the K index sets; therefore, applying (37) K times (O(i+l)K _ 0) == M
+ Fxs 1y
US
= 1 - [~J (H S + Fxs 1y) -1 [HSHS,S]
and therefore, the subsequence {f}tK }~o converges monotonically in norm to () as i ~ oc with linear rate at most
= 1-
o.
By continuity of the derivatives of
-(H s + F~sly)-lHS'S]
[~] (HS + F xSly) -1[1 O)(RS)'HRS.
Thus S
H1 M S H- 1 =1-H1RS [(H +Foxs ly )-1
~] (Rs)'H1. (39)
Since the spectral radius is bounded above by any matrix norm, the root convergence factor is bounded above by n. 0 ApPENDIX B
R+ Is
For simplicity, we now consider the case the index sets are disjoint and are chosen cyclically in the natural order, i.e., S! == s», where k == (1 + i modulo K) and {51 ~ ... ~ SK} = {I ..... p}. In that case, it follows from (39) that
M
NONE~1PTY
In this appendix. we show that the region of monotone convergence in nann R..; is nonempty for suitably regular problems. Thus, the conditions for Theorem 3 are reasonable, and the superimposed signals example in Section IV is a concrete example. First note that from (10) one can show that
(\7 110 HS)(Bs~ B) = _(\7 200 H S )(8s: B). (\l101Hs)(es~B) = 0
(cf. (3.16) of [1]). For an index set 5, define
FX51y = -(v
200
+ D- F + L-H-)1 H
(40)
(41)
where DH is a block diagonal matrix containing the diagonal blocks of H that correspond to the subsets and LH is the corresponding strictly lower block triangular matrix. We sK ... MS11I < 1 by using the can thus establish that 11l\1 T following "splitting matrix" theorem (p. 79 of (37]). Theorem 4: If H is positive definite and S is invertible, then
s».
S-lHIIT
<1
H~ if
for T
S
+ Sf :> H.
(42)
From (40), for a SAGE algorithm S = DH + DF + L H , so in light of (41), condition (42) of Theorem 4 is satisfied. Thus,
H == - \7 2
s
== 1- (D H
is block-diagonal with F xS Iy in the kth block, and
A
then from (10), one can see that the matrix F x Sly is the conditional Fisher information of X S for Os, given Y == Y and given all of the other parameters Os' Define the Hessian of the objective at (; by
H
DF
s:
···M
III -
s H )((}s; ()) A
where
SK
== -(\7 8 5 V~::i 1»(0) A
HS's = -(V 8 s V~s W}). (To simplify notation we leave implicit the dependence on ().) Note that HS is the curvature of the objective
283
IIM s K
...
M
S1
II <
1.
V sing the relationships derived above, one can establish the following result [43]. Theorem 5: Let {) be a fixed point of a SAGE algorithm, and assume that ep is strictly concave on an open set local to {j (so that H is positive definite). Then if q, and the functions ¢s are all twice continuously differentiable near iJ, there exists a nonempty region of monotone convergence in norm R+ satisfying the conditions of Definition 3 for the norm induced by T = H~.
ApPENDIX
C
FISHER INFORMATION
From (35) we see that the root-convergence factor of a SAGE algorithm is given by the spectral radius of a product of matrices M S(OS; 0) of the form (32). For an EM algorithm, this spectral radius increases towards 1 as the completedata becomes more informative, i.e., as its Fisher information increases [1], [4], [5]. In this section we demonstrate that a similar relationship holds for the convergence rate of a SAGE algorithm. Defining
we see from (39) that for 8i small
This last equation suggests that minimizing F x Sly will improve the rate of convergence of 118 i ll to O. To demonstrate this more formally, let n F 1 and D F2 be the block diagonal aggregate of the Fisher information matrices for two SAGE algorithms with DPI < DF~. Then one can use (40) with an argument similar to the proof of Theorem 2 to show that -
p(I - (D H
1 + D- F1 + LH)H)
< p(I - (DR + DF + LH)-lH). :!
Thus, less informative hidden-data spaces lead to smaller root-convergence factors and hence faster converging SAGE algorithms. In particular, once one has chosen the index sets Sk the optimal hidden-data space from the point of view of asymptotic convergence rate would simply be )( 5 = Y, since then Fxs Iy = o. But that choice will often lead to an intractable M -step. The SAGE algorithm allows one to choose hidden-data spaces whose Fisher information matrices are much smaller than that of the usual complete data of an EM algorithm. Finally, note that from (43) we see that since H is determined by cI>, once the index sets are chosen, the only design issue left is to choose the hidden-data X s. This choice should be made by considering the tradeoff between making Fxs 1y small but yet making the M -step tractable. ACKNOWLEDGMENT
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285
Unitary ESPRIT: How to Obtain Increased Estimation Accuracy with a Reduced Computational Burden Martin Haardt, Student Member, IEEE and Josef A. Nossek, Fellow, IEEE
Abstract- ESPRIT is a high-resolution signal parameter estimation technique based on the translational invariance structure of a sensor array. Previous ESPRIT algorithms do not use the fact that the operator representing the phase delays between the two subarrays is unitary. Here, we present a simple and efficient method to constrain the estimated phase factors to the unit circle, if centro-symmetric array configurations are used. Unitary ESPRIT, the resulting closed-form algorithm, has an ESPRITlike structure except for the fact that it is formulated in terms of real-valued computations throughout. Since the dimension of the matrices is not increased, this completely real-valued algorithm achieves a substantial reduction of the computational complexity. Furthermore, Unitary ESPRIT incorporates forward-backward averaging, leading to an improved performance compared to the standard ESPRIT algorithm, especially for correlated source signals. Like standard ESPRIT, Unitary ESPRIT offers an inexpensive possibility to reconstruct the impinging wavefronts (signal copy). These signal estimates are more accurate. since Unitary ESPRIT improves the underlying signal subspace estimates. Simulations confirm that. even for uncorrelated signals, the standard ESPRIT algorithm needs twice the number of snapshots to achieve a precision comparable to that of Unitary ESPRIT. Thus, Unitary ESPRIT provides increased estimation accuracy with a reduced computational burden.
T
I.
INTRODUCTION
HE recovery of signal parameters from noisy observations is a fundamental problem in (real-time) array signal processing. Due to their simplicity and high-resolution capability, ESPRIT-like subspace estimation schemes have been attracting considerable attention. Their parameter estimates are obtained by exploiting the rotational invariance structure of the signal subspace, induced by the translational invariance structure of the associated sensor array. This can be achieved without computation or search of any spectral measure [15], [17]. Unitary ESPRIT achieves even more accurate results than previous ESPRIT techniques by taking advantage of the unit magnitude property of the phase factors that represent the phase delays between the two subarrays [4]. It has been shown in [12] that constraining the phase factors to the unit circle can also give some improvement for correlated sources. For centro-symmetric sensor arrays with a translational invariance Manuscript received June 8, 1994: revised August 30. 1994. The associate editor coordinating the review of this paper and approving it for publication was Prof. Keshab Parhi. The authors are with the Institute of Network Them)' and Circuit Design, Technical University of Munich, Munich, Gcrmanv. IEEE Log Number 9410301. .
structure, Unitary ESPRIT provides a very simple and efficient solution to this task. Although Unitary ESPRIT effectively doubles the number of of data samples, the computational complexity is reduced by transforming the required rank-revealing factorizations of complex matrices into decompositions of real-valued matrices of the same si;e. Thus, we obtain increased estimation accuracy with a reduced computational load. This reduction can be achieved by constructing invertible transformations that map centro-Hermitian matrices to real matrices. These transformations have been introduced in Lee' s pioneering work on centro-Herrnitiun matrices (10). More than a decade later. her results were used to transform the complex covariance matrix of a uniform linear array (ULA) into a real matrix of the same size (8) to reduce the computational load of adaptiv...~ beamfonning schemes (91. In this paper. we use more general centro-symmetric array configurations that have been receiving increased attention lately 12~]. We derive an efficient square root version of Unitary ESPRIT that only requires real-valued computations from start to finish. by operating directly on the data instead of "squaring" it to obtain
Reprinted from IEEE Transactions on Signal Processing, Vol. 43, No.5, pp. 1232-1242, May 1995.
286
where the overbar denotes complex conjugation without transposition. Centro-Hermitian matrices of size p x q form a p . qdimensional linear space over IR [10]. To show how centroHermitian matrices can be mapped to matrices with real entries, Lee defines left II -real matrices in the following fashion. Definition 2 [10): Matrices Q E C p x q satisfying
loss of accuracy that can be compensated by combining them with Unitary ESPRIT, yielding not only improved estimation accuracy, but also completely real-valued algorithms. In [5], it is shown how Unitary Schur ESPRIT dramatically improves the performance of the new Schur-type method, an adaptive subspace estimation scheme with a computational structure and complexity similar to that of a QR decomposition, except for the fact that plane and hyperbolic rotations are used. In this case, the required rank decision, i.e., an estimate of the number r.f signals, is automatic, and updating as well as downdating are straightforward. The fully real-valued Unitary ESPRIT concept can also be extended to spatially smoothed forwardbackward estimation schemes [6], [13], [14] and is applicable to many other subspace estimation techniques (see [20] for an excellent overview). The results are comparable to the advantages obtained by operating in bearnspace [24] without the necessity of converting the data from element space to r eamspace. This paper is organized as follows. It starts with a review of the definition and basic properties of centro-Hermitian matrices. These properties will be used to derive the realvalued implementation of Unitary ESPRIT. A brief review of the standard ESPRIT algorithm is given in chapter III. It can be seen as a generalization of the matrix pencil method [7]. Chapter IV introduces the Unitary ESPRIT concept for centrosymmetric array structures. In Section IV -8 we show how all tnree required rank-revealing factorizations can be transformed into decompositions of real-valued matrices or' the same si:e yielding a completely real algorithm ..-\ new reliability test. which is a substantial improvement of current high-resolution array signal processing and spectral estimation techniques. is presented in Section IV-C. Further sunplifications of the algorithm are derived in Section IV-D. before a summary \..11' Unitary ESPRIT concludes the chapter (Section IV-E). Finally. computer simulations compare [he perfonnance of unitary ESPRIT with that of the well-known standard ESPRIT algorithm (Section V).
II.
(2) are left II -real. The unitary matrices
u; ]
-jil n
o
J2
(3)
jIn] OT
o - jTl n
(4)
for example. are left Il -real of even and odd order, respectively. More left II -real matrices can be obtained by post-multiplying a left II -real matrix Q by an arbitrary real matrix R. i.e., every matrix QR is left II-real. Now. we are in a position to state Lee' s main result, which establishes an automorphism between centro-Hermitian and real matrices. Theorem 1 [10]: Let 'I'; and U'; denote arbitrary nonsingular left II-real matrices of size p x p and 'l x 't respectively. Then. the bijective mapping
;: M ~ T~lMU(/ maps the set of all p x (j centro- Hermitian matrices onto RP x q . the set of all real matrices of the same size. This theorem can. for instance. be used to calculate the singular value decomposition (SVD) of a centro-Hermitian matrix M E C P X I/ . Corollary 1: Let M be centro-Hermitian, and assume that the SVD ~f ;;Q(M) == Q~ MQIJ E RP~-«/ is given by
o,
y(j(M) == UyE'r~V~!. where the matrices and Qq are unitary as well as left l1-real. Then, an SVD I of M is obtained as
CE~TRO-HER~lITIA~ M.\TRICES
First of all. let us introduce our notation and review the definition and the basic properties of centro-Hermitian matrices that have been derived by Lee [10]. Throughout this paper, column vectors and matrices are denoted by lower case and upper case boldfaced letters. respectively. TTl) is the p x p exchange matrix with ones on its antidiagonal and zeros elsewhere
(5) where the left and right singular vectors of M are left II -real. Proof' The first part follows from the unitary nature of QJ> and Q(]' the second from the fact that the singular vectors of a real matrix are real. • For future reference, we consider an efficient computation of a particular transformation T (.). It transforms an arbitrary complex matrix G E CP x q into a real p x 2q matrix, denoted by T (G). Notice that for every matrix G ~ the matrix
II p is a symmetric permutation matrix. it is involutorial, i.e., rI~ = II'. With this notation. we can define centroHermitian matrices in analogy to centro-symmetric matrices. Definition 1: A complex matrix M E (11 x I[ is called centro-Hermitian jf
~) ince
(1)
[G llpGII q] E
Cpx2
is centro-Hermitian. Thus, the matrix
T(G) (~f ~Q([G IIpGIIq])
== (J~{[G
II pG 1I q]Q 2q
(6)
1 Recall that the SVD of a complex matrix is unique up to a unitary diagonal scaling matrix, if all singular values are distinct.
287
is always real according to Theorem 1. Consider the case where the left II-real matrices Q p and Q2q are chosen according to (3) or (4). Furthermore, let G be partitioned as
G=
[~]
where the block matrices G 1 and G 2 should have the same size. Obviously, the row vector gT must be dropped if ]J is even. Then, straightforward calculations show that the desired real-valued matrix (6) can be expressed as
o Fig. 1. Planar array composed of n sensors (doublets).
(7) Here, Re{·} and Imj-} denote the part, respectively. Once again, if p of (7) should be dropped. Then, an T(G) E Rpx2q from the complex p x 2q real additions.
In = ,)
Subarray 2 (a)
Subarray 1 I
T • T
,Subarray 2 (b)
Consider the standard ESPRIT scenario [151, [17], i.e., an J,I -element sensor array composed of in pairs of pairwise identical, but displaced sensors (doublets). Let ..1 denote the distance between the two subarrays. Incident on both subarrays are d narrow-band noncoherent planar wavetronts
with signal propagation velocity c: and a common center frequency woo The tl impinging signals are combined to a signal vector s( t rl ). Assume, for the moment, that the two subarrays do not share any elements. i.e., they do not overlap. Then, the total number of sensors equals :.\I == 2,11, and the uncorrupted signals received at the two subarrays have the following form:
= [:~~~:n = [:fJ> ].~(tIl) = Ac;s(t,,).
(8)
AG E C A1 x d and A E C n1 x d are the steering matrices of the whole array configuration (global array steering matrix) and the first subarray , respectively. Notice that the kth columns of both array steering matrices depend on the direction of arrival (DOA) Ok of the kth source relative to the displacement between the two subarrays.' Furthermore ~ == diag{(Pk}i=l E (dxcl
is a diagonal matrix of the phase delays between the sensor doublets for the d wavefronts. Its diagonal elements, the phase factors
~
~
A. Standard ESPRIT Scenario
)
. ..
= 3 pairwise identical but displaced
Subarray 1
real and the imaginary is odd, the center row efficient computation of matrix G only requires
III. STANDARD ESPRIT
x(t n
~
=0
< »< d.
= 0 corresponds to the direction perpendicular to ~.
(9)
•
Subarray 1 I
• •
• i
Subarray 2 (C)
Fig. 2. Three different subarray choices for a uniform linear array (ULA) of M = 6 identical sensors. (a) Maximum overlap (m = 5); (b) interleaved (m = 3); (c) Mixed (m = 4).
B. More Structured Array Geometries Recall that every row of At; corresponds to an element of the sensor array. In the case of overlapping subarrays, a particular subarray configuration is described by selection matrices that choose ni elements of x( I ,,) E C .\!. where In <.\/ is the number of elements in each subarray. Let J 1 and J 2 be ni x J/ selection matrices that assign elements of x( tTl) to the subarrays one and two. respectively. Fig. 2. for example, displays three different subarray choices for a uniform linear array (ULA) of Al = 6 identical sensors. In general, the two selection matrices are chosen to be centro-symmetric with respect to one another, i.e. ( 10)
a property that plays a key role in the derivation of the fully real implementation of Unitary ESPRIT, cf. Section IV-B. Therefore, the combined selection matrix
.1 (g;f
[JJ 1] 2
E
R2m
is centro-Hermitian, i.e. "21n.J II~'I
c.
x i\/
= J,
General ESPRIT Principle
By collecting iV ~ d snapshots from each sensor, 1 ~ n :S N, measurement matrices Xl, X 2, X and a signal matrix &of)
288
A basis for the estimated signal subspace is determined from the d dominant left singular vectors according to
are formed, obeying
(11)
Then, a unitary basis for the row space of [C 1 C 2 ] can also be obtained by computing its SVD (total least squares approach). However, it is less expensive to use Prow == which corresponds to the standard least squares solution of the overdetermined set of equations
It is easy to see that every row in X corresponds to an element of the sensor array. Equation (11) implies that Xl, X 2 , and X are rank-deficient, namely rank Xl == rank X 2 == rank X == d. Thus, the d columns of
cf,
C1tJI
GL(d).
(12)
r
==
r.;« [SP
eol
form a basis for the row space of [C1
2]
if
ProwA E GL(d).
(13)
(17)
Therefore, the rank-reducing numbers of the matrix pencil
T, -
Arl
== ProwA (lP - AId)SP col
are the diagonal elements of tP (phase factors) and can be calculated as the generalized eigenvalues of the matrix pair
ir; r 1) .
Due to these observations, the ESPRIT algorithm reduces to choosing the appropriate compression matrices that define the required bases. In the absence of noise (the case discussed so far), any matrices Peal and Prow satisfying (12) and (13) will do the job. With noisy measurements, however, we are faced with the problem of estimating the signal subspace and its dimension.
for some unitary diagonal matrix A c E (d x ", Notice that the I . I ~ II 1 matrix A c:Ac/, -) IS ett -rea. Uniform linear arrays. for example. the most common arrays used in practice, are centro-symmetric. It is well known that the analogy between array signal processing and time series analysis (harmonic retrieval) can be obtained through uniform linear arrays (ULA's) by interpreting them as uniform sampling of a time series [16]. The centro-symmetry of the global sensor array Ac; and ( 10) imply that the steering matrices of both subarrays are also centro-symmetric, i.e.
Without additive noise. the Unitary ESPRIT data matrix
D. SBD-Based Subspace Estimate
The most robust way to estimate the required bases is to compute the singular value decomposition (SVD) of
admits the factorization
·
where X denotes the measurement matrix X corrupted by additive, spatially uncorrelated' noise, ~s contains its d dominant singular values, and the unitary matrices V and V are partitioned accordingly. Then, the best rank d approximation of X in the Frobenius-norm is given by X == V,I 2:s V ~. In other words. our low rank estimate of X is the matrix X satisfying Y~d
-
[XX
=
[;
/Z -
(14)
Ilx- XIIF == rankmin Ilx- YIIF'
r 1 r 2.
Unitary ESPRIT is applicable to centro-symmetric array configurations. A sensor array is called centro-symmetric [22] if its element locations are symmetric with respect to the centroid and the complex characteristics of paired elements are the same. Their global array steering matrix A c . therefore, satisfies
tPSPeol ]
e
==
A. Multiple Invariance Structure
== Prow [C 1 C2 ]
2]
(16) 1
IV. UNITARY ESPRIT
Here, GL( d) C Cd x d denotes the general linear group of all nonsingular matrices of dimension d x d. Observe that the column space or range of IX, range IX C e 2m , is usually called signal subspace. In the arne way, the d rows of
[rl
C2
followed by an eigendecomposition of tjI
form a basis for the column space of X if
sr; E
~
(15)
3If the spatial covariance matrix of the additive noise is known up to a scalar factor, the SVD can be replaced by the generalized or quotient SVD (QSVD), as described in [17].
289
1
2
II m.X 2 ]
lIrnX l
(18)
which is easily seen by using the centro-symmetry of the subarrays and the unitary nature of ~. Thus, Z is also rankdeficient, namely rank Z = d. Equations (11) and (18) show that Unitary ESPRIT essentially doubles the number of available measurements from N to 2JV. Increased estimation accuracy can, therefore, be achieved by replacing the measurement matrix X E C ~\I x IV of the standard ESPRIT formulation ( 11) by Z E C .\f x 2.V . which corresponds to forward-backward averaging of the data.
B. Real Implementation Due to the special algebraic structure of the noise-corrupted data matrix Z and the structure of the subsequent total least squares (TLS) problem, the computational complexity of Unitary ESPRIT can be reduced significantly. This is achieved by transforming the three (complex-valued) rank-revealing factorizations, • the subspace estimation step • the subsequent total least squares problem • the final eigenvalue decomposition (EVD)
Proposition 2-Total Least Squares Problem: The complexvalued SVD of size m x 2d that solves the total least squares (TLS) problem ClIP ~ C 2 , which is associated with Unitary ESPRIT, can be transformed into an SVD of the real matrix T (C 1) E Rm x 2d, where the transformation T (-) is defined in (6). Moreover, the eigenvalues ¢k of the resulting TLS solution \lJTLS E C d x d will be symmetric with respect to the unit circle, i.e., there are indices k,l E
into factorizations of real-valued matrices of the same size. Thus, real-valued computations can be maintained for all steps of the Unitary ESPRIT algorithm. The following three propositions derive the required transformations by taking advantage of the mapping between centro-Hermitian and real matrices, cf. Section II. In Remark 3, we also show how the real-valued total least squares problem can be replaced by a real-valued least squares (LS) problem. Proposition i-Signal Subspace Estimation: The principal subspace of Z E C ~\I X 2 1V (and, therefore, also the principal subspace of JZ) can be obtained through a rank-revealing factorization of the real matrix T(X) E RJ1 x2.N. where the transformation T(·) is defined in (6). Then, the complex matrices C I and C 2 • spanning the estimated signal subspace, obey
Proof: By post-multiplying the noise-corrupted matrix with a unitary permutation matrix we obtain a centroHermitian matrix in the following fashion:
- == Z-[1-
ZCH
.d}
such that
(/>~:
1
== -==-.
(22)
1)1
Proof' The multidimensional TLS problem Clt/! can be solved through an SVD of
~
C2
V~] V~ . Then, the TLS solution is obtained from the d right singular vectors corresponding to the d smallest singular values according to
(23) where we have assumed that V:!:! E GL( d). i.e.. the TLS solution is unique. For the singular case. the reader is referred to [21]. Thus. the TLS problem associated with Unitary ESPRIT can be solved through an SVD of
( 19)
Z
{1,2~···
[C l
C :! J
\ ~ ) rc l 1
1/", Cd.
Notice that this matrix has the same structure as ~ Z :::: (X II.\l X). Using, therefore, the same reasoning as in (20) and (21). the
V
(20)
TLS problem is solved by computing an SVD of the real matrix
According to Corollary 1, a rank-revealing factorization of ZCH can, thus, be obtained through an SVD of the real matrix
(24)
Its right singular vectors will be denoted by
W = which proves the first part of this proposition. Let the tl dominant left singular vectors of :.pQ ( ZCH) be denoted by EsE RJ\;! x d. Then, the d dominant left singular vectors of ZCH as well as Z are given by QA1E.'i' Therefore, the matrix
[Wll W: n
W 12 ] W 22
E
[R2dX"2d.
Then, the right singular vectors of [C 1 from
C·2 ) are determined (25)
and \VTLS is obtained from (23). Since the matrix Q2dW is left II -real, it can be written as provides a basis for the estimated signal subspace. With (10) and the left II-realness of QM we, finally, get the desired result:
II 1nC I == IImJIQ.~IEs
=HmIImJ2II1\lQAfEs == J 2 QAJ E s == C 2 ·
for some matrix VI E C d x 2c1 • cf. (2). With (25) we, therefore, conclude V 22 == V 12. Thus, if (/>, is an eigenvalue of the TLS solution \{J TLS E GL( <1), 1/ fjJi is an eigenvalue of --1
--1
\V T LS == - V 12 V 12 =
•
290
which proves (22).
WTLS
•
C. Reliability Test
Proposition 3-Eigenvalue Decomposition: The eigenvalues of the complex matrix WTLS can be determined from the eigenvalues of a real matrix of size d x d via the linear fractional transformation .r - J
f(:r) == - - . ' .t:
Proposition 2 states that the eigenvalues of WTLS, i.e., the phase factors estimated via Unitary ESPRIT, are symmetric with respect to the unit circle, since they satisfy (22). This observation gives rise to a new reliability test provided by Unitary ESPRIT without demanding additional computations. This reliability test is a substantial improvement of current high-resolution array signal processing and spectral estimation techniques since usually there is no easy way to determine how reliable the resulting estimates are. Unreliable estimation results might have been caused by a false estimate of the number of sources d or by the fact that there is no source signal at all (only noise). Remark 2-Eigenvalues with. Unit Modulus: Notice first that the eigenvalues cPk that lie on the unit circle form a subset with nonzero measure in the class of all eigenvalues ful filling (22). i.e., being symmetric with respect to the unit circle. Owing to this and the fact that Unitary ESPRIT produces consistent DOA estimates. asymptotically all the estimated phase factors (T>k will be on the unit circle. IL however. the number of snapshots .V is too small or if there is only noise present. the eigenvalues of \{J TLS might fail to satisfy
(26)
+J
Moreover, the eigenvectors of both matrices are identical. Proof' a) Assume, for the moment that the left iI-real matrix Q2 d is the one we have defined in (3). Then. (25) yields
After partitioning V and W as before, we therefore conclude .rom (23) tVTLS
== -(W 12
+ jW12)(Wl~
== - (( - W
1:2W
=f(YT LS )
221 )
-
with
- jW 21 l -
j I d )( ( - W
Y T LS
1
12 W
2} ) + j I d ) -
= -W 12W;}.
(27)
Here. f(.1") denotes the linear fractional transformation which is analytic for .1' # -j. Let TTLS
= -W12W~}
t
= TfIT- i
(26),
(30)
(28)
be the eigenvalue decomposition (EVD) of the real matrix
Y rt.s It is a well-known result from function theory that the eigenvalues of tV TLS can be obtained through the "arne linear fractional trans formation. i.e.
with
[J = Jiag{~~·~1.=t
and
.Jv'~. ~
I
-./
r
.md the corresponding eigenvectors of rt.s and tV TL~ are identical. b) An arbitrary left lI-real matrix of dimension ],(] x 2d can. obviously. be written as
which indicates that the subsequent estimates will be unreliable. Hence. no further computations should be carried out. Condition (30) implies that all eigenvalues w'J..: of Y T LS are real. cf. (26 ).~ Thus. if some of the w!}.. occur in complex conjugate pairs. the Unitary ESPRIT reliability test has failed. and the algorithm has to be restarted with an increased window length .\' or more reliable measurements. If. conversely. all eigenvalues ...JJk· are real, i.e .. the reliability test has been "passed.' all estimated phase factors c/)}.. are precisely on the unit circle.
D. Real- Valued Least Squares After replacing the real matrix W by W = R'2tlW. we invoke the same reasoning as above to prove this proposition for an irbitrary left IT-real transformation Q'2d' • Remark I-Covariance Approach: Instead of the described square root (or direct data) approach based on a real-valued SVD. cf. Proposition 1~ we can use a covariance approach based on a real-valued EVD to determine the signal subspace estimate. Then. E, E lR ,\[ x d denotes the d principal eigenvectors of (29)
First forming T(X) according to (7), followed by the computation of (29)~ is more efficient than the approach alternative suggested in [8] and [24]. There, it is proposed to compute the
Notice that the derivation of the Unitary ESPRIT reliability test is based on a total least squares solution of (16). Thus. the computation of T TLS requires an SVD (or another rank revealing factorization) of T( C 1) E jRm. x 2d. By computing the less expensive least squares instead of the total least squares solution of (16), we would, however, lose the benefits of the reliability test, since (22) would no longer be satisfied. Moreover, the complex-valued least squares problem (16) cannot be transformed into a real-valued problem of the same size. The following remark, however, sets up a different. realvalued least squares problem, which can be solved instead. Remark 3-Least Squares Estimate: After partitioning the real matrix of (24) according to
- - H
complex-valued sample covariance matrix R,"\.\ = X X E (AI x 1\-/ first. Then~~;,~ is determined from the EVD of Re{Q~iIRx,yQ.I\I}' which is computationally more expensive than using (29).
291
4 Recall that the eigenvalues of a real matrix can either be real or they occur in complex conjugate pairs.
it is easy to see that T TLS is a TLS solution of the real-valued system of equations
Now, we are in the position to summarize the described real implementation of Unitary ESPRIT, which is given in Table I. Here, the left H-real matrices Q m and QA-I are chosen according to (3) or (4). Notice that a linear estimate of the source signal matrix S (Step 7) can easily be obtained by applying the results of this section to the source signal matrix estimate derived in [4], where (without loss of generality) the additive noise is assumed to be spatially uncorrelated.
(31) To save computations, we can, therefore, solve (31) by computing its least squares solution T LS. Here, the Unitary ESPRIT reliability test is still applicable, since the resulting matrix T LS is always real. If the reliability test has been "passed", the estimated phase factors are on the unit circle. The real-valued LS or TLS problem (31) can directly be obtained from E.., by observing
T(Cd
=Q~[CI
C21[Irl
=Q~[JIQME..
V. COMPUTER SIMULATIONS In this section, we present some simulation results that compare Unitary ESPRIT with the standard ESPRIT algorithm, using the SVD implementation in all cases. Among others, we examine scenarios where the standard ESPRIT algorithm faces some problems, like low signal-to-noise ratios, short window lengths, and correlated source signals.
nJQ2d
J 2Q u E sl
1
j~n ] ~ [~nn -.JI n
v2
J2[K1Es K 2E.• ]
=
where the selection matrices K follows:
1
and K:2 are defined as
A. Signal Reconstruction
Since the matrices in braces are centro-Hermitian, K 1 and K 2 are always real, cf. Theorem I. They are even sparse if the selection matrix J 1 is sparse and the matrices Qtn and Q-'1 are chosen according to (3) or (4). This is illustrated by the following example. For the ULA with maximum overlap sketched in Fig. 2(a), J 1 is given by
1 0 0 000
o
1 o 0 0 1 000 000
J 1 ==
0 0 0 0 0 0 . 100 0 1 0
Thus, straightforward calculations yield 1 0
K
1
== 0
0 0
0 0
K:2== 0
1
0
1 1 0
0 0 0
0 0 -1 1
0 1
0 0 0 0 J2 0 0 0 1 1 0 0 1 -1 0 0 0 0 0 0 0 -1 0
0 0 0 0 1 1 -1 0 0 0
0 1
-J2
First, we examine the effect of Unitary ESPRIT on the resulting signal estimates. To this end, three impinging wavefronts are reconstructed using a single ULA of Al = 9 sensors with maximum overlap, cf. Fig. 2(a). The three uncorrelated equi-powered QPSK signals arrive from H1 = lO°. 112 =20 0 . and f}:J = 30°. respectively. Fig. 3 depicts the resulting output signal-to-noise-and-interference ratio (SNIR) as a function of the SNR and the number of snapshots .V using standard ESPRIT (dashed lines) and Unitary ESPRIT (solid lines). The values of ~V marked on the right side of the figure correspond to the solid lines, i.e., Unitary ESPRIT. The output SNIR achieved by the standard ESPRIT algorithm for a given value of ~V (dashed lines) can be found below the corresponding solid lines. For small values of .V. e.g., .V == 5 snapshots. Unitary ESPRIT achieves a significantly better performance than the standard ESPRIT algorithm. Notice that standard ESPRIT with !V == 10 snapshots attains the same performance as Unitary ESPRIT with ;.V = 5 snapshots for SNR's that are greater than 15 dB, while the performance of standard ESPRIT with N = 20 is comparable to the performance of Unitary ESPRIT with N == 10 for SNR' s that are greater than 5 dB. Thus, Unitary ESPRIT essentially doubles the number of available snapshots J.V compared to the standard ESPRIT algorithm.
B. DOA Estimation
0
0
E. Summary of the Algorithm Before presenting a summary of Unitary ESPRIT, we note an interesting relationship between the eigenvalues of the real matrix T denoted by Wk, and the estimated phase factors c/>k ei 1t k , cf. (9). Solving ei l Lk == f(Wk) for the spatial frequencies J.Lk yields the simple expression
Next, we investigate the effect of Unitary ESPRIT on the estimated phase factors 4>~. ~ 1 ::; It; ::; d. Consider a ULA with itt 6 sensors and three correlated signals impinging from £}1 20° ,()2 = 0° ~ and £}3 == 20°. Their correlation matrix is given by
= =-
=
1 (1 +
J-Lk ·.if -":= -:-In J
jWk) . 1 - JWk
= 2 arctan Wk,
k=1,2~···,d.
(33)
R;J:x
==
[~p p2i :1
2
]
(34)
The phase factors (/>1, 4J2, and q~3' estimated with the standard ESPRIT algorithm and Unitary ESPRIT, are marked by crosses (+) in the C0l11pJex plane as depicted in Figs. 4 and 5 for
292
SUMMARY l . Initializatio n ' Form the matrix X E
eM, ...
TABLE I UNITARY ESPRIT
OF
from the available measurements.
2 Signal Subspa ce Estimation . Determine the real matrix of
r(X ) (square root approach)
r eX) E RMx2N from (7), and compute the SVD
or the eigendecomposition of
r(x )r (X )H (covariance approach).
The d
dominant left singular vectors or eigenvec tors will be called E, E RM x d . Estimate the number of sources d, if d is not known a priori [22J. 3. (Total) l.eust Squares ' Solve the overdetermined system of equations
by means of least squares (or total least squares) techniques. The selection matrices K
I
and K
2
are defined
in (32) s , Eigenvalue Decomposnion, Compute the eigendecomposi tion of the resulting solution 1" =rflr - 1 ER d xd , 5 Relta bili tv Test If all eigenvalues
where
fl =diag {w .}: ~ I '
w, are real, the estimates will be reliable . Otherwise, stan again with more
measurements. 6 OOA Esnm an on : Estimate the directions of arrival (DOA's) from
k = 1,2, .. . , d ,
/-'. = 2 arctan w" according 10 (9).
7 Signa! R eCCmSlrIICI/OII : A linear estima te of the source signal matrix S E - = S
(
Dr -
I
edx N
is given by
H H)-X,
E, QM
where D E C'" denotes an arbitrary diagonal (row) scaling matrix {4].
Standard ESPRIT 90 1
:: -,
1
+
,,----- ---- ---1
180 1---t--+-+-+--~!E--+-+-+----jH~!J,l
+
Unitary ESPRIT Standard ESPRIT
10
15 SNR In dB
20
25
30
Fig. 3. Output SNIR as a function of the SNR and the number of snapshots :Ii using standard ESPRIT (dashed lines) and Unitary ESPRIT tsolid lines ) for iii 10° . Ii ~ :20°. and li l .1 0° '-' / <) sensors. 1000 trial runs ). The values of S marked on the right side of the figure correspond to the sc.lid lines, i.e.. Unitary ESPRIT. The output SNIR achieved by the standard E'iPRIT algorithm for a given value of .\' (dashed lines) can be found below the corresponding solid line s.
Fig. 4. Phase factors 0 1. 02 . and 0:1, estimated with the standard ESPRIT - 20 ° .Ih 0° . Ii:l 20° . and correlation coefficients algont hm for iii Pl2 O··) , PI .l 0..) . and (!2 1 0.2.) (.\[ 6 sensors, SNR 0 dB. S = 20. 80 trial runs).
a correlation coefficient of p = 0.5. The results of 80 trial runs with N = 20 snapshots and an SNR of 0 dB are shown. Notice that all phase factors estimated with Unitary ESPRIT are precisely on the unit circle (Fig. 5), Figs. 6 and 7 depict the estimated phase factors for a correlation
coefficient of p = 0,8. In this example, the Unitary ESPRIT reliability test has failed three times. To picture these failures, the corresponding phase factor estimates are surrounded by circles (0), cf. Fig, 7. Notice that the variance of the DOA estimates that pass the Unitary ESPRIT reliability test is much
=
=
=
=
270
=
293
= =
=
=
=
=
=
Unitary ESPRI T
Unitary ESPRIT 90 1
90
180 1--4--+-+-f-~I!E--+-+-f--r---tI
o
1
o
180 1--4--+---f--f---=;:lIiE--f--+--t--i--I
270
270
Fig. 5. Phase factors OJ, 0) , and OJ , estimated with Un itary ESPRIT for Ii, = - 20°, 8, = 0°, fh = 20°, and correlation coefficients P'2 = 0.5, Pl3 = 0.5, and Pn = 0.25 (M = 6 sensors, SNR = 0 dB , N = 20, 80 trial runs).
Fig. 7. Phase factors 0" 0), and OJ, estimated with Unitary ESPRIT for Ii, = -20°, 8, = 0°, fh = 20°, and correlation coefficients P'2 = 0.8, Pl3 = 0.8, and Pn = 0.64 (M = 6 sensors , SNR = 0 dB, N = 20,8 0 trial runs). Estimates that produced a failure of the reliability test are surrounded by a circle (0).
Standard ESPRIT
+
-e0lt- """
+
10 1 I
9~ -
8~i ,L
+
,
(/)
~
a:
3
/
I
1
, ,
,
Sf 4
-3
,/
270 Fig. 6. Phase factors 0 " 0), and OJ, estimated with the stand ard ESPRIT algorithm for Ii, = -20°, 8, = 0°, fh = 20°, and correlation coefficients P'2 = 0.8, Pl3 = 0.8, and P23 = 0.64 (M = 6 sensors, SNR = 0 dB, N = 20,80 trial runs) .
lower than the variance of the DOA estimates obtained by the standard ESPRIT approach. The advantages of Unitary ESPRIT become even more evident if the root mean squared error (RMSE) of the estimated directions of arrival is plotted as a function of the correlation coefficient p. Fig. 8 show these curves for SNR's of -3, 0, and 5 dB using 3000 trial runs. The standard ESPRIT algorithm (dashed line - - -) is compared with Unitary ESPRIT without reliability test (dotted
0 .1
.
/~
"
,
i
J
- - - -- -
SdB
a a
J
/
es
a dB +
, ,
I
,
6~ 1? i W
f----l---+--+--t--3lIIE-+--+--r------I~~
,
Unll ary ESP RIT WIth reliability test
I
+
,
Unll ary ESPR IT Without reliability test
-
"I
+
180
- - . Standa rd ESPR IT
02
0.3
04
as
corrat anon coatncrern
0 .6
0.7
0.8
a 'J
Fig. 8. Root mean squared error (RMSE) in degrees of the estimated directions of arrival as a function of the correlation coefficient P and the SNR for Ii, = - 20°, 8, = 0°, and fh = 20° (M = 6 sensors, N = 20, 3000 trial runs). The signal correlation matr ix is given by (34).
line · . .). and Unitary ESPRIT with the new reliability test (solid line -). It can be seen that Unitary ESPRIT improves the estimation accuracy considerably. In the case of low SNR' s, the estimation accuracy is improved even further, by exploiting the information provided by the new reliability test. The corresponding failure rates of the Unitary ESPRIT reliabil ity test are plotted in Fig. 9. Due to the forward-backward averaging effect. Unitary ESPRIT can separate two completely coherent wavefronts , which is demonstrated in the next example . Two correlated
294
2 5 , - - - - - - r - - - . , - - - r - - - r - - - - r- - - r - - - , - - - - r - - - - ,
20 -3 dB
15
15
10
05 300
o o
200 100
5 dB
01
02
03
Q4
03
06
correlationcoefficient
dt:c, {/I}
in degrees
08
0 .9
Fig. 11. RMS E (in degrees) of the estimated dir ections of arrival as a function of the magnitude and phase of the complex correlation coefficient p for II, = 0° and 0, = 20° using standard ESPRIT (M = 4 sensors, N = 20, 100 trial runs).
Fig. 9. Failures of the Unitary ESPRIT reliability test as a function of the correlation coefficients p for II, = -20°, ~ = 0°, and 113 = 20° (M = 6 sensors, N = 20,3000 trial run s). Onc e again, the signal correlation matrix is given by (34). These curve s correspond to the solid lines in Fig. 8.
2 ..,
1 5~
I
1Ar 13~
Unitary ESPRIT (LS)
i
Urinary ESPRIT tTLS)
i
e-
~ ~
a:
I I
Standard ESPR IT (TLS)
i
_ t
I
Standard ESPRIT (LS)
.
I
2~
I
I
I
I
I
I
I
I
1 1r
I
I
I
I
1
~
!
1~
i
o
/
J
I
-1j
9rc-_=-.0_'-.-_-___________ ___
0 8r
I
o
01
0 .2
03
0.4 0 .5 06 correlation coeffICient
0.7
0.8
09
I
signals with correlation coefficient o are impinging an aULA of ;\l = 4 sensors from lil = QOand 1i'2 = 20° . Fig . 10 shows the resulting RMS error of the estimated DOA ' s as a function of fl. The performance of Unitary ESPRIT is not effected by the correlation, while the performance of standard ESPRIT deteriorates dramatically as p increases. Notice also that the difference between TLS and LS version of standard ESPRIT is negligible, while the LS and the TLS ver sion of Unitary ESPRIT fall on top of one another. Thus, it is ad visable to use the LS ver sion of Unitary ESPRIT instead of the computationall y more expensive TL S version. Finall y, Figs . 11 and 12 show the RMS error of the estimated DOA 's as a function of the magnitude and phase of a complex -valued correlation coefficient p . confirming the concl usions drawn from Fig . 10.
~OO "
a
.tr,,{p}
100
In
degrees
Fig. 12. RMSE (in degrees) of the estimated directions of arrival as a function of the magnitude and phase of the complex correlation coefficient p for II, = 0° and ~ = 20° using Unitary ESPRIT (M = 4 sensors , N = 20, 100 trial run s).
VI. C ONCLUDING REMARKS
1
Fig. 10. RMSE (in degr ees) of the estimated directions of arriv al as a function of the correl ation coefficient p for II, = 0° and ~ eq 20° (M = 4 se nsors, N = 20, 100 trial run s). Notice that the curve s for the LS and the TLS version of Unita ry ESPRIT fall on top of on e another.
.L
~
An improved version of the ESPRIT algorithm, called Unitary ESPRIT, has been presented in this paper. Unitary ESPRIT represents a simple method to constrain the estimated phase factors to the unit circle, yielding more accurate signal subspace estimates. The computational complexi ty is reduced significantly by exploiting the one -to-one correspondence between centro-Hermitian and real matrices, allowing a tran sformation to real matrices, which can be maintained for all step s of the algorithm. Unitary ESPRIT also provides a new reliability test, which is particularly useful in extremely low SNR's. Due to the inherent forwardbackward averaging effect, Unitary ESPRIT can separate two completely coherent sources and provides improved estimates for correlated signals. Moreover, Unitary ESPRIT offers a great potential to improve the performance of approximate sig na l subspace estimation techniques, which are well suited for an adaptive implementation, since inexpensive updating strategies are known [5]. The fact that Unitary ESPRIT is efficiently formulated in terms of real-valued computations from start to finish, is
295
critically important for the extension to 2-D centro-symmetric arrays with a dual invariance structure. 2-D Unitary ESPRIT [23] provides automatically paired source azimuth and elevation angle estimates along with an efficient way to reconstruct the impinging wavefronts. Furthermore, an efficient Off beamspace implementation of Unitary ESPRIT has also been derived in [23], enabling reduced dimension processing in beamspace, if there is a priori information on the general angular location of the DOA's.
[17]
[18] [19] [20]
[21]
ACKNOWLEDGMENT
The authors would like to thank Prof. M. Viberg, Gothenburg, Sweden, for his helpful comments regarding the contents of this paper.
[22]
[23]
REFERENCES [1] C. H. Bischof and G. M. Shroff, "On updating signal subspaces," IEEE Trans. Signal Processing, vol. 40. pp. 96-105. Jan. 1992. [2] T. F. Chan, "Rank revealing QR factorization," Linear Algebra and its Applications, vol. 88, 89. pp. 67-82. 1987. [3] J. Gotze and A. J. van der Veen, "On-line subspace estimation using a Schur-type method:' IEEE Trans. Signal Processing, Nov. 1993, submitted for publication. [4] M. Haardt and M. E. Ali-Hackl. "Unitary ESPRIT: How to exploit additional information inherent In the rotational invariance structure." in Proc. IEEE Int. Can! Acoust.. Speech. Signal Processing. Adelaide. Australia. Apr. 1994. pp. 229-:!32. vol. IV. [5] M. Haardt and 1. Gotze, "Unitary Schur-type subspace estimation." Technical University of Munich. Inst. of Network Theory Circuit Design. Munich. Germany. Tech. Rep. TUM-LNS-TR-94-6, July 1994. [6] M. Haardt. P. Weismuller, and R. Killmann, "The identification of late fields: A multichannel high-resolution state space approach," in Proc. 14th GRETSI Symp. Signal Image Processing. Juan-les-Pins, France. Sept. 1993, pp. 1251-1254. [7] Y. Hua and T. K. Sarkar, "On SVD for estimating generalized eigenvalues of singular matrix pencil in noise." IEEE Trans. Signal Processing. vol. 39. pp. 892-900. Apr. 1991. [8] K. C. Huamg and C. C. Yeh, "A unitary transformation method for angle-of-arrival estimation:' IEEE Trans. Signal Processing, vol, 39. pp. 975-977, Apr. 1991. [9] _ _ . "Adaptive beamforming with conjugate symmetric weights:' IEEE Trans. Antenna. Propagat.. vol. 39, pp. 92fr-932 , July 1991. [10] A. Lee, "Cenrrohermitian and skew-centrohermitian matrices." Linear Algebra and its Applications, vol. 29. pp. 205-210, 1980. [11] K. J. R. Liu. D. P. O'Leary, G. W. Stewart. and Y. J. J. Wu, "An adaptive ESPRIT based on URY decomposition:' in Proc. IEEE Int. Conf. Acoust.. Speech. Signal Processing, Minneapolis, MN. Apr. 1993. pp. 37~O, vol. IV. [12] B. Ottersten, M. Viberg, and T. Kailath, "Performance analysis of the total least squares ESPRIT algorithm," IEEE Trans. Signal Processing, vol. 39. pp. 1122-1135, May 1991. [13] S. U. Pillai and B. H. Kwon, "Forward/backward spatial smoothing techniques for coherent signal identification," IEEE Trans. Acoust., Speech. Signal Processing, vol. 37, pp. 8-15, Jan. 1989. [14] B. D. Rao and K. V. S. Hari, "Weighted subspace methods and spatial smoothing: Analysis and comparison," IEEE Trans. Signal Processing, vol. 41, pp. 788-803, Feb. 1993. [15] R. Roy and T. Kailath, "ESPRIT-Estimation of signal parameters via rotational invariance techniques," in Signal Processing Part II: Control Theory and Applications (L. Auslander, F. A. Grunbaurn, J. W. Helton, T. Kailath, P. Khargonekar, and S. Mitter, Eds.). Berlin, Vienna, New York: Springer-Verlag, 1990, pp. 369-411. [16] R. Roy, A. Paulraj, and T. Kailath, "ESPRIT-A subspace rotation approach to estimation of parameters of cisoids in noise," IEEE Trans.
[24]
296
Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1340-1342, Oct. 1986. R. H. Roy, "ESPRIT-Estimation of Signal Parameters via Rotational Invariance Techniques," Ph.D. thesis, Stanford Univ., Stanford, CA, Aug. 1987. G. W. Stewart, "An updating algorithm for subspace tracking," IEEE Trans. Signal Processing, vol. 40, pp. 1535-1541, June 1992. A. J. van der Veen, "A Schur method for low-rank matrix approximation," SIAM 1. Matrix Anal. Api., 1994, accepted for publication. A. J. van der Veen, E. F. Deprettere, and A. L. Swindlehurst, "Subspacebased signal analysis using singular value decomposition," Proc. IEEE, vol. 81, 1277-1308, Sept. 1993. S. Van Huffel and J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis Frontiers in Applied Mathematics, Vol. 9. Philadelphia, PA: Soc. Ind. and Applied Math., 1991. G. Xu, R. H. Roy, and T. Kailath, "Detection of number of sources via exploitation of centro-symmetry property," IEEE Trans. Signal Processing, vol. 42, pp. 102-112, Jan. 1994. M. D. Zoltowski, M. Haardt, and C. P. Mathews, "Closed-form angle estimation with rectangular arrays in element space or beamspace via Unitary ESPRIT," IEEE Trans. Signal Processing, July 1994, submitted for publication. M. D. Zoltowski, G. M. Kautz, and S. D. Silverstein, "Bearnspace RootMUSIC," IEEE Trans. Signal Processing, vol. 41, pp. 344-364, Jan. 1993.
rr-
Joint Angle and Delay Estimation (JADE) for Multipath Signals Arriving at an Antenna Array Michaela C. Vanderveen, Constantinos B. Papadias, Student Member, IEEE, and Arogyaswami Paulraj, Fellow, IEEE
Abstract- We propose a novel subspace approach to estimate the angles-of-arrival and delays of multipath signals from digitally modulated sources arriving at an antenna array. Our method uses a collection of estimates of a space-time vector channel. The Cramer-Rao bound (CRB) and simulations are provided.
I.
I
INTRODUCTION
N WIRELESS communications, mobiles emit signals that arrive at a base station via multiple paths. Estimating each path's angle-of-arrival (AOA) and propagation delay is necessary for several applications, such as mobile localization for emergency services. It is in fact a classical radar problem. This work proposes a novel approach (JADE) to estimating the AOA's and delays of the multipath signals using a collection of space-time channel estimates (analogous to snapshots in subspace methods for AOA estimation) which have constant parameters of interest but different path fade amplitudes. JADE can work in cases when the number of paths exceeds the number of antennas, unlike the traditional MUSIC and ESPRIT algorithms [1], [2]. II. DATA
MODEL
We focus on the case for a single user first and show later how this approach can be extended to multiple users. The received baseband signal at the ith element of an m-element antenna array is given by
L a'i(8Z)f3l(t)r(t - TI) + ni(t) L
Xi(t) ==
(1)
signals into a vector x( t) such that L
X ( t)
==
L a (8
Manuscript received September 12, 1996. The associate editor coordinating the review of this letter and approving it for publication was Dr. Y. Bar-Ness. M. C. Vanderveen is with Scientific Computing Program, Stanford University, Stanford, CA 94305-9025 USA (e-mail: [email protected]). C. B. Papadias and A. Paulraj are with Information Systems Laboratory, Stanford, CA 94305 USA. Publisher Item Identifier S 1089-7798(97)01345-8.
(t) r (t -
Tl)
+ n (t ) .
(2)
We sample this signal at the symbol rate (i.e., instants kT). Let P be the number of the symbol-spaced samples of the channel impulse response, P == 28 + M d , where 28T is the symbol waveform duration and M d is the maximum integer path delay. We obtain
x(k) == Hs(k)
+ n(k)
(3)
where the jth element of the vector s( k) of data is b(k + j - 8) and H is an m x P channel matrix capturing the effects of the array response, delay, symbol waveform, and path fading, and taking the form:
Md
H == [a(8 1 )
...
a(BL ) ]
131 [
(4)
==: A(8) D G(r)T
where g( Ti) == [g((Md +8- 1)T - Ti ) ... g((M d +8- P )T - Ti )] is a P-Iong row vector of samples of g( t - Ti)' Since we assume the path fadings to be constant within a data burst, we have suppressed their dependence on the sampling instant k. We assume we know the number of multipaths L,l the maximum path delay M d , the modulation waveform gC), and the structure of the array response at),
[=1
where L is the number of multipaths, a, (8z) is the response of the ith antenna to the lth path arriving from angle 8l , 13l (t) is the complex envelope of the path fading, Tl is the path delay, n, (t) is the additive noise, and r (.) is the transmitted signal, given by r(t) == Li b(i)g(t - iT), where {b(i)} is the sequence of data bits, g(t) is the modulation waveform, and T is the symbol period. We collect all the sensor responses to 8l into an m-element vector a(el) == [al (ez) ... am (Oz)]T and similarly the received
l ) f3l
l==1
III. JADE Let h == vect(H) be a vector of length mP obtained by taking the transpose of each row of the matrix H and stacking it below the transpose of the previous row. Then
h == (A(O)
0
G(r)) diag(D) ==:
U((J~
r){3
(5)
where 0 denotes the Khatri-Rao (columnwise Kronecker) matrix product (that is, the lth column of U(8,r) is a(el ) @ g(TZ), where 0 denotes the Kronecker product). The m.P x L matrix U (8, r) is called the space-time matrix, and is parametrized by the ADA's and the path delays. The vector u( (), T) == a( B) Q9 g( T) is called space-time response vector to a path of unit amplitude arriving at angle 0 with delay T. As e 1 The number of multi paths can be estimated from the data matrix, but this is beyond the scope of this paper.
Reprinted from IEEE Communications Letters, Vol. 1, No.1, pp. 12-14, January 1997.
297
varies over the range of angles and r varies over the range of delays, u( 0, r) traces a multidimensional space-time manifold . In this paper, the radio channel from the mobile to the antenna array is time-slotted, modeled after the GSM standard. The channel H from the mobile to the antenna array can thus be assumed to be constant over each time slot, but it varies from one time slot to the next. This variation is due to the changing complex fadings (3z. However, the AOA's 01 and delays ri are not changing significantly from each time slot to the next, and thus we take U((}, r) to be constant over a few time slots. This assumption is reasonable in practice because we consider only a relatively small number of time slots and thus during this very short time the mobile, which is far away, appears to be almost stationary with respect to the base station . The first step in our approach consists of estimating the channel impulse response from the user to the antenna array. This can be accomplished by using training bits or blindly . We collect data from M consecutive time slots and use it to obtain estimates of H. If we let q be the time slot index, our estimates Y q of the true channel H, take the form:
q= 1, ·· · ,M
(6)
where V q is the estimation noise matrix . Applying the vecu-) operation yields, with the obvious notation,
q= 1, · ·· ,M or, if we let Y
= [Yl,"
"
YM]mPxM
Y = U((}, r)B
(7)
and similarly for B , V,
+V .
(8)
The second step in our approach consists of estimating the 2£ parameters of interest, namely 01' sand rl' s, and eliminating the £M nuisance parameters (31 'S. In other words, given the noisy estimated matrix Y and the known structure of the space-time matrix U, we seek the desired parameters (} , r according to (8). We assume that the space-time manifold does not have any ambiguities , therefore, leading to unique estimates (this is the case whenever either the array or delay manifold are unambiguous) . In order for an angle-delay subspace to exist , we also need U to be a tall matrix (i.e., L < mP). It can also be easily shown that the space-time channel matrix U corresponding to a collection of L distinct paths is full column rank. IV. ALGORITHMS AND THE CRAMER-RAO BOUND The second step in our approach consists of estimating the multipath parameters (}, -r from the estimated channel Y, using (8). Among the various ways to solve this problem, we focus on two of them.
e
30
AOA(O EGI
Fig. I.
MUSIC-like spectrum.
'Ir '~ , -
We will assume that the estimation noise V is white and Gaussian, a fact which can be readily proved for the case of nonblind channel estimation. The entries of the complex fading matrix B can be modeled as unknown deterministic quantities. Then employing deterministic maximum likelihood techniques
- - --
--,
Fig. 2. Standard deviation of estimates versus CRB.
yields the following minimization problem :
(9)
min lIY - U ((} ,r)BII}·
9 ,T,B
It is well known that this is a separable optimization problem that reduces to (O ,f) = arg max tr PURy, where P u = U(U*U)-lU* , Ry = vv- 1M, and 0* denotes the complex-conjugate transpose. This search can be done using the damped Newton method .
B. JADE-MUSIC The technique described above involves a 2L-dimensional search and may thus be computationally prohibitive. A faster, though suboptimal, approach is based on the MUSIC algorithm and involves only a two-dimensional search. We know that the true space-time channel vector u( 0, r) is orthogonal to the noise subspace En , whose columns are the eigenvectors of Ry corresponding to the mP - L smallest eigenvalues. We thus look for peaks in the two-dimensional MUSIC spectrum (u*u)/u*EnE~u. The peaks should occur close to the true (0, r) coordinates (see Fig. 1). C. Cramer-Rao Bound
The Cramer-Rao bound (CRB) provides a lower bound on the variance of any unbiased estimator. The bound for AOA estimation (without delay spread) was derived in [3] and is readily adapted to the present situation. Assuming the path fadings to be deterministic but unknown, we obtain for the model in (8) that
CRB(B ,r)
A. Maximum Likelihood
DEUt.V [T)
M
= ~ { ~real
(B;F*P&FB q )
}-l
(10)
where (j is the variance of the estimation noise, Bq = h 0 diag(,Bq), F = [A'( B)oG (r), A(O)oG'(r)], and P& = I-Pu (prime denotes differentiation with respect to the individual parameters and all matrices are evaluated at the true parameter values) .
298
V. SIMULATION RESULTS AND EXTENSIONS
Computer simulations were run to demonstrate the performance of the JADE-MUSIC algorithm. We assume a single user, three multipaths, and a two-element antenna. The classical approaches to this problem, such as MUSIC and ESPRIT, will not yield satisfactory results, since the number of antennas is smaller than the number of multipaths. However, the JADE algorithm can handle this case successfully. The AOA's are [-5, 0, 20]0 relative to the array broadside and the corresponding path delays are [1.0,0.7. 2.0]T seconds, where T is normalized to one. The collected data are corrupted by noise with inverse variance 1/ (J" = 5, 10, 15, and 20 dB. The modulation waveform is a raised cosine pulse with excess bandwidth 0.35, assumed to be zero outside the interval [-3,3). We sample at rate T /2 (the purpose of oversampling is to provide improved definition of the delay manifold). Data is collected over 20 time slots, and at each time slot the channel is estimated via least squares using 27 training bits. The experimental variance of the AOA and delay estimates is computed from 100 runs. The results are summarized in Fig. 2. The bias of the estimates was on the average 10/0-270/0 or their standard deviation (STD). Notice the STn is about 5 dB above the CRB.
A typical MUSIC spectrum for noise with inverse variance 20 dB is shown in Fig. 1. When we have more than one user in the same time slot. we can independently estimate the channel matrices H using each user' s unique (usually orthogonal) training signal. We can then proceed as above, with decoupled problems. If no training signals are available, we can still find the channel H for each user using blind methods, which exploit finite alphabet structures and oversampling [4].
REFERENCES [1] R. O. Schmidt, "A signal subspace approach to multiple emitter location and spectral estimation," Ph.D. dissertation, Stanford University, Stanford, CA, Nov. 1981. [2] R. Roy, A. Paulraj, and T. Kailath, "ESPRIT-A subspace rotation approach to estimation of parameters of cisoids in noise," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1340-1342, Oct. 1986. [3] P. Stoica and A. Nehorai, "MUSIC, maximum likelihood and Cramer-Rae bound," IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 720-741, May 1989. [4] A. van der Veen, S. Talwar, and A. Paulraj, . "Blind identification of FIR channels carrying multiple finite alphabet signals," in Proc. IEEE ICASSP, vol. 2, 1995, pp. 1213-1216.
299
Chapter 3
Performance Issues
A
consequence of the unprecedented worldwide expansion of mobile communication systems is the fact that networks are rapidly running out of system capacity. Systems such as UMTS have promised that they will provide higher value services, and as a result this ultimately means higher data rates and more capacity problems. In this scenario, new technologies and in particular adaptive antennas have emerged as strong candidates to fulfill the requirement for increased spectrum efficiency. The benefits that can be achieved with several adaptive antenna methods but also the challenges that each method faces are discussed in the overview paper at the beginning of this chapter. Each of the following papers deals with significant advances in the field and provides insight into the perfor-
mance that can be achieved when adaptive antennas are employed in different wireless communication systems. Issues considered include increased capacity and coverage, better signal quality, support of value-added services such as user location applications, more efficient handover, and so on. Nevertheless, it has to be emphasized that different communication systems will exploit different advantages or mixtures of advantages offered by adaptive antennas, depending on the maturity of the underlying system. Work that deals with important factors that affect the performance of different adaptive antenna systems, such as mutual coupling, fading correlation, multipath conditions in different operational environments, and frequency decorrelation in duplex systems, is also presented.
301
Smart antennas for mobile communication systems: benefits and challenges by G. V. Tsoulos Spatial filtering using adaptive or smart antennas has emerged as a promising technique to improve the performance of cellular mobile systems. This paper presents an overview of smart antennas in terms of key characteristics, options, challenges and benefits, in the context of current, but also with a view towards future, generation personal communication systems. 1 Introduction Over the last fewyears the demand for service provision via the wireless communication bearer has risen beyond all expectations. If the extraordinary fact that worldwide some half a billion subscribers to mobile networks are predicted by the year 2000 is put in the context of third generation system requirements (UMTS, IMT2000)I, then the most demanding technological challenge emerges: the need to increase the spectrum efficiency of wireless networks. While great effort in current (second) generation wireless communication systems has been directed towards the development of modulation, coding and protocols, antenna-related technology has received significantly less attention up to now. To achieve the ambitiou s requirements introduced for future wireless systems, new
' AEF BER
S BSRF
eDMA eIR DCS1800
I
'intelligent' or 'se lf-configured' and highly efficient systems will most certainly be required. In the pursuit of schemes that will solve these problems, attention has recently turned to spatial filtering methods using advanced antenna techniques : adaptive or smart antennas. Filtering in the space domain can separate spectrally and temporally overlapping signals from multiple mobile units, and hence the spatial dimension can be exploited as a hybrid multiple access technique complementing FDMA, TDMA and eDMA
2
Smart antenna approaches
Three main categories of smart antennas may be defined bas ed on how they produce their response: switched beam, direction finding , optimum combining. The
Abbreviations 1&-54/95
= area improvement factor = bit error ratiq = base station = Base station reduction factor
. LOS MMSE MS NLOS
= code division multiple access
carrier-to-interference ratio = digital communication system at 1800 MHz dBd = decibels above dipole D&-eDMA = direct sequence CDMA = digitalsignal processing DSP DTX = discontinuous transmission = effective isotropic radiated power EIRP FCC = Federal Communications Commission FDD =frequency division duplex =frequency division multiple access FDMA = Global System for Mobile GSM IMT2000 = International Mobile Telecommunications for the year 2000 =
PA
REF
SDMA SFIR
SINR SNR
j
TDD .TDMA UMTS
interim standard (of the American National Standards Iristitute and referring to an accepted industry. standard) for TDMA (54) or CDMA (95) -Jine of sight ='minimum mean square error = mobile station . = non line of sight = power amplifier = range extension factor = space division multiple access = spatial filtering for interference ' reduction = signal-to-interferenee-plus-noise ratio = signal-to-noise ratio . :: tim~ division duplex = time division multiple access = Universal Mobile Telecommunications System
=
Repinted with permission from Electronics and Communication Engineering Journal, G. V. Tsoulo s, " Smart Antennas for Mobile Communication Systems" Vol. II , No. 2, pp. 84- 94, April 1999. © 1999 by Institute of Electric al Engineers.
303
Table 1: Advantages and disadvantages of different smart antenna approaches Advantages
Disadvantages
Switched beams
Easilydeployed Tracking at beam switching rate
low gain between beams limited interference suppression False locking with shadowing, interference and wide angular spread
Direction finding
Tracking at angular change rate No reference signal required Easier downlink beamforming
lower overall CIR gain Susceptible to signal inaccuracies, needs calibration Concept is not applicable to small cell NlOS environments
Optimum combining
Optimum SINR gain No need for accurate calibration Performs well even when the number of elements is smaller than the number of signals (MMSE approach)
Difficult downlink beamforming with FDD and fast TDD Needs good reference signal for optimum performance Requires high update rates
reuse pattern employed (users with frequency fk in Fig. la), and through interference reduction in the spatial domain to achieve a lower cell repeat pattern (reuse distance) 2,3,8.13. With SDMAan adaptive antenna system is deployed in such a way that multiple users within the same cell can operate on the same time (tJ and frequency ifJ channel by exploiting the spatial separation of the users (see Fig. Ib)12-17. This concept can be seen as a dynamic (as opposed to fixed) sectorisation approach in which each mobile defines its own sector as it moves. The major advantages and disadvantages of these techniques are summarised in Table 22.3,12-17.
switched beam method employs a grid of beams and usually chooses the beam which gives the best SNR For direction-finding techniques (or spatial reference techniques, as they are also called) all the processing is focused on the acquisition and tracking of one parameter, the directions of the users. With optimum combining the output signal-to-interference-plus-noise ratio (SINR) is the parameter optimised. Table 1 summarises the most important advantages and disadvantages of these techniques, as presented in the literature (see, for example, References 2-13) . When employing a smart antenna with one of the above mentioned methods in the general sense of a spatial filter, one further categorisation can be recognised: spatial filtering for interference reduction (SFIR) (this concept is shown in Fig. la) and space division multiple access (SDMA) (see Fig. 1b). With SFIR the goal is, as with a traditional omnidirectional or sectorised antenna, to support one user in each of the co-channel cells of the
3 Some constraints A smart antenna system relies heavily on the spatial characteristics ofthe operational environment" to improve the output signal. For big cell structures (macrocells), there are three main scattering sources'? (Fig. 2):
\
\
/
b
a
Fig. 1 (a) SFIR and (b) SOMAconcepts
304
Table 2: Advantages and disadvantages of SOMA and SFIR
An interesting point is that in big cell environments the angular change ofthe incoming signal depends generally on the velocity and the distance of the mobile from the base scalterers remote scalterers local to the station (this is true when the multipath due to remote base station scatterers (Fig. 2) either doesn 't exist or can be ignored by the adaptive algorithm). As an example, let us address the point concerning the angular change of incoming signals in a big cell environment in the context of a GSM system application. GSM is a time division multiple access radio standard with a slotted waveform structure having a 577 ns burst period and a 4·6 ms period between consecutive bursts. In this case , even with a worst case scenario of a mobile user travelling at 200 km/h (e.g. fast train) around a circle of radius 1 km, the angular velocity is less than scalt erers local to the 3·1°s- 1, which corresponds for example to a 0·014° angular mobile change between two consecutive GSM bursts, as shown in Fig. 3. Since the angular change of the signal dictates the requir ed update rate of the main beam then, obviously, the Fig. 2 Large cell propagation environment wider the beams are, the smaller the update rates required. Furthermore, since it is possible to reduce substantially the (a) Scatterers local to the mobile: if the mobile is moving, transmission rate either in the up or the down link when these cause Doppler spread (i.e. time selectivity), and there is littleor no speech (on average each user speaks less they also cause small delay and angle spreads (usually than half of the time during a normal conversation), many less than 10°). existing systems employ discontinuous transmission (DTX) (b) Scatterers local to the base station : these contribute in order to exploit this feature". Because discontinuous multipath rays with small delay spread and large transmission effectively reduces the information rate (e.g., angular spread. in GSM,from 1burst every 4·6ms, as mentioned above - no (c) Remote scatterers: these cause independent fading on paths and contribute multipath with large delay spread DTX - to 1 burst every 120 ms - maximum DTX), this (frequency selective fading) and large angular spread. would mean that the angular change rate would increase considerably, as shown in Fig. 3 (from 0·014° per GSM burst to -0 ·4° per GSM burst, i.e. by 27 - - - withou t DTX times approximately). As a result, - - withDTX additional measures to compensate for this increase willbe necessary" 200 kmlh (e.g. periodic transmission of 100 kmlh dummy bursts) , especially for tracking fast-movingusers. In small cells on the other hand , the propagation scenario is quite ____ 200 kmlh -- - different Here, there are many 100k~------ ------- -----local scatterers in close vicinity to 10-3 L-.. -'----'----'-' the mobile and the base station 4 5 1 3 2 which result in much wider angular distance. km spread, low delay spread and moderate (pedestrian) to medium (mobile) Doppler spread. The Fig. 3 Change in direction as a function of distance and speed for large cells, previously used mapping through without OTX (1 burst/4·6 ms) and with maximum OTX (1 burst/120 ms)
--- -- -
305
the spatial response (power versus angle) or 'spatial signature' of a big cell environment to a single 'dominant' direction cannot be done anymore because there is no smart link budget balancing connection between the direction of the signals and the physical location of the user, except from I more efficient the direct paths in LOScases".The power control 'spatial signature' itself has to be ~I-----'I~ benellts used for user location, but in this ~ case there is no connection with the supportof wer redUdi00 value added services physical user location. I Another very important issue for adaptive antennas is the downlink transmission . In time division duplex (IDD) systems, the up and down links can be considered reciprocal, provided that the channel characteristics have not changed considerably between the Fig. 4 Operational benefits achieved with a smart antenna receive and transmit slots, i.e. there is limited user movement between transmission and matrix) is employed. The reason for such processing is reception. Under this condition, the weights calculated by that a subspace is a much more stable entity than a the adaptive antenna for the uplink can also be used for the channel vector and hence, instead of tracking the downlink to achieve spatially selective filtering. Application instantaneous channel, subspace beamforming of adaptive antennas in the downlink for frequency division methods track the subspace structure. duplex (FDD) systems seems to be one of the challenges related to this technology. The fundamental difference is Other methods which exploit particular characteristics of that in FDD systems the downlinkfading characteristics are the air interface method employed can also be used to independent of the uplink characteristics due to the compensate for the possible imbalance caused by the frequency difference (typically-40 MHz), while the angles downlink problem . For example, one such method could ofarrival of the multipath rays remain the same. For current be adaptive resource allocation. This effectively means FDD systems such as GSM, DCS1800, 15-54, 15-95, the allocating more radio channels (fDMA) or bandwidth processing performed in the uplink can not be exploited (CDMA)for the downlink so that the benefits are balanced directly in the downlink without any additional processing. between the two links. Furthermore, techniques which Several approaches have been proposed in the literature convert one form of diversity at the base station to another form at the mobile space/time, space/path, which attempt to solve the downlink problem; some ofthem space/coded modulation, etc.' - can also offer alternative are: solutions for the downlink problem. (a) transmit diversity", where multiple base station Different air interface techniques have different impacts antennas transmit delayed versions of the signal in on the design and the optimum approach for smart order to create frequency selective fading at a single antennas, mainly because of the different interference antenna at the receiver (mobile), which uses an scenarios. With TDMA systems, a frequency reuse pattern equaliser or RAKEreceiver to obtain diversity against is usually employed, which leads to a small number of fading strong interferers for both the up and down links (usually 2-4) . With CDMAsystems a total (1) frequency reuse plan (b) transmission with feedback from the mobile". Here feedback from the mobile to the base station is used to is assumed, which effectivelyleads to many but, due to the spectrum spreading employed, weak interferers in the estimate the impulse response of the radio channel, uplink and usually 6-12 weak interferers in the downlink. hence traditional diversity schemes or beamforming Possibly the most challenging problem related to smart can be employed. antennas is their practical implementation. Full (c) estimation of the transmit spatial correlation matrix exploitation of all the operational benefits that will be from parameterisation ofthe receive spatial correlation described in the next section would mean increased matrix. One such parameterisation can be performed complexity for the system and hence would require a fully for the angles of arrival, which can subsequently be integrated approach in an 'intelligent' system. RF and DSP calculated by direction-finding algorithms". technology will have to evolve further before this can be Cd) subspace processing" (a similar idea to c). Whereas a achieved in a cost-effective manner. Nevertheless, two and b rely on the separation of the instantaneous facts indicate that this is still a promising and valid way channel vectors, with this method separation of the forward: first, the fact that there are partial" or efficient" channel subspaces (second order spatial correlation
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implementations that can be employed in real systems to reduce complexity and, second, the fact that by the time mobile communica tion systems are ready to fully support smart antennas (most probably with 3rd generation systems) the required technology will be available and matur e enough to fully support adaptive antennas" (assuming Moore's law). Furthermore, although current implementation and installation costs are thought to be high for smart antenna systems in comparison with traditional omnidirectional or sectorised scheme s, if the range-capacity gains and the other benefits that will be discussed in the next section are included in the overall cost calculations, then it becomes obvious that this technology is rath er economical even for today's systems.
4 Operational benefits with smart antennas Fig. 4 shows an overview of the operational benefits that can be achieved from the deployment of an adaptive antenna in a mobile communication network" Each of these benefits is discussed in greater detail below. Fig. 6 BER with an adaptive antenna as a function of the achieved beamwidth and average sidelobe level
The diversity gain offered by an antenna array reduces the fading of the radio signal and hence the power control requi rements are eased. With information about the location and speed of a mobile user, the decision as to which cell to hand the user to is a much easier task. Combination of this kind of information with the handover process can ultimately lead to a 'smart' instead of 'soft' or 'hard' handover. Furthermore, in order to support the soft handoff quality enhancement technique for DS-CDMA within a mixed cell environment, all cell types have to operate on identical carri er frequencies. One possible way of achieving the RF power balancing within each area of a mixed cell scenario needed to provide seamless handover and simultaneously avoiding the near-far effect is to exploit the spatial filtering properties offered by an adaptive antenna" .
Support ofvalue added services Better signal quality - higher data rates
In noise or interference limited environments, the gain that can be achieved with an antenna array can be exchanged for signal quality enhancement, i.e. lower BER. This is demonstrated in Figs. 5 and 6. Fig. 5 shows the probability of detection for the case where a matched filter (a detector which maximises the probability of detection with additive noise) is employed for each array element. Th e probability of detection in this case is 29: (1)
wher e Q(.) is the Q function, PD is the probability of detection, PF is the threshold probability of false alarm, M is the number of elements and SNR the signal-to-noise ratio. From Fig. 5 it can be seen that the detection performance can be significantly enhanced with an
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antenna array. For example, for 0 dB SNR, there is 10% probability of detection with a single element, 20%with two elements, 35% with four, 65% with eight, 85%with twelve, 95%with sixteen and almost 100%with twenty. Fig. 6 shows another example, this time for a DS-CDMA system, where the following approximate formula for the BERis used" ;
omnidirectional antenna: power = -39 dBm RMS delay spread 24 ns
=
-
RMS delay spread, ns power,dBm
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where SF is the spreading factor (64), SIR omni is the signal-to-interference ratio with an omnidirectional antenna, and
BW + SLL (BW) k = 360 1 - 360
(3)
with BWand SLL the ideal or effective* beamwidth (degrees) and average sidelobe level (linear values) , respectively. Fig. 6 shows a plot of eqn. 2 for the case of 100 users per cell (N) in a multitier system (4 tiers) for values of a achieved beamwidth and sidelobe level between 10° and 30°, and -10 dB and -20 dB, respectively. As an example from Fig. 6, if the smart antenna that is -20 120 employed at the base station of the central cell can achieve a radiation - 25 100 c pattern with a beamwidth of 20° (ideal or 80 -0 effective), then an improvement of 1-7 '~a." orders of magnitude for the BER can be 60 >~ accomplished with average sidelobe "C 40 en levels (ideal or effective) between -10 dB :E II: and -20 dB, respectively . Obviously, the --45 20 sidelobe level achieved has such a -50 L...-_-'-_-..1._ _. l . - _ - ' - _ - - - l _ - - - l 0 profound effect on the BER results o 60 120 180 240 300 360 because it reduces the interference angle, degrees received at that region and hence improves the overall SIR, as seen from b eqns. 2 and 3. Furthermore, through spatial filtering of the multipath at th e base Fig. 7 RMS delay spread red uctio n w it h adaptive an ten nas in (a) microcells station (BS) and/or the mobile station an d (b ) picocells (MS), the RMS delay spread of the channel can be reduced, which is a positive thing if the pattern (15 dBd, 26° 3 dB beamwidth) from 0° to 360° in objective is to achieve higher data rates. This advantage steps of 5° and for each step calculating the power and the RMS delay spread. Th e results are for an LOS (line-ofincreases in environments where the angular spread of sight) point. The relationship between high power and the multipaths is wide, i.e. in small cell scenarios. Examples of RMS delay spread reduction with smart small RMS delay spread can be seen in these figures (e.g. antennas in outdoor micro- and indoor picocells are 270°-300 ° for Fig. 7a and 180°-210° for Fig. 7b), shown in Fig. 7. The method used to produce the results although it must be mentioned that it is not always shown in Fig. 7 comprised using the ray tracing tools straightforward, especially for NLOS situations" . In from References 31 and 32 to produce the impulse terms of RMS delay spread , the directional antenna has an advantage, both in LOS and NLOS (non line of sight) response of the radio channel, then steering the antenna situations, when itis steered towards the direction of high "The word effective implies that the value for the parameter under power. Whe n the main beam of the directional pattern is discussion is what can be achieved if the scattering and signal mispointed towards other than the peak power directions, mismatching effects are considered, according to Reference 30. (/)
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desired user
-e-
optimise the desired outcom e. The question here is not so much one of simply reducing the cost of the calls (which would increase the network traffic while producing the same revenu e), but instead one of improving in the most efficient manner the capacity of the system. Also, this technique enables tariff plans to be tailored to individual user needs, e.g. by providing low-cost zones when the user is in his preferred zone .
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more multipath rays are received from the base station, and this increases the RMS delay spread .
Ability to support user location for emergency calls
(911/999) The FCC has recently passed a ruling which requires that all network operators must provide this service by the year 2001, with 125 m accuracy for 67%of the area cover ed by their network (initially). Adaptive antennas can provide user location information (direction-finding methods are naturally the most suitable for this application). Th e user location concept is demonstrated with the example shown in Fig. 8, for a DS-CDMA system employing an adaptive array with eight element s. Here, the desired user is moving linearly at _20°, two interferers are moving almost linearly at _60° and 30° and two interferer s start from 0° and 60° and finish at 60° and 0°, respectively . It can be seen that the direction of the desired user can be found for the whole duration of the experiment (each point represents 1000 samples) .
Location offraudperpetrators
A significant porti on of the wirele ss oper ators' revenu es is lost every year du e to fraud . Many techn ologie s currentl y applied in ord er to solve thi s probl em are effective in identifying the fact that fraud is occurring but none of them provides the operator with a long-term solution, i.e. the ability to remove the criminal from the activity, instead of deactivating the phone numb er from th e switch. Since adaptive antennas can provide user locati on information, this now becomes feasible.
Location-sensitive billing
Adding a third dimension - location - to the curre nt two dimensions for charging rates (usage against time of day, i.e. peak, off-peak) will provide an operator with the ability to control its network by encouraging (or discouraging) any type of usag e behaviour, and ultimately
On-demand location-specific services
Such services could include roadside assistance, real-time traffic updates, tourist information and electronic yellow pages, e.g. with local entertainment and dining information .
Vehicle andfleet management
Current vehicle management techniques usually suffer from the cost disadvantage of having a single-purpose transmitter network. Location and subsequent navigation of a vehicle is something that can be done with an adaptive antenna. Possible extensions to this idea could also be package monitoring and stolen vehicle recovery.
Optimum smartsystem planning Today the planning of a network is typically theoretical and once implemented is verified through limited testing . One important element missing from this planning process is that designers never really know where wireless devices are located when a specific cell site is handling a probl ematic call. Obviously this situation can be greatly improved with the user location capabilities of adaptive ant enna s. Furthermore, with an adaptive antenna an advanced intelligent network, which will lessen the planning burd en and improve the network efficiency based on a numb er of criteria optimised through the intelligence provided by the base station antenna , becom es possible. Optimisation param eters include intra-and intercell reus e planning, link balancing requ irements, handov er requ irements, traffic den sity and many others. When such intelligence at the base station is enhanced by intelligence at the network level, self-configuration and overall optimisation becomes viable.
Coverage extension Adaptive antennas can increase the network coverage through antenna directivity and interference redu ction. Th e gain G (assuming an antenna efficiency of 100%and no mutual coupling) that can be achieved with an antenna array of M elements is (it can also be proved that this represents the SNR improvement): (4)
309
The additional gain (compared to the standard element gain) can obviously be exploited for range extension. With small angular spread and single slope path loss with exponent n, the range extension factor, REF~ can be calculated as:
REF= r2 =M 1/ n
(5)
rl
or, with the Hata path loss modef'': (6)
where rl and r2 are the default (with a single element) and extended (with multiple elements) ranges, respectively. The area improvement factor, AEF, is: (7)
The inverse of the area extension factor represents the reduction factor in the number of base stations (BSRF) needed to serve the same area as with a single element Obviously, one can get the full range extension from the directivity of the array when the angular spread of the signal is less than the antenna beamwidth (otherwise the desired signal is reduced). Fig. 9 shows that with ten antenna elements (10 dB of SNR improvement) the range can be almost doubled and the area almost quadrupled, or the number of base stations can be reduced to almost 25% of the original number. Also, it can be seen that with smaller path loss exponents (the curve for n=3 in Fig. 9) the improvement factors will be higher, i.e. the improvements will be higher in LOS as compared to NLOS scenarios. Also, the calculated range extension with the Hata path loss model is between the predictions for path loss exponents of3 and 4.
Reduced transmitpower Limitations on the maximum EIRP (effective isotropic radiated power) introduced by standards could mean that the array gain cannot be exploited for coverage or area extension. Also, recent public worries over health issues originating from exposure to electromagnetic radiation (no matter how reasonable or unreasonable this may be) will almost certainly force the governing/standardisation bodies to change the current radiation standards in the future and adopt a lower emission policy, in particular for the mobile handsets. In such cases, one could exploit the base station array gain to reduce the power transmitted by the mobile. This reduction is also beneficial because it relaxes the battery requirements and hence it can increase the talk times or reduce the size/weight of the handsets. Furthermore, if the received power requirement at the mobile remains the same with an M element array at the base station, then the output power from the base station power amplifiers (PAs) can be reduced by M -2, which willreduce the total transmitted power from the array by M - 1• The explanation for this (see, for example, Reference 34) is that the base station nowemploysMPAs and can simultaneously exploitthe directivityofthe array. Ten antenna elements will reduce the totaltransmitted power by 10dB,whilethe output of each PAwillbe reduced by 20 dB. The latter has obvious cost implications, since the high-power amplifiers are expensive hardware components ofa system.
Smart link budget balancing Examination of the up/down link budgets reveals an imbalance which originates from the difference in the power amplifiers employed at the two ends. This imbalance is usually around 10-20 dB and its actual value depends on the PA used at the base station site. As an example consider a case where there is a 13 dBW PA with a 17dBi antenna, giving 1kW EIRP at the base station (this Fig. 9 Range, area and base station reduction factors as a function of the number of elements and SNR improvement
SNR improvement
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example ignores losses from cables, connectors, mismatching, etc. and fading margins at the mobile and the base station)", Th e mobile might be able to achieve 1 W EIRP, which , togeth er with the 17 dBi from the base station antenna gain , willproduce a deficit of more than 13 dB and, hence, a link imbalance. Given that the antenna gains can be exploited at both ends , it becom es obvious that the up and down links can be balanced by using the additional directivity offered by an adaptive antenna at the base station site. For the case of a power amplifier with a dBW output, S dB gain from highsensitivity receivers and other improvements (e.g. employing diver sity at the mobile), and EIRPMS dBW at the mobile, the gain from the adaptive antenna at the base station in order to balance the two links should ideally be G (dB) = a - S - EIRPMs. By exploiting the adaptive capabilities of a smart antenna this concept can be extended and 'smart' link management can lead to dynamically optimised power budgets for different scenarios.
Increased capacity Adaptive antennas can provide capacity increases through several mechani sms. Starting with Shannon's expression for the capacity of a channel with bandwidth Wand with additive Gaussian noise, C = Wlogz (1 + SNR) (bit!s)
(8)
it can be shown" that with an antenna array which divides the power equally into M parts and sends it into M parallel independent channels (M is the numb er of antenna elements), the capacity in one channel is now:
Ca"aY=~ flOgZ(1+ S~R)]= MWlogz (1+S~R)
(bit/ s) (9)
Fig. 10 shows the spectrum efficiency (bit!s/Hz) as a function of the SNR and the numb er of antenna elements.
Apart from the increase of the spectrum efficiency as the numb er of array elements increase, it can be see n that as the numb er of elements becomes very large, the capacity of the channel becomes simply proportional to SNR. Extending this concept by multiplexing M tran smit and N receive channels can offer significant gains. In Refer ence 36 it is shown that the theoretical capacity is now (flat fading , stationary propagation environment):
where 'del' means determinant, I N is the NxN identity matrix and H "T is the conjugate transpose of the normali sed (spatial average power loss normalis ed to unity) channel matrix (the ijth element of which is the tran sfer function of the jth transmitter to the ith receiver) . In Reference 36 it is shown that with 8 tran smit and 8 receive antennas and 1% outage probability with 21 dB average SNR at each receiving element, the maximum capacity is more than 40 times that of a single antenna element at the tran smitter and the receiver. Furthermore, a layered space/ time architecture that can achieve such capacities is reported in Reference 36. With an omnidirectional antenna only a small portion of the transmitted power is actually received by the intended user , while at the same time most of the tran smitted power constitutes interference for other potential users. Hence, the problem is dual: not only is this kind of omnidirectional communication inefficient in terms of power, but also in terms of capacity. One practical way to increase capacity is to decr ease the radiated power associated with directive tran smission in combination with the lower mobile emission levels possible with directive reception. In othe r word s, by exploiting the spatial filtering that an antenna array offers, it is possible with the help of an adaptive meth od to confine the radio energy associated with a given mobile to a small addressed volume, thu s reducing interfer ence experienced from and to co-ehanne1 users. This is straightforward for CDMA systems as user s share 311
Table 3: Advantages and disadvantages of smart antennas and small cells for capacity improvement
discussed in the paper, and are summarised below:
the same bandwidth, and in a TDMA system it can be achieved through reduced reuse patterns or SDMA. Another approach to increase capacity is through dynamic reuse planning. With user location information, channels can be assigned dynamically to areas of high user density and hence better capacity can be achieved. The capacity enhancement achieved can be exploited in many ways, but one particular possible application is in reducing the need for very small cell sizes and consequently reducing the infrastructure costs. When considering the dilemma of whether to use adaptive antennas or smaller cells for capacity, the advantages/disadvantages shown in Table 3 can be mentioned'". Nevertheless, when the requirements of future generation, UMTS-type systems are considered in terms of the performance measures capacity, coverage and services, then it is almost certain that small cell structures will be deployed in order to fulfil these requirements, possibly in combination with some sort of smart antenna technology", 5
• • • • • • •
Conclusions
This paper has provided an overview of the potential benefits and challenges of applying smart-antenna technology to mobile communication systems. The basic advantages and disadvantages of two different approaches to smart antennas «a) SDMA/SFIR; (b) switched beam/direction finding/optimum combining), as they are presented in open literature, were highlighted. Then the importance of specific characteristics of the radio channel under different operational environments (e.g. large-small cells, indoor-outdoor) and interference scenarios (CDMA-TDMA) were discussed. The realisation of comparable up and down link gains along and the practical implementation of a fully integrated smart antenna system seem to be the two most challenging issues facing this technology at the moment. Although more efficient implementations will almost certainly be needed for future systems, partial implementations make the application of smart antennas to current generation systems possible. Furthermore, by the time that future generation systems are ready fully to support smart antennas, the development of RF and DSP technology will have reached such a level that complex implementations will be possible. Exploiting different characteristics of smart antennas can lead to several operational benefits for a communication system. These benefits have been
312
coverage extension increased capacity efficient power control!smart hand over support of value added services (better signal quality, higher data rates, user location) optimum/smart system planning reduced transmit power smart link budget balancing
Communication systems will exploit different advantages or mixtures of advantages offered by smart antennas depending on the maturity of the underlying system. Initially, for example, costs can be reduced by exploiting the range extension capabilities of smart antennas. Then, where there is a demand for increased capacity, costs can be further decreased by avoiding extensive use of small cells and instead exploiting the capability of smart antennas to increase capacity. Finally, more advanced systems (3rd generation) willbe able to benefit from smart antenna systems, but it is almost certain that more sophisticated space/time filtering approaches" will be necessary, especially as these systems become mature. Acknowledgments The author gratefully acknowledges the assistance of Professor J. P. McGeehan in every aspect of his research activities. Thanks are also due to Dr. G. Athanasiadou for her help and the provision of ray-tracing results. References 1 RAPELI, ]. : 'UMTS: targets, system concept, and standardisation in a global framework ', IEEE Pers. Commun., February 1995, 2 , (I) , pp. 20-28 2 TSOULOS, G.: 'Adaptive antenna technology' . lEE 3rd residential course on Digital Techniques in Radio Systems , University of Bristol, September 1997 3 MOGENSEN, P.: 'System aspects of adaptive antennas for GSM'. TSUNAMISeminar, Aalborg University, 1997 4 PAULRAJ, A : 'Smart antenna in wireless communications: technology overview'. 4th Workshop on Smart Antennas in Wireless Mobile Communications, Stanford University, 1997 5 WINTERS, L: 'Signal acquisition and tracking with adaptive arrays in the digital mobile radio system 15-54 with flat fading', IEEE Trans. Veh. Technol., 1993, vr-42, pp. 377-384 6 NAGUIB, A, PAULRAJ, A, and KAILATII, T.: 'Capacity improvement with base station antenna arrays in cellular CDMA', IEEETrans. Veh. Technol. , 1994, vr-43 , pp. 691-698
7 THOMPSON, ]., GRANT, P., and MULGREW, B.: 'Smart
antenna arrays for CDMA systems', IEEE Pers. Commun., October 1996, 3, (5), pp. 16-25. 8 ZE'ITERBERG, P., and OTIERSTEN, B.: The spectrum efficiency of a base station antenna array system for spatially selective transmission', IEEE Trans. Veh. Technol., 1995, VI'44, pp.651-660 9 KENNEDY, j., and SULLIVAN, M.: 'Direction finding and "smart antennas" using software radio architectures', IEEE Commun. Mag., May 1995, 33, (5), pp. 62-68. 10 LI, Y., FEUERSTEIN, M., and REUDINK, D.: 'Performance evaluation of a cellular base station multi beam antenna', IEEE Trans. Veh. Technol., 1997, Vf-46, pp. 1-9 11 HO, M., STUBER, G., and AUSTIN, M.: 'Performance of switched beam smart antennas for cellular radio systems', IEEE Trans. Veh. Technol., 1998, VI'-47, pp.lQ-19 12TSOULOS, G. et al.: Wireless personal communications for the 21st century: European technological advances in adaptive antennas', IEEE Commun. Mag., September 1997, 44, (9), pp. 102-109 13TSUNAMI Project Final Report, R2108/ERA/WP1.3/MR/P/ 096/b2,1996 14 DAM, H.: 'Smart antennas in the GSM system'. MSc thesis, Aalborg University, 1995 15TANGEMANN, M.: 'Near-far effects in adaptive SDMA system'. 6th Int. Symposium on Personal, Indoor and Mobile Radio Communications, 1995,Toronto, Canada, pp. 1293-1297 16 FUHL,]., and MOLISCH, A.: 'Capacity enhancement and BER in a combined SDMA/TDMA system'. Proc. 46th Vehicular Technology Conference, 1996,Atlanta, USA, 3, pp. 1481-1485 17XU, G. et al.: 'Experimental studies of space division multiple access schemes for spectral efficient wireless communications'. Proc. IEEE Int. Communications Coni 1st-5th May 1994, New Orleans, USA, pp. 800-804 ., 18WARD, C., SMITH, M., JEFFRIES, A., ADAMS, D., and HUDSON, J.: 'Characterising the radio propagation channel for smart antenna systems', Electron. Commun. Eng.]., August 1996,8, (4),pp.191-200 19 PAULRAj, A., and PAPADIAS, C.: 'Space-time processing for wireless communications', IEEE Signal Process. Mag., November 1997, pp. 49-83 20 MODLY, M., and PAUTET, M.: The GSM system for mobile communications' (Cell & Syst., France, 1992) 21 TSOULOS, G., and ATHANASIADOU, G.: 'Extrapolation of field trial results to UMTS'. TSUNAMI II Deliverable ' AC020/UOB/D1.9/DS/P/192/a1, 1998 22WINTERS, J.: 'Two signalling schemes for improving the error performance of frequency division duplex (FDD) transmission systems using transmitter antenna diversity'. 43rd IEEE
Vehicular Technology Conf., 18th-20th May 1993, Secaucus, NJ, USA,pp. 85-88 23 GERlACH, D., and PAULRAj, A.: 'Adaptive transmitting antenna arrays with feedback', IEEE Signal Process. Lett., October 1994, 1, (10), pp. 150-153 24 RAYLEIGH, G., DIGAVVI,S., JONES, V., and PAULRAj,A.: 'A blind adaptive transmit antenna algorithm for wireless communication'. Proc. IEEE Int. Communications Conf., 1995, 3, pp. 1495-1499 25 WARD, C., HARGRAVE, P., and McWHIRT'ER, ].: 'A novel algorithm and architecture for adaptive digital beamforming', March 1986, IEEE Trans. Antennas Propag., 34, (3), pp. 338-346 26 GOCKLER, H., and SCHEUERMANN, H.: 'A modular approach to a digital 6o-channel transmultiplexer using directional filters', IEEE Trans. Commun., July 1982, 30, (7), pp.1588-1613 27 GEPPERT, L.: 'Solid state', IEEE Spectrum, Analysis & Forecast Issue on Technology, January 1996, pp. 51-55 28TSOULOS, G., ATHANASIADOU, G., BEACH, M., and SWALES, S.: 'Adaptive antennas for microcellular and mixed cell environments with DS-CDMA', Kluwer Wireless Pers. Commun. f., August 1998, WPC-7, (2/3), pp. 147-169 29JOHNSON, D., and DUDGEON, D.: 'Array signal processing' (Prentice-Hall, USA, 1993) 30TSOULOS, G.: 'Approximate SIR and BER formulas for DSCDMAbased on the produced radiation pattern characteristics with adaptive antennas', Electron. Lett., 1998, pp. 1802-1804. 31 ATHANASIADOU, G., NIX, A., and McGEEHAN, J: 'A ray tracing algorithm for micro cellular wideband propagation modelling'. IEEE Vehicular Technology Conf., 1995, Chicago, USA,pp. 261-265 32ATHANASIADOU,G., NIX,A., and McGEEHAN,].: 'Anew 3D indoor ray tracing model with particular reference to predictions of power and RMS delay spread'. IEEE Personal Indoor and Mobile Radio Conf., September 1995, Toronto, Canada,pp.1161-1165 33 RATA, M.: 'Empirical formula for propagation loss in land mobile radio services', IEEE Trans. Veh. Technol., 1980, vr29, (3), pp.317-325 34 COOPER, M., and GOLDBURG, M.: 'Intelligent antennas: spatial division multiple access'. 1996 Annual Review of Communications, pp. 999-1002 (a copy can be obtained from http://www.arraycomm.com/) . 35 ANDERSEN, r. 'Capacity of parallel channels'. TSUNAMI II Seminar, Aalborg University, 1997 36 FOSCHINI, G.: 'Layered space-time architecture for wireless communication in a fading environment when using multielement antennas', Bell Labs Tech.]., Autumn 1996, PP. 41-59
313
An Adaptive Array in a Spread-Spectrum Communication System RALPH T. COMPTON, JR.
Abstract- This paper describes the integration of an LMS adaptive ar-
ray into a pseudonoise (PN) coded biphase modulated communication
system. The paper explains how these systems may be combined and presents a systems overview of the interaction between the adaptive array and the signaling waveform. An experimental system is described, and typical performance results are presented. The hybrid system requires only very modest spectrumspreading ratios, such as 5: 1, for the adaptive array to null interference. The combined system provides the interference protection of the adaptive array during the code timing acquisition phase as well as after code lockup.
I
INTRODUCTION
N A RECENT PAPER [1] , the author described an experimental adaptive array based on the LMS Algorithm [2]. That paper concentrated on the antenna pattern characteristics of the adaptive array. In the present paper, we continue Our description of this system by discussing the integration of this adaptive array into a spread-spectrum communication system. The LMS adaptive array offers the antenna designer the advantages of automatic beam tracking of a desired signal and automatic nulling of interference. The adaptive array can Manuscript received August 18, 1977; revised November 23, 1977. This work was supported in part by Contract N00014-67-0232··0009 between the Office of Naval Research, Department of the Navy, Arlington, VA, and the Ohio State University Research Foundation, Columbus, OH, and by Contract N00019-78-C-0131 between Naval Air Systems Command. Washington, DC, and the Ohio State University Research Foundation, Columbus, OH. The author is with the Electro -Scienc e Laboratory, Department of t,lectrical Engineering, Ohio State University, Colum hus, OH 43212.
operate with nonuniformly spaced antenna elements on a curved surface, such as an aircraft. There is no need for each element in the array to have the same element pattern or for uniform mutual impedances between elements in the array. Obviously, these benefits are of enormous practical importance. On the other hand, to use an adaptive array in a communication system requires a high degree of compatibility between the signaling waveforms and the adaptive array. One cannot simply put an adaptive array in an arbitrary communication system. There are several reasons for this. First, the adaptive array weights are random processes, and they modulate the desired signal. (The array is a dispersive channel [3].) The desired signal waveforms must be chosen so this modulation does not destroy the effectiveness of the communication systern. Second, there must be some difference between the desired signal and interference waveforms, so these signals can be distinguished in the array. Third, there must be some method for reference-signal generation and for acquiring system timing or frequency when the array is in the system. This paper will describe a particular communication system, a biphase modulated pseudonoise (PN) coded spread-spectrum system, in which it has been possible to integrate the adaptive array in a very successful way. The resulting hybrid system yields the full interference protection of the adaptive array (which is generally much greater than the protection afforded by the coding alone), not only after the PN-code timing is known, but during the code-acquisition phase as well. The paper will discuss the system operation at a conceptual level. That is, the objective is to present an overview of the relation-
Reprinted from Proceedings of the IEEE, Vol. 66, No.3, pp. 289-298, March 1978.
314
SIGNALS FROM OTHER ELEMENTS ~
ARRAY OUTPUT S (t )
REFERENCE SIGNAL R (t'
Fig. 2. Adaptive-array feedback loop.
Fig. 1. The LMS adaptive array.
ARRAY OUTPUT
ship between the adaptive array and the communication system. We do not include detailed theoretical analyses of the system components. It is hoped that the reader with a background in antennas will learn how the LMS adaptive array can be integrated into a practical communication system, and that the reader with a communications background will learn of the close coupling between the signaling waveforms and the adaptive array processing. It seems clear that to exploit the full potential of adaptive arrays for interference protection in communication systems, the design of the signaling waveforms and the adaptive-array processing must be undertaken together. Section II of the paper briefly reviews the feedback configuration of the adaptive array to establish terminology. (Readers wishing more information on the adaptive array are referred to the original papers by Widrow et ale [2], Applebaum [4], and to the earlier paper by the author [I] .) Section III describes the communication signaling waveforms used in the hybrid system and the method used to generate a reference signal for the array. Section IV describes the technique used for acquiring code timing and relates the code-lockup characteristics to the array parameters. Finally, Section V describes typical experiments involving interference rejection with this system.
II.
THE
lMS
ALGORITHM ADAPTIVE ARRAY
The LMS algorithm adaptive array [21 is shown in Figs. I and 2. The signal from each antenna element y;(t) is passed through a quadrature hybrid which splits it into quadrature components Xj(t). Each component xi(t) is multiplied by a weight wi and then summed to produce the array output set). The weights wi are controlled by a feedback system that minimizes the mean-square value of the error signal e(t). E(t) is the difference between the array output s(t) and a locally generated signal called the reference signal R (t).1 The feedback loops in the adaptive array have the form shown in Fig. ~. One such loop is needed for each quadrature channel in the array. This loop is often called a correlation loop, because it forms the product between the error signal e(t) and the channel signal Xi(t) and integrates the result. It was originally shown by Widrow , et ale [2) that this loop minimizes the mean-square value of the error signal. The antenna performance is controlled by what signal is used for the reference signal Rtt), We suppose the array is receiving a desired signal, thermal noise, and directional inter1
R{t) is called the "desired response" hy \Vidrow et al.
I ~ 1.
ERROR
SIGNAL E (t)
Fig. 3. Reference-signal generation.
ference. If the reference signal Rit ) is a perfect replica of the desired signal, the error signal e (r) consists of just the thermal noise and interference. Minimizing €2 (r), then, minimizes the total interference and thermal noise power in the array output. (The weights that minimize e 2 (t) result in pattern nulls toward directional interfering signals.) Furthermore, if the reference signal is a fixed-amplitude replica of the desired signal, minimizing e 2 (r) minimizes the sum of three quantities: I) the thermal noise power, 2) the interference power, and 3) the power in the difference waveform between the desired signal and the reference signal. As long as the signal-to-noise ratio (8/N) is not too low, the result is to match the desired signal to the reference signal and to minimize the thermal noise and interference powers, i.e., the array produces a maximum ratio of desired signal power to interference and noise power at the output. To apply this concept to a practical communication system, one must find some way to obtain a suitable reference signal. In general, of course, there is no way to obtain a perfect replica of the desired signal ahead of time. However, the situation is not nearly as hopeless as it appears at first glance. For some types of communication signals, it is possible to obtain a suitable reference signal by processing the array output signal as shown in Fig. 3. Whether this can be done or not depends on the characteristics of the desired signal and interference wave-forms. There must be some known difference between the two signals that the designer can exploit. To see how the signal-processing loop in Fig. 3 can be designed, we first note that the loop does not in fact have to produce a perfect replica of the desired signal. Rather, its function is to prod uce a reference signal that is 1) highly correlated? with the desired signal at the array output, and 2 In this paper we use the word "correlated" and "uncorrelated" in the sense defined by the feedback loops in the array: Two signals are uncorrelated if their product integrated over a certain time interval is zero. Th e time interval of interest here is the shortest time constant of the array feedback loops.
315
¢data (t)
ARRAY } ---I~--la.c
OUTPUT
."
2T
Td
d
2T
d
2T
TO DATA BIT DETECTION
d
¢code (t) ."
ERROR SIGNAL
« (t)
Fig.
s. The coded reference loop.
ep(t)
Fig. 4. Typical (/Jdata(t), cPcode(t), and ep(t).
2) uncorrelated with interference components at the array output. Such a reference signal will cause the array to behave in the desired way, because the feedback loops in the array are correlation loops. Only the correlation between the reference signal and signals Xi(t) affects the weights Wi. Thus the signal-processing loop in Fig. 3 should allow the desired signal component of the array output to pass through reasonably unchanged. Some distortion or delay of the desired signal is acceptable, however, so long as the referencesignal component that results remains highly correlated. In addition, the desired signal component of Rtt) should be of fixed amplitude, not dependent on the amplitude of the desired signal at the array output. On the other hand, the signalprocessing loop should alter the waveforms of the interference components. That is, it is not necessary for the reference signal to be free of interference. It is only necessary that the interference components in R (t) be uncorrelated with those at the array output. Hence, the signal-processing loop must change the interference waveforms in some manner so the correlation with the original waveforms is destroyed. For such signal processing to be possible usually requires a special type of desired signal. In the next section we describe a particular spread-spectrum communication signal, and show how a reference signal can be generated with this signal.
III.
THE SPREAD-SPECTRUM COMMUNICATION SYSTEM
Consider a spread-spectrum digital communication system Using biphase modulation. The transmitted desired signal is of the form s(t)
=A
cos [wo t + (/>(t)]
where A is an amplitude constant, Wo is the carrier frequency, and (/)(t) is a binary waveform whose value switches between o and n, The waveform c/>(t) is made up of the (modulo 21T) SUm of t\VO bit streams, (/)data(t) and (/>code(t). (/)data(t) is the useful information sent over the communication system; it has a bit rate of fd bit/so
(N is an integer), which we call the spreading ratio. Fig. 4 illustrates a typical (/)data(t) and (/)code(t) , and shows the resulting total phase modulation f!>(t). The bit transitions in 4>data (r) coincide with bit transitions in 4>code(t). It is assumed that the details of q,code(t) (i.e., the feedback connections, shift register length, and clocking rate Ie) used at the transmitter are also known at the receiving site, so a similar code can be generated in the receiver. With such a desired signal, a reference signal for the adaptive array can be generated as shown in Fig. 5. In this loop, the array output signal is first mixed against a coded local-oscillator signal '1 (t). '1 (r) is obtained by modulating a CW signal with the same PN code used to modulate the desired signal at the transmitter. This code is known at the receiving site, so it can be generated and used to produce '1 (r). (We assume for the moment that the local code timing is known. The method used to establish code timing in the receiver is discussed in Section IV.) The output from this mixer contains both desired and interference components. Because the code modulation in rl (t) matches that on the desired signal, the desired signal component at the mixer output is compressed to data bandwidth, while the interference is not. The filter bandwidth is chosen wide enough to pass the desired signal, which now has only data modulation, but not wide enough to pass the interference, which has full-code bandwidth. As a result, the filter removes all but the center portion of the interference spectrum. Next, the signal is passed through a limiter, which controls the amplitude of the reference signal. Finally, the limiter output is mixed again with the coded LO signal '1 (t), and the result is used as the reference signal. If we trace the desired signal through this loop, we find that it has the code .modulation removed at the first mixer, it passes through the data-bandwidth filter and through the limiter, which fixes its amplitude, and finally it has the code modulation put back on at the second mixer. The result is that the desired signal passes through this loop unchanged, except for the amplitude adjustment at the limiter and the envelope time delay associated with the filter. 3 An interference signal without the proper PN-code modulation, however, has its waveform drastically altered by this loop. For example, a CW interfering signal which has a single line spectrum at the array output produces a reference signal component with the full bandwidth of the PN-code modulation (and with lower power than at the array output). The correlation between the interference signal at the array output and the reference signal has been essentially destroyed by the loop. 3 The filter bandwidth has to be wider than a, conventional spreadspectrum filter to keep the envelope delay under ~ bit.
316
There are several reasons why the reference signal amplitude must be controlled. First, the amplitude of the reference signal determines the amplitude of the signals present at the array output, which should fall within a certain range for proper operation of the multipliers, etc., in the feedback loops. Second, the reference-signal amplitude should be fixed so the array will yield a maximum ratio of signal power to interference and noise power at the output, as discussed above. Third, the reference-signal amplitude cannot be linearly dependent on the array output amplitude, because then there is a problem with the operation of the array weights. For example, if the reference loop were linear and its gain (to the desir~d signal) were greater than unity, the loop would return a reference signal larger than the array output signal, causing the array weights to increase without limit. Conversely, a loop gain less than unity returns a reference signal smaller than the array output, causing the array weights to drop to zero. Thus stable operation requires a fixed reference-signal amplitude. Finally, when the reference-signal amplitude is fixed, the desired signal voltage at the array output will also be fixed, regardless of the incident power of the desired signal. This behavior is important for the delaylock loop used to track code timing (described in the next section); it makes the threshold setting for code acquisition independent of incoming signal strength. We note finally that the processing loop in Fig. 5 not only generates the reference signal, but also delivers the desired signal with the PN code removed at the output of the databandwidth filter. (The desired, signal is available at this point once the interference has been nulled.) Thus, the referencesignal loop incorporates the spectrum despreading, and the signal out of the data-bandwidth filter is ready for data-bit detection. Moreover, it is important to note that the interference protection afforded by the waveform processing (the PN coding) is available in addition to that due to array nulling. The two types of interference suppression are in cascade. A reference loop of the type shown in Fig. 5 was implemented and used to obtain the experimental data described in Section V. The desired signal was generated by using a slowspeed PN code to simulate data and a higher speed PN code for spectrum spreading. The sum bit stream was biphase modulated onto a 70-MHz carrier. The data and code frequencies (fd and Ie) could be varied (for different data modulation frequencies, different filter bandwidths were used in the reference signal loop) over a wide range up to about 5Mbit/s, so experiments could be performed with different spreading ratios. One of the goals of the experimentation was to determine the array performance that could be obtained with very modest spreading ratios. Most of the data in Section V are for a spreading ratio of 5 :I. The PN codes were obtained from a Hewlett-Packard 1930-A generator. The shift register length used to generate the codes could be varied from 3 to 20 bits. (However, as will be discussed in the sequel, PN-code sequence length appears to have no effect on array performance other than code lockup time.)
IV. THE DELAy-loCK TRACKING Loor The PN code used in the reference loop (Fig. 5) must be synchronized with the incoming desired signal code for the reference loop to operate properly . For large timing errors between the two codes, the desired signal will not pass through the reference loop. The array then regards the desired signal as interference and nulls it out. Experiments (6] - [ 8] have
shown that the array tracks the desired signal properly for timing displacements between the two codes up to about onehalf bit. Beyond one-half bit, the array nulls the desired signal. To keep the codes properly aligned, the delay-lock loop shown in Fig. 6 was used. This loop is a modified version of a conventional delay-lock loop [9] - [ 11]. (The modifications are discussed below.) This loop splits the array output signal into two channels and mixes it in each channel with a 100 MHz LO-signal biphase modulated with a PN code. The PN codes in the two La signals are displaced one bit in time. A 30 MHz zonal filter then selects the difference frequency output, and the output of this filter is shifted to 5 MHz and filtered to data bandwidth. The 5 MHz-filter outputs are squared, filtered to narrow bandwidth at 10 MHz, and envelope detected. The square root of each signal is then taken and the sum and difference of the two signals is derived. The sum voltage is used for timing acquisition, as described below, and the difference-voltage controls the veo frequency that clocks the PN code generator. The PN-code generator (not shown in Fig. 6) generates the two PN codes used in the two 100 MHz LO signals. These two codes are displaced one bit in time. It also generates a third version of the PN code timed halfway in between the two LO codes. This "in-between" code is used to provide the coded LO signal in the reference loop in Fig. 5. When the delay lock loop tracks, the code on one LO runs one-half bit ahead of the incoming signal code, and the code on the other runs one-half bit behind. The code used in the reference loop is then synchronized with the received signal code. The reason for using a delay-lock loop with squaring and square root operations was to enable the array system to lock-on and track a desired signal with data modulation present on the signal, i.e., we did not wish to send a datafree preamble for code acquisition. In a conventional delay lock loop [9] - [ 11], the incoming signal has only PN-code modulation. When this signal is mixed with a coded LO signal whose code timing differs by an amount ~T from the incoming signal code, the mixer output contains a spectral line whose amplitude is proportional to the autocorrelation function of the PN code, for delay ~T, i.e., the amplitude of this spectral line drops linearly with 6.T, up to one bit. The output of this mixer is then narrow-band-filtered to extract this line frequency, and the result is envelope detected. In the system described here, there is also data modulation on the signal. Hence, at the output of the first mixer the desired spectral line is now convolved wth a (sin x{X)2 datamodulation spectrum Therefore, it is necessary to retain an adequate bandwidth after the first mixer to pass this data modulation. However, the signal is then squared, which eliminates the biphase data modulation and collapses the spectrum. By filtering at the second harmonic, one can use a narrow-filter bandwidth to establish an adequate SIN. In this way a good estimate of the code autocorrelation function (squared) results. Finally, the square root is taken to eliminate the effect of squaring on the autocorrelation function. The major effect of using a wide-bandwidth filter after the first mixer is to increase the thermal- and self-noise power [9] - [ 11] over what they would be in a conventional delaylock loop. However, the adaptive array performance is not particularly influenced by minor amounts of code jitter, as long as the total timing error remains well below bit [6][8] . Furthermore, the amount of code timing jitter one
317
t
KEEPS
DIVISOR
FROM DROPPING BELOW AN ACQUISITION THRESHOLD
VOLTAGE
ADJUSTABLE MINIMUM
SWEEP
VOLTAGE
VOLTAGE
BIPHASE
MODULATORS
X
/
OUTPUT TO
PH GENERATOR
PH CODE I PM CODE 2 CENTER FREQUENCY
TRIM VOLTAGE
Fig. 6. Delay-lock tracking loop.
obtains is controlled by the bandwidth of the filter used at the second harmonic. Hence one can trade off this jitter against other system parameters, such as allowable frequency offsets or code-slewing speed (lockup time). Code acquisition with this delay-lock loop behind the adaptive array is rather interesting. Code timing is acquired by running the local PN -code generator faster (or slower) than the incoming signal code. This code slewing is continued until the sum channel voltage indicates that the two codes are aligned. During this slewing process, since the PN code used in the reference loop (Fig. 5) is slaved to the delay-lock loop .codes, the reference-loop code is also slewed. Thus before the -desired signal code and the reference-loop code are aligned, there is no correlation between the incoming desired signal and the reference signal. Hence the array nulls the desired signal during this time. As the two codes begin to align, the reference signal begins to correlate with the desired signal, causing the array weights to change, and the array pulls the desired signal up out of the noise. Thus the desired signal appears at the array output just as the local code timing approaches its correct value. The desired signal is present at the delay-Iockloop input when it needs to be, but not before. The sum channel output rises and trips a threshhold comparator, which removes the slewing voltage from the VCO input and allows the loop to begin tracking. This technique of slewing the PN code to acquire timing is well known, except that because the delay-lock loop interacts with the adaptive array while it slews, there are several novel features, which we discuss below. First, the array provides full interference protection during the lockup phase. That is, the array nulls interference regardless of PN-code timing in the array. (Interference nulling is not related to the local PN-code timing.) Because the array removes interference during slewing, the delay-lock loop does not have to contend with interference at all. Hence, the integration time in the loop, the slewing speed, and the time required for lockup do not need to change when interference is being received. Second, the desired signal that appears at the array output when the code timing is correct has a fixed amplitude, not
dependent on incoming signal strength. This behavior occurs because the array forces the output-desired-signal amplitude to match the reference-signal amplitude, which in turn is controlled by the limiter in the reference loop (Fig. 5). A fixed desired signal amplitude at the array output is helpful because the delay-lock loop does not have to operate over a range of signal levels. For example, a fixed threshhold value, not dependent on received signal power, can be used in the sum channel for acquisition. Also, circuit linearity problems are vastly simplified, e.g., a fixed signal level makes the squaring operation feasible. A third point of mild interest concerns how the array pulls the desired signal out of the noise when the timing approaches the correct values. Since the desired signal starts out in a null, many people have wondered how there will be any desired signal present at the array output to allow a reference signal to be generated when the timing is correct. The answer involves two factors. The first factor concerns the design of the reference loop in Fig. 5. When the array output signal is small, the voltage in the reference loop is too small for the limiter to clip the signal. Below the clipping level, the loop is linear; it produces a reference signal whose amplitude is linearly dependent on the array output signal. In this low-signal region, the loop must have a gain greater than unity. Then the reference signal will have a greater amplitude than the array output signal. This situation makes the array weight setting that nulls the desired signal a point of unstable equilibrium. In other words, if one weight changes away from this value slightly, so a small desired signal appears at the array output, a reference signal of larger amplitude will appear. This behavior reinforces the movement of that weight away from the null. In this way, the array output signal will grow until it is large enough that the limiter clips. When the clipping level is reached, the reference signal does not increase, and the array output signal settles into its steady-state value. The second factor is that the array weights in the adaptive array are random processes. They are derived from the product of two noisy signals, Xi(t) and e(t). Thus there is always a certain amount of weight jitter, so the movement away from the
318
.
I
I~--'
(a)
(a)
(b)
(b)
Fig. 8. Array interference rejection. (a) Spectrum on one array el ment. (b) Array output spectrumafter adaptation(S/Iimprovement_ 35 dB). Code rate = 2.5 Mbit/s; Data rate = 0.5 Mbit/s; Interference coherent (20 0 spatial): SIN =0 dB, S/I = -20 dB; Vertical Scale-
(c)
10 d Bjcm .
Fig. 7 . (a) Sum channel output with desired signal only (array inoperative). (b) Sum channel output with desired signal and coherent CW interference (array inoperative). (c) Sum channel output with desired signal and coherent CW interference (array adapting during acquisition).
v.
desired signal null has no difficulty starting. In practice, we have found no tendency for the array to keep the desired signal nulled once the code timing is nearly correct. The weight jitter always starts the weights away from this point. Finally, we remark that this lockup procedure requires the array speed of response to be fast enough for the desired signal to appear at the array output before the local code timing has passed by. In other words, the array time constants must be commensurate with the slewing speed . In the system used for experimentation, the adaptive array processor had time constants between 1 and 10 ms. (The array time constants depend on incoming signal power [121 .) The code slewing rate (the difference between the incoming code frequency fe and the local code frequency) could be varied . For a code rate f e of 2.5 Mbit/s and a 1023-bit code period (i.e., a lO-bit shift-register length [5)), lockup time was typically around 1 s. Of course , with this slewing technique, the code acquisition time is linearly proportional to the code length. Fig. 7 shows a typical scope trace of the sum channel output from the delay-lock loop as the array is being slewed . Fig.7(a) shows the sum channel output versus time with no interference and with the array not operating (with the weights fixed) . Fig. 7(b) shows the sum channel output when a CW interference signal 20 dB above the desired signal is added (the array is still inoperative). In this case, each channel of the delay-lock loop is saturated , and the desired signal pulse does not come through. Finally, Fig. 7(c) shows the sum channel output when the array is adapting during slewing. The interference is no longer present on the delay-lock loop, and the desired signal pulse reappears . The shape of this pulse has been altered somewhat from the shape in Fig. 7(a) because the array weights are changing during slewing (and hence they modulate the desired signal) . These figures illustrate the interference protection of the array during the code acquisition phase .
ARRAY PERFORMANCE
In this section we discuss the experimental operating char. acteristics of the adaptive array with the coded reference loop and delay-lock timing loop . The adaptive array processor in these tests was a four-element processor operating at 70 MHz, as described in [11 . In the experiments discussed here, the desired and interference signals were each split in 4-way power dividers and fed to the processor inputs after appropriate delays to simulate various arrival angles. Independent thermal noise was also included in each simulated element signal. We begin with a series of spectrum analyzer photographs illustrating typical interference-rejection experiments. Fig. 8(a) shows the power spectral density on one element of the array when PN-coded desired signal, thermal noise , and CW interference are present. The desired signal-to-thermal-noise ratio is 0 dB, and the desired signal-to-interference ratio is - 20 dB (as measured on each element). The PN-code rate is 2 .5 Mbit/s , and the data rate is 0 .5 Mbits/s, so the spreading ratio is 5 : 1. The desired signal is in-phase on each element (corresponding to a signal arriving from broadside), and the interference has a progressive phase shift of 60° between elements (corresponding to an arrival angle of 19.s° off broadside for half-wavelength element spacing) . The interference frequency is coherent with the carrier of the desired signal (70 MHz). Fig. 8(b) shows the array output spectrum after adapting. As may be seen, the interference has been nulled , and the SIN has been improved due to the array gain. Measurements taken of the output-signal powers separately, with the weights frozen , showed that the signal-to-interference ratio had been improved 35 dB in this case. Fig. 9 shows the same experiment when the interference frequency was 71.2 MHz. In this case, the improvement in signal-to-interference ratio measured 36 dB. Fig. 10 shows another case in which the interference was swept CW, swept over a 1.2-MHz band with a l-kflz triangularsweep waveform . The desired signal was modulated with a 1 Mbit/s PN code and 200 kbit/s data (5 : I spreading ratio). Fig. 10(a) shows the power spectral density on one element of
319
(a)
(b)
Fig. 9. Array interf ere nce rejection. (a) Spectrum on one element. (b) Ar ray output after adaptation (Sfl improvement = 36 dB). Code rate = 2.5 Mbit/s; Dat a rate = 0.5 Mbit/s; Interference freq. = 71.2 MHz; Signal Freq . = 70.0 MHz; SIN = 0 dB. SII = - 20 dB; Vertical scale = 10 dB /cm .
Fig. 11. Interference rejection : 31-bit PN-code sequence. (a) Spectrum on one element. (b) Array-output spectrum.
(a)
Fig. 12. Interference rejection: 7-bit PN-code sequence. (a) Spectrum on one element. (b) Array -output spectrum .
to-signal ratio , but the interference power is the quantity being varied. These curves were taken with a data rate of 250 kbit/s, a code rate of 1.25 Mbit/s (5:1 spreading ratio), and the array, and Fig. lO(b) shows the array output-power with a desired signal-to-thermal-noise ratio of 0 dB on each spectral density. In this test, no element noise was present. element . The desired signal arrives from broadside, so it is in Next we comment on the PN-sequen ce length . PN-sequence phase in all four elements. Three curves are shown , correlength appears to have no effect on the interfer ence suppres- sponding to three different angles of arrival of the interference . sion of the array. Figs. 11 and 12 illustrate the effect on the For the top curve, the interference has alSO element-tospectrum of changing code-sequence length. Fig. 11 shows the element phase shift , corresponding to a spatial angle of 4 .8° one-element spectrum and the arr ay output spectrum with for half-wavelength element spacing. The second curve shows a sequence length of 31 bit. The desired signal is modulat ed the case when the interference element-to-element phase shift at a code rate of 2 Mbitls with a spreading ratio of 10. The is 30° (a spatial angle of 9.6° for half-wavelength spacing) , interference is swept CW, swept over a 1.8-MHz band at a and the third curve shows the case when the interference 1200 Hz rate. The interference power is 6 dB above the de- element-to-element phase shift is 75° (a spatial angle of 25° sired signal power. for half-wavelength spacing) . Fig. 12 shows the same experiment repe ated with a 7 bit The trend of all three curves is similar. For low interference code sequence length. There is no measureable difference in power, the output interference increases linearly with input interference -suppr ession as a result of the change in code- interference. In this region, the interference power is small sequence length, although the effect on the spectrum may be compared with the desired signal and the thermal noise ; it seen in Figs. 11 and 12. has little effect on the array feedback . The feedback loops do Next. we show measurements of the interference-suppres- nothing to null the interference, so the output interference sion of the arra y and discuss its depend ence on various factors. power rises linearly with the input interference power . When Fig. 13 shows a typical measured curve of output inter ference the interference power increases enough, however, the interpower as a function of input intereference power , with CW ference begins to dominate the error signal, and the array interference. The abscissa is plotted as input interfer ence- feedback corrects against it . As the interference power in-
Fig. 10. Array reject ion of swept CW interference. (a) Spectrum on one element. (b) Arra y-output spectrum.
320
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creases fu rt her , the fee db ack-loo p gain in t he array increases ." This increasin g loop gain causes the ou t put int erfe re nce to dro p wh en the input inte rfe re nce is increased . Finally , fo r large enough input in ter ferenc e power , (a ppro xi mate ly 25 d B abo ve the desired signal) , th e out put inte rference begins to increas e again .t We were un abl e to ta ke dat a abo ve this point because of eq ui pme nt pow er limit ations , bu t earlier ex periments [81 have shown that the inte rfe renc e power begi ns to rise again beyond this poin t. Also , intermodulati on pro du ct s, which t ypically are not su ppressed by the arr ay fee dback , begin to appear in this rang e . Fig . 13 also illustrates how the inte rference suppressio n depend s on the angular separation between th e de sired signa l and t he interference . When the int er feren ce has very low po wer , it has little effec t o n t he array pat te rn . Th e array simply forms a be am t oward the desir ed signa l. Thus th e diff er en ce bet ween the t hree cu rves at th e lo w-po wer end (lIS < - 15 dB ) j ust reflect s t he array s pa tt ern . Th e clos er the interference is to the desired signal, the mo re in te rfe re nce po wer will be prese nt at the array out pu t . because th e in terference will be higher up o n th e main beam of th e patt ern . As the in te rferenc e po wer is inc reased int o the regio n 0 d B < l IS < 10 dB , the separation be twee n the t hr ee cur ves bec omes lar ger. This indicates that the arr ay is bettcr able to null t he interference if its angular separation from t he des ired signal is larger. The angles sho wn in Fig . 13 span the rang e where t he re is a noticeable difference in per fo rm an ce . When the int erfer en cephase an gle is 7 S° and higher , th e per for man ce is essentially inde pende nt of the angul ar separa tio n . 4 Th e fe edback-lo op gai n in th e ada pt ive array is propo rt io na l to th e power o f the signals in the arr ay (I I ) . S This behavior has be en not ed theoret ical1y b y several author s 113) -
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No t o nly the o ut put interference power , but also the output desired-signal power and de sired-signal-to -t hermal-noise ratio dep end o n t he inpu t inte rfere nce power. The desired signal o ut put power is affect ed be cause the interference pow er influe nce s the pattern . Th e feed back -loo p gain in the array depends on t he signal powers. so a change in interference power affects the loo p gain and hen ce th e pattern . When t he interference is muc h st ro nger t han t he desired signal , th e arr ay nulls t he interference regardl ess o f what happens to t he desired signal. If t he interfe rence happ ens t o be too clos e t o t he desired sign al , the interference null is clos e to t he desired signal, and there is a red uc tio n in t he de sired signal pow er at the array o ut put because of this nu ll. Fig. 14 sho ws measured curves o f ou tp u t des ire d signal power as a function of interferen ce power into the arr ay for an interference-phase angle of 30° . This is a case wher e t he interference is close en ough to t he desired signal that the interfer en ce null affects the de sired signal. (A case with wider angular separation is shown be low .) We see in this figure that raising the inte rference pow er lowers the desir ed signal power at the arr ay o ut put. Three cu rves are shown for thre e different inp ut desired signa l-to-t he rmal-noise rat ios. The di ff erent curves we re mea sured by hol ding the de sired signal power co nstant and varying the noise pow er. The top curve shows t he out put signal po wer with a +3 dB desired signa l-to-t he rmalnoise ratio on each elem ent of the arra y . Th e middle curve shows the result whe n the noise pow er has bee n inc reased 3 d B, so the ele me nt SIN is 0 dB . Th e bot to m curve shows th e case whe re the noise power has been increased another 3 dB , so the element SIN is - 3 d B. We see t hat the higher the noise into the array , the mo re dependent t he out put desired signal po wer is on the inte rference po wer. It is clear from these curves that the out put desired-signal po wer de pends on t he t hermal no ise as well as the interferen ce power. For exa mp le, consider the end s of the curves whe re the int erferen ce power is very low . With a +3-dB S IN o n th e eleme nts, t he o ut put de sired signal pow er i5 arbit rarily defined to be 0 dB . Then a 3-dB increase in the element noise results in a 0 .8-dB drop in the output desired signal power, even th ou gh the input desired signal power haS
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not been changed . A further 3-dB increase in the noise causes an additional O.8-dB drop . The reason for this behavior is as follows . The error signal is made up of three components : 1) the desired signal minus the reference signal ; 2) the thermal noise; and 3) the interference. To minimize the overall mean-square error, the array feedback makes a compromise between these three components. In general, the weights that yield minimum-error signal do not match the array-output desired signal to the reference signal exactly . Rather , they compromise between thermal noise, interference, and desired-signal contributions to the error signal. If the thermal noise in the array is increased, the noise component in the error signal becomes larger . In response to this, the array feedback lowers the weights to reduce this noise . In the process, the output desired signal is lowered as well, so the mismatch between the array output desired signal and the reference signal increases. The final weight setting compromises between decreasing the noise and increasing the desired-signal mismatch. The overall result is a reduced desired-signal output from the array. Furthermore, the interference power has a stronger effect on output desired signal power when the thermal noise is large. That is, as the thermal noise is increased , the mismatch between the desired signal and the reference signal becomes a less important contribution to the error signal. This result is seen in Fig. 14; the drop in desired signal power as the interference is increased is greater the larger the thermal noise (the lower the element SIN) . Fig. 15 shows the dependence of output desired signal-tothermal-noise ratio on input-interference power for the same experiment as in Fig. 14. Again, three curves are shown , correspondmg to element SNR's of +3, 0 , and -3 dB. Several interestmg effects may be seen in these curves. First consider the case when the interference power is low . Ideally, a four element array should provide 6 dB gam . Thus, for example, the top curve, where the element desired-signalto-thermal-noise ratio is +3 dB, should show an output SIN of +9 dB at low interference power. The measurement shows
322
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7 .8 dB . For a O-dB element SNR , the output should be +6 dB, and 5.5 dB was measured. For a -3-dB element SIN, the output should be +3 dB, and the result was 2.8 dB. We see that as the noise is decreased (or the element SIN is increased), the array performance departs more from the ideal. The reason for this is the presence of multiplier offset voltages [171, which alter the weights from their ideal values . The less noise , the more effect multiplier offset voltages have. The curves in Fig. 15 are for the case where the interference element-to-element phase shift is 30° . (The desired signal is in phase on all four elements.) Thus, Fig. 15 represents a case where the interference is too close to the desired signal for full performance from the array. The fact that the signals are too close is seen in the strong dependence of the output desired-signal-to-thermal noise ratio on int erference power.
ful suggestions of D. Townsend and D. Himes of NRL and R. Bauman of NASC.
Fig. 16 shows a similar set of curves, except that the spreading ratio has been increased to 10: 1. A comparison of Figs. 15 and 16 shows that there is little difference in the performance as a result of changing the spreading ratio. Next, Fig. 17 shows the same measurements as in Fig. 16 except that the interference phase shift between elements has been increased to 75°. (In Fig. 16 it was 30°.) With a phase shift of 7S°, it may be seen from Fig. 17 that the output desired-signal-to-thermal-noise ratio is less sensitive to the input interference power. Thus, at a 30° phase shift, the interference is too close to the desired signal; at 75°, the signals are far enough apart that the SIN is not degraded by the interference.
REFERENCES ( 1] R. T. Compton, Jr., "An experimental four-element adaptive array," IEEE Trans. Antennas Propagat., vol. AP-24, p. 697 Sept. 1976. ' (2] B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode,
"Adaptive antenna systems," Proc. IEEE, vol 55, p. 2143, Dec. 1967.
[3] R. S. Kennedy, Fading Dispersive Communication Channels. New
CONCLUSIONS
This paper has described the integration of an LMS adaptive array into a PN-coded spread-spectrum communication system. A method of reference- signal generation and code-timing acquisition were described that allow the array to distinguish the desired signal from interference. The hybrid system yields full interference protection during the code-lockup phase as well as after timing has been acquired, i.e., code lockup time is not affected by the presence of interference. Typical experimental results have been presented illustrating the interference suppression characteristics of the hybrid system. Most of the results presented are for a spreading ratio of 5: 1, a very modest value. With a 5: 1 ratio, interference suppressions of 35 dB are typical. This performance represents much greater protection than can be achieved by spectrum spreading alone with such a spreading ratio. The results presented illustrate the advantages of combining adaptive array techniques with waveform design. However, it is clear that for such systems the adaptive-array parameters and the signaling waveforms must be selected together to be compatible, because of the close interaction between the antenna and communication systems. In general, it is difficult to add an adaptive array to an existing spread spectrum system where the communication-system waveforms have been chosen independently of the array. ACKNOWLEDGM ENT
The author is grateful to Professor R. J. Huff and Professor A. A. Ksienski for many helpful discussions during this program. In addition, the author appreciates the numerous use-
York: Wiley-Interscience, 1969. (4) S. P. Appleboum, "Adaptive arrays," IEEE Trans. Antenna Propagat.; vol. AP-24, pp. 585-598, Sept. 1976. (5] J. J. Stiffler, Theory of Synchronous Communications. Englewood Cliffs, NJ: Prentice-Hall, 1971, p. 178. (6] K. L. Reinhard, "Adaptive array techniques for TDMA network protection," Section II of R. J. Huff, "Coherent multipleXing and array techniques," ElectroScience Lab., Dep. Electrical Eng., Ohio State Univ., Rep. 2738-3, Feb. 1971; prepared under Contract F30602-69-C-0112 for Rome Air Development Center, Griffis, AFB, NY. (7] K. L. Reinhard, "An adaptive array for interference rejection in a coded communication system," ElectroScience Lab., Dep. Electrical Eng., Ohio State Univ. Rep. 2738-6, Apr. 1972; prepared under Contract F30602-69-0112 for Rome Air Development Center, Griffis AFB, NY. [8] R. 1. Huff and K. L. Reinhard, "Coherent multiplexing and array techniques," ElectroScience Lab., Dep. Electrical Eng .• Ohio State Univ., Rep. 2738-9, June 1972; prepared under Contract F30602- 69- 0 1 12 for Rome Air Development Center, Griffiss AFB, NY. (9) J. 1. Spilker, Jr. and D. T. Magill, "The delay-lock discriminatoran optimum tracking device, Proc. IRE, vol. 49, p. 1403, Sept. 1961. [ 10] J. J. Spilker, "Delay-lock tracking of binary signals, IRE Trans. Space Electron Telem., vol. SET-9, p. 1, Mar. 1963. (11 ] V-'. J. Gin, "A comparison of binary delay-lock tracking loop implementations," IEEE Trans. A erosp. Electron. Syst., p. 41 S, July 1966. ( 121 R. L. Riegler and R. T. Compton, Jr., "An adaptive array for interference rejection," Proc. IEEE, vol. 61, p. 748, June 1973. (13 ) C. A. Baird, Jr., G. P. Martin, G. G. Rassweiler , and C. L. Zahm, "Adaptive processing for antenna arrays," Final Rep., Radiation Systems Division, Harris Intertype Corp., Melbourne, FL, June 1972. (141 K. L. Reinhard, "Adaptive antenna arrays for coded cornmunication systems," ElectroScience Laboratory, Dep. Electrical EnJ., Ohio State Univ., Rep. 3364-2, Oct. 1973; prepared for Rome Air Development Center under Contract F30602-72-C-0162. ( 151 B. S. Abrams, S. J. Harris, and A. E. Zeger, "Interference cancel ... lation ," RADC-TR-74-225, Final Rep., General Atronics Corp., Sept. 1974. (16) A. E. Zeger, B. S. Abrams, and C. Luvera, "Interference canc~· lation system for sensors," hoc. NRL Adaptive Antenna Systeml Workshop, Mar. 1974. [ 17] R. T. Compton, Jr., "Multiplier offset voltages in adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES-l~.. p. 616, Sept. 1976.
323
tt
On the Performance of a Polarization Sensitive Adaptive Array R. T. COMPTON, JR.
Abstract-The ability of a least mean square (LMS) adaptive array to adapt to the electromagnetic polarization of incoming signals is considered. An array of two pairs of crossed dipoles is studied. A desired signal and an interference signal are assumed to arrive from arbitrary directions with arbitrary elliptical polarizations. The output signal-to-interference-plus-noise ratio (SINK) from the array is computed as a function of the signal angles of arrival and polarizations. It is shown that as long as certain special desired signal polarizations are avoided, the array is difficult to jam with a single interference signal. To produce a poor SINK, an interference signal must both arrive from the same direction and have the same polarization as the desired signal.
INTRODUCTION
A
DAPTIVE arrays [1] -[ 3] are currently of great interest because of their ability to null interference and track desired signals automatically. Numerous papers have discussed the performance of adaptive arrays [4]. In spite of the extensive literature, however, for radio applications of these arrays (as contrasted with sonar applications), one aspect of this subject appears to have received little attention. We refer to the fact that an adaptive array can adapt to the electromagnetic polarization of signals, as well as their arrival angles. If an adaptive array uses elements responding to more than one polarization, the array feedback loops will automatically combine the signals from these elements to optimize reception, or provide a null, for particular signal polarizations. Such an array can automatically track a desired signal with one polarization while nulling interference with a different polarization. Most analytical studies of adaptive arrays have assumed isotropic elements. This assumption, although useful for certain purposes, tacitly eliminates any consideration of the effects of signal polarization on array performance. In essence one assumes all signals arrive at the array with the same polarization. If an array receives and uses more than one polarization its performance can be far superior to one that does not.. For example, an array of isotropic elements always yields poor performance if interference arrives too close to the desired SIgnal. When an array adapts to polarization, however, this difficulty occurs only if both signals have the same polarization as well as angle of arrival. When two signals arrive from the same direction, it is perfectly possible to null one signal and not the other, if their polarizations are different. The purpose of this paper is to examine the performance of a polarization-sensitive adaptive array. As a model, we will consider an array of two pairs of crossed dipoles. We will compute the output signal-to-interference-plus-noise ratio (SINR) Manuscript received February 15, 1980; revised October 16, 1980. This work was supported in part by the Naval Air Systems Command under Contract N00019-79-C-0291, and in part by the Joint Services Electronics Program under Contract NOOa 14-78-C-0049, both with the Ohio State University Research Foundation. The author is with the ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University, Columbus, OH43212.
from this array when a desired signal and an interference signal arrive with arbitrary polarizations and angles of arrivaj! We will show that in most cases interference has little effect on the array output SINR unless it arrives from the same direction and has the same polarization as the desired signal. However there are two exceptions. If the desired signal polarization is linear, oriented either parallel or perpendicular to the vertical dipoles, the array is susceptible to interference from other angles as well. These desired signal polarizations are ones that should be avoided in a system design. Finally we will find that when both signals arrive from broadside, the array output SINR is simply related to the separation between the signal polarizations on the Poincare sphere. Section II of the paper formulates the necessary equations. Section III contains the calculated results and Section IV the conclusions.
II. FORMULATION OF THE PROBLEM Consider a four-element adaptive array consisting of two pairs of crossed dipoles, as shown in Fig. 1. The signal from each dipole is to be processed separately in the array. The upper and lower dipole pairs have their centers at Z = + L/2 and Z = -L/2, respectively. Let _~ 1 (t) and x3(t) be the cornplex signals received from the upper and lower vertical dipoles, and X2 (r) and X4(t) the signals received from the upper and lower horizontal dipoles, respectively. Each signal Xj(t) is multiplied by a complex weight Wj and summed to produce the array output. We assume the weights Wj are controlled by an LMS processor [2], [5), so the steady-state weight vector, W = (wI, w2, ... , W4)T, is given by (1)
where rI> is the covariance matrix (2)
and S is the reference correlation vector S = E{X*r(t)}.
(3)
In these equations X is the signal vector
(4)
ret)
is the complex reference signal 2 used in the adaptive array feedback [2], (5), T denotes transpose, the asterisk denotes complex conjugate, and E(·) denotes expectation. Assume two continuous wave (CW) signals are incident on the array, one desired and the other interference. Let () and ¢ 1 By arbitrary polarizations, we refer to signals that are completely polarized (i.e., elliptically polarized). \Ve do not consider partially polarized signals [ 12] . 2 r(t) is called the "desired response" in [2J.
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-29, No.5, pp. 718-725, September 1981.
324
Fig. 2.
Polarization eIlipse.
x Fig. J.
Crossed dipole array.
denote standard polar angles , as shown in Fig . I . We assu m e th e: desired signal arrives from angular d irection «(J d , tPd) and th e interference from «(J i , tPi) ' Furthermore each signal is assumed to have an arbitrary electromagnetic polarization. To characterize the polarization o f each signal we make the follow ing de finit ions . G iven a transverse ele ctromagnetic (T E M) wave propagating in to the array, w e co nsid er t he polarizat ion ell ipse produced b y the tra nsve rse elec t ric fie ld as we view the inco m ing wave fro m the coo rd inate o rigin . N o te th at u n it vectors , ~, iJ , - ;, in that o rde r , fo rm a rig h t-ha nd ed coo rdi na t e sy ste m for an incorn ing wa ve . Suppose the elec t ric field ha s transve rse co rn po nen ts
I
(9a)
3 These relationsh ips are derived in [6]. Our definitions and notation correspond exactly to those in [61 if we substitute/:;q, - X, EO Y, 1) -!p.
(6)
r
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(7)
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Eq, = A cos "( Eo
(8 a)
= A sin "(e i T)
Where "( and 1/ are related to cos 2"( = cos 2Q cos 2{3
Q
(9 b)
(8 b)
(We will call Eo the horizontal co m po ne n t and Eo the vertical co m po nen t o f th e f'ield.) In general, as time progresses Eq, and E.j wi ll describe a polarizat ion ellipse as shown in F ig. 2 . Gi ve n th is ellipse we d ef ine (3 to be the o rient a ti o n a n gle o f th e m ajo r a .is o f th e ellipse w ith res pec t to Eq" as sh own in F ig. 2. T o e.irnina te amb iq uities we d efi ne (3 to be in t he range 0 ~ (3 < 71 . We also de fine the elli p tic ity a n gle Q to have a magn itude given by
= tan -
tan 1/ = tan 2Q esc 2{3.
Poincar e sphere.
The re lation ship am ong the fo ur angu lar va ria bles Q, (3, "(, a nd 1/ is m ost easily visualized by mak ing use of the Poin car e sphere co nce pt [ 6] . This t echnique represents the sta te of polariza tion by a point o n a sphere , su ch as point M in Fig. 3. For a given M, 2"(, 2(3, and 2Q form the sides o f a right sp he rical triangl e , as sho w n . 2"( is the side o f th e tr iangle between M and a poin t la be lled II in t he figu re ; H is the point represent ing h o rizo n ta l linear polar izat ion . Sid e 2(3 ex tends alo ng t he eq ua to r and side 2Q is vertical , i.e ., perpend icular to side 2(3. The an gle 1/ in (8) and (9) is the angle between sides 2"( and 2{3.3 The sp ecial case whe n Q = 0 in (6) a nd Fi g. :; co rr es po nds to linear polarizat ion ; in th is case th e po int /11 lies o n the eq ua tor. If in ad di tio n, (3 = O. only l:;1j) is nonzer o a nd the w ave is horizontall y polarized . This case de fines the point H in Fig . 3. If instead (3 = 71/ 2 , o nly c'o is non zero and the wave is vertically polarized . Point ,\If th en lies o n the equat or di ametrically behind H. The po les of th e sp he re co rr es po nd to circ u lar pol ari zat ion (Q = ±4 So) , with clo ck w ise circ ula r pol a riz ation (Q = +45°) a t th e uppe r po le . Thus a n arb itra ry pla ne w ave co mi ng in to the a rray m ay be cha racte rized by fo u r angu lar param et e rs an d a n am plitu d e . For exa m pl e t he de sir ed signal w ill be charac te rized by its ar rival a ngles «(Jd, tPd), its polarizat ion ell ipt icit y an gle Qd an d orientat ion angle (3d , and it s amplitude A d ( i.e ., A d is the value of A in (8 ) for the d esired signa l) . We will say the de sired signal is defined b y «(J d , epd, (Xd , (3d , Ad )' Similarly th e in terfe re nce is defined by «(J i, epi, (Xi , (3i, A i) ' We assu me eac h d ip ole in th e array is a short dipol e, i.e. , th e o u t p u t vo lt age f ro m eac h d ipole is pr opo rt ional to the electric field co mpo ne n t alo ng th e d ipole . Ther e fore the vert ica l and hor izontal d ipo le o ut pu ts w ill be p ro portio nal to the z- and x-com po ne n ts, res pec t ively, o f t he electric fie ld . An inc o m in g signal, wi th arb itra ry ele ct ric fiel d co m po nents EIj) and Eo, has
(5 )
Q
Fig. 3.
and (3 by [ 6]
325
x, y, z components:
vector
E = E(j)~ + E(J8
= (Eo cos (J + (Eo cos
( 17)
cos ~ - E(j) sin ~)x (J sin ~
+ Erp cos
The covariance matrix in (2) is then given by
~)y
-(Eo sin 9)2.
(18a)
(10)
where When E rP and Eo are expressed in terms of A, 'Y, and 11 as in (8), the electric field components become
E = A[ (sin 'Y cos 8 cos epejTl
+ (sin 'Y cos 9 sin
l/Je i Tl
+ cos
I
y cos l/J)Y
(12)
(-sin "1 sin (JeiTl)e-jp
rrL
A
I
I
( 18d)
with I the identity matrix. To make the LMS array to track the desired signal, the reference signal ;(t) must be a signal correlated with the desired signal and uncorrelated with the interference [2], [5]. Several techniques have been described for obtaining such a reference signal [ 10] , [ 11 ] . Here we assume
(13)
w is the frequency of the signal, 1/1 is the carrier phase of the signal at the coordinate origin at t = 0, and p is the phase shift of the signals at the dipoles due to spatial delay
=-
I
For the reference correlation vector, (3) then yields
(sin 'Y cos 8 cos epeiTl - cos 'Y sin
p
(18c)
}
(19)
(-sin 'Y sin 8e i Tl )ei P
U=
=A
E{X·*X· I I
.2U·*U· T
(11 )
where U is the vector
(sin 'Y cos 8 cos (j)eiTl - cos 'Y sin l/J)£!P
(18b)
T
and
Adding to this expression the time and space phase factors, we find that an incoming signal characterized by (9, ¢>, Q, 13, A) produces a signal vector in the array (see (4» as follows: Aei(wt+I/J)U,
= E{Xd*X d T} = Ad2Ud*UdT
e. =
- cos "1 sin l/J)X
- (sin 'Y sin 8e i Tl )Z ] .
X =
(20)
The steady-state weight vector can now be found by substituting (18) and (20) into (1). The SINR at the array output is then given by
Pd
SINR=---
r,» r;
cos 9.
(14 )
where Pd is the output desired signal power
As stated above, we assume a desired signal specified by
«(J d, l/Jd, Qd, 13d, Ad) and an interference signal specified by «(J i» l/Ji, Qi, 13i, A;) are incident on the array. In addition we assume a thermal noise voltage nj(t) is present on each signal
Xj(t). The nj(t) are assumed to be zero mean, to be statistically independent of each other, and to have power 0 2 :
(22)
Pi is the output interference power (23)
(15)
and Pn is the output thermal noise power
where [, ii is the Kronecker delta. Under these assumptions the total signal vector is given by
X=Xd+Xi+X n
= Adei(wt+I/J d)Ud + Aiei(wt+I/J ou, + X n
(21 )
(24)
(16)
where Ud and Vi are given by (13) with appropriate subscripts d or i added to each angular quantity. l/J d and l/J i are assumed to be random phase angles, each uniformly distributed on (0, 2rr) and statistically independent of the other. X n is the noise
By making use of a matrix inversion lemma, [9] , [ 13] the expression for SINR in (21) can be put in the simple form
326
(25)
10
10
30 - 45 Qj 'O - IT -/
I
0
0-
~45 15 30
a; -10-
ai -IO -e
~
0:
0:
z
'"
t/)
- 30
20, 10
H-20
~ - 20 -
-
- 30 I 30
- 40 0
I
I
60
8;
I
90
120
-40 0
180
8;
( DEGREES )
(a)
10
-45
......
10
-10 - 15
~
a j 'O Of-
I 150
15
...-!:?
~~O
45
(b)
' C
a " -45 - 30
0 '-
180
90 ( DEGREES I
-)5
~"45 o
iii - 101-e
15
30
-
;; -10-
-e
0:
0:
~-20""
~-20-
'"
t/)
-3 0
-40
-
-
-30
I
30
0
to
I 60
I
I
8;
90 ( DEGREES) ( c)
a, ' -45 , (5
120
I
I
I 30
-40 0
180
150
I 60
-15
0 , - 30
0 1-
30
i
a;-e - IOfa: ;;-20 -
45
~45 o 15 30
-10 f-
z
I
) -
I 30
0
Fig. 4.
A/
I 60
I
8,
90 ( DEGREES)
I 120
a
>-<-20 fVl
- 30 l -
I
I
-40
180
150
I 30
0
I 60
(e)
SINR versus 0;. Oil = 90°, rJ>d = 90° . Qd = 15°. I3d = 30°. SNR =0 dB. rJ>; = 90 .INR 30° . (c) 13; =60° . (d) 13; =90° . (e) 13; = 120° . (f)tJ; = 150°.
~d = - 2 = desired signal-t o- no ise ratio (SNR)
0
t (26a)
4 td and ti are the signal-to-noise ratios that will exist in a given array element if the incoming signal arrives broadside to that element ~d is linearly polarized in the direction of that element. For example, if Q
'*
180
0:
(fl
-40
I 150
(d)
a j '_3\,15 / -
-.......::::: -,L.-0"45
I 120
( DEGREES )
10
~
Of-
8;
I 90
*'
A .2
8,
I 90 ( DEGREES I
I 120
I 150
180
(0
=40 dB. (a) 13; =0° . (b) 13; =
=-i=interference-to-noise ratio (INR) , a
(26b)
The derivation o f (25) from (21) may be found in [13, ap pendix] . Calculation of the SINR from (25) is much eas ie r th an from (22)-(24) because (25) does not require calculation of the weight vector. In the next section , we show typical curves of the arr ay performance based on (25) . III. RESULTS Bec ause o f t he large number of parameters required to speci fy both the desired and interference signals, many types o f curves can be plotted . Unfortunately space does not permit an exhaustive set of curves here. However, we will show a number of typical curves, in clu di ng those illustrating the worst performance . Figs. 4 and 5 show curves of output SINR when the desir ed signal arrives from broadside (0 d = rf>d = 90°). The de -
327
10
0 0
<, < - ;
~ - 10,-
-a:
-
10
-3l
a i ' -45 """
Q
=-- 15
----
~O
d5
=
-45
z
....- 20 en
-30-
I
I
30
60
I 90 (DEGREES I
- 30
,
I 120
150
-40 0
180
180
(a) 10 a,
·-t
(b) 5
-30 ]' -1/5
10 a " -3.0
0 Or--
0
,;
-
a; -10 I-
/.
4~
30
z
....-20 Ien
I
-
-40 0
~-IO l-
I,
~-20 l-
en
I
I 60
30
ef>i
I 90 (DEGREES I
I 120
I
150
- 30 l-
I
I
-40 0
180
30
I
60
-1 5
-3~
---=
0
-'}' .15
1
I 0
~
:.---
120
I 150
180
a
I
j-45
- 3 0.~ 1 5
0
\5
30
-4[
15
0
a; -10-
a;-IO-
~-20-
~-20'-
-
-30 l-
45
30
-a:
-a:
-e
-e
en
-40 0
90 (DEGREES)
(d ) 10
a"-'j5 .15
I
I
(e) 10
-r 45
-a:
I
a:
30
15
'
- 45 .0 -15
~
-e
- 30
45
-a:'"
~-20-
-30
-30
a; -10
en
-40 0
1
0
»<>: 0 "15
VI
I 30
I 60
I 90 (DEGREES I
I
120
I
150
-40 0
180
SINR ver su s 8 i- 8 d
ef>i
90 I DEGREES 1
I
120
I 150
180
= 90° , tPd = 90° , Qd = 15°, (Jd = 30° , SNR = 0 dB , 8 i =90" , INR = 40 dB . (a) (Jj = 0° . (b) (Ji = 30°, (e) (Jj =60° . (d) (Jj = 90°. (e) (Ji = 120°. (f) (Jj = 150° .
sired signal has been chosen to have a par ticular elliptical polarization : CY.d = ISO and (3d = 30°. The SNR is 0 dB and the INR is 40 dB. The element pairs are assumed spa ced a half wavelength apart (L = 'A./2) . Fig . 4 shows the output SINR as a funct ion of the interference polar angle 8 ;, with rpj = 90° and for various interference polarizat ions . Spe cifically , F ig. 4(a)
=
I
I
60
( f)
(e)
Fig. 5.
I
30
shows the SINR for (3; 0° , Fig. 4(b) for (3; = 30°, and so forth, up to Fig. 4(f) for (3; = ISOo. Each figure shows the results for CY.; = -4So, _30° , _ 15° , 0°, 15° ,30° , and 45° . Fig. 5 shows similar results as a function of the interference azimuthal angle rp;, with OJ = 90° . Examination of these curves shows that the worst output SINR is obtained when the interfer ence arrives from the same direction as the desired signal (broadside) and has the same polarization as the desired signal. This result is not surprising, of course, because in this case when the arra y nulls the interference it also nulls the desired signal. However the interesting
thing about this case is how little difference in polarization betwee n the signals is required to allow the array to provide substantial protection. For example, it may be seen in Figs. 4(b) or S(b ) (for (3j = (3d = 30 °) that when 0; = 90 ° and rp; = 90 °, if either CY.; = 0 or CY.; = 30° (i.e., if CY.j differs from CY.d = 15° by ±ISo) the SINR out of the array is higher than -9 dB. Thus with th is small difference in polarization, the array can prov ide over 31 dB of protection against the in terference. For the special case where both signals arrive from broad side , the output SINR from the array can be simply related to the polarizations of the two signals . If
328
(27)
(13) yields (28 )
10r------------------------,
o ~ -10
~ -20
H
en
-30
-40----------. . . . - ..... o 20
and also
= 2[ 1 + cos
2'Y d cos 2'Y;
+ sin
2'Y d
• sin 2'Yi cos (17d - 11i))·
(29)
Let .-\1 d and M; be points on the Poincare sphere representing the polarizations of the desired and interference signals, respectively. Then in (29), 2'Yd' 2'Yi and the arc MdM; form the sides of a spherical triangle, as shown in Fig. 6. The angle l1d - 11; is the angle opposite side M d1\1 i- Using a well-known spherical trigonometric identity [8 j , we have cos 2"'1d cos 2'Yi + sin 2'Y d sin 2'Yi cos (17d - 17,.) (30)
so (29) can be written
and (25) becomes
If ~i -
1
60
---.l~-..a....--~_....L_ _
80
100
120
M d M i (DEGREES)
Fig. 7.
SINR =
40
~
4 cos 2 d [
2_
~; -
(Mdtfi)] --
1
+
2 . 2
.
(32)
~ 2, this result may be approximated by
(33 ) These formulas show that the SIN R obtained from the array when both signals are at broadside depends only on the separation between the two points M d and ,M; on the Poincare sphere. A plot of the SIN R versus the spherical distance ItJdM; in angular measure, as obtained from (32), is shown in Fig. 7 for SNR = 0 dB and INR = 40 dB. It is seen, for example, that a separation of MdM; = 26° on the Poincare sphere results in SIN R = -10 dB, an improvement of 30 dB over what it would be without the array. 5 This result holds regardless of the specific Polarizations of the signals, so long as they are separated by 26° on the Poincare sphere. S To reconcile the curve in Fig. 7 with the results in Figs. 4(b) and S(b), one must note that point M in Fig. 3 lies above the equator by an angle 20:. Thus, for example, a separation of M fiMi = 26 0 corre0 ,SPonds to a difference of only I G:d - O:i I = 13 , if {Ji = (3d.
SIN R versus Poincare sphere separation. (J d
140
160
_J
180
= cPd = (J i == cPi == 90° .
In general, when the desired signal arrives from some direction other than broadside, the curves of SIN R versus interference arrival angle are similar to those in Figs. 4 and 5. The worst performance always occurs when the interference arrives from the same direction as the desired signal and has the same polarization. When both signals arrive from the same direction off broadside, however, it is found that the SINR cannot be related to the polarization difference so simply as 6 in (32). In this case the SIN R de pends on the angle of arrival as well as the polarizations. The reason is that the electric field component in the y-direction is not received by the array, because the array contains only x- and z-oriented dipoles. When the arrival angle and polarization of a signal are such that there is a v-com ponent of electric field, the Sf NR is affected by the loss of power in this component to the receiving system. The amount of electric field in the y-direction depends on the angle of arrival as well as the polarization. The curves in Figs. 4 and 5 show typical performance frorn the array for an arbitrarily polarized desired signal. However it must be noted that with this array certain desired signal polarizations allow the system to be jammed over a wide range of interference angles. Namely if the desired signal excites only two of the four dipoles, then when the interference excites only the same two dipoles, the array has no ability to discriminate between signals in the azimuthal coordinate ¢. This situation leaves the array vulnerable to interference from a wide range of angles. Specifically there are two cases where poor performance occurs: when the desired signal is either vertically or horizontally polarized. For example, suppose the desired signal is vertically polarized (CXd = 0°, (3d = 90°) and arrives from an arbitrary direction 0 d . ¢d. Then a vertically polarized in terference signal (CXi == 0°, ~i = 90°) will produce a poor SIN R as long as it arrives from the same polar angle, i.e., if 8; == 8 a. regardless of ¢l' It is clear why this is so from the arrangement of elements in Fig. 1. F or vertically polarized signals the array has no ability to discriminate in the azimuthal coordinate cP. 6 When the signals arrive from the same direction off broadside with different polarizations, one can express the output SINR in a form similar to (32) by means of an artifice, as follows. For each signal, one defines an "equivalent signal" whose amplitude, phase, and polarization are chosen to make the equivalent signal produce the same voltages in a pair of imaginary crossed dipoles oriented perpendicular to the arrival angle as the voltages produced by the actual signal in the actual dipoles. If two such equivalent signals are defined, one for the desired signal and one for the interference, the output SINR will be related to the difference in polarization of the two equivalent signals as in (32). However when this procedure is carried out, it is found that the transformation equations between each signal and its equivalent are complicated enough that little additional insight is gained. For calculating SINR, it appears to be simpler just to use (25).
329
10 ..--- ----=-
-
{3, ; o-
-
-
- - - - - - -- - -- - - - - - ,
10 .---
-
-
- - - - - - -- - --
-
------.,
± 2-
o
- 4 00 30
60
30
120
150
180
( DEGREES I
Fig. 8. SINR versus ¢j. ad = 90°, ¢d = 90·. Cl.d = O' .l3d = 90·, SNR = o dB. a i =90·. Oli =0°. INR =40 dB. A particularly bad case occurs when 8 d = 90° (and
30
Fig. 9. SINR versus rl>i' ad = 90°, ¢d = 90°. Old = 0·. (Jd o dB. a i =90°. (Ji =0·. INR =40 dB.
= 0°, SNR =
Figs. 8 and 9 illustrate the vulnerability of the system to jamming when both the desired signal and the interference have either vertical linear or horizontal linear polarization. This problem occurs because in these cases the signals excite only two of the four elements and also because the dipoles in Fig. 1 are all located at x = O. That is, there is no displacement of the dipoles along the x-axis and hence the array has poor ability to provide spatial discrimination in the
330
t:ld == 1S°, (3d = 30° (see Fig. 2). In general we find that the interference has only minimal effect on output SINR unless it arrives from the same direction and has the same polarization as the desired signal. Furthermore when both desired signal and interference arrive from broadside, the output SIN R depends only on the difference in polarization of the two signals, according to (32). Finally we have found that certain choices for desired signal pol:lrization lead to poor ability of the array to reject interferenc~. Specifically if the desired signal is linearly polarized with its electric field entirely in the 8- or
111 [2
S. P. Applebaum, .. Adapti ve arrays," IEEE Trans, Antennas Propagat., vol. AP-24. pp. 5g5-598. Sept. IY76. B. Widrow , P. E. Mantey. L. J. Gnffith~. and B. B. Goode.
.. Adapuve antenna systems." Proc. IEEE. vol 55, pp. 2 \432159, Dec. 1967. [3J W. F. Gabriel. "Adaptive arrays-c-an Introduction." Proc. IEEE, vol. 64. pp. 239-272. Feb. 1976. 14J Special Issue on Adaptive Antennas. IEEE Trans Antennas and Propag at .. vol AP-24. Sept 1976. 151 R. L. Riegler and R. T. Compton. Jr.. "An adaptive array tor
interference rejection," Proc. IEEE. vol. 61, pp. 748-75g. June 1973.
G. A. Deschamps. "Geometrical representation of the polarizauon of a plane electromagneuc wave." Proc. IRE. vol. 39. pp 540544. May 195 I . [71 D. G. Brennan. "Linear diversuy combining techruques ;" Proc. IRE. vol. 47, pp. 1075-1102. June 1959. l~1 K. L. Nielsen and J. H. Vanlonkhuyzen. Plane and Spherical Trigonometry. New York: Burnes and Noble. pp. 110-119. 1954. l y 1 A. S. Householder. The Theory oj Matrices in Numerical AlltlI\'SIS. New York: Dover. p. 3. 1964. . llUI R. T. Compton. Jr. R. J. Huff. w. G. Swarner. and A. A. Ksrenski, .. Adaptive arrays for communication svstcrns: An overview of research at {he Ohio State Uruvcrvity." iEEE Trans, Antennas Propagut .. vol. AP-24. pp. 59<J-007. I <J76. 1111 R. T. Compton. Jr.. "An adaptive array in a spread spectrum communication ~ystcm." Proc. IEEE. vol. 60. pp. 289-2l)8. 197X. [121 H. c. Ko. "On the reception of quasi-rnonochromauc. partially polur izcd radio wavcs." Proc. IRE. vol. 50. pp. 1950-1957. Sept. I Y72. 1131 A. Ishrdc and R T. Compton. Jr.. "On grating null« 111 adupuvc array\.·· IEEE Trans, Antennas LInd Propugat: vol AP-2~. pp. 467-475. July 1990 [61
331
Effect of Mutual Coupling on the Performance of Adaptive Arrays INDER J. GUPTA
AND
AHARON A. KSIENSKI,
Abstract-The effect of mutual coupling between array elements on the performance of adaptive arrays is examined. The study includes both steady state and transient performance. An expression Cor the steady state output signal-to-interference-plus-noise ratio (SINR) of adaptive arrays, taking into account the mutual coupling between the array elements, is derived. The expression is used to assess the steady state performance of adaptive arrays. The transient response is studied by computing the eigenvalues associated with the signal covariance matrix. The steering vector required to maximize the output SINK of Applebaum-type adaptive arrays in the presence of mutual coupling is also given.
coupling. In Section III, the effect of mutual coupling on the transient performance of adaptive arrays is studied. It is shown that the presence of mutual coupling between the array elements reduces the speed of response of an adaptive array. In Section IV, the steering vector required to maximize the output SINR of Applebaum-type adaptive arrays in the presence of mutual coupling is found. Section V contains conclusions.
II. STEADY STATE PERFORMANCE OF AN ADAPTIVE ARRAY IN THE PRESENCE OF MUTUAL COUPLING
I. INTRODUCTION HAS BEEN shown [1], [2 J that the performance of an antenna array is strongly affected by the electromagnetic characteristics of the antenna array. An important electromagnetic characteristic of an antenna array is the mutual coupling between its elements. In the above work, mutual coupling between the antenna elements was, however, ignored. i.e., the antenna elements were assumed to be isolated from each other. In practice, elements of an antenna array have mutual I: oupling, which in turn affect the gain. bearnwidth , etc . , of the array. Mutual coupling becomes particularly significant as the interelernent spacing is decreased. I n this paper the effect of mu tual coupling on the performance of adaptive arrays is studied. It is shown that the mutual coupling does affect the performance of adaptive arrays, and these effects are significan t even for large in terelernen t spacings, i.e., for spacing of more than half a wavelength. The effect is rather drastic as the interelernent spacing drops below half a wavelength. In (Jet, for a fixed aperture with half-wavelength spaced elements, the introduction of additional elements can degrade the array performance. The failure to recognize the presence of mutual coupling will degrade the performance of Applebaum-type adaptive arrays more than that of least mean square (lMS) arrays because the steering vector has to be modified both in phase and amplitude to include the changes in the desired signal vector due to the presence of mutual coupling. In Section II, an analytic expression for the steady state output signal-to-interference-plus-noise ratio (SINR) of an adaptive array is derived. The expression takes into account the mutual coupling between the array elements and involves the normalized impedance matrix of the array elements. The expression is used to study the effect of mutual coupling on the performance of adaptive arrays, and it is shown that the output SINR of the array depends upon the mutual coupling between its elements. For strong mutual coupling between the array elements, the output SINR drops significantly below its value in the absence of mutual ·;r T
l adaptive
FELLOW, IEEE
The output SINR of an adaptive array is the most commonly accepted measure of its steady state performance, and will accordingly be derived first. The basic diagram of an adaptive array is shown in Fig. 1. The output signal from each element is multiplied by a complex weight, and then these signals are summed to produce the array output Set). The weights are automatically adjusted to optimize the output SINR in accordance with a selected algorithm. To find an expression for the output SINR, one must know the element output voltages. We will, therefore, first develop an expression for the element output voltages when the mutual coupling is taken into account. These voltages will be used as the input signals to the adaptive processor. The required expression can be obtained by considering the iV-element array as an N + 1 terminal linear , bilateral network responding to an outside source as shown in Fig. 2. Referring to Fig. 2. each port of the lV-elenlent array is shown terminated in a known load impedance ZL. The array has as its driving source a generator with open circuit voltage Vg and internal impedance Zg. Using standard notation, one can write the Kirchoff relation for the JV + 1 terminal network as
(1) .
.
J' = ijZN j + ... + ijZ N i
..
+ ... + iNZ N N + isZN s
where Zij represents the mutual impedance between the ports (array elements) i and j. Further, making use of the relationship between terminal current and load impedence, vi
;.=--, /
ZL
i==1,2,···,N.
(2)
If all the elements in the array are in an open circuit condition then
ij=O, j= 1,2,···,N, Manuscript received November 8, 1982; revised March 31, 1983. The authors are with the ElectroScience Laboratory, The Ohio State University, 1320 Kinnear Road, Columbus, OH 43212.
and from (1) vi
= uOi = Zjsis·
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP- 31, No.5, pp. 785-791, September 1983.
332
(3)
Let m + 1 continuous wave (CW) signals (one desired and m jammers) of the same frequency be incident on the array. Then the open circuit voltages at the antenna terminals are given by
stu
m
Vo =Xd
+~
kc:1
x,
(7)
where
Xd
x,
CONTROL SIGNAL
Fig. 1.
ZL
ZL
ip
ZL
=
[2(8 d, fJ>d,Pd)e iPd2
(10)
ZL
Antenna array as aN + 1 terminal network.
where (ed, ¢Jd) defines the desired signal direction. Pa is the polarization of the desired signal, fj(8, fJ>, p) is the pattern response of the jth element to a signal incident from direction (e, ¢J) with polarization P and Pdf is the desired signal phase at the jth element, measured with respect to the coordinate origin.
Substituting (2) and (3) into (1) one gets
Z11 1+_
(9)
f 1 ((J d , fJ>d' Pd) e d 1
BILATERAL N+I PORT NETWORK
Ud
Fig. 2.
«v.:
coordinate origin, l/J ik is the carrier phase of the kth jammer at the coordinate origin and Ud, Ui k are, respectively, the desired signal vector and the kth jammer vector defined as follows;
OUTSIDE SOURCE
__----...10..-------'---------. } A~~~~~A LINEAR
Aikei(wot+1/J
A 7k is the average power in the kth jammer, Wo is the carrier frequency, l/J d is the carrier phase of the desired signal at the }
_
=
(8)
In (8) and (9), A~ is the average power in the desired signal,
Basic adaptive array.
Vi •
= Adei(wot+l/Jd)Ud
Z 12
ZlN
ZL
ZL
vi
II (Oik, ¢Jik' Pik)e i Pik J Uik
J I
VON
f2COik' ¢Jik' Pik)eiPik2
(11)
NC8ik, ¢Jik' Pik )e i PikN
where the notation is analogous to that for the desired signal vector. Using (6) and (7), the input signal to the adaptive processor will be
(4)
Or, more compactly
V=ZOl(Xd+i;xik)'
(5)
In (5), Zo is the normalized impedance matrix and Vo represents the open circuit voltages at the antenna terminals. Since Zo is nonsingular, one can find the element au tput voltages from the open circuit voltages. The element output voltages will be given
v = Zo 1 Vo.
=
k=l
If thermal noise is also added to each element of the array then the total input signal to the processor will be X= V+X n
(6)
::=
It should be noted that the matrix Zo is a normalized impedance matrix, normalized to the load impedance. It acts like a transformation matrix, transforming the open circuit element voltages to the terminal voltages. What is normally assumed in analyzing adaptive antenna systems is that the element spacing is large enough .so that the mutual coupling between the elements is small and consequently the matrix Zo becomes diagonal. If one further assumes that the self-impedances (Z;;, i = 1,2, ... ,N) are equal, the input signal vector will be just the open circuit voltage vector multiplied by a trivial scaling factor involving the self and load impedance terms. Thus the array performance will be the same as calculated using the open circuit voltages as the input signals to an adaptive processor.
(12)
z; 1 [Xd+ i; xikJ + X n
(13)
k=l
where X n is the noise vector defined as Xn
= (n 1( t), n2 (t),
'.., nN (t)) T .
(14)
In (14), T denotes transpose. In the case of an adaptive array, the signal xi(t) from the jth element is multiplied by a complex weight wi(t). The signals are then summed to produce the array au tpu 1. Using the LMS algorithm [3], the steady state weight vector W of the array is given by W=-IS
(15)
where is the covariance matrix
333
= E {X* X T }
(16)
and S is the reference correlation vector
S
gets
= E{X*R(t)}.
(17)
ZT
~-l = a~ (R; 1 - iR; 1 ~U'SR; 1 X(Zo J )*)- I
In (16) and (17), R(t) is the complex reference signal in the adaptive array [3], [4], the asterisk denotes complex conjugate, and E{ •} denotes expectation. From (13) and (16)
where (26)
~=EI [ZoJ (xd+~ Xik) +XinT · [ZOI
(Xd+
= (ZOI)*£
I[(
tl Xik) +XnJTl
x, + ~ X ik) + ZoX n
The array will acquire and track the desired signal if the reference signal is correlated with the desired signal and is uncorrelated with interference signals. Assuming that the reference signal R(t) is given by
r
R(t)
S = A,.Ad(ZO 1 )*U~.
[a Z*Z T + ~ A ~ u: UT + A U*U 0
0
~
k=l
Ik
Ik
d2
Ik
d
T] d
W = KZ&R;; 1 ~
(19)
-vhere 0 is the thermal noise power. From (19)
+
~ ~
T
~
k=l
AA K=~ (1- r UTR-l~) 2 d n d a
Pd
SINR= --m
(21)
k==l
~ Pi k +Pn
where Pd is the output desired signal power
Pik is the output interference power due to the kth jammer
Pi k
= ~ E{I(Z Ol X i k)TWI 2 } A2
2 = ~I 2 Ur(ZIk 0 1)TW1 ,
0
+ ~dUaUJXZo 1 )T.
(22)
Note that R n is the normalized (with respect to the thermal noise power) covariance matrix of the undesired signals(jammers and the thermal noise). To find the steady state weights (15), ~1>-1 must be computed. The following matrix inversion lemma [5] is used to compute <1>-1: (A-aU*UT)-l =A- 1-{jA- 1U*U T A -
1
(23)
where A is a nonsingular N X N matrix, U is a N X 1 column vector and Q, t3 are scalars related by
a-I +/3-1 =UTA-1U*.
(24)
Using the matrix inversion lemma to invert in (22) one
(33)
and Pn is the output thermal noise power 2
Pn=-IWI 2 . 2
then = (Zo 1 )*a (R n
(31)
k==l
where ~d is the ratio of the desired signal power to the thermal noise power and ~ik is the ratio of the kth jammer power to the thermal noise power. Let
2
(30)
(32)
(20)
s, = z3zl; -+ ~ ~ikU!kUhc
(29)
is a constant. The weights given in (29) will lead to the maximum output SINR in the presence of multiple jammers (see Appendix) . Knowing the steady state weight vector, one can compute the output SINR of the array which is given by
t v ,,*UT]eZ-1)T + C;d d d 0
m
(28)
where
(Z-l)T 0
2
"" = (Zo-1)*a 2 [Z*OZOT 't'
(27)
Using (25) and (28) the steady state weights (15) of the array are given by
Assuming that the thermal noise voltages from the array elements are Gaussian with zero mean and are uncorrela ted with each other, and the carrier phases of the narrow-band signals .ire uniformly distributed on (0, 2n') and are statistically independent of each other and of the thermal noise voltages, the covariance matrix ~ is given by 2
= A,.ej(wot+ wd)
and using (13), (17) yields
(18)
= (Z-l)* o
(25)
(34)
Using (3l)-{34), (31) yields
SINR = ~d UJR;; J U;.
(35)
Equation (35) is used to compute the steady state output SINR of an adaptive array consisting of N half-wavelength, center-fed dipoles. All the dipoles are assumed to have similar radiation characteristics and are spaced at a distance d apart (Fig. 3). Note that H-plane arraying of the dipoles is done. The desired signal and all jammers are assumed to be theta polarized (Fig. 3). For the results presented in this 'paper, [k(f) , 4>, p) = 1.28142 (10). Fig. 4 shows the output SINR of an adaptive array of six dipoles as a function of the desired signal direction. The dipoles
334
o
N'"
..,;
.,.
o
o
N
Fig. 3.
o
An array of N half-wavelength, center-fed dipoles. Fig. 5.
o
2 3 IN WAVELENGTHS
4
5
Mutual impedance between two half-wavelength , center-fed thin dipoles versus the spacing between the dipoles.
-
CD
-e
a: CD z
CD
-e
H
U)
a:'" ....z
NO MUTAL COUPLING - - WITH MUTUAL COUPLING
U)
NO MUTUAL COUPLING WITH MUTUAL COUPLING
04-.,....,.,....,....,.,....,...........,......,....,....,.....,-.....,........--,.......,....,...,
-1.0
-0 .5
0
0.5
1.0
SIN (epd)
Fig. 4. Output SINR of an array of six half-wavelength, center-fed dipoles versus the desired signal direction (<1>d)' (d = 5 dB, 9d = 90°, d = 0.51\ , ZL = Zii'
are spaced at a distance of half a wavelength and each dipole is terminated in a load impedance equal to the complex conjugate of the self impedance of a half-wavelength, center-fed dipole . 1 The input signal-to-noise ratio (~d) is 5 dB , and the output SINR is computed in the absence of all jammers. The continuous curve in the figure shows the output SINR when the mutual coupling between the array elements is taken into account while the broken curve represents the output SINR when the mutual coupling between the arra y elements is ignored . Note that the presence of mutual coupling changes the array performance and the output SINR of the array depends on the angle of arrival of the desired signal. The dependence of the array output SINR on the angle of arrival of the desired signal can be explained as follows . Mutual coupling changes the desired signal component of the element output voltages (6) . The array illumination due to the desired signal is no longer uniform and depends on the angle of arrival of the desired signal , while the noise being internal is not affected by the mutual coupling between the array elements. The output SINR of the array, therefore, changes with the angle of arrival of the desired signal. In the above example , the interelement spacing was large (A/2) and we found that the mutual coupling does nevertheless affect the performance of the array. For small interelement spacings , the mutual coupling between the array elements will be large (Fig. 5) and, therefore, the array performance will be affected mo re. This is evident in the plots of Fig . 6, where the output SINR of the array is plotted as a funct ion of the interelement spacing . The desired signal is incident from the broadside 1 The self and mutual impedance between the dipoles were computed by evaluating the expressions given by Schelkunoff and Friis [6] .
0+rr-T'T-r-r""""''''''''''''''''''''''''T"T.,....,....,.....,...rr-rT"l a 2 3 4 5 o IN WAVELENGTHS Fig. 6. Output SINR of an array of six-half-wavelength, center-fed dipoles versus the interelement spacing. td = 5 dB, ZL = Zii' (9d, <1>d) = (90° ,0°).
direction (90° , 0°) and jammers are assumed to be absent. The broken curve in the plot shows the output SINR in the absen ce of mutual coupling while the continuous curve shows the output SINR when the mutual coupling is taken into account. Note tha t the mutual coupling between the array elements affects the array performance even for large interelement spacing (d > A/2) . The effect is more pronounced for small interelement spacing '(d < A/2) , where the output SINR drops below the expected value (in the absence of mutual coupling) by a significant amount. The smaller the interelement spacing , the larger the drop in the SINR. Note that the SINR curve is similar to the gain curve of a broadside array of six half-wavelength , center-fed dipoles [7] . The drop in the output SINR for small interelement spacings can also be rela ted to the reduction of the total incident energy . As the interelement spacing is decreased, the total aperture of the antenna decreases and so does the total incident energy due to the desired signal. Since the receiver internal noise remains unchanged . the signal-to-noise ratio drops . For a similar reason, the introduction of additional elements into a fixed aperture with half-wavelength spaced antenna elements can degrade the adaptive array performance. The total aperture is fixed and so is the total incident ene rgy . The introduction of additional elements adds to the thermal noise without increasing the available signal power and that degrades the array output SINR. Fig . 7 shows the output SINR of an adaptive array as a function of the num ber of antenna elements. The array is a linear array of half-wavelength , center-fed dipoles . The total aperture is fixed at 21.. and the desired signal is incident from the broadside direction. Again jammers are absent. The output SINR is computed with and with out mutual coupling. In these plots only the indicated points are
335
by SINR = lD
-e
-lD
a:: Z
H
en
...
o-o-e ti-6-ii
o
NO MUTUAL COUPLING WITH MUTUAL COUPLING
2 4 8 NUMBER OF ELEMENTS
10
12
Fig. 7. Output SINR of an may of half-wavelength, center-fed dipoles of fixed aperture versus the number of elements. td = 5 dB, ZL =Z1i. (8 d, ¢d) =(90·, 0·), total aperture = 2 A.
meaningful (the total number of antenna elements is always an integer) . Note that in the absence of mutual coupling the output SINR increases with the introduction of additional elements. It is consistent with the previous work of Compton [8], but in the presence of mutual coupling, the array output SINR can decrease with the introduction of additional elements. One can see that the output SINR reaches a maximum for a four-element array and the array performance degrades with the introduction of additional elements. The reason for this is that one needs a minimum number of antenna elements to receive all the energy incident on a given aperture . But beyond this point the aperture becomes overcrowded and leads to a worse performance . In the examples given so far, the array performance was computed in the absence of jammers. The presence of jammers degrades the array performance. As pointed out in our earlier work [1] , [2]. the degradation in the array performance can be computed using the unperturbed pattern of the array. The unperturbed pattern of an adaptive array was defined to be proportional to the receive pattern of the array responding to a single desired signal in the absence of interfering signals. In the absence of jammers. the normalized noise covariance matrix (21) becomes R tl = ZOZT.
(36)
Substituting (36) in (29), the steady state vector of the array will be
W = K(Zo IUd )*.
(37)
Thus . the weight vector for the unperturbed pattern will be (Z~ 1 Ud )* and the value of the unperturbed pattern in the direction (0. cp) for polarization P will be (38) where U is the signal vector of the array in direction (0, cp) for polarization p. Substituting (36) into (35), the output SINR c-f the array in the absence of all jammers will be SINR
= ~d(ZO I Ud)T(Zo J Ud)* (39)
and is proportional to the value of the unperturbed pattern (38) in the desired signal direction. Further, following the same procedure as given in [1]. it can be shown that the output SINR of the array in the presence of one jammer will be given
dr
~ ~(O
cP d »
2
d,Pd
) _IE(Oil' CPi! ,PiJ) 1
(ZoJuiJl(ZoJuiJ)*
]
(40)
where E(0 iJ , CPiJ, Pi!) is the value of the unperturbed pattern in the jammer direction . The same can be done for multiple jammers [2J . Thus , the degradation in the array performance can be computed using the unperturbed pattern . In this section, the effect of mutual coupling between the array elements on the steady state performance of an LMstype adaptive array was presented. It was shown that though LMS adaptive arrays produce the maximum obtainable output SINR (see Appendix), their performance is affected by mutual coupling . One should, therefore, take mutual coupling into account to compute the true output SINR of the array. In the next section , the transient response of an adaptive array in the presence of mutual coupling will be studied. III. TRANSIENT RESPONSE OF AN ADAPTIVE ARRAY IN THE PRESENCE OF MUTUAL COUPLING The speed of response of an adaptive array is controlled by the eigenvalues of its signal covariance matrix [4] . As pointed out in the last section , the presence of mutual coupling between the array elements affects the input signals to the adaptive processor and thus the covariance matrix. The eigenvalues of the covariance matrix will, therefore, be different than those in the absence of mutual coupling. In this section , the transient response of an adaptive array in terms of the eigenvalues of its covariance matrix will be studied. From (20), the covariance matrix 4> can be written as
4>
= 0 2 [I +
i: ~ik(ZO
ke J
J Uik)*(Zo J Uik)T
+ ~d(ZO J Ud)*(Zo I Uik)TJ .
(41)
In the presence of m + 1 signals (one desired signal and m jammers), 4> has, at least, N - m - 1 eigenvectors (N is the total number of antenna elements) having unity eigenvalues (assuming 2 0 = 1) and the rest of the eigenvectors have eigenvalues larger than one . The presence of mutual coupling between the array elements will affect these eigenvalues. We will, therefore, compute the nonunity eigenvalues to study the effect of mutual coupling on the transient response of adaptive arrays . Fig. 8 shows the nonunity eigenvalue of a six half-wavelength , center-fed dipole adaptive array in the presence of one signal only (no jammer). The desired signal is incident from the broad side direction (90°; 0°) and is 10 dB stronger than the thermal noise. The eigenvalue is plotted as a function of the interelement spacing . Note that the mutual coupling between the array elements affects the eigenvalues even for large interelement spacing, but the effect is more severe when the spacing is less than half a wavelength. For small spacings, the eigenvalue drops significantly below the value obtained in the absence of mutual coupling (broken line) . The drop in the eigenvalue indicates a reduction in the speed of response of the adaptive array. In other words , the array will take more time to adapt to the changes in the desired signal parameters, The main feature of an adaptive array is nulling the undesired signals Gammers). The
336
the array elements affects the eigenvalues even for large interele. ment spacings, but the effect is more severe for small interelement spacing (d ~ "A/2). For such spacing, the eigenvalues drop substantially below the values that they would have had in the absence of mutual coupling (broken curve). The smaller the eigenvalues, the longer will be the transient, and the array will take more time to null the jammers, which may be undesirable. The strong fluctuations in the eigenvalues for large interelement spacings are due to the fact that one or more signals (jammer as well as the desired signal) are incident from grating lobe directions. In this section, the effect of mutual coupling on the transient response of an adaptive array was studied. It was shown that the presence of mutual coupling between the array elements affects the transient behavior of the adaptive array and slows its response to both desired as well as jamming signals for closely spaced elements. In the next section, it is shown that mutual coupling affects the performance of Applebaum-type adaptive arrays and that the steering vector for these arrays must be modified to account for mutual coupling in order to maximize the output SINR.
--WITH MUTUAL COUPLING - - N O MUTUAL COUPLING
-~
o
I
234 DIN WAVEL ENGTHS
5
Fig. 8. Nonunity eigenvalue (A) of an array of six half-wavelength, centerfed dipoles in the presence of a desired signal versus the interelement 0 spacing. td = 10 dB, (8d, tPd) = (90 , 0°) ZL = Zii' no jammer.
--WITH MUTUAL COUPLING - - - NO MUTUAL COUPLING
- -
~--.----.--....""..--~--
IV. APPLEBAUM ARRAYS In the case of an Applebaum adaptive array. one uses a steering vector or initial weights instead of a reference signal as the control signal (Fig. 1). The steady state weight vector for this type of adaptive array [9] is given by
o
2 3 4 DIN WAVELENGTHS
5
W= [
Fig. 9.
Nonunity eigenvalues (A) of an array of six half-wavelength, center-fed dipoles in the presence of a desired signal and a jammer versus the interelement spacing. td = 10 dB, (8d, epd) = (90°, 0°), ~i1 = 20 dB, (8il,
1.25
o
IN
MUTUAL COUPLING
2.5
3.75
WAVELENGTHS
(42)
where Us is the steering vector, G is the loop gain, and J is a N X N identity matrix. Note that (42) contains the signal covariance matrix <1>. As pointed out earlier, the presence of ITIUFlal coupling between array elements will change (}> and thus will affect the steady sta te performance of Applebaum-type adaptive arrays. The eigenvalues of the signal covariance matrix can trol the speed of response of Applebaum -type adap tive arrays too. The presence of mutual coupling between array elements will, therefore, affect the transient response of Applebaum-type adaptive arrays in the same fashion as it affects the LMS array discussed in the last section. We will now find the optimum steering vector to maximize the output SINR of Applebaum-type adaptive arrays in the presence of mutual coupling. Assuming that the loop gain G is large, the steady state weights (42) become
- - - NO MUTUAL COUPLING
o
1 J-J Us G 1+
5
Fig. 10. Nonunity eigenvalues (A) of an array of six half-wavelength, center-fed dipoles in the presence of a desired signal and two jammers versus the interelement spacing. td = 10 dB, (Od, tPd) = (90 0 , 00 ) , til = 20 0 0 dB, ti2 = 30 dB, (8 iI, cPil) = (90 , 30°), (8 i2,
Z;'..
transient response of an adaptive array in the presence of jammers, therefore, will be considered next. Figs. 9 and 10 show the nonunity eigenvalues in the presence of one and two jammers, respectively. The angles of arrival of 0 0 0 0 the two jammers are (90 , 30 ) and (90 , _45 ) , respectively, and the jammers are 10 dB and 20 dB stronger than the desired signal. Again the eigenvalues are plotted as a function of the interelement spacing. Note that the mutual coupling between
W = -IUS'
(43)
Using (25) in (43) one gets
W --
Z6 (R-n 1 -
(44) rR-1U*UTR-1)Z*U n d d nOS' 02 If the steering vector is chosen to steer the beam in the desired signal direction, i.e., Us = Ud then the steady state weight vector will be given by
ZT
W == ~ (R- 1 - rRJ U*UTR- 1 )Z~U* 02 n n d d nOd·
(45)
Comparing (29) and (44) one can see that the two are not the same, and thus this choice of the steering vector would not give the optimum SINR. If instead of the "open circuit" desired
337
that obtained when mutual coupling is ignored and the speed of response of the array is reduced. APPENDIX
-
From (21)
CD 'a
s; =Z3!I+ Let U';k
f
k=l
~ik(ZOlUik)*(ZOlUikl}ZoT.
(49)
= ZC;l Ui k .
Then (49) becomes
o
2 3 4 IN WAVELENGTH 5
5
Fig. 11. Output SINR of any array of six half-wavelength, center-fed dipoles versus the interelement spacing using Applebaum algorithm. td = 5 dB, Z L = Z7;, (8 d» tPd) = (900 , 09 ) .
signal vector (U d)' the complex conjugate of the desired signal component of the element output voltage is used to generate the steering vector, Le.,
Us = (ZOJUd)*,
(46)
then from (44)
W=KtZ'{;R;;JUd,
(47)
where 1 T _ J * K 1 = 2" (l-iUdR n Ud ) ·
a
(48)
s; = Z~ {I + ~ ~ikU;;U;r} Z'[ = Z6R~Z'[
(50)
where
R~ is
m
= I + ~ ~ikU;k* U;f. k==l
(51)
Using (50) in (29) the steady state weight vector of the array
w= K(R~)-l(Zo 1 Ud )* = K(R~)-ll!d*
(52)
where
lid = Zo) ti;
(53)
Comparing (52) with the optimum weight vector [9, eq. (4.1)] one can see that the two are similar. Thus the weight vector given by (29) will lead to the maximum output SINR in the presence of multiple jammers.
Comparing (29) and (47), one can see that the two weight vectors differ only by a scale factor. The choice of the steering vector as given in (46) will, therefore, lead to the optimum perREFERENCES formance of the array. [I} I. J. Gupta and A. A. Ksienski, "Dependence of adaptive array perFig. 11 shows the output SINR of an adaptive array of six formance on conventional array design." IEEE Trans. Antennas Propagat.; vol. AP-30. pp. 549-553. July 1982. half-wavelength, center-fed dipoles as a function of the interele[2] - , "Prediction of adaptive array performance," IEEE Trans. Aeror.ent spacing for the two choices of steering vectors. The desired space Electron. Syst .. vol. AES-19, no. 3, pp. 380-388. May 1983. signal is incident from the broadside direction and the load [3J 8. Widrow, P. E. Mantey. L. J. Griffiths. and 8. B. Goode. "Adaptive antenna systems, Proc. IEEE. vol. 55, no. 12, pp. 2143-2159, impedance is equal to the complex conjugate of the self impedDec. 1967. ance of a half-wavelength, center-fed dipole. Again jammers are [4J R. L. Riegler and R. T. Compton, Jr .••. An adaptive array for interassumed to be absent. Note that when the steering vector is ference rejection." Proc . IEEE. vol. 61. no. 6, pp. 748-758. June 1973. chosen according to (46). the adaptive array gives a better per[5] A. S. Householder, The Theory of Matrices in Numerical Analysis. formance for small interelemen t distances where the mutual New York: Dover, 1964, p. 123. coupling is the strongest. Comparing Figs. 6 and 11, one can [6] S. A. Schelkunoff and H. T. Friis, Antenna Theory and Practice. New York: Wiley, July 1966, p. 409. see that the output SINR of the properly excited Applebaum C. T. Tai, "The gain of uniform arrays of isotropic sources and a.ray is the same as that of the LMS array. In the presence of ' [7] dipoles," Ohio State Univ. Antenna Lab. Tech. Rep. 1522-2, Mar. jammers, the array output SINR will be the same as that of the 1963. [8] R. T. Compton, Jr., "A method of choosing element patterns "in an LMS array and can be predicted using the unperturbed pattern. adaptive array," IEEE Trans. Antennas Propagat., vol. AP-30, pp. Thus, Us = (ZOI Ud )* , leads to the optimum performance 489-493, May 1982. of an Applebaum-type adaptive array. [9] S. P. Applebaum, Adaptive arrays," IEEE Trans. Antennas Propato
lO
gat., vol. AP-24, pp. 585-599, Sept. 1976.
V. CONCLUSION In this work, the effect of mutual coupling between array elements on the performance of adaptive arrays was studied. I t was shown that mutual coupling affects the performance of adaptive arrays even for large interelement spacings. The effect is particularly serious for small interelement spacing where the steady state output SINR of the array is significantly lower than
338
Optimum Combining in Digital Mobile Radio with Cochannellnterference JACK H. WINTERS,
Abstract-lbis paper studies optimum signalcombining for spacediversity reception in cellular mobile radio systems. With optimum combining, the signals received by the antennas are weighted and combined to maximize the output signal-to-interference-plus-noise ratio. Thus, with cochannel interference,space diversity is used not only to combatRayleigh fading of the desired signal (as with maximal ratio combining) but also to reduce the power of interfering signals at the receiver. We use analytical and computer simulation techniques to determine the performance of optimum combining when the received desired and interferingsignals are subject to Rayleigh fading. Results show that optimum combining is significantly better than maximal ratio combining even when the numberof interferers is greater than the number of antennas. Results for typical cellular mobile radio systems show that optimum combining increases the output signalto-interferenceratio at the receiver by severaldecibels. Thus, systemscan require fewer base station antennas and/or achieve increased channel capacitythrough greater frequency reuse. We also describe techniques for implementing optimum combining with least mean square (LMS) adaptive arrays. I.
S
INTRODUCTION
PACE diversity provides an attractive means for improving the performance. of mobile radio systems. With space diversity, the signals from the receiving antennas can be combined to combat multipath fading of the desired signal and reduce the relative power of interfering signals. Previous studies of mobile radio systems (e.g., [1]) have considered space diversity only for combating multipath fading of the desired signal. Interference at each receiving antenna is assumed to be independent in these studies. Under this condition, maximal ratio combining! [1, p. 316] produces the highest output signal-to-interference-plusnoise ratio (SINR) at the receiver. However, in most systems (in particular, cellular mobile radio systems [1]) the same interfering signals are present at each of the receiving antennas. Thus, the received signals can be combined to suppress these interfering signals in addition to combating desired signal fading and thereby achieve higher output SINR than maximal ratio combining. The output SINR can be maximized by using adaptive
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array techniques at the receiver (e.g., [2]-[4]). We will not analyze the performance of the various adaptive array techniques in this paper, but only study the performance of the optimum combiner that maximizes the output SINR. Although the adaptive array (i.e., optimum combiner) has been studied extensively, it has not been previously analyzed with the fading conditions of digital mobile radio. This paper studies the performance of the optimum combiner in digital mobile radio systems. We assume flat Rayleigh fading across the signal channel and independent fading between antennas. The average bit error rate (BER) of the optimum combiner is studied for coherent detection of phase shift keyed (PSK) signals. Analytical and computer simulation results show that optimum combining is significantly better than maximal ratio combining even when there are more interferers than receive antennas. Results for typical cellular mobile radio systems show that the optimum combiner can increase the output SINR several decibels more than maximal ratio combining. In Section II we describe the optimum combiner. Section III studies the BER of the optimum combiner when the desired and interfering signals are subject to Rayleigh fading. We discuss analytical results for one interferer and Monte Carlo simulation results for multiple interferers. In Section IV we consider the optimum combiner performance in cellular mobile radio systems. Section V discusses the possible methods for implementing the optimum combiner in mobile radio with a least mean square (LMS) [3] adaptive array. A summary and conclusions are presented in Section VI. II.
OPTIMUM COMBINER
A. Description and Weight Equation
Fig. 1 shows a block diagram of an M element space diversity combiner. The signal received by the ith element y;(t) is split with a quadrature hybrid into an in-phase signal xI;(t) and a quadrature signal xQ,(t). These signals . Manuscript received February 25, 1983; revised November 23, 1983. are then multiplied by a controllable weight WI (t) or This paper was presented at the International Conference on Communica- wQ;(t). The weighted signals are then summed to form the tions, Boston, MA, June 1983. The author is with the Radio Research Laboratory, AT&TBellLabora- array output so(t). tories, Holmdel, NJ 07733. The space diversity combiner can be described mathe1 I!1 maximal ratio combining, the received signals are weighted promatically using complex notation [5]. Let the weight vector portionatelyto their signal-voltage-to-noise-power ratios and combinedin phase. w be given by Reprinted from IEEE Transactions on Vehicular Technology, Vol. VT-33, No.3, pp 144-155, August 1984.
339
signals are uncorrelated, we can show that R nn ARRAY OUTPUT
and the received signal vector x be given by
(2) The received signal consists of the desired signal, thermal noise, and interference and, therefore, can be expressed as L
L »,
(3)
j=:l
where x d ' x n ' and x j are the received desired signal, noise, and jth interfering signal vectors, respectively, and L is the number of interferers. Furthermore, let Sd(t) and Sj(t) be the desired and jth interfering signals as they are transmitted, respectively, with
(4) and L.
(5)
L UjSj(t)
(6)
for 1 ~ j
~
Then x can be expressed as L
(8)
(9)
where a is.a constant.? and the superscript -1 denotes the inverse of the matrix.
(1)
= UdSd(t)+x n +
j=l
w = aR-1u* nn d
Fig. 1. Block diagram of an M element space diversity combiner.
X
L
"i: E[u~u~] J J
where 0 2 is the noise power and I is the identity matrix. In (8) the expected value is taken over a period much less than the reciprocal of the fading rate (e.g., several bit intervals). Note that we have assumed that the fading rate is much less than the bit rate. Finally, the equation for the weights that maximize the output SINR is (from [6])
WEIGHT GENERATION
X=Xd+X n+
= 0 2 1+
j=l
where Ud and u j are the desired and jth interfering signal propagation vectors, respectively. The received interference-plus-noise correlation matrix is given by
where the superscripts * and T denote conjugate and transpose, respectively. Assuming the noise and interfering
340
B. Discussion The mobile radio environment is quite different from the signal environment in which adaptive arrays (i.e., optimum combiners) are usually employed. In a typical adaptive array application in a nonfading environment, at the receiver there are only a few interfering signals, and their power is much greater than that of the desired signal. The adaptive array places nulls in the antenna pattern in the direction of these interferers, greatly suppressing these signals in the array output. In general, an M element array can null up to M - 2 interfering signals and still optimize desired signal reception. The output SINR is, therefore, substantially increased by the array. In mobile radio systems, on the other hand, at the receiver there can be several interfering signals whose power is close to that of the desired signal, and numerous interfering signals whose power is much less than that of the desired signal. Therefore, the number of interfering signals may be much greater than M, and the array may not be able to greatly suppress every interfering signal. Thus, the array output SINR may not be markedly increased by the array. To be useful in mobile radio systems, however, the adaptive array does not have to greatly suppress interfering signals or vastly increase the output SINR. Interfering signals need only to be reduced in power by a few decibels so that their power is below the sum of the power of other interferers. Furthermore, a substantial output SINR improvement is not required because a several decibel increase in output SINR can make possible large increases in the channel capacity of the system. Thus, although the signal environment in mobile radio systems is quite different from that of the typical adaptive array system, adaptive array techniques can still offer significant advantages. One other major difference is the ability of the array to resolve closely spaced transmitters. In a nonfading environment the adaptive array cannot suppress an interfering 2 Note that a does not affect the array output SINR (i.e., the array performance) and, therefore, we will not consider its value.
signal if the angular separation between the interfering and desired transmiters is too small. In this case, the desired signal phase difference between receive antennas is nearly the same as that for the interfering signal. Therefore, the array cannot both null one signal and enhance reception of the other. The use of additional antennas results in a small decrease in the required angular separation, but the problem remains. In mobile radio, because of multipath, the signal phase at one antenna is independent of the signal phase at another antenna when the antenna separation is greater than half of a wavelength (several inches at 800 MHz) [1, p. 311].3 Therefore, the adaptive array antenna pattern is meaningless. Similarly, at the receive antennas, the signal phases from two different transmit antennas are independent when the transmit antennas are more than half of a wavelength apart. Therefore, for all practical purposes, the received signal phases are independent of vehicle location. Thus, the resolution of interfering and desired signals does not depend on how closely the vehicles are located. Instead, for all locations, there is a small probability that the array cannot resolve the two signals. This occurs when the phase differences between the antennas are nearly the same for both the desired and interfering signals. With moving mobiles," however, the period of time with unresolved signals is very brief, and the performance of the adaptive array can be averaged over the fading. Furthermore, since the signal phase differences between the antennas are independent, the probability of unresolved signals with M antennas is approximately equal to the probability of unresolved signals with two antennas raised to the M - 1 power. Thus, each additional antenna greatly decreases the probability of unresolved signals, and this probability becomes negligible with only a few antennas. In summary, in mobile radio the adaptive array cannot resolve desired and interfering signals a small percentage of the time, rather than over a given angular separation. Therefore, we need not be concerned with vehicle location for the resolving of the signals. In the following analysis we study the optimum combiner performance averaged over the Rayleigh fading.
III.
biner, first determining the output SINR distribution and then the BER for coherent detection of PSK. All results are compared to those for maximal ratio combining. In cellular mobile radio, the interference plus noise at the receiver consists primarily of cochannel interference. In a typical system, there are numerous cochannel interfering signals, each of which affects the performance of the optimum combiner. An exact analysis of the performance is, therefore, quite complicated, especially since, with fading, each of these signals has a random amplitude. Therefore, in the analysis in this paper, we consider only the strongest interferers individually. The remaining interfering signals are combined and considered as lumped interference that is uncorrelated between antennas. Since with Rayleigh fading the in-phase and quadrature components of each of the received interfering signals have a Gaussian distribution, the components of the sum of these signals also have a Gaussian distribution, and the sum can be considered as thermal noise. Thus, under this assumption the combiner cannot suppress the lumped interference, and we are therefore analyzing a worse situation since the actual combiner performance will be better. Therefore, although this analysis does not show the maximum improvement (over maximal ratio combining) with optimum combining, it does show most of it. Note that if we consider all the interference as lumped interference, the results are identical to those for maximal ratio combining. The analysis involves several parameters which are defined as follows:
r = mean received desired signal power per antenna mean received noise plus interference power per antenna
r = d
(10)
mean received desired signal power per antenna (11) mean received noise power per antenna
I'. = mean received jth interferer signal power per antenna } mean received noise power per antenna
OPTIMUM COMBINER PERFORMANCE WITH
(12) _ local mean desired signal power at the array output YR mean noise plus interference power at the array output
FADING
This section studies the performance of the optimum combiner when the received signals are subject to fading. For the analysis we assume flat fading across the channel, with the fading independent between antennas. We assume that the received signal has an envelope with a Rayleigh distribution and a phase with a uniform distribution. We study the steady state performance of the optimum com3In our analysis we assume the antennas are spaced far enoughapart so that the received signal phases are independent. ..If all mobiles are stationary, channel reassignment can be used to
eliminate the problem.
341
and
(13)
_ local mean desired signal power at the array output Ylocal mean noise plus interference power at the array output
(14) In the above definitions, mean is the average over the Rayleigh fading, and local mean is the average over a period less than the reciprocal of the fading rate (e.g., several bit durations). It is useful to note that
f=
fd
l+I:IJ
---_
5x10·1n-----~---~---
(15)
L
L~'1
10- 1
j=l
- r 1-0 --- r 1 ~ 1 ........ r 1 s 2
(OdS) (3d8)
A. Analytical Results with One Interferer
We first consider 'YR with one interferer. YR can be determined from [6, weight equation (9)]
(16) where from (8)
10-'----.....a..-~-..I6.:-...:~
. 20
(17)
We note that with our fading model the components of Ud and u1 are complex Gaussian random variables that vary at the fading rate. Thus, to determine 'YR' the expected value in (17) must be averaged over the Rayleigh fading. It can also be seen that 'YR will vary at the fading rate. The probability density function of 'YR can be calculated to be given by [7] e-YRlfd(
P(YR)=
t)
YR
..5
Fig. 2. The cumulative distribution function of 'YI( versus YRIf for optimum combining when the desired and interfenn~ signals are subject to fading. Results are shown for one interferer WIth several values of M and I'J: The distribution function for YR with fixed average received SINK is shown to decrease as the power of the interferer becomes a larger proportion of the total noise plus interference power. The decrease 1S even larger as M increases.
(20)
P(YR)=l-e- YR / t) M-2 dt,
(18)
rd
M
E
k-l
(~
d
r-
1
(k-l)!
(21)
which agrees with [1, p. 319]. With a high power interferer f 1 = 00, and the cumulative distribution function is given by
) =lYR/fde-XxM-l(l+ Mf1 ) 0 (M -2)! ·fe-XMfll(l-t)M-2dtdx.
0
YR/r(dB)
fAM-2)!
Thus, the cumulative distribution function is given by
P(
..L.L_....L----:-.---JJ
" - 10
or
M-\1 + Mf1)
• ~l e-«YRlfd)Mfl)I(I-
- 15
(22) (19)
In the above equations it is seen that YR can be normalized by rd. Therefore, from (15) we can also normalize YR by r and compare the performance of optimum combining to that of maximal ratio combining for fixed average received SINR. Fig. 2 shows the cumulative distribution function of YR versus YR/f with optimum combining for several values of M and fl. The f 1 = 0 distribution curve is also the distribution curve for maximal ratio combining. Fig. 2 shows that for fixed average received SINR the distribution function decreases as the interference power becomes a larger proportion of the total noise-plus-interference power. The decrease becomes even greater as M increases. Thus, optimum combining improves the receiver's performance the most when the interferer's power is large compared to the thermal noise power and there are several antennas. As in [7] let us now consider the effect of a very high power interferer on optimum combining. Without interference (or with maximal ratio combining) f 1 = 0, and the cumulative distribution function is given by
342
or
Thus, optimum combining with an infinitely strong interferer gives the same results as maximal ratio combining without the interferer and with one less antenna. In other words, optimum combining with a strong interferer and an additional antenna will "always do better than maximal ratio combining without the interferer. For digital mobile radio this has the following impact. Let us consider a system where the required system performance can be achieved with maximal ratio combining at the base station receiver and adaptive retransmission- (see Section V-B) for base-to-mobile transmission. Then, another mobile can be added per channel per cell by using optimum combining and adding one antenna at the base station, Thus, optimum combining provides a relatively simple means for growth in a system. The BER for coherent detection of PSK is given by
(24)
L"1 - 1 ' 1 "0 - - - 1' 1 a1 ( OdBI
From (18) and (24) the BER for optimum combining with one interferer can be calculated to be given by [7]
········1'1"2 (3dBI
>'"
10- 2
II:
0
II: II:
'" >-
1<,3
lD
=---_ _
10- 4 - 10
M -2
- L
k: =1
i=1
-I)!! )])
(2i i!(2+2fd ) i
0
r
5
10
__=l
15
20
(dBI
Fig. 3. The average BER versus the average received SINR for optimum combining with one interferer. Results are shown for several values of M and fl ' The improvement with optimum combining (in decibels) is shown to be nearly independent of the average received SINR for BER's less than 10- 2 •
(-Mf 1)k
.[1- Vr-rz (1 + t ~
L....._ _L....."---'L...-lL.....--.::-\....II-.._---.:~ _
-5
(25) 1 2 . - - - - - - r- - - - . - - -- - - - r - -- - - "
where
10
(2i -I)!! = 1·3·5· . .. . (2i -1).
(26)
iii
~
>-
8
z
Similarly, for maximal ratio combining the BER can be determined from (21) and (24) as [81
'" ~
6
~
II:
~ 4 2
-5
o
5
10
1' 1 (dBI
Fig. 3 shows the BER versus the average received SINR (f) for optimum combining with one interferer. Results are shown for several values of M and fl ' The results for f 1 = 0 are the same as those for maximal ratio combining. Let us define the optimum combining improvement as the decrease (in decibels) in the average received SINR required for a given BER as compared to maximal ratio combining. Fig. 3 shows that the improvement is nearly independent of I' for BER's less than 10- 2 . Thus, for most systems of interest, the improvement is independent of f d and depends only on f 1 and M. That is, the improvement depends on the interferer's power relative to the combined power of the other interferers and not the power of the desired signal. In Fig. 4 the optimum combining improvement is plotted versus f 1 for one interferer and several values of M. The results are shown for a BER of 10- 3 , but as discussed above, similar results can be obtained for other BER values (less than 10- 2 ) . Results show that as f 1 and M increase, the improvement also increases. Fig. 4 also shows the maximum improvement that can be achieved if the interferer is completely nulled in the array output. The difference between the maximum and the actual improvement for a given M (M > 2) is shown to be constant for large fl' From (23) it can be seen that this
Fig. 4. The improvement of optimum combining over maximal ratio co~p ining versus f l with one interferer for several values of M and a 10 BER. Results show that as f l and M Increase, the improvement becomes significant.
difference is the increase in the required received SINR with the loss of one antenna with maximal ratio combining. For example, the required received SINR with maximal ratio combining is 2.3 dB for M = 5 and 4.0 dB for M = 4. Thus, for M = 5 the optimum combining improvement is 1.7 dB less than maximum for large fl'
B. Simulation Results with Multiple Interferers With two or more interferers, it is extremely difficult to determine analytically the optimum combiner performance. Therefore, in this section we use Monte Carlo simulation to determine the performance of optimum combining. For the simulation we consider y rather than YR as in the previous section. For a given bit duration, the array output SINR is given by
(28) where Pd is the power of the desired signal and Pi + n is the
343
power of the interference plus noise. The desired signal power is given by
(29)
nents are also independent complex Gaussian random variables. Thus, the probability density function of y can be determined analytically to be given by [1, p. 367]
,
where the superscript t denotes complex conjugate transpose. The power of the interference plus noise is given by
p(y)=
M(~)M-l
r(l + ~)
(37)
M+l
(30) and the cumulative distribution function is given by where u, is the vector of the thermal noise vectors at the receiver. The components of u, are independent complex Gaussian random variables with a variance corresponding to the noise power. The array output SINR is then given by
(31)
y=
The weight vector for the optimum combiner is given in (9). As can be seen from (8), R n n is Hermitian, and therefore, from (9)
(32)
y=
The cumulative distribution of y is plotted versus y/ r in Fig. 5. Simulation results with 100 000 samples are shown for optimum combining with two interferers that have 3 dB higher power than the noise, and analytical results are shown for maximal ratio combining. With 100 000 samples there are small deviations in the simulation results only for very small values of the distribution function. Fig. 5 shows that optimum combining significantly decreases the value of the distribution function as compared to maximal ratio combining. This decrease becomes even greater as M increases. The BER can be determined from the cumulative distribution function by the equation
111 cos-
Thus, the SINR can be expressed as
tu~R;~u~12
(38)
BER=(33)
",
0
1
(!y ) p( y ) dy
(39)
or
BER=.!.11cos-l(!y)(dP(Y))dY. With Rayleigh fading, the components of U d, un' and uj are complex Gaussian random variables with zero mean and variance r d , 0 2, and f j , respectively. Therefore, through Monte Carlo simulation, the probability distribution of y can be determined. We now discuss the distribution of y with maximal ratio combining so that a comparison to optimum combining can be made. For maximal ratio combining, the weights are given by W=U~
Thus, the BER for optimum combining was determined from the simulation results using the above equation. Since the cumulative distribution function can be determined for y normalized by I', from one simulation run we can determine the BER over a wide range of L's, Similarly, the BER for maximal ratio combining is seen from (37) and (39) to be given by BER =
(34)
or
(40)
dy
'IT 0
-1 M ",
Ijf
0
cos " (vTX)
XM - 1 M
(1+ x-)
+
1
dx
(41)
which is numerically equivalent to (27).
For one interferer, the BER results obtained using the
Therefore, from (31) the SINR is given by y=
tu~u~12
(35) above equations (with a simulation using 100000 samples) agree with the analytical results shown in Fig. 3. (36)
Since the components of u, + EJ=o:luj are the sum of independent complex Gaussian random variables, the compo-
For two interferers, the BER results are shown in Fig. 6. The simulation used 100 000 samples per data point. Fig. 6 shows that there is a marked improvement with optimum combining as the number of antennas increases. For example, for a BER of 10- 3 and M equal to 5, optimum combining requires 4.2 dB less SINR than maximal ratio combining. Thus, in this case, optimum combining with five antennas (which requires -1.9 dB for a 10- 3 BER) is
344
L"2
......
--r,"rZ"o - - - r , " I' z "2 (3dfl)
10-'
,
/
,
I
-15
-10
-5 YII' (dBI
,I
I
I
I
I
I
o
,
/
,-
,//"
5
10
Fig. S. The cumulative distribution function of y versus Y/f for optimum combining with two interferers that have 3 dB higherllower than the noise. Analytical results for maximal ratio combining (T, = f 1 = 0) are also shown. Optimum combining is seen to significantly decrease the distribution function. 5 . 1 0 · ' , . . . - - - , - - - - , - - - - r - - - - - , . - -----r-----,
L-2
--I',-rz"o
--- r," I'z
o
-5
r
5
10
"213dBI
15
20
(dBI
Fig. 6. The average BER versus the average received SINR for optimum combining with two interferers . Results are shown for optimum combining with two equal ~ower interferers (T, = f z = 2) and for maximal ratio combining (f1 = f. = 0). There is a marked Improvement with optimum combining as the number of antennas increases. 12 g:----- - - - r - - - - - , - , - - - - , - - - - - - : : I I
L-2
BER - 10- 3
10
I'z -2 13dBl
...z
~ 6
\oJ
o> ::: 4
!
2
o "=-10
IV.
--.",__- -
r - --'-5
--'--
r,
o
---J'-
5
one interferer when there is also a second interferer with a 3 dB signal-to-noise ratio. The simulation used 100000 samples per data point. The results are shown for a 10- 3 BER, but as seen in Fig. 6, these results are similar to the results for other BER's less than 10- 2 • Fig. 7 also shows the maximum improvement possible if both interfering signals are completely nulled in the receiver output (i.e., the difference between the maximal ratio combiner performance with and without interference). The improvement is within about 2 dB of the maximum with six or more antennas. Fig. 8 shows the improvement versus the number of antennas with one to six equal power (fj = 3 dB) interferers. Again, 100000 samples per data point were used. The improvement is shown to be between 1-6 dB as M varies from 2 to 8. Thus, optimum combining has some improvement over maximal ratio combining even with a few antennas, and the improvement greatly increases with the number of antennas. Although the results of Fig. 8 are for equal power interferers with a particular value of Ij, they demonstrate the following characteristics of optimum combining that apply to other interference cases as well. First, when the number of antennas is much greater than the number of interferers, the improvement is limited. That is, in this case there is little improvement (relative to maximal ratio combining) with additional antennas. This can also be seen from Figs. 4 and 7. Second, except for the above case, the increase in the improvement (in decibels) with each additional antenna is approximately constant (about 0.6 dB for f J = 3 dB). Finally, the most interesting characteristic is that there is a large improvement even when the number of interferers is greater than the number of antennas. This implies that in analyzing systems we must consider many interferers individually even if there are only a few antennas. For example, consider the case of five antennas with six interferers , each with f J equal to 3 dB. From Fig. 8, the improvement is 2.7 dB. However, if only five interferers are considered individually, and the power of the sixth one is combined with the thermal noise, f J is - 1.8 dB and the improvement is only 1.6 dB. Thus, we must consider individually as many interferers as possible to determine accurately the actual optimum combining improvement.
...=tl
10
(dB)
Fig. 7. The improvement of optimum combining over maximal ratio combining versus the signal-to-noise ratio of one interferer when there is also a second interferer with a 3 dB signal-to-noise ratio. The improvement is within about 2 dB of the maximum improvement with six or more antennas.
better than maximal ratio combining with nine antennas (which, from (27), requires -1.7 dB for a 10- 3 BER). Fig. 7 shows the optimum combiner improvement over maximal ratio combining versus the signal-to-noise ratio of
PERFORMANCE IN TYPICAL SYSTEMS
This section studies the performance of optimum combining in typical cellular mobile radio systems. Using the techniques of Section III, we study optimum combining when the signals are subject to Rayleigh fading.' Optimum combining is studied only at the base station receiver because multiple antennas and the associated signal processing for optimum combining are less costly to implement at the base station than on numerous mobiles. (Adap tive retransmission with time division [1], [9] can be used to
345
s In an actual mobile radio system, the signals are also subject to shadow fading [9] which greatly complicates analysis. We therefore only consider Rayleigh fading so that system comparisons can easily be made.
10
r--,----,----,----,----,....----.
TABLE I
COMPARISON OF OPTIMUM AND MAXIMAL RATIO COMBINING IN TYPICAL MOBILE RADIO SYSTEMS- THE NUMBER OF ANTENNAS REQUIRED AND THE SINR MARGIN FOR A 10 - 3 BER
BER • 10- 3
...
B
r
J (j ·1 .LI·2(3dBl
iii
!Z....
ea.. 6
Bale Station
::IE ....
Geometry
> o
'"::IE Q.
Combininl
Reuse
Decoy Exponential
Number o(
SlNR
1
3
14 15
0.5
1.1
6 7
0.0 1.7
4
12 13 3 4 2 3 II 69
0.1 0.7 0.3 2.9 1.0 5.5 0.0 1.0
5 6 3 4 1 3 17 II
1.9 4.5 2.6 5.1 3.0
55 69 7 I 5 6
0.0 1.0 0.3 1.2 1.6 2.9
II 12 5 6 4 5
0.5
Frequcnq
)-corner
3
7
3 4
o :-2----;-----:--~:----L.--...l--~ 3 4 6
Milimal Ratio
B
1 Centrally
3 4
Loealed 3
J
4
improve reception at the mobile with multiple base station antennas only (see Section V-B).) As before, all results for optimum combining are compared to maximal ratio combining. Analysis of optimum combining with numerous interferers requires a substantial amount of computer time. It is therefore nearly impossible to determine the average performance of the adaptive array in the typical cellular system with random mobile locations. Therefore, in this section we consider a worst case scenario only, i.e., the mobile transmitting the desired signal is at the point in the cell .fart~est from the .b ase station, and the interfering mobiles In the surrounding cells are as close as possible to the base station of the desired mobile. Furthermore, in the ~alysis we consider only the six strongest interferers individually. The power of the other interferers is combined and considered as thermal noise. .The systems studied involve two different cell geometries WIth hexagonal cells. In one geometry the base stations are located at the cell center, and in the other geometry the base stations are at the three alternate comers of the cell and are equipped with sectoral horns. In the latter geometry, each of the base station's three antennas has a 120° be~width and serves the three adjoining cells. We also consider both frequency reuse in every cell and the use of three channe~ sets. Furthermore, because in the typical system the SIgnal strength falls with the inverse of the distance raised to between the third and fourth power we ' also consider these two extremes," The performance of optimum combining and maximal ratio combining in typical mobile radio systems is shown in Table I. For each of the systems described above Table I lists the number of antennas required to achieve a 10- 3 BER and the average output SINR margin. We also show the margin with an additional antenna. The results show that with three-comer base station g~~etry and frequency reuse in every cell, optimum combining more than halves the required number of antennas. Furthermore, the increase in margin with an additional .6Th e calculation of the power of the signals in these cellular systems WIll not be descnbed here. The method is similar to that described ill [10J.
Antennas
Mlrsin. a (dB)
Optimum
Number or Intcnnu
Combinin•
SINR Matpn a (dB)
1.6
0.4 0.9
1.1
0.1 2.4 2.7 5.3
a Margins are a~urate to within a few tenths of a decibel and were determmed from Simulation results using 100 000 samples .
antenna is much greater. With the same geometry and three channel sets, even though only a few antennas are required with maximal ratio combining, optimum combining increases the margin by 2-3 dB. With centrally located base stations and frequency reuse in every cell, optimum combining substantially reduces the number of antennas. As few as 11 antennas are required with optimum combining as compared to more than 50 with maximal ratio combining. Finally, with three channel sets, optimum combining requires one less antenna and has higher margins. Thus, the improvement with optimum combining is the largest in systems where a large number of antennas is required because of low received SINR. However, even with high SINR and few antennas, the improvement is 2 dB or more. Therefore, the results for typical cellular systems agree with those of Section III (i.e., Fig. 8). In an actual system we would expect the optimum combining improvement to be even greater than that shown in Table I because of the following three reasons. First, all the channels in all the cells may not always be occupied. Thus , the total interference power will be less, and the power of the strongest interferers (when transmitting) relative to the power of the sum of the other interferers r will be higher. As shown in Section III, as r increases, the optimum combining improvement increases, Second, with random mobile locations rather than the worst case, the total interference power will be lower. Thus, rJ for the • strongest mterferers (those closest to the desired mobile's base station) will be higher, and therefore, so will the improvement. Third, for the results in Table I only the six strongest interferers were considered individually, and thus the results are somewhat pessimistic. Finally, we note that in actual systems the fading can be non-Rayleigh with direct paths existing between an interfering mobile and a base station (i.e., the fading might not be independent at each antenna). Under these conditions, the performance of maximal ratio combining can be significantly degraded while optimum combining can still achieve the maximum output SINR.
346
v.
IMPLEMENTATION
In this section we discuss the implementation of optimum combining in mobile radio. We consider the use of an LMS [3] adaptive array at the base station receiver and adaptive retransmission with time division for base-tomobile transmission. For the LMS adaptive array, we discuss the dynamic range, reference signal generation, and modulation technique.
'Oll)
ARRAY
OUTPUT
A. The LMS Adaptive Array 1) Description: Of the various adaptive array techniques [2]-[4] that can be used in mobile radio, the LMS technique appears to be the most practical one for mobile radio because it is not too complex to implement and it does not require that the desired signal phase difference between antennas be known a priori at the receiver. Fig. 9 shows a block diagram of an M element LMS adaptive array. It is similar to the optimum combiner of Fig. 1 except for the addition of a reference signal r( t) and an error signal eel). As shown in Fig. 9, the array output is subtracted from a reference signal (described below) r( t) to form the error signal e(t). The element weights are generated from the error signal and the x/,et) and xQ/(t) signals by using the LMS algorithm which minimizes the power of the error signal. The reference signal is used by the array to distinguish between the desired and interfering signals at the receiver. It must be correlated with the desired signal and uncorrelated with any interference. Under these conditions the minimization of the power of the error signal suppresses interfering signals and enhances the desired signal in the array output. Generation of the reference signal in digital mobile radio systems is described in Section V-A3). We now consider the weight equation for the LMS adaptive array in a mobile radio system. In the typical system the bit rate is 32 kbitsjs, and the carrier frequency is about 840 MHz. With the signal bandwidth 1.5 times the
+ r(t) REFERENCE SIGNAL
Fig. 9. Block diagram of an M element LMS adaptive array.
desired and interfering signals. However, the weights must also change much more slowly than the data rate so that the data modulation is not altered. It has been shown [11] that for PSK signals the maximum rate of change in the weights without significant data distortion is about 0.2 times the data rate. For the typical mobile radio system, the maximum fading rate is about 70 Hz (for a carrier frequency of 840 MHz and a vehicle speed of 55 mijh), and the code rate is 32 kbitsjs. Thus, the permissible range in signal power at the array input is given by . R _ 0.2x32xl0 3 D ynannc ange 70 ~
bit rate, the relative bandwidth of the mobile radio channel is only 0.006 percent, and we can consider the signal as narrow band. For narrow-band signals, the weight equation for the LMS array is given by [6, eq. (9)], i.e., the LMS adaptive array maximizes the output SINR. However, these are the steady state weights, and in mobile radio the signal environment is continuously changing. Therefore, we must consider the transient performance of the array. That is, because the weights are constantly changing, the performance will be degraded somewhat from that of the optimum combiner. (Analysis of the transient performance is not considered in this paper.) Also, we must consider the dynamic range of the LMS adaptive array. 2) Dynamic Range: One limitation of the LMS adaptive array technique is the dynamic range over which it can operate. In an LMS adaptive array, the speed of response to the weights is proportional to the strength of the signals at the array input. For the array to operate properly, the weights must change fast enough to track the fading of the
20 dB.
(42)
The received signals in a mobile radio system vary by more than 20 dB, however, and therefore automatic transmitter power control (which could add significantly to the cost of the mobile radio) is required to control the power of the strongest signals at the receiver. With this power reasonably fixed, the dynamic range determines the power ratio of the strongest to the weakest received signal that the array can track. A 20 dB dynamic range is certainly not large, but it is more than adequate for mobile radio for the reasons described below. In the mobile radio systems studied in this paper (see Section IV), the average received SINR at each antenna is relatively small. This is because an adaptive array is not needed when the received SINR is large. For example, for maximal ratio combining with two antennas, an average received SINR at each antenna of 11 dB [1] is required for coherent detection of PSK with a 10- 3 BER. For optimum combining the required SINR is less with two antennas and, of course, even lower with more antennas. Thus, the received SINR is much less than 20 dB for all cases of interest. (It is typically between - 5 and 5 dB.) A small received SINR affects array operation as follows. First, if the power of an interfering signal is more than 20 dB below the desired signal's power at an antenna, the array need not track the interfering signal at that antenna because it has a negligible effect on the output SINR. Second, if the power of an interfering signal is more than 20 dB higher than the desired signal's power at an
347
antenna, the array need not track the desired signal at that antenna because the resulting weight for the antenna will be almost zero. Thus, because the received SINR is small in the systems where the LMS adaptive array is practical, a 20 dB dynamic range is adequate. Note that if the received SINR is large' (e.g., greater than 20 dB, as in a lightly loaded system), the LMS adaptive array will have the same performance as maximal ratio combining. 3) Reference Signal Generation and Modulation Technique: The LMS adaptive array must be able to distinguish between the desired signal and any interfering signals. This is accomplished through the use of a reference signal as discussed in Section V-AI). The reference signal must be correlated with the desired signal and uncorrelated with any interference. A reference signal generation technique that allows for signal discrimination is described in [12] and involves the use of pseudonoise codes with spread-spectrum techniques. To generate the spread-spectrum signal the pseudonoise code symbols, generated from a maximal length feedback shift register, are mixed with lower speed voice (data) bits, and the resulting bits are used to generate a PSK signal. The code modulation frequency is an integer multiple of the voice bit rate, and this multiple is defined as the spreading ratio k. The reference signal is generated from the biphase spread-spectrum signal using the loop shown in Fig. 10. The array output is first mixed with a locally generated signal modulated by the pseudonoise code. When the codes of the locally generated signal and the desired signal in the array output are synchronized, the desired signal's spectrum is collapsed to the data bandwidth. The mixer output is then passed through a filter with this bandwidth. The biphase desired signal is therefore unchanged by the filter. The filter output is then hard limited so that the reference signal will have constant amplitude. The hard-limiter output is mixed with the locally generated signal to produce a biphase reference signal. The reference signal is therefore an amplitude scaled replica of the desired signal. Any interference signal without the proper code has its waveform drastically altered by the reference loop. When the coded locally generated signal is mixed with the interference, the interference spectrum is spread by the code bandwidth. The bandpass filter further changes the interference component out of the mixer. As a result, the interference at the array output is uncorrelated with the reference signal. Thus, with spread spectrum, a reference signal is continuously generated that is correlated with the desired signal and uncorrelated with any interference. Furthermore, since pseudonoise codes are used, every mobile can be distinguished by a unique code. Unfortunately, spread spectrum increases the biphase signal bandwidth by a factor of k and therefore increases both the total cochannel interference power and the number of interferers in cellular mobile radio. For example, with frequency reuse in every cell, the cochannel interference power and the number of interferers from surrounding cells are increased by factors of k and 2k -1,
ARRAY OUTPUT ADAPTIVE ARRAY
REFERENCE SIGNAL
Fig. 10. Reference signal generation loop with the adaptive array. When the desired signal is a biphase spread-spectrum signal, the reference signal is correlated with it but not with any interference.
respectively. This increase in interference power is canceled by the processing gain of spread spectrum, but the increased number of interferers degrades the performance of the LMS adaptive array. Furthermore, 2(k -1) cochannel interferers are now present within the desired mobile's cell. Thus, even with a small spreading ratio (e.g., 5 or less) the performance of the LMS adaptive array with the biphase spread-spectrum signal can be worse than that of maximal ratio combining, making the LMS system impractical. The bandwidth increase with spread spectrum and its associated problems can be overcome in the following way. The biphase spread-spectrum signal is combined with an orthogonal biphase signal modulated by the voice bits only (see [13]). The data modulation rate of the orthogonal biphase signal is the same as the code modulation rate of the biphase spread-spectrum signal. The resulting fourphase signal therefore has a bandwidth determined by the data rate only, i.e., the bandwidth is not increased by the spreading ratio. Furthermore, a reference signal for the four-phase signal can be generated from its biphase spread-spectrum signal component using the loop described earlier. As shown in [14], the performance of the LMS adaptive array with the four-phase signal is close to that with the biphase signal. Therefore, with this system, we can generate a reference signal without any increase in interference power or the number of interferers and achieve an improvement with an LMS adaptive array close to that for optimum combining which is shown in Sections III and IV. We now describe the modulation technique in detail by describing three possible ways to modulate the four-phase signal. The simplest technique is for the voice bits to modulate only the orthogonal biphase signal. The biphase spread-spectrum signal then contains the code plus data bits for transferring information from the mobile to the base station. With this first technique, the signal bandwidth corresponds to the voice bit rate r (e.g., 32 kbitsz's). However, the energy-per-bit-to-noise (interference) density ratio Eb / No is half that of a biphase signal. Thus, the improvement with an LMS adaptive array is 3 dB less than that shown in Sections III and IV. A data channel is also available, however, with an r/ k data rate. Furthermore, since the Eb / No for the data bits is k times that for the voice bits (because of the spread spectrum), the BER for the data bits is very low. If a data channel is not required, then voice bits can
348
TABLEII
replace the data bits. With this second technique, the voice bits are split into two channels, one modulating the biphase spread-spectrum signal and the other modulating the orthogonal biphase signal. The bit rate for the latter channel is k times that for the biphase spread-spectrum signal. The signal bandwidth is reduced by k I( k + 1) as compared to the first technique. However, the Ebl No of the voice bits on the biphase spread-spectrum signal is k times that on the orthogonal biphase signal. Through appropriate coding techniques, this difference can be used to improve the overall BER. We can equalize the BER for both channels by decreasing the power of the biphase spread-spectrum signal by II k. With this third technique the Ebl No for the voice bits is just k /tk + 1) times that for a biphase signal. For example, with k equal to 5, the improvement with an LMS adaptive array is 0.8 dB less than that shown in Sections III and IV. Table II summarizes the above results for the three modulation techniques. A block diagram of the four-phase signal generation circuitry for the three modulation techniques is shown in Fig. 11. The code symbols of duration ~ are mixed with either voice or data bits of duration kA. The resulting symbols modulate a local oscillator to generate a biphase spread-spectrum signal. As shown in the lower portion of Fig. 11, voice bits, also of duration ~, modulate the local oscillator signal shifted by 90° to generate the orthogonal biphase signal. This signal is then combined with the biphase spread-spectrum signal to obtain the four-phase signal. By adjusting the biphase spread-spectrum signal level with fJ and modulating this signal with either voice or data bits, we can generate any of the three four-phase signals listed in Table II.
FOUR-PHASE SIGNAL PARAMETERS FOR THREE MODULATION
TECHNIQUES IN AN Technique No.
Relative Biphase Silnal Powen
LMS ADAPTIVE ARRAy SYSTEM
Spread-Spectru m Biphue Sian_l Bit a Information E,,/N o b Biu Rile
Ortholonal Bipbuc Silnal Information Bit Bill Rate
1
1:1
Data
r fk.
2
1:1
Voice
rj(1e +1)
k /2
Voice
3
ljle :1
Voice
rj(1e +1)
lej(1e +1)
Voice
a The code modulation rate is b Relative to biphase signals.
le/2
Voice
E"INo b
r
0.5
Ie
(kF)r
O.S
(k+1 )r
k/(Ie+l)
k times the bit rate.
CODE
------.,,+ ) - - - - - - - - e . ( Y l - - - - l (~)
DATA
OR
of.
VOICE BITS
(A.6)
+
FOUR - PHASE
I,.......---.. SIGNAL
VOICE BITS
(td
Fig. 11. Block diagram of the four-phase signal generation circuitry for the LMS adaptive array. A biphase spread-spectrum signal, modulated by code symbols plus data or voice bits, is combined with an orthogonal biphase signal, modulated by voice bits, to generate the four-phase signal.
B. Base -to - Mobile Transmission
As we have shown, the LMS technique can significantly improve signal reception at the base station. This improvement is, of course, also desired at the mobile. However, since there are many more mobiles than base stations, it is economically desirable to add the complexity of the LMS technique (particularly multiple antennas) only to the base stations. Adaptive retransmission with time division (1], [9] can be used to improve reception at the mobile with multiple base station antennas only. With adaptive retransmission, the base station transmits at the same frequency as it receives, using the complex conjugate of the receiving weights. With time division, a single channel is time shared by both directions of transmission. Thus, with the LMS technique, during mobile-to-base transmission the antenna element weights are adjusted to maximize the signal-to-noise ratio at the receiver output. During base-to-mobile transmission, the complex conjugate of the receiving weights are used so that the signals from the base station antennas combine to enhance reception of the signal at the desired mobile and to suppress this signal at other mobiles. Therefore, by keeping the time intervals for transmitting and receiving
much shorter than the fading rate (e.g., transmitting in 10 bit blocks), we can achieve the advantages of the LMS technique at both the mobile and the base station. With adaptive retransmission using the LMS technique, each base station transmits in a way that maximizes the power of the signal received by the desired mobile relative to the total power of the signal received by all other mobiles. Thus, at the mobiles, interfering base station signals are suppressed and the improvement in the performance with the LMS technique as compared to maximal ratio combining should be similar to that at the base stations. The actual improvement for a given mobile, however, depends on the interference environment of every base station. Because of the complexity of the analysis, we will not study this improvement in detail. It should be noted, though, that for base-to-mobile transmission, spread spectrum on the signal is not required because a reference signal is not generated at the mobile. Therefore, without the degradation with the modulation scheme in the mobileto-base transmission (see Section V-A-3), the BER at the mobile may be lower than that at the base station.
VI.
SUMMARY AND CONCLUSIONS
In this paper we have studied optimum combining for digital mobile radio systems. The combining technique is optimum in that it maximizes the output SINR at the receiver even with cochannel interference. We determined the BER performance of optimum combining in a Rayleigh fading environment and compared the performance to that of maximal ratio combining. Results showed that with cochannel interference there is some improvement over maximal ratio combining with only a few receiving antennas, but there is significant improvement with several
349
antennas. With optimum combining, the typical cellular system was seen to have greater margins and require fewer antennas than with maximal ratio combining. Finally, we described how optimum combining can be implemented in mobile radio with LMS adaptive arrays. Thus, we have shown that optimum combining is a practical means for increasing the channel capacity and performance of digital mobile radio systems. REFERENCES
[10] Y. S. Yeh and D. O. Reudink, "Efficient spectrum utilization for mobile radio systemsusingspace diversity," IEEE Trans. Commun., vol. COM-3D, R. 447, Mar. 1982. [11] T. W. Miller, ' The transient response of adaptive arrays in TDMA systems," Electrosci. Lab., DeP. Elec. Eng., Ohio State Univ., Columbus, OR, Rep. 4116-1, p. 287, June 1976. [12] R. T. Compton, Jr., "An adaptive array in a spread-spectrum communicationsystem," Proc. IEEE, vol. 66, p. 289, Mar. 1978. [13] J. H. Winters, "Increased data rates for communication systems with adaptive antennas," in Proc. IEEE Inter. Coni. Commun., June 1982. [14] - , "A four-phase modulation system for use with an adaptive array," Ph.D. dissertation, Ohio State Univ., Columbus, OH, July 1981.
[1] W. C. Jakes Jr. et al., Microwave Mobile Communications. New York: Wiley, 1974. [2] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. . [3] B. Widrow, P. E. Mantey, L. 1. Griffiths, and B. B. Goode, "Adaptive antenna systems," Proc. IEEE, vol. 55, p. 2143, Dec. 1967. [4] S. R. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-24, p. 585, Sept 1976. [5] B. Widrow,1. McCool,and M. Ball,"The complex LMS algorithm," Proc. IEEE, vol. 63, p. 719, Apr. 1975. [6] C. A. Baird, Jr. and C. L. Zahm, "Performance criteria for narrowband array processing," 1971 Coni. Decision Contr., Miami Beach, FL, Dec. 15-17, 1971, p. 564. [7] V. M. Bogachev and I. G. Kiselev, Optimum combiningof signals in space-diversity reception," Telecommun. Radio Eng., vol. 34/35, p. 83, Oct. 1980. [8] P. Bello and B. D. Nelin, "Predetection diversity combining with selectively fading channels," IRE Trans. Commun. Syst., vol, CS-10, p. 32, Mar. 1962. [9] P. S. Henry and B. S. Glance, "A new approach to high-capacity digital mobile radio," Bell Syst. Tech. J., vol. 60, no. 8, p. 1891, Oct. 1981. U
350
On Optimum Combining at the Mobile RODNEY G. VAUGHAN,
Abstract-Optimum combining for diversity antennas at the mobile is discussed. Effectively, the aim is to add the wanted signal vectors in a maximum ratio sense, while interferers are weighted so that their resultant is in a permanent deep fade. Even if there are not enough degrees of freedom available to accomplish this fully, an optimum solution can still be found. Many interpretations from conventional array technology do not apply to the mobile communications case and the mechanism of optimum combining for array branch signals rather than discrete spatial signals is reviewed. Physical interpretation of the formulation is emphasized tbroughout. Problems with the adaptive algorithm and its implementation are also identified. Sample matrix inversion is shown to be a likely algorithm to apply in vehicular mobile communications receivers. A worst case example gives an idea of the required computation rates.
MEMBER, IEEE
z r-'" t' ""'---X-" ~ \
\
\
\
/;"'------~ /
I
\
I
,/
\,'
I_-ARRAY PATIERN
I
Y
I I r
ARRAY
S
\ \
I
'
I
Fig. 1. Scenario for conventional array. In simple scenario. array has enough degrees of freedom to steer ., nulls ., of array pattern toward all interferers while maintaining finite gain toward wanted signal.
I. INTRODUCTION
S
IGNAL COMBINATION in mobile communications is generally discussed (cf. [8], [9]) in terms of pre- and postdetection classes of selection, switched, equal-gain, and maximum ratio. To date, adaptive techniques have not been extensively discussed in the area of mobile communications. In the antenna literature, the interpretation of the adaptive techniques revolves around antenna patterns. In particular. the adaptation of nulls toward interferers has been the explanatory mechanism, which accounts for terms such as "null steering." The wanted signals and interferers are considered as discrete. usually resolvable, points in real space, as indicated in Fig. 1. This is not the case in a typical mobile communication scenario, where wanted sources and interferers easily outnumbered the degrees of freedom of the diversity array and, furthermore, are distributed and generally unresolvable, as suggested by Fig. 2. From the point of view of adapting antenna patterns, the situation appears impossible. Lee [9, p. 451] brietly discusses adaptive techniques for mobile communications but bases the discussion on antenna patterns and interferer directions. Bogachev and Kiselev [3] seem to be the first to have discussed optimum combining with respect to (space) diversity antennas. They derive curves for the probability of the signal-to-noise ratio (SNR) in the presence of Rayleigh fading. Winters [17] presents a good discussion of optimum combining at the base station and also provides simulation results. For the returns offered by optimum combining in the mobile communications case, the reader is referred to Winters' article. The best improvements (over other combining methods) occur when the interference Manuscript received April 1, 1987; revised August 20, 1988. The author is with the Physics and Engineering Laboratory, Department of Scientific and Industrial Research, P.O. Box 31313. Lower Hutt, New Zealand. IEEE Log Number 8927107.
0)
WANTED
88
UNWANTED
Fig. 2. Multipath scenario for array at mobile. Wanted signal and interferers arrive from many directions. Array pattern. after convergence to optimum combining solution, does not offer meaningful interpretation mechanism as in simple scenario case.
levels are high relative to the wanted signal, and when the number of diversity antenna elements is large. As far as the author is aware, there is no discussion of optimum combining applied at the mobile, where, as at the base station, it is necessary to consider branch signals rather than discrete spatial sources. The application is in high-density cellular systems where the cochannel interference is high. A theoretical worst case occurs in a corner of a hexagonal cell where the average interference power is less than 5 dB below average signal power [19]. Local shadow effects such as caused by large buildings can make this figure worse still, and Winters [17] suggests that a signal-to-interference-plus-noise ratio (SINR) of + 5 dB to - 5 dB is typical. The number of diversity antenna elements should be large, typically five or more, to obtain significant improvement (i.e., several dB in SINR) over maximum ratio combining. Glance and Greenstein [5] discuss up to six base station elements and Yeh and Reudink [19] go as far as discussing a system with 20-element mobile antennas with 24element base station antennas, each using predetection maxi-
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 33, No.4, pp. 181-188, November 1988.
351
mum ratio combining. A disadvantage of optimum combining at the mobile rather than at the base station (using adaptive retransmission) is that the cost and complexity of implementation at every mobile is much greater over-all than that of implementing only at each base station. An advantage of many-branch systems at the mobile, however, is that the antennas can be realized in a compact manner. At the base station, even a six-element array becomes very expansive. In the following, the mobile diversity antenna is treated as an array antenna. The terminology is also array antennaoriented. For example, the multipath environment of mobile communications is referred to as a multiple source, or distributed source scenario since the distribution of sources effectively approaches a continuum in urban environments. The scenario is referred to as stationary, which is here taken to mean that its statistics are unchanging with time and the position of the mobile. In some sections, the scenario is referred to as static, indicating that the source positions are constant relative to the mobile. A real-world mobile scenario is rarely stationary or static, but the latter simplification does not seem too drastic for the short periods over which each optimum array solution is sought. Section Il contains the basic complex formulation for the array signals. The differences between the more familiar conventional array case and the mobile communications case are discussed with specific attention given to the physical interpretation of the formulation. Section III follows an approach similar to that in Section II but covers the signal combining aspects. The optimum antenna array (specific antenna elements are not within the scope of the article) is addressed in terms of conventional array parameters. Section IV contains a brief look at adaptive algorithms for the mobile communications case, and Section V looks at aspects of implementation. A worst case example gives some idea of the computational requirements and the lirnitations of adaptive algorithms. II.
source is defined as a single point source; any incident signal
which is derived (e.g., diffracted, reflected, or delayed) from the point source, is considered as a separate (point) source. In the mobile communications case, N is limited to less than, say, six. M is effectively unlimited since the sources are considered distributed. Many of the M sources bear the same signal because of the multipath. If there are P different signals (one wanted signal and P - 1 interferers) , then each of the P signals can be considered randomly allocated to many of the M sources. With the time factor ei wt understood, the complex envelope of the signal conveyed by the nth branch from the mth source is (1)
where mm(t) is the signal modulation of the mth source and
(j)nm is the carrier phase of the mth source, in the nth branch.
The total signal in the nth branch is thus .W
xn(t) = }:; xnm(t).
(2)
m=l
is real and represents the signal amplitude (or rather its mean over the modulation) in the branch from a particular source. The static scenario is characterized by the Q and cP being independent of time. Some physical insight is offered for the mobile communication case if a and cP are thought of as functions of distance moved by the mobile. After the antenna has (hopefully) adapted to a set of Q and (j), the mobile has moved, and adaptation to a new set of Q and q, is required.
Q nm
Array Branch Signals Define the total (sum over all sources) branch signals in the usual way
BASIC FORMULATION
The distributed source scenario around the mobile in an urban environment must be considered unchanging for each adaptation cycle of the combining algorithm. With this in mind, this section considers constant (in location) sources only. In particular, signal correlation measurement intervals or a sequence of them-to establish the covariance matrix with sufficient accuracy during convergence-are taken over a static scenario. In practice, this will not be the case (unless the mobile and its surroundings are still): the adaptation algorithm will be chasing a changing solution, and its performance will be correspondingly degraded. If the adaptation is fast compared with the rate of change of the sought solution, the degradation will not be too much. Some practical points are discussed in Section V. Where possible, the notation here follows that of Hudson [7]. However, the multipath scenario of mobile communications calls for a different formulation, and some designations common to this article and Hudson's are not interchangeable. The array antenna has N branches with M incident signals from M sources in the field of view of the array. Here, a
352
(3)
and a vector of the RF (or IF) phase terms weighted by the amplitude of the source signal in the branch
T m are the vector contributions of the mth source to the array. When only the phases are included (Qnm = 1, all n) Tm = Sm is the source vector to the mth source, a term from conventional array technology. Now let all the sources be included in the total weighted source vector
(5)
Note that a column of Q corresponds to a given source signal in all branches and a row corresponds to all the source signals in a given branch. Finally, define a vector containing the source modulations
AT = [ml (t) m2(t) · .. m.w-(t)]
(6)
products occur in R. Denote a source modulation correlation
so that X(t)
= QA (t)
(7)
is the vector formulation of (1). In the mobile communications case, let there be P different (uncorrelated) signals, with each signal conveyed by subsets of the M sources. The subsets contain MJ, M 2 , •• " M; sources. Now
Pmq
= m :(t)mq(t)
(15)
in which the normalization is understood. Obviously, PmQ = P * and Pmm = 1. Now the inner term of the covariance matrix (14) is
of
Pl2
pi2
A*AT=
••• PIM
1
(8)
(16)
i= I
and (2) can be written as xn(t) =
L L anmmM;(l)ej(j>nm P
,Wi
(9)
i=l m= I
L anMjmAfi(t)eJcPnMi p
~
1=
(10)
I
where the summation is over i for all Mi. In (9), the inner summation is over each source bearing the ith signal. and the outer summation is over the difference source subsets. i.e., over the different signals. The subscript M; serves as a reminder that the (subscripted) signal quantity is derived from several sources rather than an individual source in the scenario. Equation (10) shows that each branch can be considered to support P signal-bearing vectors, each of which correspond to the vector sum of the signals received from the appropriate sources. This is an important maxim as it allows branch signals to be combined such that certain signals are maximized and/or others minimized. The latter effect is the extension of maximum ratio combining to optimum combining. Equation (10) is used in the following in a vector formulation of the covariance matrix for the mobile communications case.
In the case where all the sources bear different (uncorrelated) signals, A *A T becomes the identity matrix. In the mobile communications case, the off-diagonal elements will be randomly placed (other than complying with the Hermitian construction) ones or zeros. Each source adds one to the rank of A *A T which makes it of mathematical interest only except in simple scenarios, i.e., those with a manageable number of sources. In the mobile communications case, not even the rank of A * A T is known. Inclusion of the weighted source vectors is facilitated by denoting
=
7r nmqpcPqpnm'
(Q)nm is the nmth element of Q. cPqpnm is the phase difference between the pth source in the qth branch and the mth source in the nth branch. 1r nmqp has the dimensions for power and denote 1r nmnm = 1r nm . From (10), (14), and (18),
L p
(R)nq
=
1r nM;qlvf;cPqMinMi
and (20)
The output signal is usually written
= WT X(t)
i= I
(11 )
where W is the column vector of weights. The output power is the Hermitian form (12)
in which the covariance matrix is defined as R=X*(t)XT(t)
(19)
i= I
The Covariance Matrix
y(t)
(18)
(13) (14)
The overbar means time averaging in the presence of the static scenario. It is evident that R must contain all cross products of both the weighted space vectors and modulations. Some physical insight can be obtained by examining where these cross
353
The off-diagonal elements are seen to be a summation of vectors, each vector corresponding to one of the P signals from one of the subsets M, of the M incident sources. Little can be remarked about simplifying the physical interpretation of R, its form being too involved. However, some progress is possible by performing a similarity transformation of the desired-signal-only covariance matrix (see the later discussion of prewhitening, below). In practice, the covariance matrix will also contain terms due to noise. The noise is uncorrelated between branches (any unwanted signal that is correlated between branches is defined as interference) so that only the principle diagonal in the covariance matrix becomes augmented. This is convenient because it ensures that R is always positive definite and the existence of R - I is assured. Conversely, if the noise level is very low relative to the source signal levels, numerical problems may sometimes arise in seeking R- I from measured
The optimality criteria can be expressed as the Wiener solution
values of R. Techniques for ill-conditioned R include taking the generalized inverse instead of the conventional inverse. III.
MWopt=kTZ,o
THE OPTIMUM ANTENNA
In public service mobile communications. the average (over several fades) power of interferers would rarely be above that of the wanted signal. In each antenna branch or a single port antenna, however, the instantaneous level of the interferer could be a few tens of decibels above the wanted signal owing to the Rayleighlike fading. The optimum antenna (here the antenna is regarded as the elements, weights, and summing network) is defined to maximize the SINR. If covariance matrices R, and M are defined to embody the wanted signals only and the interference-plus-noise, respectively, then the adaptive algorithm seeks an optimum weight vector Wopt which is well-known to satisfy
where k is a scaling constant ( =golk ' ) which has no effect on the SNIR. If M is invertible (which can be arranged, if necessary, by adding noise), then the scaled weights of the optimum antenna are given by (28)
Equations (27) and (28) are the generalized version (applicable to the mobile communications case) of the results for conventional arrays in which TMO is replaced by roSo. The difference is that the conventional array employs purely phase weighting. Here, both phase and amplitude weighting are necessary. In practice, o must be known a priori, or else it is measured, as is the covariance matrix. In mobile communications, a priori knowledge of 110 is not possible and instead, knowledge of a characteristic of the wanted signal is used to estimate To measu~e M, the wanted signal must first be removed from each branch. Measurement of R, which includes the wanted signal, is clearly more practical but its unentangled inclusion in the optimum solution of (28) requires special conditions. Specifically, if
Ttt
(21)
where k' is a constant which is, in fact. the maximum SINR. Denote a weighted space vector for the wanted source To; then
TZt .
(22)
In the mobile communications case, the wanted signal and interferers are distributed across many source vectors and the small size of the array will normally prevent the isolation of only one of these, the To above. However, R, can always be expressed as a dyadic when the sources are considered branch sources rather than spatial sources. Each branch source is the summation of the contributions from the actual sources which bear the same signal. Equation (19) depicts the situation. The weighted source vector of (4) becomes reformulated as branch signals, (23)
in which M;
TnM ; = ~ t;
(27)
(24)
(29)
then (21), (26), and (29) can be combined to yield the optimum scaled weights solution
Wopt=R-ITZto
(30)
which is known to be equivalent to (28) (e.g., Hudson [7, p. 74]) but features the more readily available total covariance matrix. The validity of (29) is the key resource: the desired signal and the interferers must be uncorrelated (Pmp = 0, m =1-= p) or else orthogonal in the sense
TTMm T*Mp =0 ,
m=l
m e p,
(31)
In the conventional array case, the analogy of (31) is the spatial orthogonality relationship
and then the form (25)
ST·~*=O fTtI p ,
is a valid representation even in the case of a spatially distributed set of (e.g., wanted) signal sources. Substituting (25) into (21) gives
k' MWopt = TttoT~oWopl =goTt,o
(26)
where go is a constant representing the amplitude gain of the array to the wanted signal or to the wanted source in the case of an isolated source. In the latter case, the right side of (26) is go1roSri . So is called the steering vector, and 1ro is the total power (in all branches) of the desired source, quantities familiar from the conventional array case.
m e p,
(32)
which is sought in null steering by retrobeams (eigenbeams for subtraction from the quiescent array pattern). In the mobile communications case, eigenbeams for preprocessing can be formulated mathematically (only at the expense of juggling signal powers since the TM; are not orthonormal), but the patterns are not in real space.
Prewhitening Preprocessing can offer increased convergence rates for adaptive combiners in conventional arrays. The power of the prewhitening is in the equalization and decorrelation of all interference (-plus-noise) in each branch. The combining algorithm can then just maximize the output power. White [15]
354
antenna. For example, in a conventional array, the modulus part of complex weighting allows the suppression of array space near-field sources while maintaining a response to farfield sources in the same direction. In the mobile communications case, the N complex weights provide 2N nOF, but each interferer must be dealt with in both amplitude and phase. Cancellation of an interferer thus requires two real OOF. In the context of mobile communications, it would be appropriate to consider the complex weighted N-branch array to have N complex OOF, each available one of which can be used to suppress at least one interferer.
discusses adaptive beam-forming in simple scenarios but does not consider transient performance. Accelerated convergence techniques (see, e.g., Monzingo and Miller [12, p. 188-192]) can help considerably for antennas with a relatively small number of elements, as would normally be the case in mobile communications. Preprocessing, in particular Gram-Schmitt orthogonalization (e.g., Monzingo and Miller [12, pp. 364383]), seems to offer considerable gains, but again, the scenario is assumed to be unchanging during adaptation. Since the rate of convergence is important in mobile communications, an investigation of the applicability of preprocessing is worthwhile. Since M is always positive definite, it can be uniquely factored using the Hermitial matrices CC = M, and since M is of full rank, C- 1 exists, so C-IMC- 1 =/.
Array Patterns The array pattern for a conventional array is defined as the amplitude gain for a given set of weights
(33)
g({}, ¢) = WTs
The operation is seen to decorrelate the interference signals between all branches (e.g., Hudson [7, p. 62]). The transformed covariance matrix of the desired signal is
B=C- IR sC- l •
(34)
Maximizing the SINR is now just a question of finding the maximum eigenvalue of B. The transformation from R, to B is between array space and eigenspace. However, the eigensources cannot be related to array space sources as is possible with conventional arrays. In fact, the eigensources are difficult to relate to the branch sources (see (4)) as well. The problem again lies with the weighted source vectors TM; not being orthonormal. In the conventional array case with a simple static scenario featuring M < N spatial sources, the array with a beam-former C- 1 provides a (transformed) covariance matrix which is diagonal and contains the powers of each of the sources as its element values. In practice, much effort is required to implement the beamformer of changing scenarios. (For constant scenarios, the beam-former can be hard-wired; indeed, most applications are for this case.) Generating C-I is not trivial because C is the square root of M. M must first be measured, its eigenvalues found numerically, their square roots taken (positive radicals are understood in the unique factorization of M), and then assembled into a diagonal matrix. Finally, the inverse taken to yield C- 1. The continuous measurement and processing required to implement the prewhitening is formidable, and any acceleration of convergence as a result of preprocessing will be well mitigated by the time taken for the preprocessing. Moreover, it is shown in Section V that the gradient search techniques are not very suitable for single-receiver systems in the vehicular mobile communications case.
(35)
where S provides the directive information. If the scaled weight vector corresponds to maximizing the array amplitude gain towards a source 51, then (in the absence of interferers) W = S and S I is the steering vector. If a second source 52 is introduced, the amplitude gain toward 52 using weights W is given by
i,
g({}, cP)=S~5~.
(36)
The array pattern then is a mapping of an inner product of 51 and S2' S1 could be referred to as a testing source, in keeping with the position of a probe antenna in pattern measurements. The nulls of g «(J, q,), if they exist, occur when the source 52 does not affect the array output, i.e., when the sources are spatially orthogonal,
Degrees of Freedom (DOF) Clearly, an N-branch conventional array with N phase-only weights has N DOF; or if the weights are complex, there are 2N DOF. In general, not all the DOF are available for suppression of an interferer, and the number available depends on both the scenario and the configuration of the array
355
(37)
Array space patterns are difficult, if not impossible to interpret for the mobile communications case. A signal amplitude gain quantity can be defined analogously to (35) (cf. (26)) by g = T'f.,; W. TM; carries the "directive" information but the direction is not in real array space. A test weighted source vector could be defined as above (cf. S2 above), but again there seems little point because the inner product analogous to (35), T'f,T2 , does not readily offer a mapping which is physically meaningful. The traditional use of array patterns to illustrate antenna adaptation does not apply to mobile communications. IV.
ADAPTIVE ALGORITHMS
The adaptive algorithms well-known from conventional arrays are all applicable for complex rather than phase-only weights. Some caution must be exercised with some conceptual and explanatory interpretations as noted in the previous section. The suitability and implementation feasibility of the various algorithms in the mobile communications case is the remaining issue. Ricardi [14, p. 6.2] suggests three classical
methods, under which all adaptive algorithms can be classified. These are power inversion, least mean squares (LMS), and direct sample matrix inversion (SMI). SMI is conceptually the simplest technique. The covariance matrix and weighted steering vector are estimated (in this case) and the weights are calculated directly from (38) where the hat denotes estimate. This offers the quickest technique for finding the weights and is independent of the makeup of R, as long as it is well-conditioned with respect to the word size of the implementation. The covariance matrix estimate is formed with K samples 1 K R(k)=- }: X*(k')XT(k')
K
(39)
k'=l
where X(k) is a sample of X(t) at the kth sample time. The estimate for the weighted steering vector is similar (cf. Monzingo and Miller [12, p. 300]):
where d(k) are the samples of d(t), which is a reference signal well-eorrelated with the desired signal. In some ways, SMI is not truly adaptive because it is an open loop system. However, it is considered adaptive in the sense that it will produce a new solution for a new scenario. As in any open-loop system, the accuracy of the implementation must correspond to the desired accuracy of the solution. SMI is discussed further at the end of the subsection. The power inversion and LMS algorithms employ a closedloop system that works to converge towards the solution. Power inversion is simply a minimization of output power, often with a constrained gain to the wanted signal (or spatial source, in simple scenarios). The weighted steering vector needs to be known a priori so the power inversion algorithm is not applicable to the mobile communications case. When the weighted steering vector is estimated with the aid of a reference characteristic of the wanted signal, the algorithm effectively becomes the LMS technique. Despite their differences in implementation, theLMS and power inversion techniques are very similar. Both algorithms are steepest descent gradient search techniques. The convergence rate of the gradient search techniques is very complicated to predict accurately, except in simplified situations. The convergence rate is known to be sensitive to the spread of eigenvalues in the covariance matrix. As far as the author is aware, no simulations have been reported for complex vectors are weights. Simulations for conventional arrays in simple scenarios usually involve very low signal-to-interference ratios, a situation which would be unlikely in the mobile communications case. Winters' [17] base station simulations include Rayleigh fading but assume complete convergence (no combiner losses) although no transient analysis is included. From conventional array simulations (see, e.g., Monzingo and
Miller [12]) convergence to within 3 dB of the optimum solution rarely occurs in less than several hundreds or even thousands of iterations. The eigenvalue spread of the covariance matrix in these simulations is generally more than would be found in mobile communications, so results will be pessimistic. On the other hand, the interference-to-signal ratio is much higher than would be found in the mobile communications case, which would make the results optimistic. Results from conventional array simulation can only give a rough guide. Some conceptually derived time constants allow some conclusions regarding implementation. An upper limit on the rate of change of the weights is given by the data rate. Winters ([17], p. 152], after Miller [11]) notes that the maximum rate of change in the weights is about 0.2 times the data rate before significant data distortion occurs for phase-shift keyed (PSK) signaling. 1 For a data rate rd = 16 kbits/s, the maximum rate of weight update for PSK signals is then about every 0.06 ms, and the weights cannot change completely within 0.3 ms. For good adaptive performance, it is best to try to attain this rate. In terms of a Rayleigh fading envelope, this upper limit is quite severe. A worst case fading rate, given by a vehicle traveling at, say, 140 kmlh with a 9OO-MHz carrier frequency and omnidirectional antenna, is about 120 fades/so If the ,'branch scenario" (the scenario describing the "positions" of the P signals in each diversity branch) can be assumed uncorrelated at intervals of half of the fading period, i. e. , every 2.1 ms, then there can be a maximum of less than 2.1/ 0.06 :: 35 iterations between independent scenarios. This worst case situation does not look favorable for the gradient search techniques. However, once the weights are close to optimum, the required number of iterations are less, but still well over 30. A dual receiver system appears necessary for the gradient search techniques. One receiver would be for establishing the weights and another, using only periodically updated weights, dedicated to data throughput. The 8MI technique offers a possible single-receiver solution. The processing required can be couched in the traditional units of number of multiplications. The covariance matrix is formed with K independent samples, and clearly, the larger K is, the better the estimate and the more accurate the solution since a static scenario is assumed. For an ensemble average solution to be within 3 dB of the optimum solution for 50 percent of the estimates, Reed et al. [13] suggest K > 2N 3. The usual role of thumb is to take K > 2N. Boronson [2] notes that K > 3N and K > 4N ensures that the solution will be within 3 dB of the optimum for 98 and 99.68 percent of the estimates, respectively. Formation of R(k) and Mo(k) require KN(N + 1)/2 and KN complex multiplications, respectively. Inversion of R(k) and calculation of W(k) require (N3 /2 + N2) and N2 complex multiplications respectively [13]. Thus for a three-element array, there are at least 86 complex multiplications. A real 16-bit multiplication can be performed within about 150 ns with current off-theshelf hardware. For the three-element array, the multiplica-
356
t
I In fact, the weights must respond slower than 0.2 times the data rate. In practice, the weights can be updated at the data rate, but cannot change completely in less than five data bit intervals (Winters [18]).
4. 63
6 .17
18
24
7 .71 30
TIME 1" \
)
9 .16
10. 80
12.3 4
13 .89
1\ . 43
36
41
48
\4
60
DIS TANCE TRAVELLED AT 140 \:mj h - 00
i ~ i:i
20
1&
SO !It IA"PLE NU"UIl:
40
after the processing is complete and the weights are calculated [18] .
~ cm )
V.
60
-' 0
- 10
- 10
tI
iI 1
/
Mea s ur e
"-
We lgh ts set Ca lc ul e t e
and set we t qnt. s
We Ights us ed
I i n tm s in te rva l e r i v ed f rom mea sureeent s he re
~
1 UPDATE CfCL[
Fig. 3. Example of worst case fading (carrier frequency = 900 MHz. vehicle speed 140 krn/h) with maximum weight response rate (for l o-kbit/s data rate and single receiver system) of about every 0.3 ms. At 140 km/h , weights respond about every 1.2 cm . Limitation in manageable dynamic range of signal is illustrated in fade 10 left. where installed weights correspond to "out of date" measurements. Limitation in realizing this with sufficient accurac y. maximum update rate is in estimat ing
Two
tions alone would take about 52 JJ.S, easily within the upper limit of 0 .3 ms for the weight updating . A six-element array requires 504 complex multiplications, taking about 0.3 ms, indicating the need for a second multiplier to maintain the maximum weight update rate. It is probable that an exponential deweighting for the covariance matrix update would be more suitable for tracking the solution . Hudson ([7, p. 125] after Lunde [10], and Monzingo and Miller [12, p. 314]) give formulas for directly updating the inverse of the deweighted covariance matrix and the steering vector, respectively . A fundamental limitation lies with the required measurement times . The K independent samples must be taken within a short enough time that the scenario appears static. On the other hand, the period taken to retrieve the K samples necessarily extends long enough for the correlations to be correctly detected . This places some lower limits on the crosscorrelation bandwidths of the reference signals. If pilot tones are used, for example, a correlation time interval equal to the maximum weight response rate of 0.3 ms calls for pilots to have a theoretical minimum spacing of about 3 kHz . (In practice, there would be a much larger pilot spacing requ ired.) This is rather inconvenient; for eight cochannels, for example, an entire 25 kHz bank is used only for pilots! A point of academic interest is that pilots should be spaced by more than the upper Doppler limit of about 120 Hz ; otherwise , an interferer may often be singled out as the wanted signal and vice versa . If it can be assumed that the weighted steering vector can be estimated accurately in less than about 0.3 ms, then the maximum weight response rate can be realized using SMI. At the worst case fading rate (corresponding to a vehicle traveling at 140 km/h), the response occurs about every 1.2 ern. The measured signals are taken over the preceding interval , or maybe even two intervals before weight update, if slower processing is used . The situation is depicted on a Rayleigh envelope in Fig. 3 . A way around dealing with "out of date" measurements is to delay the received signal and operate on it
IMPLEMENTATION
Winters ' [17] base station implementation proposal uses a spread-spectrum system with an LMS algorithm. It is based on Compton 's [4J description . While the system is claimed to be practical, the transient analysis of the algorithm is not addressed. A possible system could use Winters' spreadspectrum proposal with the SMI algorithm. The spreading factor should be kept modest, not only to keep the processor bandwidth down, but also because the cochannel interference increases with channel bandwidth in a cellular system, and the returns from the optimum combining system diminish . However, the spreading factor must be sufficient to get a pseudonoise sequence completed and preferably repeated within the time that the branch scenario changes, i.e ., about every 0 .3 ms in the worst case . With implementation at the mobile rather than the base station, shorter pseudonoise sequences are possible since there are far fewer base situations than mobiles . For example , for a sequence length of 16 bits with an information rate of 16 kbit/s , a spreading factor of at least three is required to attain a rate of one sequence per 0.3 ms . The implementation of optimum combining is obviously much more complicated and expensive compared to conventional combining . Switched or selection combining are the easiest techniques, especially when there are just two branches. Postdetection equal-gain combining is also easy to implement, since there is no need for cophasing the signals . Postdetection maximum ratio combining is more complicated because an amplitude weighting must be included, and calculating the weights requires measurement of SNR in each branch. Predetection maximum ratio combining requires both amplitude weighting and cophasing of the signals . This demands much extra hardware for each branch. Maximum ratio combining is more often used as a theoretical performance benchmark because much progress is possible in calculating the diversity returns. A loss budget must be introduced for imperfect combining . For example, the system proposal of Yeh and Reudink [19] assumes maximum ratio combining and allows I-dB .. modem loss " for the theoretical output SINR . Optimum combining, in the implementation considered here, requires not only the measurement and weighting hardware for predetection maximum ratio combining , but also powerful digital computation hardware for calculating the weights . Signal digitization would be one of the expensive aspects of an implementation. A possibility which has not received much attention is bandpass sampling, in which the IF (or RF) signal is digitized directly to baseband. The cost of the extra hardware necessary for optimum combining is very high indeed when compared to a switched , selection , or equal-gain system . Nevertheless, integrated packages for this type of signal processing are becoming increasingly available and digital hardware costs are still decreasing . If high spectrum efficiency becomes sufficiently important in cellular mobile systems, then optimum combining of diversity antennas at the mobile, with its significant
357
improvement in the channel capacity compared to conventional combining types, forces itself into consideration for the system architecture. V.
CONCLUSION
Optimum combining for diversity antennas at the vehicular mobile has been discussed in terms of traditional array parameters. The emphasis has been on physical interpretation of the mathematical formulation. A worst case example of a fast-moving vehicle operating with cellular communication frequencies indicates that the LMS, or other truly adaptive algorithms are not particularly suitable owing to the potentially short times available for adaptation. Also, it has been shown that preprocessing is not likely to be useful for accelerating convergence. However, for slower mobile speeds such as pedestrians and/or lower carrier frequencies, gradient techniques become feasible. Using a sample matrix inversion algorithm, the signal-processing requirements are not so severe and can be implemented using current hardware. In a system with conventional frequency divided 25 kHz bands for each channel, the measurement time for estimating the weighted source vector imposes serious limits on the situations for which the optimum combining is useful. Conversely, the rate of change of the scenario (effectively the speed of the vehicle or its immediate surroundings) limits the returns because of the required measurement time for the weighted source vectors. To derive the benefits of optimum combining, it is necessary to use wide-band pilots such as the pseudonoise codes in a spread-spectrum communications system.
digital mobile radio with diversity combining," IEEE Trans. Commun., vol. COM-31, no. 9, pp. 1085-1094, 1983. [6] P. W. Howells, "Intermediate frequency sidelobe canceller," U.S. Patent 3 202 990, 1965. [7] J. E. Hudson, Adaptive Array Principles. London: Peregrinus, 1981. [8] »: C. Jake~, Ed Microwave Mobile Communications. New York: Wiley, 1974. [9] W. C. Y. Lee. Mobile Communications Engineering. New York: McGraw-Hill, 1982. [10] E. B. Lunde, "The forgotten algorithm in adaptive beamforming," in Aspects of Signal Processing, G. Tacconi, Ed. Reidel, 1977. [11] T. W. Miller, "The transient response of adaptive arrays in TDMA systems," ElectroSci. Lab., Dep. Elec. Eng., Ohio State Univ., Columbus, Rep. 4116-1, 1976. (12] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. [13] I. S. Reed, J. D. Mallet, and L. E. Brennan, HRapid convergence rate in adaptive arrays," IEEE Trans. Aerospace Electron. Syst., vol. AES-IO, no. 6, pp. 853-863, 1974. [14] L. F. Ricardi, "Adaptive and multiple-beam antenna systems," in
Proc. Summer School on Satellite Communication Antenna Technology. The Netherlands: North Holland/Elsevier, ch. 6. [15] W. D. White, "Cascade preprocessors for adaptive antennas," IEEE Trans. Antennas Propagat., vol. AP-24, no. 5, pp. 670-684, 1976. [16] B. Widrow, P. E. Mantey, L. J. Griffiths. and B. B. Goode, "Adaptive antenna systems," Proc. IEEE, vol. 55, 1967. [17] J. H. Winters. "Optimum combining in digital mobile radio with cochannel interference," IEEE Trans. Veh. Technol., vol. VT-33, no. [I 8) [19]
REFERENCES
S. Applebaum, "Adaptive arrays." IEEE Trans. Antennas Propagat.. vol. AP-24. pp. 585-598. 1976. [2] D. M. Boronson. "Sample size considerations for adaptive arrays:' IEEE Trans. Aerospace Electron. Syst., vol. AES-16. no. 4. pp.
[I]
446-451, 1980. V. M. Bogachev and I. G. Kiselev, "Optimum combining of signals in space-diversity reception." Telecommun. Radio Eng., vol. 34/35, no. 10, p. 83, 1980. (4] R. T. Compton, HAn adaptive array in a spread spectrum communication system," Proc. IEEE. vol. 66, pp. 289-298, 1978. [5] B. Glance and L. J. Greenstein, ·'Frequency selective fading effects in
[3)
358
3, pp. 144-155, 1984. - - , private communication', Mar. 1988. Y -S. Yeh and D.O. Reudink, "Efficient spectrum utilization for mobile radio systems using space diversity, " IEEE Trans. Commun., vol. COM-30, no. 3, pp. 447-455, 1982.
The Performance of an LMS Adaptive Array with Frequency Hopped Signals
LEVENT ACAR R.T. COMPTON. JR., Fellow, IEEE The Ohio State University Electroxcience Laboratory
The performance of an Ll\-tS adaptive array with a frequency hopped, spread spectrum desired signal and a CW interference signal is examined. It is shown that frequency hopping has several
effects on an adaptive array. It causes the array to modulate both the amplitude and the phase of the received signal. Also, it causes the array output SINR (signal-to-interference-plus-noise ratio) to vary with time and thus increases the bit error probability for the received signal. Typical curves of the desired signal modulation and the time-varying SINR at the array output are presented. It is shown how the array performance depends on hopping frequency, frequency jump size. interference frequency, signal arrival angles. and si2nal powers.
Manuscript received January 19. 1984: revised December 27, 1984. This work was supported in part by the Department of the Navy, Naval Air Systems Command. Washington, D.C. under Contract NOOO 19-82C-0190 with the Ohio State University Research Foundation.
Adaptive arrays based on the least mean square (LMS) algorithm [1] are very effective for protecting communication systems from interference. These antennas can automatically track desired signals while also nulling interference [2]. Methods have been developed for using adaptive arrays with ordinary amplitude modulated (AM) signals [3], binary frequency shift keyed (FSK) signals [4, 5,61, binary phase shift keyed (PSK) spread spectrum signals [4, 7], and quadriphase PSK spread spectrum signals [8]. These techniques have all been demonstrated experimentally. In this paper we study the performance of an adaptive array with another type of spread spectrum signal, a frequency hopped signal [9]. Frequency hopping is a widely used method of spectrum spreading. Its purpose is to make a communication system less vulnerable to interference. For some applications, it may be desirable to combine adaptive arrays with frequency hopped signals to obtain the interference protection of both. However, very little information is available on the performance of adaptive arrays with frequency hopped signals. As we shall show in this paper, frequency hopping has several adverse effects on an LMS array. First, it causes the array to modulate both the amplitude and the phase of the received signal. Second. it makes the output SINR (signal-to-interference-plus-noise ratio) vary with time and hence increases the bit error probability for the demodulated signal. If an LMS array is to be used with frequency hopped signals. these effects must be taken into account in the design of both the array and the signal modulation. In this paper we consider an ordinary LMS adaptive array with continuous feedback loops. We do not consider various modifications of the LMS array (such as weight storage and recall algorithms) that might be used to improve its performance with frequency hopped signals. Our purpose here is to determine when the basic LMS array has problems and to characterize the array behavior with frequency hopped signals. We use a simple model to study this problem. We consider an array with three elements. and we assume the frequency hopped signal has only a few frequencies. Such a model is adequate to illustrate the effects of frequency hopping on the array, and it allows us to explore the interaction between the hopping characteristics and the array transients. Section II of the paper defines the problem and formulates a method for calculating array behavior with frequency hopped signals. Section III describes numerical results based on the technique in Section II. Section IV contains the conclusions.
Authors' current addresses: L. Acar, The Ohio State University,
II. FORMULATION
Electro-Science Laboratory. Department of Electrical Engineering. Columbus. OH 43212: R.T. Compton, Jr., Department of Electrical Engineering, Ohio State University, 2015 Neil Avenue. Columbus, OH 43210.
Consider an LMS adaptive array [1 J with three elements, as shown in Fig. 1. Let the clements be
Reprinted from IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-21, No.3, pp 360-371, May 1985.
359
hopped with a periodic hopping pattern. We suppose the hopping sequence repeats after p hops. We model the desired signal as a CW signal with constant frequency Wh on each hop interval Th - l :5 t < Ti; where h is an integer denoting the hop interval (1 :::; h s p) and This the time at the end of interval h. The duration of hop interval h, TIr - Th - 1 , is called the dwell time. We assume the dwell time is the same for each h. We refer to the separation between two Wit that are adjacent in frequency as the frequency spacing, and we assume the Wh are equally spaced across the band. (Two Wh that occur sequentially in a given hopping pattern are not necessarily adjacent.) We define the hopping frequency iii to be the number of hops per second (the reciprocal of the dwell time), j~ = (T 1 - To) - 1, and the pattern frequency f p to be the number of complete hopping periods (or .. patterns' ') per second ~ i.e. ,fp = (TfJ TO)-l = fH/P, We define the center frequency We of the desired signal to be the arithmetic mean of the hopping frequencies Wh. (The antenna elements in Fig. 1 are assumed to be a half wavelength apart at frequency we') We denote the difference between a specific signal frequency Wh and the center frequency We by .lw".
ARRAY
OUTPUT
i(U
REFERENCE SIGNAL r(t)
Fig. 1
Three-element adaptive array.
isotropic, noninteracting, and a half wavelength apart at the center frequency of the signals. The analytic signal y)(t) from the jth element is mixed with a local oscillator (LO) and then passed through a narrowband filter (NBF). The purpose of the LO and NBF is to dehop the desired signal, as discussed below. The filter output xJ(t) is the input to the jth channel of an LMS processor [1]. This processor multiplies each signal x)(t) by a complex weight w) and then sums the result to form the array output get). The weights wJ in an LMS processor are obtained by correlation feedback loops that minimize the average power in the error signal e(t) [3]. e(t) is the difference between the reference signal ret) and the array output set). The reference signal determines which signals are to be retained in the array output and which are to be nulled. Received signals correlated with ret) will be retained and signals uncorrelated with ret) will be nulled. In practical communication systems, ret) is usually derived from the array output by nonlinear signal processing operations [4-8]. In this paper, we do not address the problem of refernece signal generation. We simply assume ret) to be a signal correlated with the desired signal. Let yet) be a vector containing the element signals.
(5) Finally. we define the relative bandwidth B, of the desired signal to be its total bandwidth divided bv its . center frequency.
max{wh} - min{wh}
To dehop the desired signal. we assume that the LO in Fig. 1 is hopped synchronously with the received desired signal. 1 The LO signal is
(7) where WL is the center frequency of the LO. We assume that WL < We and that the NSF has a center frequency We WL' The bandwidth of the NSF ~ which we denote bv B,-. is assumed smaller than the separation between . adjacent hopping frequencies. All the NBFs in Fig. 1 are assumed to be identical. With this model, the vector YJ(t) is
(1)
and let X(t) be a vector containing the signals at the LMS processor input (2) where T denotes the tranpose. We assume below that the array receives a desired signal and an interference signal, and that there is also thermal noise in the element outputs. Thus, yet) and X(t) may be written yet) = Yd(t) + Yi(t) + Yn(t)
(6)
Yd(t)
=
AJ
eXP{j f(Wc + ~Wh)t + 4JJ)} ] exp{j[ (we + ~Wh) it - Tel) + wdl} [ exp{j[(w c + UWh)(t - 2Td) + l1J J 1}
,
(3)
and (4)
'We do not address the issue of timing synchronization here. Also. if the desired Signal amves from a direction other than broadside. its hopping will have a different timing on each element. because of the propagation time delay between elements. Thus. strictly speaking. the LO hopping cannot be synchronized exactly with the desired signal hopping on every element. However. we assume the interelernent propagation time to be very small compared with the dwell time. In this case, differences in desired signal timing on different elements may be neglected.
where Yd(t), Y,(t), and Yn(t) are vectors containing the desired, interference, and thermal noise signals from the elements, and Xd(t), Xi(t), and X'l(t) are the corresponding vectors at the processor input. Now let us define the signals and determine these vectors. First, we assume the desired signal is frequency 360
where Ad is the desired signal amplitude, 4Jd is the desired signal carrier phase angle, and Td is the propagation time delay between two adjacent elements,
Finally, we assume the element signals Yj(t) contain white, Gaussian noise. After mixing and filtering, the iJ(f) then contain narrowband Gaussian noise signals. The noise vector Xn(t) is
(9)
(19) where the ii/t) are zero mean, Gaussian random processes, each with variance tr', We assume the flj(t) are statistically independent of each other and of l/Jd and l/Ji' Once the signal vector X(t) is defined, the array weights may be found as follows. The weights satisfy the system of equations [1],
ad is the desired signal arrival angle with respect to
broadside. (8 is defined in Fig. 1.) $d is assumed to be a random variable uniformly distributed on (0, 21T). After dehopping, the desired signal vector Xd(t) is }(d(t) == Adexp{j[(Wc-WL)t+WJJ} Ud(h),
dW(t) dt
where
- - + k¢(t) ( 11 )
== kS(t)
(20)
where Wet) is the weight vector
and
(21)
( 12)
k is the LMS loop gain, ¢(t) is the covariance matrix,
Next, consider the interference. Suppose the interference is a CW signal at frequency Wi incident on the array from angle Sf' The interference signal vector f,(t) is
(22)
and Set) is the reference correlation vector,
Set) == E [X* (t)r(t)]. In these equations, E ( .) denotes expectation, and * complex conjugate. Since Xd(t), Xi(t), and Xn(t) are uncorrelated with each other, ¢(t) reduces to
( 13)
where Ai is the amplitude, W, is the carrier phase angle, and T, is the propagation time delay between two adjacent elements.
¢(t) == E[X;r (t)XJ (t)] + E[Xt (t)Xr (t)]
+ E[X:(t)XJ(t)]
( 14)
== A~Ud(h) U~(h)
The carrier phase angle 4Ji is assumed uniformly distributed on (0, 211") and statistically independent of Wd' After mixing and filtering, the interference signal vector X,(t) is .X,(t) == A;(h) exp{j[(w i
-
WL -
We -
To find the reference correlation vector Set), we must define the reference signal ret). As discussed above, r(t) must be a signal correlated with Xd(t) and uncorrelated with Xi(t) and Xn(t). We shall assume the reference signal has the same form as the desired signal on channel 1 of the processor, but with amplitude An
.lwJr)t + Wi]} V,.
otherwise
uWhl <
8(12
f(t) == A, exp{j[(wc
-
WL)t + l/Jd]},
(16)
and VI == [1, exp( - j;) , exp( - j2
+ A: 2(h)Ut U] + (J"2/, (24)
where
Iw, -
(23)
Then, from (23), Set) is ( 17)
Set) == A,A d VI (h).
with 4>; the interelement phase shift,
(26)
Also, we note that (t) and Set) depend only on h, because they are constant during each hop interval. Hence we denote their values during interval h by (h) and S(h). Thus, for one period of the hopping pattern, Wet) satisfies the sequence of equations,
( 18)
Note that the frequency hopping has converted the CW interference signal at the antenna element into a pulsed signal at the processor input. 361
d:?) + k
both periodic functions of time. Thus, W(t) satisfies a differential equation with periodic coefficients and a periodic driving term. The solution to such an equation will also be a periodic function of time after any initial transients have died out. 3 In this paper we concentrate on the periodic steady-state behavior of W(t). We do not consider initial transients. Once W(t) has settled into its periodic steady state, the initial weight vectors W(T o), W(T I ) , ••• , W(Tp - 1 ) may be found by invoking the periodicity of W(t) to note that W(Tp ) must be the same as W(To). Thus, using (28) to compute W(t) at the end of each interval, we have the following relations among the initial vectors W(T h ) ,
To:S t < T1
d:?) + k
d:?) + k
Tp_l:S
t < T;
(27)
Suppose W(Th - L) is the value of W(t) at the end of interval h - 1. Then, since W(t) is continuous at each hop, W(Th - l ) is also the initial value of W(t) for interval h. Hence, the solution to this sequence of equations is2
W(t) = exp[ - k
WI
+ -I(l)S(l),
= exp[ -k
W2
W(t) = exp[ - k
1
Wo
+ -l(p)S(p),
Tp - 1 ~ t < T1
WO] Wi
[W:_
1
_
-
To)]
<1>-1(2)5(2)
1-
(p - I )5( p - I)]
exp[-k
+
Tp -
2) ]
[W(Tp _ 2 )
+ - I (p [Wp -
1) ]
- 1)5( p - 1)
1
ct>-l(p)S(p)
where. to simplify notation. we have used W, = Wel,). Note that in the last equation we have replaced Wp by Wo· If the initial vectors Wi in (29) are regarded as unknowns, (29) has as many equations as unknowns. Rearranging (29) gives the following system of equations for the W,:
-exp[ -k
I
o
-exp[ -k
o
o
[[I-
I
o
!
+
(29)
where W(To), W(T1) , ••• , W(T -1) are the initial weight vectors for each interval. If these initial vectors were known, we could calculate W(t) at any other time from these equations. To determine the W(Th ) , we proceed as follows. Because the hopping pattern is periodic, (t) and S(t) are
exp[ -k
=
-
(28)
<1>-1(1)5(1)
= exp[-k
- -
W(t) = exp[ -k
+
exp] -k
+ -1(2)S(2), Wp-
- -l(p)S(p)]
=
To)J [W o
-
T1)]
-exp[ -k
exp[ -kcll(p)(Tp - Tp_1)]]-1(p)S(p) [! - exp[-kcll(1)(T1- To)]]-I(l)S(l)
[I - exp[ -k
~ Tp-2)]]-I(p -
I
]
(30) 1)S(p - 1) .
3This statement can be proven in the same manner as in [II. eqs. (24)-(29)]. A general proof of this property of differential equations may be found in D'Angelo [12J.
2Matrix exponentials, such as exp[ -k
[10].
362
This system may be solved numerically" for the initial vectors Wi = W(T;), and W(t) may then be found for other times from (28). Once the weights have been found , the array performance may be calculated . First, the time-varying weights cause the array to modulate the desired signal. The desired signal at the array output is
and
or , from (10), Sd(t)
=
I ~
«
r..
(32)
To characterize the desired signal modulation , we define the envelope modulation ad(t) and the phase modulation TJd(t) by
I
aAt) = Ad WT(t) Ud(h)
I
Th -
I
-s t <
r,
In section III we refer to ~i as the " input INR." The reader will understand that ~, is actually the INR on each processor channel only for those hopping intervals when the interference apears in the filter output. In Section III we use these equations to calculate the array performance with frequency hopped signals.
III. RESULTS
AdWT(t)Ud(h) exp{j[(wc-wdt+l!JdJ},
Th -
is (41)
(31)
Sd(t) = WT (t)Xd(t)
~,
(33)
and (34)
The output signal powers also vary with time. The output desired signal power is
Using the equations above, we have computed the signal modulation and the SINR for a variety of cases . We pre sent these results as follows. First, in Section IlIA, we show typical curves of the desired signal envelope and phase modulation and the output SINR as function s of time . To characterize these time -varying quantities in a simple way, we also define an envelope variation, a phase variation, and a bit error probability. Then , in Sections IIIB-F, we describe how each signal parameter affects the envelope and phase variations and the bit error probability. Section IlIB discusses hopping frequency, Section IlIe frequency jump size, Section IIID interference frequency, Section IIlE arrival angles, and Section IlIF signal powers.
(35)
A. Typical Curves
(36)
First we consider envelope modulation. Fig. 2 is a typical curve , computed for 6d = 15°, 6, = 30°, ~d = 6 dB, ~, = 20 dB , B, = 0.1, and p = 2. The figure
the output interference power is
and the output thermal noise power is
0.....-------------------,
co -
o
3 ELEMENT LMS AAAA T
(37)
The output SINR is SINR =
Pd P, + P"
IW(tl1 2 + ~ :(h)IWT(tlU,12 Th -
where
~
I ~ (
< T,
(38)
c w
a:
is the input SNR per element,
~ =A/ la
V>
2
~: (h) is the input INR in each processor channel during
interval h,
~: (h ) = {t
IWi -
We -
otherwise
~wlrl
<
Bf l2
Fig . 2.
..j.,.,..................."T"'.-rrl~"T"'......,:""""' .................I """"..................I""""...,.............r"""orj
'l'_o.J C>
(40)
0 .0
0. 1
0 .2
0 .1
o.~
O.S
TIME
0 .6
0 .7
0 .8
0 .9
= 15°, 6, = 30°, ~d 6 dB, ~ , = 20 dB, P = 2, B, = OJ, w, = WI '
Desired amplitude versus time. 6d =
' To solve (30), one must evaluate matrix exponentials such as exp(k
w
c
(39)
shows the output desired signal envelope versus time . The envelope is plotted in dB, relative to the envelope that would exist with no frequency hopping and no interference. The time axis is normalized so a complete hopping period begins at t = 0 and ends at t = 1. The desired signal is on frequency WI = 0.95wc for 0 < t < 1/2 and on frequency Wo = 1.05w c for 1/2 < t < 1. The interference is on frequency WI (i.e ., Wi = 0.95wc ) . As
363
1. 0
1. 1
may be seen, there is significant envelope modulation on the output desired signal. Moreover, step discontinuities occur in the envelope when the frequency jumps. The reason for this behavior is as follows. A jump in desired signal frequency, at a given arrival angle, is electrically equivalent to a jump in desired signal arrival angle with no change in frequency, because either situation causes a jump in the interelement phase shift. In Fig. 2, the desired signal arrives from ~ == 15° with frequency 0.95w c for 0 < t < 1/2 and 1.05wc for 1/2 < t < 1. This situation is electrically the same as if the desired signal were always on a frequency 1.05wc and arrived from a = 13.54° for 0 < t < 1/2 and from a = 15° for 1/2 < t < 1. Fig. 3 shows the pattern of the array computed at frequency 1.05wc for several times during
(42)
m is the fractional modulation, since it is the total excursion of the envelope normalized to its peak. We refer to m as the envelope variation. Next we consider phase modulation . Fig. 4 shows a typical curve of output desired signal phase versus time ~~..,.-------------,-----.-,
; ELEHENT lHS ARRAY
o
-:::: ~ + ..................,.,~T"""TTT
..........'T".....,........"T'"...................,.,~T"""......,.............l
0'-0.10.00. 1 0. 2 0 . 3 0 .•
fig .~ .
- 70 - 60
-se
-~o
-30 - 20 -10 0
DB
0 . 5 0• •
TlHE
0.7
0.'
0.9
1.0
Desired signal phase angle versus ume . tl" = 15°. fl, ~, = ~O dB . p = ~. B, = 0 .5. w , = W"
L = 6 dB.
=
1.1
~5 °.
for the case ad == 15°.1:1; == 45°. ~J == 6 dB. ~, == 20 dB. B, == 0 .5. P == 2. and W , = WI ' As may be seen. there is substantial phase modulation on the output desired signal. Fig. -+ is typical of what usually happens: the desired signal phase jumps up or down at the beginning of each hop interval and then decays back to zero. This phase modulation is due to the frequency hopping on the desired signal and not to the presence of interference . Fig. 5 shows the output desired signal phase versus time for the same situation as in Fig. 4 but without
10
Fig. 3. Array patterns during hopping period. ed = 15°. e, = 30°. ~ d = 6 dB, ~ , = 20 dB , p = 2, B, = 0.1. W , = WI' Patterns computed at W 2.
the hopping period. The interference is at a = 30° on frequency 0.95w" which is equivalent to interference arriving from a = 26.9° at frequency 1.05w c ' As may be seen in Fig. 3, at the end of interval 1 (at t = 1/2), the array has formed a null on the interference at a = 26.9°. In the second hop interval, the interference disappears (it is filtered out) and the desired signal angle jumps from 13.54° to 15°. Because this jump is toward the existing null, an instantaneous drop in the desired signal amplitude occurs at the beginning of interval 2. During interval 2, the desired signal amplitude increases as the array adapts. Then at the end of interval 2, the desired signal angle jumps back from 15° to 13.54° (away from the null), so the desired signal amplitude jumps up instantaneously. After this jump, the desired signal amplitude drops as the weights form the null at 26.9° again, since the interference has reappeared at 26.9° during interval 1. In order to characterize such a time-varying waveform in a simple way for use below, we define an envelope variation as follows. Let am ax be the largest and am," be the smallest (absolute) value of the output desired signal envelope during the hopping period. Then let
~ ~-y--- ------------------., o 3 ELEMENT
LMS ARR!H
!oJ ...J
~
z'" a:. r-,
'-----------
- ..
"'~
....ex: C
~: ~~..,...,....."T'"TT"O'"T""""....,~,...,... , iT. ''''''''''''1"1 ,.....,.i~'.-T'"'.,,"'''''i~'11""'.'...,.... , .f""T"...,.,........j
C"O.IO .O
Fig. 5 .
0.1
0 .2
0.3
0 ••
T">'
0 .5
TI"E
0 .6
0 .7
0.'
0 .9
Desired signal phase angle versus lime. ed = 15°, p = 2. B, = 0.5 . no interference.
tJ
1.0
1.1
= 6 dB.
interference. Note that almost the same phase modulaton occurs in both cases. The phase modulation occurs because the desired signal interelement phase shift jumps when the frequency hops (unless the signal is incident from broadside) . Hence, immediately after each hop, the 364
array weights no longer have the correct phasing to maximize desired signal response . As the weights respond after each hop, the phase shift of the array seen by the desired signal changes with time. To characterize phase modulation , we define TIm., and TJmin to be the maximum and minimum phase angles of the output desired signal over the hopping period ( - 1T -s TJmtn ::s TIm., -s 2.. ) and !3 to be
!3
= TIm., - TJm,n .
(43)
21T
We refer to !3 as the phase variation . Finally, we consider the output SINR from the array. Fig. 6 shows a typical curve of SINR versus time over one hopping period. This curve is computed for the same ~
........- - - - - - - - - - - - - - - - - ,
3 ELEMENT
LM5 RRRfH o
-
(44) where E, is the signal energy per bit and No is the onesided thermal noise spectral density. For our purposes, we may replace Eb/ No by the signal-to-noise ratio,
Eb = PdTb=~= SNR No No (NjTb )
where P, is the desired signal power and T, is the bit duration; i.e., since T,-I is the effective noise bandwidth, N)Tb is the noise power. Hence P, may be written P, =
P, =
.
2 +"'~~'TT""....-r........y.,..,..""""'"T'r'""""""""""""""""T'r'"TT""...........-n-.'T""""""; '·P.IO.O 0.1 0.2 0 .3 o.~ 0 .5 0 .6 0.7 0 .8 0.9 ).0 1.1 Tll'IE Fig. 6.
SINR versus time . ll" = 15' . ll, = 30°. ( , dB . p = 2. B. = 0.1. W, = W "
= 6 dB. S, =
20
parameters as in Fig. 2. Note that the SINR drops approximately 16 dB at the beginning of the hopping period. Although the SINR recovers quickly from this drop, such a drop can nevertheless greatly increase the bit error probability for the received signal, since the number of bit detection errors is larger by orders of magnitude during this short interval than it is when the SINR is higher. To characterize such a time-varying SINR curve, we define an average bit error probability. We arbitrarily assume that the desired signal, in addition to being frequency hopped, has binary differential phase shift keyed (DPSK) modulation [14] .5 The bit error probability for a DPSK signal in white noise is [15] 'Adding biphase modulation to the desired signal does not change the array weight behavior as long as the bandwidth of the phase modulation is small enough to pass through the dehopping filters. With small bandwidth. the covariance matrix <1>(1) is the same as for a CW signal because the phase modulation terms cancel out. (Also. aside from the dehopping filters. it has been shown [16 J that desired signal bandwidth has almost no effect on array performance anyway . even if the bandwidth is large .) Moreover. as long as the reference signal carries the same DPSK modulation as the desired signal. which we assume. the reference correlation vector S(n is also unchanged . Since both <1>(1) and 5(1) are the same , the weight behavior will be the same with this signal as with a CW signal. (Some examples of how digital phase modulation can be transferred to the reference signal may be found in [4-8] .)
21 exp(-
SNR).
(46)
In addition, for this analysis we assume that interference power has the same effect on detector performance as thermal noise power, so
I---
~
(45)
21 exp(-
SINR).
(47)
Finally, we assume that the SINR transients and the desired signalphase modulation produced by the array are slowcompared with the bit length Ti. In this case the SINR and the signalphase may be consideredconstant over a time interval of 2Tb • (Two adjacent bits are used to detect a DPSK signal.) We define the effective bit error probability Pe as the average of P, over one hopping period, - = ~ 1 IT: '12 exp[ - SINR(t) ] dt. P,. [ ,.
10
(48)
0
We use (48) below for comparing different SINR curves.' Now let us consider the effect of the various signal parameters on the signal modulation and the bit error probability. B. The Effect of Hopping Frequency
The envelope and phase variations m and f3 are large at low hopping frequency and drop as the frequency increases. Bit error probability, on the other hand, is low at low frequency and increases with frequency. Both m and Pe may have local peaks at intermediate frequencies. Fig. 7 shows typical curves of m versus (pattern) frequency. This set of curves was computed for ad = 15°, a, = 45°, ~d = 6 dB, ~i = 40 dB, p = 2, and Wi = WI ' The figure shows several curves for different hopping bandwidths B,. As may be seen, for all but the smallest bandwidths, m has a complicated behavior at intermediate frequencies. This behavior occurs because of "We recognize that (48) glosses over many subtleties that will affect detector performance in an actual system. Our intent here is simply to reduce each SINR curve to a single number to compare different SINR curves. In the presence of a specific system definition , (48) will do as well as anything.
365
:------------------, \ - ' , 3 ELEI1ENT
~--......
~
-
,
C>
II:
o
:--~ __ :__ ~I1S rRA!-
..:-
...
I
-- ' >-
; Sr = I :
. .J'"
co a: co
0"
cr:' ncr:
~=r
- - - - - -- - - - -- - -- - - - .. - ._. :. .~ --;'-:-:- --.....-.. _ ~ _ .i. _ ....:... _
---
ec w
... '", -I
g. 7.
0
1
2
3
~
PATTERN FREOUENCY . HZ XIO"
5
-I
6
Envelope variation versus pattern frequency . e" = 15°. ll, 45°. ~., = 6 dB .~. = 40 dB. p = 2 . '», = w, .
Fig. '}.
=
the way the desired signal envelope changes as the hopping frequency varies. In particular, the time at which am m occurs is in one hopping interval for low values of [p and in the other hopping interval for high values of [p. Typically the smallest envelope in one interval increases with [p, while the smallest envelope in the other interval decreases with [p. At the value of [p where the two minima become equal, the location of am rn in time changes from one hop interval to the other. At this change, the slope of am m versus [p reverses, so the slope of m changes in Fig. 10. Fig. 8 shows typical curves of phase variation f3 versus [p . These curves were computed for the same 0-r-
~
:.
'
~---~-~--~----~--
~~ ':c-_~:i},--:;;~l:l:1:~-c_=C '"'2
-I
0
I
2
3
PATTERN FREOUENCT. HZ
~
X\O"
5
3
~
5
6
Bit error probability versus pattern frequency . 9" = 15°. ll, = 45°. ~" = 6 dB. ~. = -to dB . P = 2. w, = w, .
intermediate J;, and then drops at higher f;, . However.
The larger the frequency jumps encountered by the array. the larger the variations m and f3 and the greater the SINR reduction . In a frequency hopped system, the size of the frequency jumps depends not only on the frequency spacing (the total bandwidth divided by the number of frequencies) , but also on the hopping pattern. For the same spacing. different hopping patterns will produce different frequency jumps. Moreover, bit error probability is affected not only by the size of the frequency jumps. but also by how oftell the jumps occur. since it is an integrated quantity. In general. to minimize 15" one should choose a hopping pattern that minimizes the number of large jumps and also reduces the frequency with which large jumps occur. The effect of frequency jump size may be seen in Figs. 7, 8 and 9. These figures each show several curves for different bandwidths Br • Since there are only two frequencies i p = 2), the total bandwidth is the same as the frequency spacing and the frequency jump size. As may be seen, as bandwidth increases, the variation m and 13 and the bit error probability Pe all increase . This behavior is easily understood . As the desired signal frequency jumps become larger, the jump in interelement phase shift at each hop becomes larger. A larger jump means that the array weights are farther from their optimal values at the new frequency . Thus, a larger weight transient is required after the jump. More
, . .
~
2
C. The Effect of Frequency
LMS AnnA T
\
I
PATTERN FREOUENCY . HZ X\O"
behavior is similar to what happens when an adaptive array receives pulsed interference and a desired signal with no hopping [Ill . (Frequency hopping converts the CW inteference into pulsed interference . The two problems differ, however. because frequency hopping also causes jumps in the desired signal interelement phase shifts .)
--,
~~
0
: 0.001 ,
P, is always higher at Iarge j, than at low!" . This
3 ELEMENT
\
t
: 0 .2 : 0 .1
6'
Fig. 8. Phase variation versus pattern frequency. e,/ = 15°, e, = 45°, ~d = 6 dB, ~; = 40 dB, p = 2, W; = W I '
parameters as in Fig. 7. In general, phase variation is highest at low hopping frequency and drops to a constant as the hopping frequency increases. At large Jp, the array weights are too slow to track the hopping. The nonzero asymptotic phase variation is caused by the jumps in interelement phase shift when the frequency hops. Finally, Fig. 9 shows typical curves of bit error probability Pe versus [p, again for the same param:!ers as in Fig. 7. As may be seen, for higher bandwidths P, simply increases with [p. At lower bandwidths, P, peaks
366
or-------------------,
equivalent angle will be closer to 6, if the interference is on W I than if the interference is on wz, since W3 - W I is greater than W 3 - Wz. Hence, with interference on w" the desired signal falls farther into the interference null, the SINR is reduced more and a higher P, results than with interference on Wz. Note that the edge of the band that is worse depends on the signal arrival angles. In the example above, we have 0 < 6d < 6" and the worst performance is obtained with the interference on W I ' If instead we have 0 < 6j < 6d , then interference on W 3 , the other band edge, will give the worse performance.
,
E. The Effect of Arr ival Angles
envelope and phase modulation is produced and the SINR is lower after the jump. D . The Effect of Interfer en ce Frequen cy
Interference near the edge of the hopping bandwidth is more harmful to the array than interference near the center of the band. Figs. 10 and 11 illustrate this point. These figures 15°. 6j 45°, ~d = 6 dB, show P, versus f p for 6;, z
o x_
.... ''" CD
The envelope variation and the bit error probability increase as the interference arrival angle approaches the desired signal arrival angle. Interference arrival angle has almost no effect on phase modulation except when the interference signal is extremely close to the desired signal. Figs. 12 and 13 show the envelope variation m as a function of pattern frequency for 6d = 15°, ~ d = 6 dB,
a:
CD
o~
~'
Q..
~
0' ~
.
.... ~
... '", CD
-I
Fig. 10.
R
o
O
Il
3
~
PA TTERN FREQUENC Y, HZ XIO·
5
Bit error proba bility versus patte rn freq uency. ij./ = 15' . = ~5 ° . ~.J = 6 dB. ~ , = ~O dB. p = 3. w , = w ,
°
0,
6
3 ELEHENT LHS ARRAY
E
3 ELEHENT LHS ARRA Y
x...... l
.... '"' . -CD
_
~
.
-
, 6 r=l.4
ILl"
;
...J
._
.
~
:
._.-
,
... '", ,- --~.-
...
._
._
._
._
:
.~
i
o
._
ILl
.
ILl
3
~
-,
., i l
3
~
PATTERN FREQUENCY, HZ X10"
Fig. 12. E nvelope variation versus pattern frequ enc y. 6" = 15°, ~ ,/ 6 dB. ~ , = 40 dB , p = 2, B, = 0.1, W , = W "
CD
Oil
,
O
-I
0 .00 1 /
PATTERN FRE QUENCY , HZ XIO"
.,
>'" 20
,
.... r ·0 .2
-1
... _._i
D.o
~: _:> ; ~-'~-l~-,]~t:=Fig. II .
---..
ij ,
=
5
0-.=-."-=,...,,..::-::-.,..-.,.....,..,...0-:-:-------------,
Bit erro r prob ab ility vers us patte rn freq uency . 6" = 15°. 6, = ~5 ° . ~.J = 6 dB. ~ , = ~O dB. p = 3. w , = w , .
E
= 40 dB, and p = 3. Fig. 10 is for W , = W I and Fig. 11 is for W = W :! (where WI < Wz < w 3)' The performance in Fig. 10, when the interference is at the edge of the hopping band. is much worse than that in Fig. 11 , when the interference is at the center of the band. The reason for this difference may be understood in terms of the equivalence between desired signal frequency and arrival angle discussed earlier. Suppose 0 < OJ < 6, as in Fig. 10. The array will produce a null in the pattern at 6j on the interference frequency, either W I or w, . Since o < 6J < a., the equivalent desired signal arrival -angle. as seen on the interference frequency, will be closest to 6, when the desired signal is on frequency w, . The ~,
j
"
ILl ' 0.0
o...J
ILl",
>. Zo
ILl
-I
O
i l
3
~
PATTERN FREQUENCY. HZ XIO"
Fig. 13. Env elop e variat ion ver sus pattern frequen cy. 6" 6 dB , ~, = 40 dB , p = 2, B, = 0.1, W , = W "
367
5
= 15°, ~ " =
~d = 40 dB, B, = 0.1, P = 2, W, = WI. and for different interference angles. Fig. 12 shows 6; = 0°, 30°, 45°, 60°, and 90°, and Fig. 13 shows 6; = 5°, 10°, 13°, 17°, 20°, and 25°. It may be seen that m increases as 16, - 6d l decreases. For 6, very near 6d , the variation m is quite large. The phase variation 13 is small unless 6, is very near 6". Fig. 14 shows a typical case, for 6" = 15°, ~d = 6 dB, ~; = 40 dB, B, = 0.5, p = 2, and W; = WI' Note
probability is primarily to shift the value of the hopping frequency for a given 15e : Bit error probability is very sensitive to the input SNR. Figs. 16 and 17 show the envelope variation and the bit error probability versus jp for 6J = 15°, 6, = 45°, ~ = 6 dB, P = 2, w, = WI ' B, = 0.1, and for several 0-r-
E
0_-----------.,----,-------, 3 ELEHENT
--,
3 ELEHEN T lHS ARRAY ,
..
zci
o
I-
L.HS ARRA Y
~
!
I !
..
:
irci
>
- -
;
. :-:--:- . .
jj ...
.
~
:/
t::-= ::.::---A
~25°
-J
' - - 130 I
2
3
PATTERN FREQUENCY , HZ
~
X10 N
Fig: . 16.
..o
30~ I
i
!
L....
cr
CD 0'"
a: '
"-
a: ~=r a:
....
;
1
,
/
I
. .# . . . , . . . . . " .",
2
3
.
~
PATTERN FREQUE NCY, HZ XIO N
S
2.8,
=
0.1. w ,
=
ad =
15°.
W"
e,
=
20
'
6~~ ·
II>
o
I
2
3
PATTERN FREQUENCY, HZ
~
X10 N
Bit error probabil ity versus pattern freq uency
=
45 °. ~"
=
6 dB. f'
=
2.8,
=
a"
0 .1 . w, = W"
S
=
6
15°.
a,
values of INR. Fig. 17 illustrates how the hopping frequency at which P. peaks varies with the INR. Fig. 18 shows 13 versus j~ for the same parameters as in Figs. 16 and 17 except that B, = 0.5 . Fig. 19 shows the bit error probability versus f p for aJ 15°,6, = 45° "~i = 40 dB, P = 2, Wi = WI' B, = 0.1, and for several values of input SNR. As may be seen, Pe is extremely sensitive to the SNR, as it would be even in a simple DPSK communication system without an adaptive array.
!
60 0
".
=
~ I N R = 10dS'
r ig . 17.
•
- ---- ----------
.
6 dB . P
~
-I
; : 45 i ___'__s> 0
t, =
6
S
... ,
I
,
~
X10 N
Envelope variation versus pattern frequency .
-' '" ' CD
• 0° ............
I
3
,
~----;--~- ~-
'Fig. 15.
I
3 ELEHENT LHS ARRAY
3 ELEHENT LHS ARRAY
=
2
o
0,-.--------------------. ; 8i l
J
0
)(-
also that 13 is much larger for 6, just above 6d than for 6, just below 6d • The reason is as discussed above: in one case the desired signal hops into the null left by the interference whereas in the other it hops away from the null. Fig. 15 shows curves of the bit error probability for the same parameter values as in Fig. 12. It is seen that P, is largest when 6, is near 6d •
o
i
i !
L...
20
.
t
,
PATTERN FREOUENCY , HZ
45 °.
6
5
Fig. 14. Phase variation versus pattern frequency. e" = 15°, I;d = 6 dB , 1;; = 40 dB, P = 2, B, = 0.5, w, = WI'
-I
;
!
. 50
- - -~--:-:~-----------j o
j.
> ,'""- ~/ . z 0 ~.~ w . IO
_ . -~'--....., .... "",,-=-. ~- .' -=--:-";':"';"=' '' -=-=-,-
~ IOO
·1
I
, INR i=4 0 dB l·~ ~ ·
~~ ..........--
.iI
.. .
j
~
6
Bit error probability versus pattern frequency. ed = 15°,1;" = 6 dB, 1;, = 40 dB, p = 2, B, = 0.1, w, = W I '
F. The Effect of Signal Powers
IV. CONCLUSIONS
The input INR has almost no effect on the phase modulation and very little effect on the envelope modulation. The effect of the INR on the bit error
A frequency hopped desired signal has several effects on an LMS array. It causes the array to modulate both the envelope and the phase of the output desired signal.
368
----,
o~
Also , it causes the array output SINR to vary with time below its optimal value and increases the bit error probability for the received signal. The signal parameters affect the desired signal modulation and the bit error probability as follows:
3 ELEMENT LMS ARRAY
z o
C)
1- ...
~.; et:
(1) Envelope and phase modulation are large for low hopping frequencies and drop as the frequency increases. Bit error probability is low at low hopping frequencies and increases with frequency . Both the envelope variation and the bit error probability may have local peaks at intermediate hopping frequencies. (2) Envelope and phase modulation increase with the size of the frequency jumps in the hopping pattern. Bit error probability is increased as the frequency jump size increases. (3) An interference frequency at the edge of the hopping bandwidth is more harmful to the array performance than an interference frequency at the center of the band . (4) Envelope modulation and bit error probability incre ase as the interference arrival angle approaches the desir ed signal arrival angle. Phase modulation is not affected by interference arrival angle unless the interference is extremely close to the desired signal. (5) Input INR has almost no effect on phase modulation and very little effect on envelope modulation. Input INR affects bit error probability by shifting the value of the hopping frequency required for a given bit error probability. Input SNR has a very large effect on bit error probability, as it would in any DPSK system, even without an adaptive array.
>r
UJO
::I: '"
ll.';
.
I----~/_ · _
Fig, IX ,
30
. 40
.
Phase va riation versus pattern frequency. B" .J5° . f."
=
fldB .!, = 2 .8, = 0 5 .w. =
o ;---
><... I
'
.
o I 2 3 ~ PATTERN FREOUENCY. HZ XIO"
-I
~
INR =: 10dB 20 '
/ '
:
-.::-
-----'_ _
""-SNR: - IO dB
~
-
15 , H,
" " " .. " ",r
, ~t..I..J'_=CJ>U._ i
LMS ARRA y -:--
~O
6
W,
'-
~
...
5
_
:
~ '"
eo '
a:
al
0"
a:' Q..
a: o~ a: a:
....
.r "
... ,
- - ,
\~
10
\
~
\
\
-I
Fig , IlJ
o I 2 3 ~ PATTERN FREOUENCY . HZ XlO"
5
6
Btl arm probability versus pattern rrcqucncv ~ " = 15' . H = .J5 ' . ~, = .JOdB . !' = 2. 8 , = 0 l . uJ ~ w !
369
[8]
REFERENCES [1]
[2]
[3]
[4]
[5]
[6]
[7]
Widrow, B., Mantey, P.E., Griffiths, L.J., and Goode, B.B. (1967) Adaptive antenna systems. Proceedings of the IEEE, 55 (Dec. 1967),2143. Compton, R.T., Jr. (1976) An experimental four-element adaptive array. IEEE Transactions on Antennas and Propagation, AP-24 (Sept. 1976), 697. Riegler, R.L., and Compton, R.T., Jr. (1973) An adaptive array for interference rejection. Proceedings of the IEEE, 61 (June 1973), 748. Compton, R.T., Jr., Huff, R.J., Swarner, W.G., and Ksienski, A.A. (1976) . Adaptive arrays for communication systems: An overview of research at the Ohio State University. IEEE Transactions on Antennas and Propagation, AP-24 (Sept. 1976),599. Hudson, E.C. (1980) Use of an adaptive array in a frequency-shift keyed communication system. Report 712684-1, Ohio State University ElectroScience Laboratory, Columbus, Aug. 1980. Ganz, M.W. (1982) On the performance of an adaptive array in a frequency shift keyed communication system. . M.Sc. thesis, Department of Electrical Engineering, OhIO State University, Columbus, 1982. Compton, R.T., Jr. (1978) An adaptive array in a spread spectrum communication system. Proceedings of the IEEE, 66 (Mar. 1978),289.
[9] [10]
[11]
[12]
[13]
[14]
[15]
[16]
370
Winters, J.H. (1982) Spread spectrum in a four-phase communication system employing adaptive antennas. IEEE Transactions on Communications. COM-30 (May 1982), 929. Dixon, R.C. (1976) Spread Spectrum Systems. New York: Wiley, 1976. Bellman, R. (1970) Introduction to Matrix Analysis. New York: McGraw-Hill, 1970. Compton, R. T., Jr. (1982) The effect of a pulsed interference signal on an adaptive array. IEEE Transactions on Aerospace and Electronic Systems, AES-18 (May 1982), 297. D'Angelo, H. (1970) Linear Time-Varying Systems: Analysis and Synthesis. Boston: Allyn and Bacon, 1970. Hildebrand, F.B. (1952) Methods of Applied Mathematics. Englewood Cliffs, N.J.: Prentice-Hall, 1952. Ziemer, R.E., and Tranter, W.H. (1976) Principles of Communications. Boston: Houghton Mifflin, 1976. Lindsey, W.C., and Simon, M.K. (1973) Telecommunication Systems Engineering. Englewood Cliffs, NJ.: Prentice-Hall, 1973. Rodgers, W. E., and Compton, R. T., Jr. (1979) Adaptive array bandwidth with tapped delay-time processing. IEEE Transactions on Aerospace and Electronic Systems, AES-15 (Jan. 1979), 21.
l. INTRODUCTION
An LMS Adaptive Array for Multipath Fading Reduction
YASUTAKA OGAWA, Member. IEEE
MANABU OHMIYA KlYOHIKO ITOH, Member, IEEE Hokkaido University
Multipath fading often poses a serious hindrance in radio communication. The application of a least-mean-square (LMS) adaptive array to the problem of multi path fading reduction is discussed. However, it is known that multipath components are in general correlated with one another. \\ore examine the effect of the correlation on the performance of the L~IS adaptive array. When the correlation coefficient does not equal or approximate I, the L~IS
adaptive array suppresses the multipath signals significantly
by nulling. On the other hand. when the correlation coefficient nearly equals 1. the L~IS adaptive array prevents the output sianal power from decreasing. Therefore. the L~lS adaptive array may reduce the multipath fading effectively for any correlation coefficient value. A reference signal in the
L~IS
adaptive array is
also discussed. It is shown that synchronization in the reference signal generation must be extremely accurate. Moreover, a processor configuration is proposed which may generate the reference signal with the required accuracy.
Radio communications suffer from multipath fading. It has been reported that only a few multipath components are often dominant in strength and play an important role in the multipath fading phenomena [I]. Thus, an adaptive array [2] has a potential to reduce the multipath fading. A least-mean-square (LMS) adaptive array automatically tracks a desired signal and nulls interference signals. The LMS adaptive array. however. requires a reference signal in order to control each weight. Let us assume that the desired signal contains a deterministic component which is fully known at the receiver. Then, the deterministic component may be used for the adaptive array reference signal. An example of the deterministic component is a pilot signal which is added to the transmitted communication signal [3, 6]. According to the Ii terature [3], the LMS adapti ve array may eliminate the undesired multipath signals by nulling, in a case where a modulated pilot signal has a sufficient bandwidth to discriminate between multipath propagation modes. This means that the LMS adaptive array may suppress the undesired multipath components when the incident components are not correlated with one another. In mobile communications. however, we do not know the time delay differences between rnultipath components. Then. the required bandwidth of the pilot signal is in general unknown. Even though we know the time delay differences, an unrealistically wide bandwidth might be required. Thus, in order to reduce the multipath fading by the adaptive array, we must consider the effect of the correlation between multipath components on the performance of the adaptive array. The literature [4] proposed a preprocessing scheme for the adaptive array which may suppress the coherent signals. A disadvantage of this scheme is that it needs more antenna elements than the conventional adaptive array. We show that an LMS adaptive array may reduce the multipath fading effectively for any correlation coefficient value between multipath signals. First. we examine the behavior of the LMS adaptive array in the presence of the correlated multipath signals. Second, we find required synchronization accuracy in reference signal generation. Third, we propose a processor configuration which generates the reference signal. II. FORMULATION OF THE PROBLEM
ManUSCript received September 11. 1985: revised March 29. 1986. ,A~lhors' address: Department of Electronic Engineering. Faculty of Engineering, Hokkaido University. Kita 13. ~ishi S. Kita-ku. Sapporo 060, Japan.
We consider the N-element linear LMS adaptive array shown in Fig. 1. We assume that two multipath components set) and m(t) are incident on the array from angles 8I and em relative to broadside, respectively. The antenna elements are assumed to be isotropic and a halfwavelength apart. \Ve represent both signals on the kth element by Sk(t) and mk(t) (k == 1--- N). Thermal noise nk(t) is assumed to be present on each element signal.
Reprinted from IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-23, No.1, pp. 17-23, January 1981.
371
Information Signal
P'i Lo t Signal
\nl'ular F"requency
Fig. 2. Fig.!.
LMS adaptive array.
Then, the complex-valued element signal is given by ik(t) = Sk(t)
+
mk(t)
+
nk(t).
(1)
We assume that the thermal noise components on different elements are independent and that they are also independent of the signals set) and m(t). We define an iV-dimensional signal vector as
(2) where T denotes transpose. Furthermore. we express the complex weight for the kth element as Wk and the LVdimensional weight vector as W, i.e., W ::::
We assume that the parameter G in Fig. 1 is determined adequately so that the deviation of the weight vector W from the ensemble average is negligibly small. Thus, we consider the weight vector W to have the ensemble average. We represent the power of Sk(t), mk(t), and iik(t) by Si' M i, and N;, respectively. Namely, we have < Sk(t) 12 ),
=
N,
I
(4)
for k = 1 --- N
where (.) denotes the ensemble average. It is assumed that till (r) is delayed from S I (t) by Then, rill (1) is expressed as mt(t) = \iM,ISiSl(t-T)
is almost the same as that of the information signal. For simplicity, we consider only the pilot portion. We express the reference signal as (6)
When Tr = 0, the reference signal coincides with s(t). Similarly, when Tr = 1', it coincides with met). Now, we represent the complex envelopes of 51 (t) and ml (t) by S1 (r) and M1(t), respectively. Then. we have
(7)
m1(t )
e-J'lI'
(5)
where '1" is a phase delay which occurs by a reason other than the propagation delay difference. In order to generate a reference signal ret) in the LMS adaptive array, a pilot signal is always transmitted together with an information signal [3, 6]. The power spectrum of the signal is illustrated in Fig. 2. The pilot signal is modulated by a signal which is fully known at the receiver. Both bands are located very closely to each other. Namely, let We and w; be the center angular frequencies of the pilot and information portions, respectively. Then, IW e - w;l/w c « 1 holds. The reference signal which is generated by the reference signal processor is a replica of the pilot portion of the signal. We assume that the bandwidth of the pilot signal
j W c1
(8)
•
(9) We define the normalized autocorrelation function of as
S1(t)
( 10)
where * denotes the complex conjugate. Moreover. we define the covariance matrix Rxx and correlation vector Vxr as Rxx = (X*(t) XT(t) ) Vxr
'T.
= Ml(t) e
From (5), (7), (8), we may obtain
(3)
[WI l-V2 . . • WN]T.
s, ::::
Spectrum of signal.
=
(X *(t)
r (t)
( 11 ) ( 12)
).
Here, we assume that the bandwidth of the pilot signal is narrow enough that the interelement delay does not change the envelope. From these results, the i p ,q )th element of Rxx is given by
(~i; (t)~iq(t) )
= S; eJ(p-q)Q>s + + \is; M i
M; eJ(P-q)
P*(T)
+ N, bp q
ej{(p-I)
where
= 1T sin
as
( 14)
(15) and t1p q is the Kronecker's S.
372
70...--- --
Similarly, the kth element of Vxr is given by ~
CD
( i:(t),(t) ) :==
5; p r
e
"vIf'T , M ; P I u
-( ~ - T)
r
e
j {' k - ) )
(s n r)m) (t) )/ Y5,M, = p * (i ) e - /IW,T+ ,v' l
(\6 )
( 17)
This means that p * (T) e- j (W,.T + 'V · ) is the complexvalued correlation coefficient between .~ ) (t) and iii)( r) . As stated previously. we consider that the weight vector has the ensemble average . Thus. the steady-state weight vector is given by the Wiener solution. i.e .. Rxx -
--,
., 3 0
.
~
.:- 2 0 ~
:==
-
~c::J 4 0F-""7~~~::::
j{Ck - I }
o
Here. it should be noted that the following equation holds.
W
-
~ 50
- * (T)
-I-
-
S0
I
Vxr .
(18)
This is true whether the incident signals are correlated with or not. We represent the array output of .i'(tl. 171(£ ). and the thermal noise by so(t ) . 1710(£)' and lio(rl. respectively. Moreover , we represent the power of them by So. Mo. and No. Namely. we have .\-10
, - I' \1/ -1
= ( l //l o(t ) -
00
0 .S
1 .0
c
Output DCR versus C. N = 2. 8, = 0°, 8.. = 30°, Tr = 0 , S,IN, = M,IN, = 20 dB.
Fig. 3.
provided that C does not equal or approximate I. The output OUR is above 20 dB when C ~ 0 .9. Fig. 4 illustrates the effect of C on the array pattern. It is apparent that when C ~ 0 .9. the null is pointed almost exactly toward InU). However. when C > 0,9. the null is shifted or lost. Now we discuss the case where C nearly equals I. When C = I. i is significantly less than the reciprocal of the frequency bandwidth of the signal. In this case, the fad ing is not frequenc y-selective but freque ncy -tlat. Furthermore . we may regard d ;)(r) defined by (2 1) as the approx imate output desired signal. d ;l (t) :== .~I) ( t)
([ 9)
When So 2:: Mo. we consider that .W ) is the desired signal and ,nU) is the undesired one. On the other hand. when .Vl o > So, we consider that ,nUl is the desired signal and sU) is the undesired one. Here. we represent the desired-to-unde sired-signal-ratio by OCR. Note that the output OUR is given by So;Ml) when Sf) 2:: Mo and that it is given by M ol 50 when Mf) > So. Moreover. we add that all of the numerical results which are shown later are computed values.
10
Let D ;\
.i,
In ,)(t) .
D:, denote the power of ii ;\(t).
i.e .. (22)
: a~ ([) I ~ ') , ~ ,
:== (
The output O':\R (D ' I)/ .Y,)l represents the approximate output desired-signal-to-noise-ratio when C = I. Fig. 5 shows the output O':-JR versus C tor several values of '1' . From these curves. it is seen that the LMS adaptive array prevents the output signal power from decrea sing. Namely. the frequency-nat fading is reduced , When C = I, the weights are determined in such a way
III. MULTIPATH FADING REDUCTION
Now we discuss the steady-state performance of the LMS adaptive array. In this section we assume that Tr:== 0 holds. Namely . we assume that the reference signal coincides with s( t) . In order to simplify the notation. we introduce the real-valued symbols C (0 ~ C ~ 1) and 'I' which satisfy (20).
--" I I J~
Fig.
.i
(20) From ( 17), it is seen that C and '1' are the magnitude and phase delay of the correlation coefficient of sd t) and rn\ (t ) , respectively. Fig. 3 shows the output OUR versus C for several values of '1'. Since Sf) 2:: Mo holds for these parameters. i( f) is the desired signal and 1iI(t) is the undesired one . It is seen that the output OUR depends on the correlation Coefficient (C e - )'1') . The undesired signal is, however. Suppressed significantly by the LMS adaptive array
Array pattern. N = 2, 8, = 0°, 8.. = 30°, i' = O°, Tr =O. S,IN, = M,/N, = 20 dB . 5 0 , --
-
-
- -- - ,
~ 4B
. 95
Fig. 5.
373
. 99
C
. 999
. 9 9 99
Output D'NR versus C. N = 2, e, = 0°, 8", = 30°, Tr = 0, S,lN, = M,/N, = 20 dB.
that the weighted signals Wk {sk(t) + mk(t)} (k = I - N) are added in-phase at the array output. This means that the LMS adaptive array realizes space diversity when C = I. Here we consider the physical reason why the weighted signals are added in-phase at the array output. As is shown later, the waveform distortion of the output 1 Wk{Sk(t) + mk(t)} is signal component do(t) = negligibly small . In other words, when C = 1, the output signal component has almost the same waveform as that of the reference signal f(t) = SI(t). The LMS adaptive array shown in Fig . I realizes the weights which min imize the mean-square error ( je(t)12 ). We see that the error is given by
Lr=
e(t) = r(t) -
y(t) tv
2.:
k =1
wk,ik(t) ·
The last term L~= 1 IVk'lk(r ) is the thermal noise . If the weighted signals are added in-phase at the array output , the output signal component do(r) coinsides almost perfectly with the reference signal keeping the weight . norm YWw small value (t denotes complex conjugate transpose) . This means that the thermal noise power :V, W+W in the error e(t) has a lower value . Thus. the meansquare error is minimized by add ing the weighted signals in-phase at the array output when C = I . According to the literature [4], the signal cancellation phenomenon occurs when the desired signal is correlated with one or more interfering signals . When the incident signals are correlated with one another. the desired signal is canceled in adaptive arrays other than the LMS adaptive array shown in Fig . I. Even though the incident signals are not correlated with one another, the desired signal may be canceled (7) . Under some circumstances . the weights do not converge to the Wiener solution . The "non-Wiener" effects cause signal cancellation [7] . However. as stated previously, the parameter G in Fig. is determined in such a way that the deviation of the weight vector from the ensemble average is negligibly small. Namely, the steady-state weight vector is given by the Wiener solution (18). Thus, the signal cancellation phenomenon due to the non-Wiener effects does not occur in the LMS adaptive array discussed In this paper. The LMS adaptive array does not cancel the desired signal even when the incident signals are correlated with one another. This is because the signal cancellation increases the mean-square error in the LMS adapti ve array. It is shown analytically in the Appendix that the weighted signals Wk{sk(t) + ,nk(t)} (k = I -N) are added in-phase at the array output when T = 0 (C = I) and that the signal cancellation phenomenon does not occur in the LMS adaptive array . Moreover, we investigate the distortion contained in do(t) . We define the distortion power Eo as Eo
= m~n(
Ido(t) - a 5 1(t) 12 )/2.
(23)
Fig. 6 shows the output E'NR (EoIN o) as a function of C for several values of 'l' . We see that the output E'NR is less than 0 dB. Namely, Eo is less than the output thermal noise power. Therefore, we may say that although the multipath signal is not suppressed when C = I. the waveform distortion is negligible . IV. REFERENCE SIGNAL GENERATION
Thus far , Tr has been assumed to be a. Namely . we have assumed that the reference signal coincides exactly with s(r) . In this section we discuss the problem of the reference signal generation . In the remainder of this paper, we assume that the pilot signal is biphase modulated by a pseudonoise (PN) sequence with a long period . Then , we express the normalized autocorrelation function p(r) as pU) =
{I
for It I :oS T elsewhere
-ltI IT,
a,
(24)
where T denotes a clock pulse duration. A local PN sequence generator at the receiver modulates the carrier which is recovered from the array output. However, the reference signal in general does not coin cide with the incident signal in time . i.e . . Tr rf 0 and Tr rf r . Thus we must synchronize it to the time of arrival of s(t) or ,n(tl. A. Synchronization Accuracy
From the above assumptions . we examine the effect of Tr on the steady-state performance of the LMS adaptive array. Fig . 7 shows the output DUR versu s t nt: When , iT =: a.5 , sU) and ,nU) are correlated with each other
_ 21!r --
-
-
-
---,
OJ
~
10
. 95
Fig. 6.
.99
C
Output E'NR versus C. N
. 999
= 2.
. 9999
0, = 0°. 0,,,
S,IN , = M ,IN , = 20 dB . 51! r - -OJ -0
Q:
= 30
0
•
T,. = O.
- - - ---,
40 30
::J
020
., , 10 .,a.
o"
0
-I 0 '--~--'--~-'-~-' -I 0
Tr /T
Fig. 7.
Output OUR versus Tr lT. N= 2. a,= 0°.0", = 30°. '!' =0°. S,INi = M, IN, = 20 dB .
374
according to (17) and (24). When TIT = 5, they are independent of each other. When Tr IT = 0 , the reference signal coincides with s( t) . Similarly. when TIT = 0.5 and tnr = 0.5. it coincides with m (t ) . It is seen that satisfactory output OUR is obtained around Tr iT = 0 or TrlT = 0.5 for TIT = 0.5 . On the other hand , the output DUR has a steady and satisfactory value for - 1 < Tr iT < I when TIT = 5. Fig. 8 shows each output power normalized by N;l2 versus Tr lT for TIT = 0.5 . The period for which the output undesired signal power is suppressed less than No is about 0 .2 T around Tr tT = 0 or Tr IT = 0.5 From these results, we say that the synchronization of the reference signal must be extremely accurate in the case where the input multipath components are correlated with one another. B. Reference Signal Pro cessor
Now we propose the configuration of the reference signal processor . The reference signal generation consists of two parts just like the synchronization process in a spread spectrum receiver 15). One is acquisition and the other is tracking. The acquisition is implemented by a sliding correlator [5] which performs the search process and calculates the correlation between the received pilot signal and reference signal. The sliding correlator makes the reference signal coincide with the pilot signal within T. During the initial acquisiuon. the reference -ignal is not correlated with the input signal and all ~) f the weights are driven to O. if the LMS adaptive processor operates . Thus, until the initial acquisition IS achieved . we do not make it operate. Namely. the weights are frozen in fixed = \\\.= O. values, for example. W I = I. \t.: = \\', = . Now we discuss the tracking process which is the second part of the synchronization. The tracking circuit operates in such a way that the reference signal coincides with the transmitted pilot signal as precisely as possible. When the multipath components are correlated with each other, the correlation function between the array output signal and generated reference signal is not a symmetric triangular function. Thus. a delay-lock loop 15] may not be employed . Also, it is difficult to achieve an accurate synchronization bv use of the conventional tau-dither clock-tracking loop [5]. Thus. we must configure the new tracking circuit. 3 0 ....-- - --
'" -10 'I .
-
-
-
Fig . 9 shows the normalized MSE (mean-square error) in the LMS adaptive array versus Tr IT. Here , the MSE is defined as (25)
It is seen that when the reference signal coincides with s( t) or nl (t) (Tr IT = 0 or Tr IT = 0 .5), the MSE has an extremely low minimal value. Then, the MSE may be used for the recognition of the synchronization. After the initial acquisition is achieved by the sliding correlator, we make the LMS adaptive processor and tracking circuit operate. The configuration of the tracking circuit is shown in Fig. 10 . Note that each signal in Fig. 10 has a real value. The yeO (voltage-controlled oscillator) is controlled in such a way that the MSE has a minimal value . This circuit is analogous to the tau-dither clock-tracking loop used in a spread spectrum receiver. Fig . II shows an example of waveforms in the tracking circuit. We set the duration time (T') of the rectangular wave a U) lonzer than the converzence time of the weights. The ~mplitude of a(t) is~ V 'N,I'2 . Then, a (t ) = ::: Y N/ 2 holds. At a leading edge of a U ), the clock phase of the reference signal is shifted back by a fraction (.~ T) of the clock pulse duration. It is shifted forth by the same amount at a trailing edge of a(t). At the output
Fig 9 .
Array
L
5-20
a..
-30
-1
h it )
ri g. 10.
~~-
Fig. 8 .
L_ _---'
1 U--L -~
~:
~:
l 0
T racking c ircuit.
8~ i '--~T- !'~i
--,
. :",- ,
':~O ;'\ l
Adaptive Process or
Local Os ci ll a to r
. '.
.
LMS
Ou t p u t
';; 0 :'. , _~ --:' Il
\ lSE 'N, vervus r- t .\' = 2. fl , = 1)0 . fl" = 30'. 'I' 7.T = 0 .5 S. .v, = ,W,. N, = 20 JB
I"
,
!
nr t ,
i
Il ---_ _
Tr / T
8, = If. em= 30". ' V = 0°. TI T = 0.5. S,' S , = M ,: N, = 20 dB .
Power ratio versu s Tr I T. N
= 2.
ri me
Fig. II .
375
Waveform s in tracking circuit.
= ()o .
from the LPF (low-pass filter) placed behind the squaring device, we may obtain the waveform b(t) which is almost proportional to the MSE « le(t)l2 ». We assume that b(t) = V2 (le(t)1 2 )/vMholds. Then, we have h(t) = ± ( I e(t) 12 ). The LPF placed in front of the vco extracts the DC component v(t) from h(t). Since the yeO is controlled by v(t) in such a way that the MSE has a minimal value, the reference signal coincides with s(t) or m(t) in time. By using the tracking circuit, we may generate a highly accurate reference signal. We assume that the transfer function of the LPF in front of the yeO is K 1/ (1 + sT I ). Since the phase of the yeO-output signal is proportional to the integral of the control signal v(t), we may have
1) The behavior of the LMS adaptive array depends on the correlation coefficient of the incident signals. However, if the reference signal is generated properly, the LMS adaptive array may reduce the multipath fading effectively for any correlation coefficient value. 2) The synchronization in the reference signal generation must be very accurate in the case where the multipath components are correlated with one another. 3) We proposed a processor configuration which generates the reference signal. We showed satisfactory performance of the tracking circuit.
-- =
7=0 (C=1)
d(Tr) dt T
(26)
Kv(t).
Fig. 12 and 13 show the variations of Trt'T and the output OUR, respectively ~ in a case where the tracking circuit operates. It is seen that TrlT reaches the range from -0.005 (- :lTI2T) to 0.005 (~TI2T). This means that the reference signal is synchronized with the time of arrival of s(t). It is also seen that the output OUR takes on satisfactory values when TrlT approaches o. The output DUR ranges from 32.3 dB to 38.6 dB even after convergence. This is because Tr is vibrated.
v.
CONCLUSIONS
We have examined the fading reduction performance of the LMS adaptive array. Moreover ~ we have discussed the problem of the reference signal generation and we have the following results.
APPENDIX.
THEORETICAL ANALYSIS FOR
We assume that 7 = 0 holds. From (10) and (20)~ we obtain C = 1. Namely, m(t) is perfectly coherent with s(t). In this case. the following equations hold. (AI)
(A2)
Then, the combined signal on each antenna element is given by
+
Sk(t)
mk(t) =
b,
5\ (r) e'":',
(k
= I---N)
(£~3)
where b
k
=
e -}(k-1)tP s + Vlvl,1Sj e -J{(k- l)
(A4)
We define an N-dimentional vector as (AS)
05,....-------~
From these results, we have
Rxx = s, B* BT +
l-
::
00
Fig. 12. Tr/T versus time. N = 2, f\
(A6)
= S, B*.
(A7)
Even when the signals &(t) and met) are correlated with each other ~ the steady-state ensemble average of the weight vector is given by the Wiener solution Rxx - 1 Vxr. Then, from (A6) and (A7), we may obtain
am = 30°. 'V = 0°, TIT = 0.5, S,IN, = M;lN, = 20 dB, ~T/T = 0.01. T' = IO/(G'N,), T 1 = lOO/(G·N,), K( = V2iFi;, K = - SV2Gv'N, X 10- 5 .
= 0°.
W
50~------,
"0
I
where I denotes an N x LV identity matrix. Also, assuming that Tr = 0 holds, we obtain Vxr
£D
iV;
40
=
(5 iB* BT + lV,-J)-I 5 iB* Si - - - - B*. SiBT B* + s,
~ 30
o
~ 20
o 10
(A8)'
Furthermore, from (A3) and (A8), each weighted signal is given by 6 Time
x
~~~~
as
Fig. 13. Output OUR versus time. N = 2, = 0°, em = 30°. \{1 = 0°. TIT = 0.5, S;lN, = M,llv, = 20 dB, ~TIT= 0.01, T ' = lO/(G·Nr ) T) = lOO/(G·N,), K) = y'2IN j , K = - 5V2Gv'N, X 10- 5 .
376
Wk {Sk(t)
+
1nk(t)} (k
= 1-- N).
(A9)
We see that the phase of each weighted signal has the same value. Therefore, it may be said that the weighted signals are add~d in-phase at the array output and that the signal cancellatIon phenomenon does not occur.
Adaptive beamfonning for coherent signals and interference.
[5)
REfERENCES (1]
[4]
Ikegami. F., and Yoshida. S. (1980) Analysis of multi path propagation structure in urban mobile radio environments.
[6]
IEEE Transactions 011 Antennas and Propagation, AP-28. 4 (July 1980),531-537.
(2]
Monzingo, R.A., and Miller, T.W. (1980)
[3]
Introduction to Adaptive Arrays. New York: Wiley, 1980. Hansen. P.M .. and Loughlin, J.P. (1981)
Adaptive array for elimination of multipath interference HF.
IEEE Transactions on Antennas and Propagation, AP-29, 6 (Nov. 1981).836-841. Shan, T.J., and Kailath, T. (1985) IEEE Transactions on Acoustics. Speech, and Signal Processing, ASSP-33 , 3 (June 1985), 527-536. Dixon, R.C. (1984) Spread Spectrum Systems (2nd ed.)
New York: Wiley, 1984. Ogawa, Y.. Ohmiya, M., and Itoh, K. (1985) An LMS adaptive array using a pilot signal.
IEEE Transactions on Aerospace and Electronic Systems, AES-2/, 6 (Nov. 1985). 777-782.
(7]
Widrow, B.. Duvall, K.. Gooch. R.P., and Newman, (1982)
w.e.
Signal cancellation phenomena in adaptive antennas: causes and cures.
IEEE TranSaC!lVI1S on Antennas and Propagation, AP-30. 3 (May 1982).469-478.
at
377
Optimum Combining for Indoor Radio Systems with Multiple Users JACK H. WINTERS,
Abstract-This paper studies the use of optimum combining to increase the capacity of narrow-band in-building radio communication systems with multiple users .. We consider systems consisting of a base station with numerous remotes in a Rayleigh fading environment and study tbe problem of more users requiring channels than the number of channels available. A system is described that, with multiple antennas at the base station but only one antenna at each remote, uses optimum combining to suppress interfering signals. We sbow that this system, with M antennas at the base station, can achieve an M-fold increase in the number of users or tolerate M - 1 interferers from other systems. Thus, with optimum combining, radio communications can be used in high-density, multipleuser environments, such as within buildings, even when only limited bandwidth is available.
I.
MEMBER .. IEEE
age) fading. Narrow-band channels are assumed, i.e., the channel bandwidth is assumed to be much less than the coherence bandwidth [9] . These results show that such a system with one antenna at each remote and M antennas at the base station can achieve either an M-fold increase in capacity (over systems without optimum combining) or tolerate M - 1 interferers from other systems. Section II describes the multiple-user system proposed in this paper and calculates the interference tolerance of the system without optimum combining. In Section III, we describe optimum combining and calculate the increase in capacity and interference tolerance with optimum combining in the system. A summary and conclusions are presented in Section IV.
INTRODUCTION
II. A BASIC SYSTEM FOR MULTIPLE USERS Fig. 1 shows the system to be analyzed in this paper for inbuilding radio communication in a multiple-user environment. Multiple remotes communicate with a base station via radio, with the radio channel characterized by multipath (Rayleigh) and shadow fading. Each user uses a single frequency channel, i.e., frequency-division multiple access (FDMA) is used in multiple channel systems. (As discussed in Section Ill, the system can also have multiple users per frequency channel by using a form of space-division multiple access, i.e., through the use of optimum combining since the remotes are physically separated.) As described in detail in Section III, the base station has multiple antennas (antenna diversity), while each remote has only one antenna. As discussed below, dynamic channel assignment and transmit power control are also used. Let us first consider dynamic channel assignment [9] to increase the average number of users in a multiple-user system and transmit power control [9] to reduce adjacent channel interference, and determine the interference tolerance of such a system without optimum combining (i.e., without antenna diversity). For a multiple-user system with multiple channels, dynamic channel assignment [9] is required for efficient channel usage. With this method, before transmission begins, the channels are scanned to find a quiet channel (one with little or no interference) for channel assignment. Furthermore, during transmission, the assigned channel is continuously monitored for interference, and the channel assignment is changed to a quiet channel when the interference becomes too strong. The latter process must occur because the signal environment is constantly changing as the user moves, the environment changes (e.g., doors are closed or opened), or as other users move or begin transmission. Thus, with dynamic channel assignment, interference does not affect the outage performance of the System as long as there are quiet channels available. Another technique to reduce interference among users is power control. Within the coverage region, the signal attenuaPaper approved by the Editor for Radio Communication of the IEEE tion between the transmitter and receiver can vary widely, by Communications Society. Manuscript received December 15, 1986; revised as much as 80 dB or more. Thus, a system with a base station May 18, 1987. This paper was presented at the International Conference on and multiple remotes, all transmitting at the same power level, Communications, Seattle, WA, June 1987. can have received signals differing in power by as much as 80 The author is with AT&T Bell Laboratories, Holmdel, NJ 07733. dB at the base station, which creates an adjacent channel IEEE Log Number 8717084.
W
IRELESS in-buildi.ng. commu~~ation allows. t~e user to be mobile and ehnunates wmng and rewmng when adding or moving phones, terminals, etc., and reconfiguring networks. In-building radio propagation [1]-[6] is hard to predict and continuously changing, however, which makes interference management with multiple users difficult. Furthermore, since bandwidth must be shared by all users within the coverage areas (which could overlap), the capacity of a multiple-user system can be much less than that required in many office buildings. One technique for interference reduction is optimum combining [7]. With optimum combining, the signals received by several antennas are weighted and combined to maximize output signal to interference plus noise ratio (SINR). Thus, interfering signals are suppressed and the desired signal is enhanced. Optimum combining has been shown to substantially reduce interference in mobile radio [7] where multipath fading is present and in systems without fading [8]. For inbuilding radio communication, there is multipath fading as in mobile radio, but the fading rate is much slower. This makes it possible to use optimum combining in combination with other techniques to further reduce interference. In addition, optimum combining can be implemented as an adaptive technique [7], so that detailed a priori knowledge of a building's radio environment is not required and changes in the environment are automatically tracked. In this paper, we describe a digital in-building radio communication system that allows a large number of users in a small area. We consider a system consisting of a base station with numerous remotes and show how optimum combining, in combination with other techniques, can be used to increase the maximum number of users and eliminate interference from other systems. Computer simulation results are shown for a digital system with phase-shift-keyed (PSK) modulation and coherent detection with Rayleigh and shadow (due to block-
Reprinted from IEEE Transactions on Communications, Vol. COM-35, No. 11, pp. 1222-1230, November 1987.
378
I
REMOTE
~
· l1
•
\ / •
REMOTE
0 L-1 EQ.4
2
tg
iii
~
.....
(-1 SIN)
(1)
where S IN is the signal-to-noise ratio . Thus, a 6 .8 dB S IN is required for a 10- 3 BER. Next , consider the effect of a PSK interfering signal with a phase difference (J from the desired signal. The worst case interference occurs when the bit timing for the interferi ng and desired signals are equal. In this case, the received demodulated signal is modified by the factor I + -1 I I S cos (J where II S is the interference to desired signal power ratio, I and therefore, the BER is given by I
BER=- erfc (-1 z(O» 2
(2)
where z(O) = SI N(l
+.JUS cos
0)2.
(3)
The phase difference (J changes with the modulating bits and varies slowly with time for small frequency offsets between the two signals . We therefore assume that (J has a uniform probability distribution. Thus, the BER averaged over (J is given by
BER=:" [" 1f
~ erfc (-1 z(O»
Jo 2
dO
}EQ.(A-11
-10
GAUSSIAN EQ. (A-31 BOUND EQ.(A-4)
BER-10- 3 -15
interference problem. The problem can be reduced by adaprively controlling each remote 's transmit power so that the received power is equal for all signals at the base station. Furthermore, to reduce adjacent channel interference at the remotes, the base station can transmit all signals with equal power. We now consider the effect of interference on a digital communication system using PSK modulation and coherent detection. In general, for voice communications, good voice quality can be ma intained at a bit error rate (BER) less than 10- 2 . In this paper, we conservatively consider a 10- 3 BER. For data communications, we assume coding could be used to reduce the error rate to a more acceptable value . The BER for cc .ierent detection of a PSK signal in white Gaussian noise is given by [10, p . 381]
I
4
5
H
The multiple-user radio system .
BER = - erfc 2
3
-5
--s--
Fig. l.
I
(4)
where z«(J) is given by (3) . Thus, from (4), we can determine the maximum IIS that can be tolerated for a given BER. Fig . 2 shows IIS versus S IN for a 10 - 3 BER . (Multiple interferer results are discussed in the Appendix .) That is, the figure shows the maximum IIS that can be tolerated for a given SIN and a 10 - 3 BER. For the single interferer case , the maximum IIS increases from - 20 to - 5 dB with a 5 dB I Note that we are assuming perfect phase synchronization at the receiver . This is discussed further in Section III-D .
-20 '-----'--'"-- - - - - - ' - - - - - - - - - ' 5 10 20 SIN (dBI
Fig. 2.
The interference to desired signal power ratio versus signal-to-noise ratio for a 10 - l BER.
increase in S IN (from 7 to 12 dB) . Thus, by increasing transmitter power, we can significantly increase the interference tolerance . However, this works only up to a limit since the single antenna system cannot tolerate an interferer stronger than the desired signal no matter how high the S IN. Because the signal propagation in buildings varies substantially with position , it is a very real possibility that interfering signals from nearby systems could be stronger than the desired signal. Thus, even if the capacity of a single antenna system were adequate for an office, interference from nearby systems could easily block channels . thereby reducing capacity or abruptly terminating transmissions. Thus, from both a capacity and interference standpoint, a single antenna system is inadequate for offices . III.
MULTIPLE ANTENNA SYSTEMS
A. Optimum Combining 1) Overview: Interference at the receiver can be reduced
with optimum combining. With this technique, the signals received by several antennas are weighted and combined to maximize output signal to interference plus noise ratio . Thus , diversity (e .g ., space [9 , p. 310], direction [9, p. 311 , II], polarization [9, p. 311 , 12], or field [9, p. 148] [see Section III-OJ) is used to suppress interfering signals and enhance desired signal reception. Optimum combining has been shown to substantially reduce interference in systems both with [7] and without [8] signal fading . Our proposed indoor radio system falls somewhere between the se two cases because, although there is fading , we compensate for it by adjusting the transmit power (see Section II) .
Without fading, optimum combining can null M - 1 interferers with M antennas if the angular separation of the desired and interfering signals is large enough. With fading, as in mobile radio, the angular separation no longer matters because of the multipath. In fact, the receiver can suppress interfering signals and enhance desired signal reception as long as the received desired signal powers and phases differ somewhat from the received interfering signal powers and phases at more than one antenna. Thus , in a system using several antennas for space, direction, polarization, and/or field diversity, the probability of being unable to suppress an interfering signal is very small. Furthermore, since with dynamic channel assignment the channel can be changed if the interference cannot be suppressed , systems with optimum combining can overcome most interference problems. As discussed in [7], optimum combining need only be used
379
at the base station receiver. Adaptive retransmission with time division [9], [13] can be used to improve reception at the remote without requiring multiple remote antennas. With adaptive retransmission, the base station transmits at the same frequency as it receives, using the complex conjugate of the receiving weights. With time division, a single channel is time shared by both directions of transmission. Thus, with optimum combining, during transmission from the remote to the base station, the antenna element weights are adjusted to maximize the signal to interference plus noise ratio at the receiver output. During transmission from the base to the remote, the complex conjugate of the receiving weights are used so that the signals from the base station antennas combine to enhance reception of the signal at the desired remote and to suppress this signal at other remotes. Thus, we can achieve the advantages of optimum combining at both the remote and the base station with multiple antennas at the base station only. 2 As discussed above, a system with optimum combining can suppress interfering signals with a high probability even if their power is equal to or greater than that of the desired signal. Therefore, with optimum combining, several signals can use the same channel simultaneously, thus increasing capacity. Also, signals from other systems can be suppressed even if they are stronger than the desired signal. These topics are discussed in detail in Sections III -B and III -C. 2) Description and Weight Equation: Fig. 3 shows a block diagram of an M antenna element diversity combiner. The signal received by the ith element y;(t) is split with a quadrature hybrid into an in-phase signal X/j(f) and a quadrature signal XQ;(I). These signals are then multiplied by a controllable weight W/j(f) or WQi(f). The weighted signals are then summed to form the array output 50(1). Let the received interference-plus-noise correlation matrix be given by L
R nn =0 21+ ~ u-«: ~ J J
ARRAY OUTPUT
WEIGHT GENERATION
Fig. 3.
signal propagation vector is given by (7)
(5)
J=l
where 0'2 is the noise power, I is the identity matrix, L is the number of interferers, Uj is the jth interfering signal propagation vector, and the superscripts * and T denote conjugate and transpose, respectively. In (5), the correlation is over a period much less than the reciprocal of the fading rate, i.e., Uj and Ud [in (5)-(10)] are assumed to be reasonably constant over the period in which the bit error rate is calculated. Note that we have assumed the fading rate is much less than the bit rate. The equation for the weights that maximize the output SINR is then (from [14]) (see [7])
- R-nn1 u d* w-a
Block diagram of an M antenna element diversity combiner.
(6)
where w is the complex weight vector, a is a constant.! the superscript - 1 denotes the inverse of the matrix, and Ud is the desired signal propagation vector. 3) Preliminary Assumptions and Analysis: In this study, we will assume independent Rayleigh fading (due to multipath) at each antenna with the same shadow or obstruction fading at each antenna for a given signal. Of course, the fading produced by multipath may not be Rayleigh in all locations in all buildings. However, it must be stressed that optimum combining always maximizes the signal to interference plus noise ratio, even if the fading is not Rayleigh. With independent Rayleigh fading at each antenna and transmit power control as discussed in Section II, the desired 2 Note that for adaptive retransmission to be completely effective, all systems within range must use optimum combining and adaptive retransmission with synchronized time division (see Section III-D). 3 Note that Q does not affect the performance of the optimum combiner, and therefore we will not consider its value.
where the Ud; are independent complex Gaussian random variables and P rd (= u~u;) is the total received desired signal power. Note that because of transmit power control, the components of Ud are not independent. Although the phases of the components are independent, the amplitudes (and, therefore, the powers) are dependent. The interfering signal propagation vectors (the Uj's) for the interfering users in a multiple users per channel system have the same characteristics as Ud in (7). For interference from other systems, the characteristics of the Uj'S can vary widely, however. In Section III-C, we study the system performance with fixed total received power for each interferer, i.e., the Uj'S have the same characteristics as Ud in (7), but with a total received power (Prd ) that can be different from the desired signal. 4) SINR and HER: We are interested in achieving the lowest possible BER for the digital system. The optimum combiner, however, maximizes the SINR. With Gaussian interference and noise, maximizing the SINR does indeed minimize the BER. However, in our system, the interference is one or more PSK signals. Therefore, maximizing SINR does not necessarily minimize the BER, although it substantially reduces the BER. Thus, since no simple formula currently exists for determining the weights that minimize the BER [from (3), (4), (A-I), and (A-2) note that the BER is a complicated function of SIN and liS], 4 optimum combining is used. As discussed above, interference has a different effect from noise on the BER. In fact, the effect of interference depends on the noise and vice versa, as shown in Fig. 2. Thus, in our analysis, we first determined the weights that maximize SINR and then determined the liS and SIN at the optimum combiner output. The BER can then be determined from (4) for L = 1 and (A-1) for multiple interferers. For the diversity combiner of Fig. 3, it can be shown that the interference to desired signal power ratio liS and the desired signal-to-noise ratio SIN at the array output are given by L ~ IW t u j*\2 (8) I1S=i=\wt u; 12 4 Note that optimum combining does minimize the upper bound on the BER given in (A-4) and the BER approximation for the interference considered to be the same as Gaussian noise (A-3).
380
and
SIN==
Iw tu;1 2 t
2
awW
'
number of cases (corresponding to randomly positioned remotes) were generated, and the probability was calculated by determining the proportion of cases in which all signals had a HER less than 10- 3 . Thus, for each case, the following procedure was employed. First, signal propagation vectors were generated for each signal by
(9)
respectively, where w is given by (6) and the superscript t denotes complex conjugate transpose. Note that without interference (L = 0), from (5) and (6), a
W=2 a
and therefore, .noting that P rd
u;,
=
U
1) generating independent complex Gaussian random numbers, and 2) calculating Ud from (7).
(10)
Second, with these signals vectors, it was determined whether the desired signal at the output of every optimum combiner had a BER less than 10- 3 by, for each signal,
~u;, from (9),
P rd SIN=-2 . a
1) designating the signal as the desired signal and all others as interfering signals, 2) calculating the optimum weights (6), 3) calculating SIN and liS [(8) and (9)], and 4) determining if SIN and lIS were below the appropriate curve of Fig. 2.
(11 )
interference, optimum combining causes the SIN to be slightly less than that of (11), while the liS is substantially less than that received at each antenna. Assuming an acceptable channel unless the BER e~ceeds 10- 3 , we are interested in the probability that the HER ~s less than 10- 3 (and not interested in the average HER). That IS, we are interested in the probability that a given channel can be used. This is, of course, the probability that SIN and 1/ S are below the curves of Fig. 2. In Sections III-B and III-C, we calculate this probability and from it determine capacity and interference tolerance.
Wit~
B. Multiple Users Per Channel As discussed previously, because optimum combining can suppress signals even when their power is equal to or greater than that of the desired signals, multiple users per channel are possible. Thus, a much higher capacity than that for sin~le antenna systems can be achieved. In this section, this capacity is determined. The proposed system with multiple users per frequency channel has one base station with M (M > 1) antennas and multiple remotes with one antenna each. The base station ~as, for every remote's transmitted signal, an optimum combiner that uses the signals received by each of the M antennas. Thus, the designation of the desired and interfering signals depends only on which optimum combiner is being considered. All the signals are, of course, desired at the receiver. The capacity of multiple users per channel systems ~as calculated by first using Monte Carlo simulation to determine the probability that (for a given received signal-to-noise ratio and number of antennas) a given number of users can use the same frequency channel simultaneously. From this probability, we then calculated the probability that, with a given number of simultaneous users, another user can be added to the channel. Finally, these results were used to determine the capacity of systems with a 0.01 blocking probability (i.e., 99 percent availability was considered in our study). The analysis uses the following notation. Let K be the number of simultaneous users per channel (all with BER < 10- 3) . Also, let r d and r j be the average received signal-tonoise ratio per antenna for the desired and jth interfering signals, respectively. Thus, r d = Prdl Ma 2 , and for the multiple users per channel system, I', = I' d for j = 1, L a~d !-= K - 1. Our results are given as a function of rd' This IS because T d determines the required transmit power of the remotes or, alternatively, with fixed maximum transmit power, the maximum range. Note that a 6.8 dB SIN is required for a 10- 3 BER, and assuming a cubic law of signal strength falloff with distance, a 9 dB increase in required r d with fixed transmit power implies a 50 percent range reduction. The probability P K that K users can simultaneously use the sanle channel was determined by computer simulation. A large
Figs. 4-7 show the probability that K users can use the same channel simultaneously versus the average received desired signal-to-noise ratio per antenna with two-nine antennas. Ten thousand cases per data point were used. To conserve computer time, only up to six simultaneous users were considered. The figures show that one user per channel is always possible if T d is greater than 7-10 log 10 M dB, and that for K > 1, the probability of accommodating K simultaneous users increases with rd. M users per channel with high probability are possible if I' d is increased by up to 20 dB, with higher values of K possible only at a much lower probability. Note that as the number of antennas increases. smaller increases in I' d are required for multiple users at a high probability. For example, with nine antennas, an increase in I' d of only 10 dB is required for a six-fold increase in capacity. For fixed transmit power in a typical building, this represents about a 50 percent reduction in maximum range. We now consider the probability P K/ K - 1 of being able to add the Kth user (with BER < 10- 3 for all K users). That is, P K / K -1 is the probability that one more user can use the same channel given that K - 1 users are using the channel. This probability can be derived from the previous results by noting that the BER for each of the existing K - 1 users can only be increased (not decreased) by adding an additional interferer. Thus the cases where BER < 10- 3 with K users are a subset of the cases where BER < 10- 3 with K - 1 users, and the probability of adding the Kth user is PKIPK - 1• Fig. 8 shows the probability that a Kth user can be added to a channel versus the received desired signal-to-noise ratio per antenna for six receive antennas. This probability is similar to the probability for K simultaneous users (Fig. 6) because the probability of adding the Kth user successfully is usually much less than that for the K - 1 user. Similar results were obtained for two, four, and nine receive antennas. The blocking probability for a single channel with capacity K is defined here as the probability that a K th user cannot be added to the system, 5 i.e., for a one-channel system (N = 1),
B= 1-PK / K -
1e
(12)
Thus, the call blocking probability for a single channel can be calculated directly from the above results. Fig. 9 shows the capacity (maximum number of simultaneous users) versus I' d for a single-channel system with a 0.01 blocking probability. The figure shows that the increase in r d required for each additional user becomes smaller as the
381
5 This is actually the worst case blocking probability for the capacity K system since the blocking probability is substantially less when there are fewer than K - 1 users.
1.0
K-1
1.0
O.B
('{;"
,.,
I : I :
I:
O.B foM=2
0 .6
0 .6
x
I:
-------10
0
1.0
, I
--"----
,
I
I
l« - 3
0 .6
M-4 0 .6
Q.
,,
I I
I
,
0 .2
I
I
I
I
10
r,
20
0 -5
30
K-1
O.B
0 .6 K-5 I
0 .2
I
· I
I
I
I
I
I
I
I
I
"
I
I
I
I
I
I
I
I
I
K-6
r
20
30
N=1
M-9
. B- .01
;
M-6
~ M=6
r ···········
4
....>-
u
..
.........
~
u
K '" 6
r------ M=4 I I
...j ,..
2
I
I
J
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I
I I
.,'/ 0
"
6,--...,----r---r-----,------,
I
0 .4
I
, , I
I
I
, ,,
I
I
I
I
I
Fig. 8. Probability that a Kth user can be added to a channel that already has K - I users with the BER less than 10-J for all K users versus received desired signal-to-noise ratio per antenna for M = 6 receive antennas .
,"
I• I I
x
o
(dBI
Fig. 5. Probability that K users can simultaneously use the same channel with a BER less than 10 - J versus received desired signal-to-noise ratio per antenna for M = 4 receive antennas .
Q.
" I
.'
," ,, ,
• I
04
I
0
30
. I
x, x
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20
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, ,
0 .2
10
0
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I
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,I'
,
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I:
1.0
..,
:
:
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Fig. 7. Probability that K users can simultaneousl y use the same channel with a BER less than IO-J versus received desired signal-to-noise ratio per antenna for M = 9 receive antennas .
,-
.'
,,
I:
0 -5
30
(dBI
,"
I
x
0 .0 -5
1
J
I I
K-4: 'K-6
K=4
20
r,
·,, ·
O.B
1.0
,:
K::~I :
0.2
K-3
Fig. 4. Probability that K users can simultaneously use the same channel with a BER less than 10 - J versus received desired signal-to-noise ratio per antenna for M = 2 receive antennas.
0 .0 -5
J
I: ,: ~K-5
0 .2
0 .0 -5
, ,
,,
J
'. r:
0 .4
M=9
I
I: I:
Q.
0 .4
I
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, ,
I :
': K=2f r : ':J:
x
Q.
,
,:
10
r,
20
o
30
20
30
(dBI
Fig. 6. Probability that K users can simultaneously use the same channel with a BER less than IO- J versus received desired signal-to-noise ratio per antenna for M = 6 receive antennas.
382
Fig. 9. The capacity (maximum number of simultaneous users) versus r d for a single-ehannel system with a 0.01 blocking probability for several values ofM.
Dumber of antennas increases. For example, five users with six antennas require r d = 17 dB, while with nine antennas, only 5 dB is required. Also, the results show that close to M users are possible, but only with a substantial increase in r d as compared to the single-user system. However, multiple users with a small I' d penalty are possible if the capacity is much less than M. \Ve now study the capacity of multiple channel systems (N > 1) where N is the number of channels. Because of dynamic channel assignment, the capacity for a given blocking probability is greater than just N times the capacity of a singlechannel system. In fact, with dynamic channel assignment, there may be many users in one channel and only a few in another. However, to simplify the analysis, we will assume that all channels have K users before any have K + 1 users. This is a worst case model since the capacity is greater if the number of users in each channel is more unevenly distributed. OUf results are, therefore, somewhat pessimistic. Consider an N-channel system with N - (I - 1) channels with K users per channel and I - I channels with K + I users per channel (0 < I ~ N). Then the total number of users is NK + (I - 1), and the blocking probability for the next user is given by
B = (1 - P K + 1/ K ) N -
(I- I)
(1 - P K + 2/ K + I ) 1- I •
(
40
32
>- 24
~
u
u
16
B
0
-5
0
10
r d (dB)
20
30
Fig. 10. The capacity (maximum number of simultaneous users) versus I'd for an eight-channel system with a 0.01 blocking probability for several values of M.
13)
That is, (13) is the call blocking probability for a system with capacity NK + t. Thus, from the previous results in this section and (13), the capacity (maximum number of users) for a given blocking probability can be determined. As an example, consider an eight-channel system. Fig. 10 shows the capacity versus r d with a 0.01 blocking probability for several values of M. This figure shows that an M-fold increase in capacity can be achieved with M antennas if r d is increased by as much as 20 dB (for M = 2). However, the required increase in I'd decreases with more antennas. Furthermore, for less than an M-fold capacity increase, the r d penalty is significantly less. For example, with nine antennas, a fivefold increase in capacity is possible with only a 3 dB increase in rd. Note that as the number of channels increases, for the same blocking probability, the required I' d decreases. The results can be generalized as follows. In systems with Rayleigh fading, an M-fold capacity increase is obtained because M - 1 signals are nulled by each optimum combiner. Thus, the number of signals that can be nulled is the same as that in a nonfading environment (M - 1). We might therefore expect that our results would be valid even if the fading were not Rayleigh and/or there were more than nine antennas. Hr:wever, such results need to be verified in a practical system. C. Interference In this section, we determine the number and power of interfering signals that can be tolerated by the optimum combiner. We first describe how the results were generated and discuss the effect of interference on the optimum combiner. Next, results are shown for the maximum level of interference for a 0.01 blocking probability with L equal power interferers and M antennas. Finally, we determine the maximum number of interferers at any power that can be tolerated. The probability that L interferers of equal average received POwer (rj ) block a channel for the desired signal was detennined by computer simulation. A large number of cases (corresponding to randomly positioned remotes) were generated, and the probability was calculated by determining the proportion of cases in which the single desired signal had a BER greater than 10- 3. The method used was the same as that described in Section 111-B, except that there is only one desired Signal and the power of the interferers is not necessarily equal
to the desired signal power. Results for the maximum interference power for a given blocking probability were obtained by increasing the interference power (in 1 dB steps) until the blocking probability exceeded the given value. The weights are affected by the power of the interference as shown in (5) and (6). If T, < 1 (i.e. ~ the power of the interference is less than that of the noise), the interference has little effect on the weights, and the interference-to-noise ratio at the optimum combiner output is close to that at the input. However, when I', > 1, the weights are adjusted to suppress the interference in the output to a level far below the noise. In this case, increasing the received interference power decreases the interference-to-noise ratio at the optimum combiner output. The optimum combiner can greatly suppress (far below the noise level) interferers and not greatly suppress the desired signal if the received desired signal phases differ somewhat from the received interference signal phases at more than one antenna. With multiple antennas and multipath, it is very unlikely that the phases will be the same. Therefore, the probability of the optimum combiner being unable to null the interference is negligible. However, interference nulling does reduce the output desired signal-to-noise ratio. Thus, call blocking occurs when SIN is reduced to less than 7 dB (i.e., BER > 10- 3) with high received interference power. The optimum combiner can therefore tolerate interference at any power" with high probability if I' d is large enough. These points are illustrated in Fig. 11 for M = 4. This figure shows the maximum rjlr d versus I' d for a blocking probability of 0.01 with eight channels. Thus, the probability of call blocking in one channel is 0.56 [(0.56)8 = 0.01]. Results show that the system can tolerate M - 1 (= 3) interferers at any power if r d is 7 dB greater than that required without interference. With M or more interferers, the optimum combiner can only tolerate interference that has power approximately equal to that of the desired signal even with very high rd' Similar results were obtained for M = 2 and 4 with N = 1 and 8. From the above results, the r d required for the system to tolerate L interferers at any power can be determined. Fig. 12 shows the maximum number of interferers at any power versus r d for a blocking probability of 0.01 with one channel. The figure shows that close to M - I interferers can be tolerated with large increases in rd. 6 In a hardware implementation, the maximum interference power that can be tolerated is usually limited to 40-80 dB.
383
6
20 L'1
M-4 N'8 8'.01
10
'"
4
."
....~ .....
C
-,----- -
20
:;
..J
L·4
.-
~
__ -L'5
:::>
x
4
~
~
x
:;
M·4
~
:::> 4
N• 8 B • .01
M'6
2
-10
M·2 _20L-
o
Fig. I I.
---'
---L
10
--.J
20
30
10
The maximum rJ /r d versus r d for a blocking probability of 0 .01 with eight channels and four antennas .
r d (dBI
20
30
Fig. 13. The maximum number of interferers at any power versus r d for a blocking probability of 0.01 with eight channels and M = six. four, and two antennas .
6,----- - . - - - - - - - - , - - - - - - - , M'6
..J
N'1
8' .0 1
4
,------M·4
2
M'2
o L-_L-_.1.-..L.o
r,
[dBI
...I-_---'_ _...J 30
Fig. 12. The maximum number of interferers at any power versus r d for a blocking probability of 0 .01 with one channel and M = six. four. and two antennas .
Fig . 13 shows the maximum number of interferers at any power versus r d for a blocking probability of 0.01 with eight channels. M - 1 interferers can be tolerated with M = 2, 4, and 6 and increases in r d of only 3, 7, and 8 dB, respectively . Thus, the results in this section show that M - 1 interferers at any power can be tolerated with a several dB increase in r d if M s 6. Since these results are similar to those for a nonfading environment (where up to M - 1 interferers can be nulled), we might again expect that our results would be valid, even if the fading were not Rayleigh and/or there were more than six antennas .
D. Implementation For the system with optimum combining to be practical, the antenna array at the base station must not require a large area. The separation for (nearly) independent fading at each antenna is one-quarter wavelength (A/4, e.g. , 8 em at 900 MHz and 1.5 mat 50 MHz) . Thus, with space diversity [9, p. 310] , an array of M antennas requires a A/4(M - 1) by A/4(M - 1) area. Furthermore, direction [9, p . 311, uj, polarization [9, p. 311, 12J, or field diversity [9, p. 148J can also be used. With these diversity schemes, antennas can be added without increasing the physical size of the antennas array . For example, with polarization diversity in addition to space diversity, the number of antennas can be tripled (three orthogonally polarized antennas for each space diversity antenna) without any change in the area of the array. Thus,
with a mixture of diversity techniques, a large number of antennas can be placed in a relatively small area. Optimum combining can be implemented for in-building systems in the same way as in mobile radio [7] . The optimum combiner can be implemented with an LMS [15], [16] adaptive array . Signals can then be distinguished at the base station by different pseudonoise codes, with these codes added to the biphase PSK signal with an orthogonal biphase PSK signal (see [17]) . The pseudonoise codes that are used to distinguish signals are also useful for carrier recovery . The received signal can be mixed with the code to generate a narrow-band signal for carrier recovery . Because of the processing gain with the code, the narrow-band signal will have a high signal to interference plus noise ratio. even when /IS at the receiver output is high . Therefore, the receiver can track the signal phase with little phase jitter even when /IS at the receiver output is close to 1. A major difference between in-building systems and mobile radio is the fading rate . In mobile radio, the fading rate is about 70 Hz . Thus, the weights must adapt in a few mill iseconds. In buildings, however, the fading rate is much less . For example. a 1.5 mls velocity (i.e., walking with the remote) produces a 4 .5 Hz fading rate at 900 MHz and a 0.25 Hz fading rate at 50 MHz. Thus, the weights can be adapted much more slowly, making implementation of the LMS algorithm on a chip much easier. Furthermore, because the fading rate is less, the dynamic range of the LMS adaptive array is greater. That is, the receiver can operate with higher interference to desired signal power ratios . Using the analysis of [7], we can show that the maximum interference to desired signal power ratio is on the order of 30 dB for a 4.5 Hz fading rate as compared to 20 dB for mobile radio. If greater dynamic range is required, other (more complicated) techniques [8] may be used because rapid adaptation is not required. As noted in Section Ilf-Al ), for adaptive retransmission to be completely effective (i.e., same BER at the remote as at the base station) , two requirements are placed on the systems. First, all transmissions must be synchronized. That is, all remotes must transmit at the same time , and all base stations must transmit at the same time . With one base station and multiple remotes, synchronization is not a problem. However, with multiple base stations, there should be synchronization between systems within the same building. A second requirement is that all base stations use optimum combining with adaptive retransmission . If another system did not use this technique, it could interfere with the base-to-remote transmis-
384
than integration. Unfortunately, the series has convergence problems (on a digital computer) for most of the cases of interest in this paper. Thus, (A -1) was used to calculate the BER, but only for L -s 5. Fig. 2 shows the results. Note that for large SIN with L = 5, there appears to be some error in the curve. (For L = 5, the error could not be determined because of the extensive computer time required.) However, this error does not affect our results for the reasons discussed below. We also considered two other BER equations. First, for large L, the interference can be considered to be the same as Gaussian noise [18], and therefore, the BER is given by
sio ns 7 of other systems on a channel. However, the system without optimum combining could suffer interference on both transmission paths. Therefore, in high-density multiple-user environments, systems could not operate without optimum combining, and would be required to use optimum combining with adaptive retransmission. In this paper., we have studied only the steady-state performance of the optimum combiner. In an actual system, the base station receiver must track both the desired and interfering signals. Although the dynamics of in-building radio communications are slow, the movement of the remotes will affect the performance of the LMS adaptive array (or any other implementation of the optimum combiner). Thus, the transient performance of the system should also be studied. Finally, in this paper, we have studied the performance of the base station receiver only. A brief analysis (not presented in this paper) shows that the BER at the remote should be similar to that at the base station (for adaptive retransmission with time division). Computer simulation is needed, however, to verify that when the BER is less than 10- 3 at the base station, it is also less than 10- 3 at the remote. IV.
1 BER=-2 erfc (
)
(A-3)
Results using this approximation are given in Fig. 2. Second, an upper bound on the BER with interference for any number of interferers is given by [20]
BER~exp [_
SUMMARY AND CONCLUSIONS
In this paper, we have studied multiple-user in-building radio communication systems. We described a multiple-user system and showed that optimum combining can be used to increase the capacity and interference tolerance of the system. Computer simulation results showed that with optimum combining, a system with one antenna at each remote and M antennas at the base station can achieve either an M-fold increase in capacity or tolerate M - 1 interferers. Finally, we discussed implementation of the system and showed that the system was practical for the office environment. ApPENDIX
Extending the results of Section II, we can see that with L interferers, the BER is
where
+···+-JIL/Scosf)L)2
1
(S/N)-l+//S
(A-2)
and Iii S is the interference to desired signal power ratio of the ith interferer. Note that the total interference to signal power ratio liS is ~f=l L/S, There are two problems with (A-I) and (A-2), however. First, the BER depends not only on the total interference to signal power ratio, but on the individual interference powers as well. However, it was concluded (although not proved) in [18] and [19] that for fixed total interference power, the highest BER is achieved with equal power interferers, i.e., Ii/S = (I/L)IIS for i = 1, L. Therefore, we considered equal power interferers as a worst case and generated an approximate lower bound for maximum liS versus SIN for a 10- 3 BER. A second problem is that for numerical evaluation of (A -1), computer time grows exponentially with L, and therefore, calculations are only practical for small values of L. Another formula for the BER is given in [18], which uses a series rather
1
(S/N)-l+//S
]
(A-4)
Results using this upper bound are also shown in Fig. 2. Note that this bound is not very tight for small liS; from this bound, the SIN is 8.4 dB at a 10- 3 BER (without interference, liS = 0), while the actual SIN required [from (1)] is 1.6 dB less. Fig. 2 shows that the maximum II S varies significantly with the BER equation used. (Equations (A-I) and (A-2) with equal power interferers were used for the results presented in Figs. 4-13.) However, our results for the optimum combining system (with M antennas and L interferers) for L < M in Figs. 4-13 and our conclusions do not depend on the BER equation used. This is because, for L < M, the number of degrees of freedom in the adaptive array using optimum combining is greater than or equal to the number of interferers, and therefore, the array can usually greatly suppress the interferers without affecting the desired signal. Therefore, the liS at the array output is small, and, if the SIN is large enough, the BER is less than 10- 3 • Thus, the array usually operates in the small liS region where the required SI N is about the same for all the BER equations (except for the upper bound (A-4) where the required SIN is 1.6 dB higher). We verified that our results for L -s M in Figs. 4-13 were not significantly changed by the liS curve used, except that the SI N was 1.6 dB higher for the liS curve from (A-4). For L ~ M, the number of degrees of freedom in the array is less than the number of interferers, and therefore, the array cannot greatly suppress all the interferers in most cases. Thus, the variation in maximum 1/S at high SIN has a dramatic effect on the results. As noted above, the results in this paper are based on (A-I) with equal power interferers, and thus, our results should be conservative for L ~ M. However, our conclusions (an M-fold increase in capacity and suppression of M - 1 interferers) are based on the L < M case, and therefore, do not depend on which BER equation is used.
7 It would not interfere with remote-to-base transmissions of systems with optimum combining, however, as optimum combining suppresses any mterference.
385
REFERENCES
[1]
K. Tsujimura and M. Kuwabara, "Cordless telephone system and its propagation characteristics," IEEE Trans. Vehic. Techno/., vol. VT26, pp. 367-371, Nov. 1977. [2] K. Yamada, S. Naka, A. Nishiyama, and T. Miyo, "2 GHz-band cordless telephone system," in Proc. 29th IEEE Vehic. Techno/. Conf., Arlington Heights, IL, Mar. 1979, pp. 159-163. [3] S. E. Alexander, "Radio propagation within buildings at 900 MHz," Electron. Lett., pp. 913-914, Oct. 14, 1982. [4] P. S. Wells, "The attenuation of UHF radio signals by houses," IEEE Trans. Vehic. Technol., vol. VT-26, pp. 358-362, Nov. 1977. [5] A. A. M. Saleh and R. A. Valenzuela, "A statistical model for indoor multipath propagation," in Proc. Int. Conf. Commun., 1986, pp. [6]
27.2.1-27.2.5. D. M. J. Devasirvatham, "The delay spread measurements of
[7]
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
wideband radio signals within a building," Electron. Lett., vol. 20, pp. 950-951, Nov. 8, 1984. J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE J. Select. Areas Commun., vol. SAC-2, pp. 528-539, July 1984 (also IEEE Trans. Vehic. TechnoI. , vol. VT33, pp. 144-155, Aug. 1984). R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. W. C. Jakes, Jr., Microwave Mobile Communications. New York: Wiley, 1974. H. Taub and D. L. Schilling, Principles of Communication Systems. New York: McGraw-Hill, 1971. K. H. Awadalla, "Direction diversity in mobile communications," IEEE Trans. Vehic. Technol., vol. VT-30, pp. 121-123, Aug. 1981. R. T. Compton, Jr., "On the performance of a polarization sensitive adaptive array," IEEE Trans. Antennas Propagat., vol. AP-29, pp. 718-725, Sept. 1981. P. S. Henry and B. S. Glance, "A new approach to high-capacity digital mobile radio," Bell Syst. Tech. J., vol. 60, pp. 1891-1904, Oct. 1981. C. A. Baird, Jr. and C. L. Zahm, "Performance criteria for narrowband array processing." in Proc. Con! Decision Contr., Miami Beach, FL, Dec. 1971, pp. 564-565. B. Widrow, P. E. Nantey, L. J. Griffiths, and B. B. Goode. "Adaptive antenna systems," Proc. IEEE, vol. 55, pp. 2143-2159, Dec. 1967. B. Widrow, J. McCool, and M. BaH, "The complex LMS algorithm." Proc. IEEE, vol. 63. pp. 719-720, Apr. 1975. J. H. Winters, "Increased data rates for communication systems with
adaptive antennas, " in Proc. Int. Coni. Commun., Philadelphia, PA, June 1982, pp. 4F.3.1-4F.3.5. [18] A. S. Rosenbaum, "Binary PSK error probabilities with multiple cochannel interferences," IEEE Trans. Commun. Techno/., vol. COM-18, pp. 241-253, June 1970. [19] V. K. Prabhu, "Error rate consideration for coherent phase-shift-keyed systems with co-channel interference," Bell Syst. Tech. J., vol. 48, pp. 743-767, Mar. 1969. [20] G. J. Foschini and J. SaIz, "Digital communications over fading radio channels," Bell Syst. Tech. J., vol. 62, pp. 429-456, Feb. 1983.
386
The Performance Enhancement of Multibeam Adaptive Base-Station Antennas for Cellular Land Mobile Radio Systems SIMON C. SWALES, MARK A. BEACH, DAVID J. EDWARDS, JOSEPH P. McGEEHAN, MEMBER, IEEE
Abstract- The problem of meeting the proliferating demands for mobile telephony within the confinements of the limited radio spectrum allocated to these services is addressed. A multiple beam adaptive basestation antenna is proposed as a major system component in an attempt to solve this problem. This novel approach is demonstrated here by employing an antenna array capable of resolving the angular distribution of the mobile users as seen at the base-station site, and then using this information to direct beams toward either lone mobiles, or groupings of mobiles, for both transmit and receive modes of operation. The energy associated with each mobile is thus confined within the addressed volume, greatly reducing the amount of co-channel interference experienced from and by neighboring co-channel cells. In order to ascertain the benefits of such an antenna, a theoretical approach is adopted which models the conventional and proposed antenna systems in a typical mobile radio environment. For a given performance criterion, this indicates that a significant increase in the spectral efficiency, or capacity, of the network is obtainable with the proposed adaptive base-station antenna.
T
I.
INTRODUCTION
HE FREQUENCY SPECTRUM is. and always will be. a finite and scarce resource, thus there is a fundamental limit on the number of radio channels that can be made available to mobile telephony. Hence, it is essential that cellular land mobile radio (LMR) networks utilize the radio spectrum allocated to this facility efficiently, so that a service can be offered to as large a subscriber community as possible. Indeed, a major consideration of the second generation cellular discussions in both the US and Europe has focused on this point. However, present and proposed future generation cellular communication networks which employ either omnidirectional, or broad sector-beam, base-station antennas, will be beset with the problem of severe spectral congestion as the subscriber community continues to expand. A measure often used to assess the efficiency of spectrum utilization is the number of voice channels per megahertz of available bandwidth per square kilometer [11. This defines the amount of traffic that can be carried and is directly related to the ultimate capacity of the network. Hence, as traffic demands increase, the spectral efficiency of the network must also increase if the quality and availability of service is not to be degraded. At present this is overcome in areas with a Manuscript received April 18. 1989. This work was supported by UK
SERe.
The authors are with the Centre for Communications Research, Faculty of Engineering. University of Bristol, Bristol, 858 lTR. UK. IEEE Log Number 9034227.
AND
high traffic density by employing a technique known as cell splitting. However, the continuing growth in traffic demands has meant that cell sizes have had to be reduced to a practical minimum in many city centers in order to maintain the quality of service. As well as increasing the infrastructure costs, the number of subscribers able to access these systems simultaneously is still well below the long-term service forecasts due to the reduced trunking efficiency of the network. This places great emphasis on maximizing the spectral efficiency, or ultimate capacity, of future generation systems. and thereby fulfilling the earlier promises of performance. There have already been significant developments in terms of spectral efficient modulation schemes, e.g., the proposed US narrow-band digital linear system [21, [3) and the second generation Pan-European cellular network [4}. Also. in the area of antenna technology. the use of fixed coverage directional antennas has been considered [51. In particular. the use of fixed phased array antennas. with carefully controlled amplitude tapers and sidclobe levels for the enhanced UK T:\CS network (ETACS) [6]. are currently under evaluation. However. the application of adaptive antenna arrays in civil land mobile radio systems has hitherto received little attention. in spite of the significant advances made in this field for both military and satellite communications. In this paper a multiple beam adaptive base-station antenna is proposed to complement other solutions, such as spectrum efficient modulation, currently being developed to meet the proliferating demands for enhanced capacity in cellular networks. The feasibility of such a scheme is demonstrated. and a comparison made with existing conventional antennas in a realistic mobile radio environment. Geometrical and statistical propagation models are used and a unique insight is given into the benefits of utilizing adaptive base-station antennas in a cellular radio system. Finally, the concept of such a scheme is discussed and the integration of adaptive antenna array technology into a mobile communications environment considered. II.
ADAPTIVE ANTENNA ARRAYS
An adaptive antenna array may be defined as one that modifies its radiation pattern, frequency response, or other parameters, by means of internal feedback control while the antenna system is operating. The basic operation is usually described in terms of a receiving system steering a null, that is, a reduction in sensitivity in a certain angular position. toward a
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 39, No.1, pp. 56-67, February 1990.
387
source of interference. The first practical implementation of electronically steering a null in the direction of an unwanted signal, a jammer, was the Howells-Applebaum sidelobe canceller for radar. This work started in the late 1950's, and a fully developed system for suppressing five jammers was reported in open literature in 1976 by Applebaum [7]. At about the same time Widrow [8] independently developed an approach for controlling an adaptive array using a recursive least squares minimization technique , now known as the LMS algorithm. Following the pioneering work of Howells, Applebaum, and Widrow, there has been a considerable amount of research activity in the field of adaptive antenna arrays , particularly for reducing the jamming vulnerability of military communication systems. However, to date, there has been little attention to the application of such techniques in the area of civil land mobile radio. Adaptive antenna arrays cannot simply be integrated into any arbitrary communication system, since a control process has to be implemented which exploits some property of either the wanted. or interfering, signals. In general. adaptive antennas adjust their directional beam patterns so as to maximize the signal-to-noise ratio at the output of the receiver. Applications have included the development of receiving systems for acquiring desired signals in the presence of strong jamming. a technique known as power inversion [91 . Systems have also been developed for the reception of frequency hopping signals [10), [II). TDMA satellite channels [12) and spread spectrum signals [131. Of particular interest for cellular schemes is the development of adaptive antenna arrays and signal processing techniques for the reception of multiple wanted signals [14] .
A. Fundamentals of Operation The adaptive array consists of a number of antenna elements. not necessarily identical. coupled together via some form of amplitude control and phase shifting network to form a single output. The amplitude and phase control can be regarded as a set of complex weights, as illustrated in Fig. I. If the effects of receiver noise and mutual coupling are ignored . the operation of an N element uniformly spaced linear array can be explained as follows. Consider a wavefront generated by a narrow-band source of wavelength A arriving at an N element array from a direction fh off the array boresight. Now taking the first element in the array as the phase reference and letting d equal the array spacing. the relative phase shift of the received signal at the nth element can be expressed as ,T. 'l'nk
=
27rd(n - I)
A
.
Sin
(J
k·
(1)
Assuming constant envelope modulation of the source at Ok. the signal at the output of each of the antenna elements can be expressed as
(2) and the total array output in direction fh as
YkCt) =
L wnej("'I +'~n.) N
n ~1
(3)
Far Field Signa l Source Array Outpu t
Antenna Arra y
Fig. I.
An adaptive antenna array..
where wn represents the value of the complex weight applied to the output of the nth element. Thus by suitable choice of weights, the array will accept a wanted signal from direction I and steer nulls toward interference sources located at fh, for k # I. Likewise. the weighting network can be optimized to steer beams (a radiation pattern maxima of finite width) in a specific direction. or directions. It can be shown [15] that an N element array has N - 1 degrees of freedom giving up to N - I independent pattern nulls. If the weights are controlled by a feedback loop which is designed to maximize the signalto-noise ratio at the array output. the system can be regarded as an adaptive spacial filter. The antenna elements can be arranged in various geometries . with uniform line. circular and planar arrays being very common. The circular array geometry is of particular interest here since beams can be steered through 360 0 • thus giving complete coverage from a central base-station. The elements are typically sited A/2 apart. where A is the wavelength of the received signal. Spacing of greater than A/2 improves the spatial resolution of the array. however. the formation of grating lobes (secondary maxima) can also result. These are generally regarded as undesirable.
o
B. Adaptive Antenna Arrays for Cellular Base-Stations Multiple beam adaptive antenna arrays have been considered by Davies et al. [16] for enhancing the number of simultaneous users accessing future generation cellular networks. It is suggested that each mobile is tracked in azimuth by a narrow beam for both mobile-to-base and base-to-mobile transmissions. as shown in Fig. 2. The directive nature of the beams ensures that in a given system the mean interference power experienced by anyone user, due to other active mobiles, would be much less than that experienced using conventional wide coverage base-station antennas. It has already been stressed that high capacity cellular networks are designed to be interference limited, so the adaptive antenna would considerably increase the potential user capacity . This increase in system capacity of the new base-station antenna architecture was evaluated [17] by considering the spatial filtering properties of an antenna array. The results show that this type of base-station antenna could increase the spectral efficiency of the network by a factor of 30 or more. These results were obtained for a hypothetical fast frequency hopping
388
cellular systems. However, some well-established trends are becoming apparent in the quest toward higher spectrally efficient modulation schemes [I] for the systems of the year 2000 and beyond . It is thus vital during the initial stages of research to develop antenna architectures which are, in essence, modulation scheme independent, so that a figure of merit can be obtained for the rnultibeam base-station antenna .
Base-station
III . REDUCTION OF CO-CHANNEL INTERFERENCE USING ADAPTIVE ANTENNAS
Fig. 2.
Tracking of mobiles with multiple beams .
code division multiple access cellular network (18) . assuming uniform user distribution and complete frequency reuse for the omnidirectional antenna case. i.e .. adjacent cells are cochannel cells. Complete frequency reuse is then assumed for each of the beams formed by the adaptive array . i.e.. adjacent beams are co-channel beams . Further. it was shown that a similar enhancement of efficiency can be obtained for either an idealized multibeam antenna . or a realizable 128 element circular array (19] . It was recognized in the analysis. but not fully assessed. that this approach would greatly increase the level of co-channel interference. It was. therefore. suggested that this problem could be overcome using dynamic channel allocation to eliminate the so called common zones. This again introduces additional hand-offs , reducing the trunking efficiency and available capacity of the network. as the mobile circumnavigates the cell. The only study previous to the work discussed above considering the use of an adaptive antenna array in land mobile radio was by Marcus and Das (201 in 1983. The analysis assumed that the base-station. or repeater. sites could be placed closer together if an antenna array formed 20 dB nulls toward co-channel sites. This effectively reduces the amount of co-channel interference at the output of the base-station as explained in Section II . It was suggested that in this system the beam steering information could be derived from the squelch tone injection which is presently used in the US FM land mobile radio. In contrast with the null steering technique considered by Marcus and Das, here the ability of the adaptive array to steer radiation pattern maxima toward the mobiles is considered . In the limit it can be envisaged that individual beams will be formed towards each mobile as illustrated in Fig . 2. It has already been mentioned that adaptive antenna technology cannot be simply integrated into an arbitrary communication system. and at present no one particular modulation scheme , or access technique, has been selected for the third generation of
In this paper the integration of an idealized adaptive array into an existing cellular network is considered . In order to ascertain the benefits of this class of antenna system compared with that of conventional omnidirectional base-station antenna systems, the following network topology has been assumed . 1) A cellular network consisting of hexagonal cells. with channel reuse every C cells (C is the cluster size) . 2) The base-station transmitters arc centrally located within each hexagonal cell . 3) There is a uniform distribution of users per cell. 4) There is a blocking probability of B in all cells. 5) The omnidirectional base-station antenna has an ideal beam pattern. giving a uniform circular coverage . 6) The adaptive base-station antenna can generate any number. m, of ideal beams. with a bearnwidth of 2;r !m. and a gain equal to the omni-antcnna . 7) Each adaptive beam will only carry the channels that are assigned to the mobiles within its coverage area . 8) Any mobile (or group of mobiles) can be tracked by the adaptive base-station antenna. 9) The necessary base-station hardware is available to enable bcarnforrning and tracking. 10) The same modulation scheme can be used with each antenna system. The blocking probability of B in assumption -i) is the fraction of attempted calls that cannot be allocated a channel. If there are "a" Erlangs of traffic intensity offered. the actual traffic carried is equal to at 1 - B) Erlangs. The Erlang is a measure of traffic intensity. and measures the quantity of traffic on a channel or group of channels per unit time . This gives an outgoing channel usage efficiency (or loading factor) (211 of (-i) 1) = a ( l - B) /N where N is the total number of channels allocated per cell. Assumptions 6). 7). and 8) imply the deployment of a somewhat hypothetical adaptive antenna system. This approach can be justified since a uniform user population has been assumed for both categories of antenna system. It is recognized that the dynamic. nonuniform. user distribution will have a significant effect on the results presented here. This will be considered in a subsequent more rigorous study. Also, in the analysis which follows only the base-to-mobile link has been studied. however, it can be shown that the analysis is also valid for the mobile-to-base link. Two different categories of co-channel interference models are used as the basis for the study presented here. The first is the geometrical model adopted by Lee l51. followed by a more rigorous statistical analysis (211-(23) .
389
o
= Base station
Wanted cell
Interfering cell Region of interference
Region of no interterenc/ dw
0-------------- -------------------------------Wanted
Fig. 3.
IV.
Base-station
D
...
_ Interfering Base-station
Worst-case position
Two co-channel cells.
GEOMETRICAL PROPAGATION MODEL
This approach considers the relative geometry of the transmitter and receiver locations, and takes into account the propagation path loss associated with the mobile radio channel. A. One Co-Channel Cell
Consider one co-channel cell which forms part of a cellular network as shown in Fig. 3. By definition both the cells have the same channel allocation . and a reuse distance of D separating the base-station transmitters. The co-channel reuse ratio is defined as
Q
==
D iR.
(5)
This ratio has also been termed the co-channel interference reduction factor [5j since the larger it is (i.e ... the further apart the cells) the less the co-channel interference for J given modulation scheme. The level of acceptable co-channel interference governs the value of this parameter and the overall spectral efficiency of the network. The area mean signal level experienced at the mobile is assumed to be inversely proportional to the distance from the base-station raised to a power -y. With the advent of smaller cells . the propagation path loss is close to the free-space value [24 J.. however. it is envisaged that the proposed base-station will initially operate in larger cells. Therefore . as a starting point for the comparison to follow, the commonly used approximation that the received signal power is inversely proportional to the fourth power of range will be used [251. Hence . the area mean signal level (in volts) received from the wanted base-station at a mobile a distance d w from the transmitter is
(6) Similarly, the area mean signal level from the interfering basestation transmitter at a distance d, is (7)
assuming in each case identical radiated transmitter powers and signal propagation constants, as denoted by the constant k.
Co-channel interference will occur when the ratio of the received wanted signal envelope, s.,; to the interfering signal envelope, s., is less than some protection ratio . Pr, i.e.:
(8)
Fig. 4.
Contour defining interference regions.
The protection ratio is defined by the modulation scheme employed [1]. Considering only the propagation path loss, the received signal envelopes are equal to the area mean signal levels, hence: mw
d~
d~ S Pro w
(9)
So.. for a given protection ratio . a locus given by
d, [d ; == V!jJ;
( 10)
can be drawn. This defines a region where no interference will occur.. and where it will always occur, as illustrated in Fig. 4. For the worst-case position, which is in a direct line between the transmitters as shown, the co-channel reuse ratio is
Q ==D!R == 1 +di/dw == 1 + JJJ;.
(11)
For a given protection ratio and modulation scheme, this defines the minimum spacing between co-channel cells in order to avoid interference, and the maximum spectral efficiency obtainable. In this discussion it is assumed that the same modulation scheme is employed for both antenna systems under evaluation. This implies that the protection ratio and reuse distances are identical in both cases. Therefore, there would appear to be no apparent benefit from employing adaptive antenna technology at the base-station site. However, the occurrence of co-channel interference is a statistical phenomena. Hence, when comparing omni- and adaptive antennas, it is necessary to introduce the concept of the probability of co-channe/ interference occurring, i.e., Pis.; :::; p,Sj). This is often called the outage probability, which is the probability of failing to obtain satisfactory reception at the mobile in the presence of interference. If the cells are considered to be identical, i.e., have equal blocking probabilities, then on average, there will be N1J active channels in each cell (71 is as defined in (4». So, in the case of the omnidirectional antenna, given that the wanted mobile is already allocated a channel, the probability of that channel being active in an interfering cell is the required outage
390
0 ··· ··········· 0
J/ ~O
t~
i "Z i -o
,: c0 : u ,
11>
\U1
.0
o o L a.
.0
Wan ted ce ll
g
QC9
Q)
0'
o .....
'\~
:J
o
Fig . 6.
lnt er ter inq cel ls
~
Hexago nal cellular layout showin g tiers of inte rferers .
a ( I - B ) active channels (or user s). Th is is only rea lly valid if a ( I - B ) > m and that the use rs are uniformly dist ributed
5
10
15
20
25
30
Nu m b er of beams Fig . 5.
35
40
45
within the cell. If this were not the case, and the numbe r of beams formed was less than m, the outage probability would be reduced even further since the wanted mobile will not be covered by a co-channel beam all the time . This situati on will not be pursued further since thi s analysis can be regarded as worst -case situation.
50
m
Outage as a function of the number of beams .
probability. He nce. when the wanted mobile is in the region of co-c hannel interference the outage probability is given by P (s w
< Pr Si) = -
numbe r of active channels . total number at channels
NT] N
=
B . Six Co -Channel Cells
1) .
( 12)
Now consider the case of the adaptive antenna as previously desc ribed. with m beams per base-station providing coverage of the whole cell, and wit h N T/ 1m channels per beam. give n a unifo rm distribution of user s. T he same regions of co-channel interfe rence can be defi ned . however , whe n the wanted mobile is within the region where co-cha nnel interference may occ ur, the outage probability is reduced . The wanted mob ile is always covered by at least one beam from the co-channel cell. hence. the outage probability is equal to the probability that one of the channels in the aligned beam is the corresponding act ive co-channel ' and is given by
Pts; < P rs ; ) -
number channels per beam total number of channels
= ----,----.,.-'---NT/ 1m N
=
m
The previous ap proach can now be simply extended to JS sess the effect of six co-channel interfer ers. i.e .. the first tier of co-channel cells in a con ventional cellular scheme JS shown in Fig . 6 . It is co nsidered that further tiers of interfere rs will not significantly af fect the results except when reu se distances become small. Equat ion ( 9 ) can now be rewritten for this mor e reali stic representation o f the cell ular network s". s/
I The " active co-cha nnel " is the chan nel that has also been allocated to the wanted mobile .
In /
=
d,:c II
Ld,-c
'5:
o,
( 14)
1= 1
where the total mean signal le ve l from the interfering cell s. m, , is the sum of the mean level from each active cell. Thu s in a fully loaded system. the number of active users is six ( i.e . , n = 6). If all the d , are assumed to be equal and the wanted mobile is at the edge of a cell bou nda ry. as for the case described in Lee [5). then the co-c hannel reuse factor ca n be expressed as
(13)
where the omnicase is given by m = I . These results are pre sented graphically in Fig . 5, and show the strong influence of the number of beams, m , on the outage probability. The influence of the loading factor, 1] , is as expected, i.e ., the less the loading , the fewer the numbe r of active channels , and hence, a red uced chance of co-c hannel interfe rence . This assumes that there are still m bea ms formed even thoug h the re are only
I n ".
Q = [6(5 w I5dl" "'I .
( IS)
Subjective tests showed that over a mobile radio channel
swl5/ 2:: 18 dB (i .e. , PR = 18 dB) gave good speec h trans-
mission for a 25-kHz FM channel operation . A value of Q can now be calculated to defi ne the minimum cluster size, C. Using sim ple geometry it would be possible to evaluate the actual swls/ in the worst-case locations. From this, a conto ur may be d raw n defi ning region s with and without interference . For bo th classes of antenna sys tems the outage probab ility is still ze ro within the contour (i .e .. when s.. .. ls, > p,) , but outside,
391
in the region of interference: P(Sw 5: Pr S / ) = ( : )
6
(16)
where s I is the total co-channel interference and the omnicase is given by m == 1. Since it is assumed that all m beams per cell are formed, there are six beams aligned onto the wanted mobile at any time. The outage probability within the region of interference is then found by considering the probability that the active co-channel is in each of these beams. C. Analysis of Results The use of adaptive multiple beam-forming base-stations would, based upon the analysis presented so far, appear to give an improvement in performance with regard to the reduction of the probability of co-channel interference. The improvement depends on the degree of adaptivity used, i.e .. the number of ideal beams formed. However . the above approach is over simplistic and gives a rather optimistic view of the situation. Firstly, the beams are assumed to be ideal. giving an equal gain over the whole beamwidth. In practice this would not be the case. Also. a hypothetical situation could be envisaged where. if In is large enough to satisfy a given outage critcrion.? it would appear that the ultimate reuse distance (D R == 2) is possible for any modulation scheme. Hence. adjacent cells arc co-channel cells . the radius of which is decided by the rcqui red coverage area of the base-station site. In spite of this though. the analysis has been useful in introducing some of the important factors that affect the performance of a mobile radio network which exploits frequency reuse as a means of increasing spectral efficiency.
v . STATISTICAL
PROPAGATION ~loDEL
In the previous analysis only the path loss associated with the mobile radio environment was considered when calculating the level of co-channel interference. This was useful in demonstrating the principle benetits to be offered by adaptive antennas . although it is an over simplified approach and totally unrealistic of many land mobile radio environments. It was shown that a single contour defining regions of operation where co-channel interference would occur can be drawn., however. it is known that the signal levels fluctuate rapidly generating small isolated pockets of interference in an operational system. In some adverse environments these areas may be quite close to the base-station antenna. There is seldom a line of sight path between the base-station and the mobile . and hence, radio communication is obtained by means of diffraction and reflection of the transmitted energy. This produces a complicated signal pattern causing the field strength to vary greatly throughout the cell, and the received signal at the moving mobile to fluctuate very rapidly. This is generally attributed to the superposition of two different classifications of signal fading phenomenon: fast fading (or just fading) due to the multipath nature of the received signal. and slow fading (shadowing), the slower variations of 2
the received signal due to variations in the local terrain. In areas experiencing this type of signal variation, the area mean signal level is essentially constant. In order to model these propagation effects, the are included in a statistical fashion, the fading and shadowing described above being represented by Rayleigh and log-normal type distributions, respectively.
A. One Co-Channel Cell Various studies [26]-[28] have been undertaken to analyze co-channel interference originating from a single co-channel interfering cell in an attempt to characterize the mobile radio environment. In particular the rigorous analysis presented by French [22] has been adopted here. The fast fading is the rapid fluctuation of the signal level 5 about the local mean s (s == (s)), and is usually described by a Rayleigh type probability density function (pdf) . i.e.:
7rS
[ _ ;rs~ ] .
Pts Is) == ---:-;- exp 25-
( 17)
4s~
Shadowing of the radio signal due to the terrain, i.e., by buildings and hills, causes the local mean level s to fluctuate about the area mean. It has been generally accepted that this variation is log -normally distributed about the area mean md, where md == (Sci) . the mean of s in decibels. (Note, a subscript I".d' indicates that a signal is in decibels.) The area mean level is approximately proportional to the inverse of the distance from the base-station raised to the power 1, as described in Section IV-A. Hence. the log-normal shadowing pdf is given by ( 18) The standard deviation. (J . describes the degree of shadowing. This parameter typically varies from 6 to 12 dB in urban areas ~ the larger value being associated with very built up inner city areas. The combined pdf can now be expressed as
If there arc ten beams (m = 10) a loe;{J outage criterion could be satisfied
in a fully loaded system (13).
392
P(S)
= i:P(S/S).P(Sd)dSdo
By substituting s == IcY d / 2o (from Sd == 20 the combined pdf becomes P(s)
==
J
7r 18(J2
J
x
.
-x
S -dj'O
lOS ..
exp
. exp
(19)
10gIO s)
into (17),
2] [7rS -d no 4 x lOS
- (Sd -
.,
2(J-
[
1-
m d)2]
dSd.
(20)
1) Outage Probability With Fading and Shadowing: The outage probability with fading and shadowing is derived in French [22] and the resulting integral is
Pis; 5: PrSi) where, Zd = mdw internal variable.
-
1
=..Ji mdi -
j'oo
-00
PR,
exp ( _u 2 ) 1 + 1O(zr2aul/ 1O du (J
=
O"w
=
(Ji,
and
(21) u
is an
In many situations it is possible to greatly reduce the fading, e.g. , antenna diversity at the mobile, and a similar result to that above can be derived [22) for shadowing only. Note that the result in (21) is for the case of the omnidirectional basestation antenna . 2) Outage Probability with an Adaptive Antenna: The outage probability for an adaptive antenna can be simply ex pressed as
rts; :::; p-s] , m)
=
a = 6 dB n' = 0 .7
-,
10
-0
o o ....
et»; :::; P,Si)
-0
Q..
probability of an active co-channel) (
Q)
Q'l
...,o
in the aligned beam
= Pis; :::; p ,s,) ' ( ; )
:J
o (22)
i.e ., the probability that the ratio of the wanted signal to the interfering signal is less than some protect ion ratio (21) and the probability that the aligned beam actually contains the ac tive channel (13) . Again, the outage probability is redu ced by a factor m. This is illustrated graphically in Fig . 7. The loading factor is fixed at 70lJD (,., = 0.7) and the fading and shadowing case is considered for a = 6 dB . This represents a typical urban environment. The se results have been obtained by solving (21) and (22) numerically with In = l. 2. ..J.. 8. 16, and 32. Note that In = I gives the ornnicase . The outage probability varie s as expected with :::<1 and . for the omnicase is consistent with French . Note . however . that for a given Zd the outage is reduced by a factor of 111 when an adaptive antenna is considered . 3) Calculation of the Reuse Distance: When the fading and shadowing characteristics of the mobile radio channel are considered, it can be shown that no definite boundary ex ists between regions of interference , and regions of no interference. Co-channel interference can even occur dose to the wanted transmitter if. for example. the wanted signal s fades and the interfering signal peaks as illustrated in Fig . 8. Now. since co-channel inte rference is a statistical phenomena. it can be described by contours of outage probability. Using the definition for Zd and (9) yields
10
Fig . 7 .
30
20
50
~o
60
Outage probabiliry tor on e co- cha nne l cell.
fa)
' n t er fer en c e
a .n ter f eri nq
Bo se -s tat ion
:~ = JZii; = J IO(~d 'P" J ,20 .
(23)
So for a given Zd, protection ratio p R , and outage probabil ity. contours can be drawn as shown in Fig . 9 . If an adaptive antenna is used , the value of each contour is reduced by a factor of m, hence , for a given outage cr iterion, the service area is increased. This can be represented graphically by substituting (23) into (11) and expressing the co-channel reuse ratio as (24)
From this formula the outage probability against the reuse ratio for a given protection ratio can be obtained in a manner similar to that of French . 4) Calculation of the Cluster Size: In a cellular network with a hexagonal layout, the cluster size is related to the reuse
393
(b)
Fig . 8.
Contour defirung regions of inter fe rence . (a) With no shadowing or fadin g . (b ) W ith shadowing and fading .
distance by
C = Q:'/3.
(25 i
Note that only certain value s of C are possible in a hexagonal cellular network [25), i.e. , C = (3, 4, 7 , 9, 12. 13. 16. 19. 21, · · .). Using (24) this can be expressed as
C = 1/3[1
+V
IOIZd +h )/ :'O):' .
(26 )
Again the outage probability can now be evaluated for variou s cluster sizes for a given protection ratio .
Out age Proba bility contours~
· · O.~l:.~.
-.
.....
.
Adaptive service area (m = 10)
.
a Interf ering Base-station
....... ..
." Fig .
l} .
Om ni- service ar e a
Outage probability contours and service areas for a 10'; chance of interference .
essary to consider interference originating from multiple cochannel cells. Several different studies [21], [29]-[31] have pursued this goal. but of particular interest is the work by Muammar and Gupta [23] . This has been adopted here since the analysis follows directly from the previous discussion . However, a few alterations have been necessary in order that a more meaningful comparison could be presented. Fig. 6 shows the cellular layout of a mobile radio network for an arbitrary cell cluster size of C. It is recognized that there are many tiers of co-channel interferers present, but only the first. i.e. , cells at a distance D from the wanted base-station. it considered here. This assumption was shown to be valid in similar studies [29], [31]. The wanted mobile in the central cell receives a signal envelope Sw from the wanted base-station . It also receives unwanted signals from the cochannel cells s., i = 1, 2, . .. .n . where n is the number of active interfering co-channel cells (the maximum number being six in this case) . The total co-channel interference is thus given by n
5) Calculation of Spectral Efficiency: To gain a more
meaningful interpretation when comparing different system architectures in a cellular network . the spectral efficiencies [II of the various schemes are usually considered . This gives an unbiased measure of spectrum utilization. and is usually expressed as the number of channcls/Ml-lz of bandwidth /krrr' . I. e.. I efficiency, E = B<~ = B,C.4 B,lCAl
i ~1
When the wanted signal does not exceed this value by the protection ratio. co-channel interference will occur. In order to calculate the total probability of co-channel interference (or simply the outage probability). it is necessary to consider the probability of there.being co-channel interference and n interfering co-channel cells . Using conditional probability theory this can be expressed as P; (co-channel interference) Ii tn active co-chunnelsj)
where
B,
Be
C
A
=P(S,. SPrs, !n) ·P(n).
total available bandwidth . channel spacing in megahertz. number of cells per duster. cell area (krrr') .
C nmm C a<1apl '
(30)
Pirn is the PDF of nand Pts ; S p.s, I n ) is the conditional outage probability (the probability of co-channel interference given that there are n active interfering cells) . Hence. the total
To enable a simple comparison to be made between ornniand adaptive antenna systems. it is necessary to assume that an identical modulation scheme will be employed in both cases . Thus E 'X l /C. and the relative spectral efficiency can be expressed as
E a<1apl E mnOi
(29)
S/=LSi.
outage probability is given by
(28)
Equation (22) was solved numerically and then the cluster size C. given by (26). was calculated for a fading and shadowing (6 dB variation) environment with a loading factor of 0.7. An outage criterion of I II.. is used and. although this value is quite low, it serves to give some idea of the advantages that can be obtained by using this new class of base-station antenna. Two values of protection ratio. 8 and 20 dB, are considered in order to cover a variety of modulation schemes [I]. Then. using (28). the relationship between the relative spectral efficiencies was calculated for m = I. 2. 4, 8, 16. and 32, and is shown in Fig. II .
P(Sw S Prs,)
In order to present a more realistic comparison between omnidirectional and adaptive base-station antennas, it is nee-
394
PrS' ! n ) . Pen)
(31)
n
since all possible values of n must be taken into account. Here only the first tier is considered. so the maximum number for n is six . The pdf of the signal envelope s is as given by (20) and from here the conditional probability of co-channel interference for multiple interferers. when considering both fading and shadowing. can be derived. This result is simply quoted here without proof as details [23] can be found elsewhere.
where 20 loglo K(X, u) =
B . Six Co-Channel Cells
= L ri»; S
Zd
+C· In(4/ 1rn:2) 1
~
2
+oX - aNYu - 4C (a"Nx - aNY)
(32)
and where a~x and a~y are defined by Muammar and Gupta [23]. The variable Zd is as defined previously, and u and X are internal variables. This integral can be solved using various numerical techniques and the results are presented later. 1) PDF of n, Ptn): P(n) is the probability that the number of active interfering co-channel cells is n and so if the channels are assumed independent and identically distributed, this has the form of a binomial pdf:
a = 6 dB ~
= (:) p"(l
-1
..Q
o
-2 10
L
o
where p is the probability of finding one interfering cochannel active. Using the loading factor 11, as defined before. the probability p that a single co-channel cell has an active co-channel, given that the wanted mobile has been assigned that channel already, is number of active channels
p
== total number of channels
a(
1 - B) == 11· N
-+J
::J
~
-3
10
) -== )
L..-
p(SH,'
S Pr S ' n i
(34)
(35)
- YJ}
{) n -~
Hence, the origination probability is given by
60
(dB)
Outage probability \\ uh vix co-channel (db.
Y1 \
.
-
number of active channels in beam total number of channels 11 m
so
40
n
(probability that the interfering co-channel is in the) beam pointing at the wanted mobile
N
z,
30
The overall outage probability can now be expressed as
This can now be calculated for a given outgoing channel usage efficiency over the range of ;'r/. Alternatively, as befor;, the outage probability can be considered against the cochannel reuse ratio Q, or the cluster size C. 2) Integration of Adaptive Antennas: With an ornnidirectional antenna the probability of an active interfering cochannel cell was given by n . the outgoing channel usage efficiency or cell loading factor. Since it is assumed that at any one time all In beams per cell are formed. there will always be six beams aligned onto the wanted mobile. Hence. for the adapti ve antenna:
a(l-B)jm
20
10
(36)
p
.7
-4
n
===
~
v
~
10
Hence. giving the total outage probability as
Pi s ; < Pr S ' -
~
~
~
1S'
~
~
o
Fig. 10.
11"(1 -11)6-11,
~
~
The origination probability [21 J. or the probability that n co-channel interfering cells are using the same channel as the wanted mobile. can then be expressed as Pen) = ( : )
~
~
0... OJ CJl
~
~ ~
o
(33)
_p)6-"
0.7
10
..Q
Pen)
=
(37)
,;;)
(')-fl
(39)
where the omnidirectional case is given for /11 -=- I. As before. when only one co-channel cell was considered. a comparison can now be made between the two base-station technologies. Fig. 10 shows the variation of the total outage probability against ~d. The case with both fading and shadowing is considered here. hence. the results are obtained by num;rically solving (32) and applying (39) for m == 1.2.4.8. 16. and 32. From (26) and (28) the cluster size and the relative spectral efficiency can now be calculated for a given outage criterion (1 (,{), and is shown in Fig. 11. An outgoing channe: usage efficiency of 0.7 and a log-normal shadowing standarc deviation of 6"dB has been assumed. Protection ratios of 8 and 20 dB have also been considered. 3) Analysis of Results: It can be seen from Fig. 11 that for a given outage criterion and modulation scheme. the introduction of an adaptive array capable of forming eight tracking beams, into an existing network. would produce at least a threefold increase in efficiency. This can be equated to be three times the number of channels per megahertz per square kilometer, or simply as three times as many users in each cell. This result has been obtained by considering the co-channel interference originating from single and multiple co-channel cells. The propagation model employed here considers both the fading and shadowing characteristics of the mobile channel. Although the application of power control of the individual tracking beams has not been considered to date, this may
395
12
l'.
"" -
20 dB, 6 co-ch annel cell • •
o
P R
8 dB, 6 co-channel cell••
•
P
a
III -
R
::::I
20 dB. one co-channel cell. 8 dB, one co-channel cell.
>.
u
C CIl
u
~l
6
0404-
W
o Bose-station
o
L
+'
U
CIl
a.
Mob iles
4
V'l CIl
·3o
2
CIl
a:::
o
o
5
10
15
20
25
30
35
Number of Beams m Fig . II .
Relat ive spectral efficiency as a tuncu on of the number of beams formed .
prove to be essential for future generation networks . It has been shown [211. [301 that base-station power control can substantially reduce co-channel interference for omnidirectional antenna systems. It can thus be envisaged that power con trol of the multiple beam adaptive base-station antenna would greatly reduce energy "overspill " into neighboring cells . The combination of the two adaptive techniques will be considered in a later study. VI.
F.ig. 12.
Optimal beam forming .
Rx/T x Antenna Array
THE "SMART" BASE-STATION AI'TE:-;NA
Adaptive antennas operate by exploiting some property of the signal environment present at the array aperture [71. [81. and it is due to this ability that they are often aptly referred to as "smart" arrays (32). In the previous theoretical analysis it was assumed that the base-station antenna could track any mobile . or group of mobiles. within its coverage area . Therefore. on reception. the array must be capable of resolving the angular distribution of the users as they appear at the basestation site. Armed with this knowledge, the base-station is then in a position to form an optimal set of beams, confining the energy directed at a given mobile within a finite volume . This concept can be further illustrated by considering the sequence of events illustrated in Fig . 12. The scenario depicted is realistic of many operational systems where there are lone mobiles , or groups of mobiles, dispersed throughout the cell. Using the spatial distribution of the users acquired by the arrayon reception , the antenna system can dynamically assign single narrow beams to illuminate the lone mobiles, and broad beams to the numerous groupings along major highways. It can be seen that by constraining the energy transmitted toward the mobiles, there are directions in which little or no signal is radiated. It is this phenomenon which gives rise to
396
N Signa l
ports
SOURCE ESTIMATION PROCESSOR
!
BEAMFORMER
Mobile Location Data F+g. 13.
The " smart " antenna .
the reduction in the probability of co-channel interference occurring in neighboring cells, and thereby increasing the spectral efficiency (or capacity) of the network as illustrated in the previous section. The realization of such an adaptive base-station antenna requires an architecture capable of locating and tracking the mobiles, and a beam-forming network thus capable of producing the appropriate multiple independent beams . The former requirement can be broadly classified as that of a direction finding, or a spatial estimation problem. These two tasks are illustrated in Fig. 13 as a source estimation or direction find-
in spectral efficiency. In addition to the other advantages that can be gained, as outlined in the previous section, the infrastructure costs incurred by this base-station antenna must also be considered. The rnajority of these costs are associated with the acquisition of the base-station site, construction of various buildings and antenna masts. When compared with existing schemes, such as cell splitting, the overall cost is less, since fewer base-station sites arc required for an equivalent user capacity. Also, unlike the techniques of cell splitting and cell sectorization, the multiple beam adaptive array would not irnpair the trunking efficiency of the network.
ing (DF) processor and a beamformer. In recent publications [33], [34] this concept was further extended to consider the implementation of such a base-station antenna. Also the results from some initial computer simulations were presented and these demonstrated the ability of the antenna array to resolve multiple mobile users in the signal fading conditions typical of the LMR scenario. Many of the popular superresolution DF algorithms used in radar were evaluated, and some beam-forming techniques which could generate the optimal beam set using the knowledge of the mobile distribution were discussed. Finally, a proposal was put forward for a fully adaptive base-station antenna test rig to demonstrate the principles of operation and show how such an antenna could be incorporated into the existing cellular network. The results of this work will form the basis for a future paper. Intelligent antenna systems have also been considered for numerous other applications. Sandler and Kokar [35] have described the use of an adaptive antenna in conjunction with an artificial intelligence system as an antijamming antenna for radar. Here the antenna has a wealth of stored data which is used to ·'teach the system about the various scenarios it will encounter. In this way it can adapt readily to every new situation as it is presented and "learn from its mistakes. It is intended to exploit the synergy that exists between this application and that of the proposed cellular mobile radio basestation. Initially, the knowledge of the mobile locations within each cell could be utilized to provide an elegant hand-off mechanism as the mobiles cross cell boundaries. Also. combined with a knowledge of the local terrain and shadowing characteristics, it should be possible to extend this technique and provide a cellular network with dynamic cell boundaries. This would thus allow the optimal usage of the available system capacity.
ACKNOWLEDGMENT
We are extremely grateful to our colleagues at the Bristol University Centre for Communications Research for their stimulating intellectual support. REFERENCES
H
H
VII.
DlSCLSSION
The full potential of adaptive antenna technology in the future generation of ubiquitous portable communication networks is yet to be realized. The goal is to be able to provide universal pocket sized communications by the year 2000. This implies that the system must make very efficient use of the radio spectrum if it is to be made available to a large consumer base: thus making the portable equipment relatively cheap. Also, it is highly desirable that the portable communicator has a long duty cycle between battery recharging implying power efficient modulation. The role of adaptive antennas has already been discussed in terms of the former requirement. however, it must be emphasized that spectrum efficient modulation is still a vital parameter in the design of these systems. The potential enhancement of power efficiency obtainable using spatial filtering has not been fully assessed to date, and the merits of this technique in a rural service area are of particular interest. The study presented here has demonstrated the feasibility of an adaptive base-station antenna for cellular communications networks. A comparison made between the conventional and proposed schemes has shown that a marked improvement in spectral efficiency and capacity can be obtained, e.g., an idealized eight beam antenna could provide a threefold increase
[1] H. Hammuda, 1. P. McGeehan and A. Bateman, "Spectral efficiency of cellular land mobile radio systems," in Proc. 38th IEEE Veh. Techno!' Con]., Philadelphia, PA, pp. 616-622, June 15-17, 1988. [2] J. A. Tarallo and G. I. Zysman, "Modulation techniques for digital cellular systems," in Proc. IEEE Vehicular Technology Con]., pp. 245-248, June 15-17, 1988. [3] J. Uddenfeldt, K. Raith and B. Hedberg, "Digital technologies in cellular radio," in Proc. IEEE Vehicular Technology Conf., pp. 516-519, June 15-17, 1988. [4] F. Lindell, 1. Swerup and J. Uddenfeldt, "Digital cellular radio for the future," The Ericsson Rev., no. 3, 1987. [5] W. C. Y. Lee, "Elements of cellular mobile radio systems," IEEE Trans. Veh. Technol., vol. VT-35, pp. 48-56, May 1986. [6] P. C. Carlier, "Antennas for cellular phones," Commun. Int., pp. 43-46, Dec. 1987. [7] S. P. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 585-598, Sept. 1976. [8) B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode, "Adaptive antenna systems," Proc. IEEE, vol. 55, pp. 2143-2159, Dec. 1967. [9] R. T. Compton, "The power inversion adaptive array: Concept and performance," IEEE Trans. Aerospace Electron. Syst., vol. AES15, pp. 803-814, 1979. [10) L. Acar and R. T. Compton, "The performance of LMS adaptive array with frequency hopped signals," IEEE Trans. Aerospace Electron. Syst., vol. AES-21, pp. 360-370, May 1985. [11] K. Bakhru and D. J. Torrieri, "The maximum algorithm for adaptive arrays and frequency-hopping communication," IEEE Trans. Antennas Propagat., vol. AP-32, pp. 919-928, Sept. 1984. [12] R. T. Compton, R. 1. Huff, W. G. Swarner, and A. A. Ksienski, "Adaptive arrays for communication systems: An overview of research at The Ohio State University," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 599-607, 1976. [13] R. T. Compton, "An adaptive antenna in a spread spectrum communication system," Proc. IEEE, vol. 66, pp. 289-298, Mar. 1978. [14] M. A. Beach, A. J. Copping, D. J. Edwards, and K. W. Yates, "An adaptive antenna for multiple signal sources," in Proc. lEE Fifth Int. Con! on Antennas and Propagation, University of York, 1987. [15J J. E. Hudson, Adaptive Array Principles, lEE Electromagnetic Wave Series No. 11. Stevenage, U.K.: Peter Peregrinus, 1981. [16] Telecom Australia, "Base-station antennas for future cellular radio systems," Rev. Activities 1985/1986, pp. 41-43,1985-1986. [17] W. S. Davies, R. 1. Lang, and E. Vinnal, "The challenge of advanced base station antennas for future cellular mobile radio systems," presented at IEEE Int. Workshop on Digital Mobile Radio, Melbourne, Australia, Mar. 10, 1987. [18] G. R. Cooper and R. W. Nettleton, "A spread spectrum technique for high capacity mobile communication," IEEE Trans. Veh. Techn., vol. VT-27, pp. 264-275, 1978. [19) D. H. Archer, "Lens-fed multiple beam arrays," Microwave J., pp. 171-195, Sept. 1984. [20) M. J. Marcus and S. Das, "The potential use of adaptive antennas to increase land mobile frequency reuse," presented at lEE 2nd Int. Con! on Radio Spectrum Conservation Techniques, CP224 , Birmingham, UK, Sept. 6-8, 1983.
397
[21] K. Daikoku and H. Ohdate, "Optimal channel reuse in cellular land mobile radio systems," IEEE Trans. Veh. Technol., vol. VT-32, pp. 217-224, Aug. 1983. [22] R. C. French, "The effects of fading and shadowing on channel reuse in mobile radio," IEEE Trans. Veh. Techno/., vol. VT-28, pp. 171-181, Aug. 1979. [23] R. Muammar and S. Gupta, "Co-channel interference in high capacity mobile radio systems," IEEE Trans. Commun., vol. COM-3D, pp. 1973-1978, Aug. 1982. [24] J. H. Whitteker, "Measurements of path loss at 910 MHz for proposed microcell urban mobile systems," IEEE Trans. Veh. Techno/., vol. VT-37, pp. 125-129, Aug. 1989. [25] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [26] W. Gosling. "Protection ratio and economy of spectrum use in land mobile radio," Proc. Inst. Elect. Eng., vol. 127, pt. F, pp. 174-178, June 1980. [27] M. Rata, K. Kinoshita, and K. Hirade, "Radio link design of cellular land mobile communication systems," IEEE Trans. Veh. Techno!., vol. VT-31, pp. 25-31, Feb. 1982. [28] A. G. Williamson, "Coverage, co-channel interference and outage probability calculations for mobile radio systems," in Proc. IREE 20th Int. Electron. Convention and Exhibition, Melbourne, pp. 224-227, Sept./Oct. 1985.
[29] D. C. Cox, "Co-channel interference considerations in frequency reuse small-coverage-area radio systems," IEEE Trans. Commun., vol. COM-3D, pp. 135-142, Jan. 1982. [30] V. Palestini and V. Zingarelli, "Outage probability computation and cellular coverage for mobile radio," in Proc. 37th IEEE Veh. Technolo Conf, Tampa Bay, FL, pp. 468-476, 1987. [31] Y. S. Yeh and S. C. Schwartz, "Outage probability in mobile telephony due to multiple log-normal interferers," IEEE Trans. Commun., vol. COM-32, pp. 380-388, Apr. 1984. [32] W. F. Gabriel, "Adaptive arrays-An introduction," Proc. IEEE, vol. 64, pp. 239-272, Feb. 1976. [33J S. C. Swales, M. A. Beach, and D. J. Edwards, "Multi-beam adaptive base-station antennas for cellular land mobile radio systems," in Proc. 39th IEEE Veh. Techno/. Conf, San Francisco, CA, pp. 341-348, Apr. 29-May 3, 1989. [34] S. C. Swales, M. A. Beach, D. J. Edwards, and 1. P. McGeehan, "A multi-beam adaptive base-station antenna for cellular land mobile radio systems," in Proc. 1989 IEEE Workshop on Mobile and Cordless Telephone Communications, University of London, UK, pp. 55-61, Sept. 25-26,1989. [35] S. S. Sandler and M. Kokar, "Intelligent antennas," in Proc. URSI Int. Symp. on Electromagnetic Theory, Budapest, Hungary, pp. 159161, 1986.
398
Combination of an Adaptive Array Antenna and a Canceller of Interference for Direct-Sequence Spread-Spectrum Multiple-Access System RYUJI KOHNO,
MITSUTOSHI HATORI,
MEf\.1BF.R, lEEE~
MEMBER~ IEEE. A~D
HIDEKI IMAL
Abstract-In the realization of code division multiple access based on a spread-spectrum communication system, i.e., spread spectrum multiple access (SS~IA). reduction of cochannel interference is an i01portant problem. This paper proposes an adaptive array antenna system including a canceller of cochannel interference. which can improve performance by a cornbinatiun of temporal and spatial filtering. While ~ he adaptive array suppresses interference sources w ith arrival angles different from that of the desired user. the adaptive digital filter-canceller rejects those whose arr-ival angles are the same a~ that of the desired user. The proposed system can achieve stable acquisition and low error rate of demodulated data even in a heuvy interference channel where a conventional array antenna s~ stem cannot achieve sat isfactury acquisition.
T
1.
INTRODUCT1()~
HE demand for spread-spectrum (SS) communication techniques in commercial applications has been increasing recently ..~ SS communication technique has advantages such as robustness against narrow -band interference as well as noise and realizability in the form of code division multiple access. i.e .. spread spectrum multiple JCCCSS (SSMA) [1]. However. if enough inherent pro.essing gain of a SS system cannot be obtained (e.g .. due ;'0 restricted transmission bandwidth in a channel). cochannel interference (that is. an interfering SS signal from an undesired user due to cross-correlation aITIOng pscudonoise (PN) sequences assigned to different users in SSMA) cannot be suppressed completely Then it will be difficult to achieve initial acquisition and phase tracking and to increase the number of sirnultaneously accessing users. This is known as a near-far problem. In order to reduce cochanncl interference. we have proposed an adaptive canceller of cochanne! interference as well as the design of a set of PN sequences with good correlation characteristics 12}. 13]. However. since the adaptive canceller demodulates and resprcads undesired SS signals from every user in order to generate a replica Manuscript received February 17, I9R4: rev iscd November I), I 4X4, R, Kohno JnJ H. lmai arc with the DIV\"IOn of Electrical and Computer Eng inccnng , Yokohama Nauonal Univcrsuy. l)() Tok rwad.u. Hudo gaya-ku. Yokohama 240, Japan. M, Haiori is with the Department of Electrical Engineering. Uruvcrsuy of Tokyo, Hongo , Bunk yo-ku , Tokyo II J. Japan S, Pasupathy is with the Department of Electrical Engmccnng , Uruvcrsity of Toronto. Toronto. Ont. M5S I A~. Canada IEEE Log Number 9()~~ 792,
SENIOR MEMBER. IEEE,
SUBBARA Y AN PASUPATHY,
SENIOR ~1EMBER, IEEE
of them by using an adaptive digital filter and then subtracts the replica from received signals, the amount of hardware will be considerably large corresponding to an increase in simultaneously accessing users. On the other hand. an adaptive array antenna is useful in suppressing interfering signals because it can adaptively control directivity of the antenna even if the desired SS signal's arrival angle is unknown [4]. [5]. However. if there is a high-level interfering signal from an undesired user with the same arrival angle as that of a desired user, an array antenna cannot suppress it. In this paper. in order to minimize these problems inherent in an adaptive canceller and an array antenna. an adaptive array including a canceller of cochannel interference for SSMi\ is proposed and the performance is investigated. The proposed system can suppress interfering SS signals. i .e .. cochannel interference. with arrival angles different from that of a desired user by using a null steering array antenna and eliminate by means of a canceller the residual interference and cochannel interference having an arrival angle the same as that of the desired SS signal. The canceller consists of a SS demodulator and modulator for interfering 5S signals and an adaptive digital filter which can generate replica of interfering SS signals even in a time-varying channel. The weights of array elements are adaptively updated by using more reliable reference signals which are obtained by the canceller. In particular. even if there is considerable high-level interference from an arrival angle same as that of a desired SS signal. the proposed canceller can eliminate it more effectively. Therefore. the proposed system can achieve stable demodulation and improve the error rate of decoded data even in a heavy interference channel where a conventional array antenna s~stetn cannot achieve acquisition. This is an approach based on a combination of spatial and temporal filtering for rejection of interference in a SSMA system (61. In Section II. a conventional adaptive array antenna for a direct-sequence (DS) SSMA system is briefly explained. In Section III. the structure and controlling algorithm of the proposed system are described in detail. Finally, computer simulation results that evaluate the proposed system in comparison to a conventional system are presented in Section IV.
Reprinted from IEEE Journal on Selected Areas in Communications, Vol. 8, No.4, pp 675-682, May 1990.
399
Desired
[ 1T denotes transpose and
Interference
8
= d, (k) c, (k)
r; (k) ~--
LN
+ Y(kl
OUTPUT
,
y(k) =
Fig. 1. Adaptiv e array antenna with N clements.
II. AN
ADAPTIVE ARRAY ANTENNA FOR A
oW
L: h;r,(k) exp ;=0
{-j(21rL,,/'A) sin
el} +.
= 11-==\ 2:
= [hro, n.;
Re {xn{k)* W,,(k)} [Re
+ IIII
{x
11 (
k)} Re { W"( k) }
{x" (k ) } 1m { W" ( k ) }] .
(2)
R"" .... l.j and lrn tf ....- lJ arc a real and an irnauinarv ... ., parts of a complex value ::.. respectively. * denotes the complex conjugate. Let the Oth user's 55 signal r., ( k) be the desired one . i.e ... the reference signal. The error signal e( k) is defined as the difference between y( k) and an in-phase or rea1 pa rt 0 f ro ( k I. i. e .. e ( k) = y ( k) - Re {ro ( l:) ~ . which comprises of the interference and noise components. If mean square error E{ e t k: ):! 1 is employed as a criterion for optimizing complex weights W" ( k) (11 = I. 2. . . . . N ).. such optimum weights W" which minimize E[e(k)2] can be obtained by solving the well-known Wiener-Hoff equation. given by 1 :lIl
{
-
N,,(k)
(3 )
( 1)
where let a real and an imaginary pans of W" be W'R" Rc { W" } and Will = Irn { W" }. respectively. then r W = [WR I' W/I' ~vR 2. W· f 2 • • • • ~VR,\' ~t'f.\'] •
where h, and T/ (k) which are vectors of a channel impulse response for the i th user and a S5 signal from the i th user r. ( k), respectively, are given by h,
N
L: n=1 .v
DS-SSMA
Fig. 1 shows the structure of a conventional adaptive arrav"'" antenna with tv elements. Assume for simplicity that each element is omnidirectional and that all mutual impedances are zero. Though a quadrature hybrid is used for each element in order to split the received signal into quadrature components in a practical system. it is omitted in Fig. 1. Throuzhout the paper.. analytic signal notation with ~omplex weights is used [5]. .r, (k) is the sample of received signal in the 11th element at instant t = kT where T is a sampling interval as well as the duration of a chip of PN sequence: k is an integer. XII (k) consists of the desired SS signal. interfering 55 signals from other users . and a noise component. The COITIplex signal .r, ( k) is expressed by x,,(k) =
{j (wekT + o,)} .
M + 1 is the number of simultaneously accessing users, and d, (k) and c, (k) are the binary ( + I or - I ) datum and PN sequence signals of the i th user at instant k'T, respectively. L II is the distance between the first and the nth elements where L I = O. 0; and 0; are the arrival angle and the phase shift of the SS signal from the i th user, respectively. We is the carrier frequency. A is the free-space wavelength . and Nil ( k) is the noise component in the nth element. Each XII (k) is multiplied by a complex weight WIl (k) in the 11th element and then summed to produce the array output y (k) • as
-4--=--~
e(kl
Z lkl
exp
..... him] T
'I
R\\(O) and R,,.(O) are the 2N x 2N covariance matrix
and the 2N x 1 cross-correlation matrix .
.\ R I ( "
x,
dk
- i )
XR
dk
- j)
XR ,
(k - i ) Re {
i ) X /I
("
-
I)
\ R I (k -
x" (k - i )
X/I
(k - I)
'\/1
(k -
X /I
(k r i )
X H ~ (" -
XH~
x,~(k - i) XR1(J... - j)
-
; )
x J2( k - i) x,,{k - I)
i ) .\ H ~ ("
(k - i )
\H
i)
.\H I ( "
~ (J... -.I)
'\/1 ( "
I ).\ H ~
-
(J... -
i)
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.\H ~ (J...
-
-
-
i ) X f ~ (" I ) XI
-
I )
~ (J... - I )
i ).\ f ~ (J... - j )
X/~(" - i) \,~(" -.I)
"0 (k ) }
x" i k - i ) Rc { "u (k ) } R" (i )
=E
x H ~ (k - i ) Rc { r"( k ) } x, ~ (k - i) Rc
(~h)
{I"" (k ) } 400
llXFl 0
OATA~
l..5ffi 1
CATA
'lHITE G'<J5S1Nl
l~SS 'ffi l~ ::d::=>
Xl
"'CIS(
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_0
.-
_
IITEl' - - , . ; . :' - CHm'a --;,...,- - - -- - fUl' 1'9l -
Fig . '
=
W,, (k ) - /ldk).r,, (k) *
( 5)
where /l is the step size . If every complex weight is o ptim ized. the adapti ve a rray will yie ld automatic beam tracking o f the de sired SS sg na l and adequate su ppressi o n o f interfering SS s igna ls because o f d irecting its null s at them . As a result. th e interfering si g na ls will be attenuated by th e nulls w h ile the de sired SS signal will not he a null . How ever . an a rray antenna sys te m can rej e ct co m p lete ly o nly narrow -h and signa ls . In the adapti ve arra y antenna using DS -SSMA . the null depth ma y not be suffi cient to ac h ie ve the de sired interference rejection unless the syste m is modified to u peate over wide bandwidth s . Hen ce . there w ill be so me residual inte rfe re nc e [41. [51 . More o ver. the adapti ve ar ray antenna cannot suppress th e interfering SS sig na l who se arrival angle is the same as that o f the desired 55 signa l.
III. AN ADAPTIV E ARR A Y INTERF EREN CE FOR
I NCL UDIS G A C 'SCELLE R OF A
-
- - - - --
--
A DS ,SSMA sy ste m w ith an adapt ive arra y antenna incl udi ng a ca nce lle r of interference (pa ra lle l ca nce lle r struc tu re) .
for integer i , l- xR,,(k) = Re { x,,(k )} and xl,,(k) = 1m ( .r, (k) }. Since matrices R,,(O) and R" (0) cannot be obtained from the observed signal. LMS algorithm can be used in order to update complex weights chip by c h ip . suc h a s
W,, (k + I )
-
D5-SSMA 5 YSTE\1
Thi s sect io n propose s a D5 -5SMA rec eiving system using an adaptive arra y which can demodulate a de sired 55 sig na l robustl y even if he a vy inte rfe ring 55 signal s have the sa me arrival angle as that o f the de sired S5 s ig na l in a time-varying channel.
A. The Structure of the System Fig . 2 sho ws the structure o f the D5-55MA syste m with an adaptive array including a canceller of interference : it is po ssible to extend it to other S5 modulations . A prim itive 55 receiving sy stem ha ving neither an array antenna nor a canceller cannot demodulate the desired S5 sig na l even by using the inherent processing ga in of the 55 sy stem when the desired S5 s ig na l power is much lower than the interfering S5 signal power. The adaptive array antenna system (mentioned in the previous section) is effec tive in suppressing cochannel interference with arrival angles different from that of a de sired user. but the residual
interference and cochannel interference having the same arrival angle as the desired 55 signal are a major problem in S5MA . In order to solve thi s problem. we propose an adaptive array syste m including a canceller of interference . which can eliminate the interfering 5S s ig na ls having the sa me arri val angle as that of the de sired 55 s ignal by using ad apti ve digital filters (A D F ·s ). The proposed sys te m can al so ca nc e l the residual co channel interference having arrival angles different from that of a de sired user which a n adapti ve arra y antenna cannot completel y suppress. When th e int erfering si g nals from M undesired users rema in at the o utput of the arra y antenna. the ad apt ive canceller has ( W O typ e s of struc tur e. suc h as the parallel structure sho w n in F ig . 2 and the se ria l o ne in Fi g. 3. H for each antenna e le m e nt in F ig . 2 is a qu adrature hyb rid splitt ing the recc iv ed s ig na l into quadrature components . In the ca nc e lle r. th e int erfering 55 s ig na l from the ith user (i = I . 2 . . .. M ) is demodulated and respread by 55DEMi and 55MODi . respecti vel y . ADFi ( i = I. 2.. . . M ) is used to identify (he entire channel characteristics of both the channel for the i th user and the array and to generate a repl ica of the di storted inte rfe re nce component in the a rra y o utput. Then every replica of interference is subtracted from the delayed output s ig na l of the array . The o utput signal from the c a nce lle r is then fed to ADFO. whi ch compen sates the di stortion of the desired S5 signal from the Oth user. and demodulated by 5SDEMO . The final o utput data are respread by SSMODO in order to produce the reference s ig na l for the arra y and the ADF' s. In the se ria l st ruc tu re o f the canceller. interfering S5 s ig na ls are cancelled in the order of decreasing receiving po wer be cause it is ea sy to achieve acquisition, demodulati on , and ca nc e llat io n of an interfering 55 s ig na l ha ving g rea te r power and its can cellation makes it possible or ea sy to cancel o the r interfering 55 signals . Therefore . the se ria l structu re may perform more robust cancellation than the parallel one . However, the latter, shown in Fig. 2, can achieve more stab le adaptability of the array than the former. because time-delays within the feedback loop updating the array weights will result in instability unless the loop gain or the array speed of response is reduced . When there are a few strong interfering 5S signals, cancelling only those strong signals is sufficient to achieve
401
~=r::::r
' EIC>
Fig . 3. Serial canceller structure .
stable acquisition and reliable demodulation of the desired SS signal. Then the amount of hardware will not be very large. Note that this system can achieve more stable acquisition , demodulation. and cancellation of the interfering SS signals than a conventional array system without a canceller even when the D /1 ratio (of the desired to undesired SS signals' power) at the array output is small. This is because interfering SS signals can be demodulated more correctly in the case of such a small 0 /1 ratio . On the other hand . when the D / 1ratio is so large that the system can achieve stable acquisition and reliable demodulation without such a canceller. the funct ion of cancellation of ADFi (i = I. 2.... M) in Figs . 2 and 3 may be stopped . Moreover. it is possible to control ADFO so as to reject other intentional jamming and narrow-band interference similar to an interference rejection filter [7].
= /,2.: Re =0
-4l<-- - - '
.IV
I. = "~I
Re {h , W" exp { -j (21fL,, /A) sin (),}} .
and
I, = [J.o .}; I '
. ..
1;11I (.
Let B, and T, (k) be the tap coefficient vector of ADFi and an estimate of r, i k ), i.e .. the reference vector for i = I . 2. . M. Then the output signal of the canceller II (k) is lI(k)
1/
L
= y(k )
, - I
Re:
where
In the proposed combination of an array antenna and a canceller. it is important how one updates the array weights and the tap coefficients of ADF ·s. If the array weights are updated independently or the canceller by using the array output y (k) and its demodulated and rcmodulated SS signal as shown in Fig . I . time-delays within the feedback loop will be small. However. unless PN acquisition is achieved at the array output and a proper reference signal is obtained, the array and the following canceller cannot be correctly controlled . In order to improve the performance. the canceller output can be used in updating the array and the ADF's as shown in Figs . 2 and 3. Even if there are strong interfering S5 signals at the array output. the canceller system can accompl ish acquisition of those signals and correctly demodu late and cancel them . Thus , it will be easy to achieve acquisition of the desired SS signal and obtain a correct reference signal at the canceller output. Since the time-delays within the feedback loop increase, it may be necessary to reduce the loop gain and make the array slow in order to avoid instability if channel characteristics vary quickly with moving of an undesired user. Assume that interfering SS signals from M undesired users still remain at the array output. They consist of the interfering SS signals having the same arrival angle as that of the desired SS signal and the residual interfering SS signals with different arrival angles which the array cannot completely eliminate by its directivity . Then the output signal of the array y (k) becomes
y(k)
-
where the Oth user is the desired user. /; is an impulse response vector of the entire channel which consists of the channel for the i th user and the array antenna and can be written as
B. Controlling Algorithm
M
L[
- - - -- -
{fTr;(k)} + n(k)
B.
=
i, ( k)
=
[B ,II' B
i [ •
{B ,'r,(k)}
(7 )
r
•
•
B""I .
•
Ii , ( k ) . i, (k
T
-
I ) . . . . . ':i(k - m)] .
In order to update: both the: complex weights of the: array antenna and the tap coefficients of ADF's adaptivcly . we:
can use the error t'( k) which is defined as the: difference between the: II (k) equalized by ADFO and the reliable reference Re { ':11 ( k) J: that is. e( k) = CTu ( k) - Re:
{,oil (k ) }
( R)
where C is a tap coeffic ient vector of ADFO
C =
I Co.
t
C ,. . . . Cd .
and u (k)
= [1I(k) . u i k:
-
T
I) . . . . lI(k - L)] .
When the mean square error J = E 1(' ( k )21 is used as a crite rion. the optimum tap coefficients of C and B, (i = I , 2 ... M) can be obtained by so lving the following equations :
(6)
402
aJ
-
se, =0 aJ
ac
==
for i
=
I. 2 . . . M .
o.
(9a)
(9h)
Therefore. the optimum B, is equal to the channel impulse response/; i i = I. 2 .. . M) . i.e . . B, =
t.
for i = I. 2.. . . . M.
( 10)
and the optimum C is expressed by
C
= R I~/l R,If
( II )
where Ruu and Ru r are the (L + 1) x (L + 1) covariance matrix and (L + 1) x 1 cross-correlation matrix . WTRn(O) W
WTR\.\( 1) w
WTR,.\(L) W
WTR.\.\" ( -1) w WTR,x(O) W
RII II
W TRu
( -
WTRu(L-l)W
L ) W W TR.\_, ( - L + 1) W
WI}?\!. (0)
WTR,r (1 )
R
II ,.
W 1Rx,.(L)
T is transpose and R n (i - j ) ( for i . j == 1.. 2....... L + 1) and Rx r (i ) are defined by (-la) and (4b). respectively. Assuming that there is no noise . the optimum C will be
of the desired user by a nulling operation . its suppression will facilitate tracking and demodulation of the desired SS signal.
equal to the inverse characteristic of It). Thus . the optimum transfer function of .A.DFO C ( :.) is given by C(.:) ==
IV.
This section shows computer simulation results in order to evaluate the performance of the proposed system in comparison with a conventional system . i.e .. an adaptive array antenna system without a canceller. Ins i111 U Ia t ion s. eve ry use r in 0 S-SSM A use sad iffe rent A1-sequence. i.c .. maximal linear feedback shift register s~quencc with a period of 31. A model of the channel impulse response shown in Fig. 4 is used. Complex weights of the antenna and tap coefficients of ADF"s are updated by using known training signals during the beginning 500 data hits and after that by using demodulated data. and each result is calculated for 10 000 data bits. A sliding correlation SChCIl1e is used for the acquisition of PN sequence». If the corrclator output is greater than the
-1//--
L
,---=0
J~)I:'
I
In order to obtain these optimum values rccursivclv. algorithm can be used such as
L~IS
for i == I. 2. . . . ,'vI C(k
1"
1) == C(k) -
J.1,l'(k)ll(k)
( 12a) ( 12h)
where C ( k) and 8, ( k : (i == 1. 2 . . . "'-I I are the tap coefficients vectors C and HI at instance k T. J{, ( k , IS J L X III matrix given hy Rc
RI (/\)
Re
{r, (k) } {r, (k -
I )}
Re {PI (k
I) ),
RC {F, (k - L) }
Re r;: (k
:2) }
Re {p/ (k
J
t '
Re {p/ t k - I1z) } Re {PI (k Ilh and Ill" are the step sizes . and e ( k
SI\1ULATION RESULTS
-
) is defined by (8). In a controlling process of the proposed array system including a canceller . the array weights should be fixed (e.g ... only a weight is one and the others are zero) until the canceller can achieve PN acquisition for a desired SS signal because the array does not have a proper reference signal during the timing search, when the local timing is in error. In order to achieve the acquisition for the desired SS signal, at first, acquisition for the undesired SS signals at the array output is performed and then those undesired SS signals are eliminated by updating tap coefficients of ADF·s. After the acquisition for the desired S5 signal is accomplished . the array weights will be updated by using a reliable reference signal which can be obtained by the interference cancellation. Since the array can suppress interfering SS signals with arrival angles different from that
Re lf P (k
I )}
/11 -
!
1 - L)}
-
111
- L)}
dcsi red threshold level (i , e ... a lock level) more than three consecutive times during an interval of a PN sequence period. the phase of the local PN sequence will be locked: the lock level is normalized by the maximum value of the autocorrelation of the PN sequence. In the simulation, the number of multiple accessing users is three: the SS signal of one undesired user arrives from the same angle (J = 45 with respect to broadside as the desired user and that of the other undesired user arrives from a different angle (J == -45 in Figs. 5-8. The number of antenna elements is two, i.e ... N == 2. The element spacing is one-half wavelength. The number of taps in ADFO, L is 21 and the stepsize ~ is 0.01. The received signal-to-noise power ratio (SIN) in each element is 10 dB. In Figs. 5-9, the D /1 ratio for the same arrival angle (that is, the power ratio of the desired signal to the undesired signal with the same
403
0
0
\
CONVENTlOpw'SI'STIM (WITH NOCANCELlBlI
PROPOSED SYSTEM
!WIIlICA/aIiEfll
o
o
-2T
on • 1.0
o
10- 1
+2T
Fig . 4 . A model of the ch annel impulse response: T = sampling inte rva l.
10'
l§
ffi 10'
\
~ z
~
2
4
6
10
I.(JJ(lEVEll x .lI .
..------r-------r------.
on• ~6
~
011 .10
PROPOSED SYSTDA IWIrH CANCllifAI 011 •
10- 1
~
-
I
o
Fig. 8. Demodulated chip error rate versus lock level for PN acquisiton.
10' COIM1mONAL 6YSIDA IWllIlNO CANcaJBll
SIN • ' Od8 ANGIf• !II'......._ \ __'__ _....._ _~ I _ _ ~ .
L:..:.~_"_
O~
& 10
SIN • 11Id8
10. '
o
lJICl( lEVEl· O.' ANGlE .!K7' fTlRATlONI x1llllOi
Fig. 5. Convergence property of mean square error as a function of the D/I ratio.
Fig. 9. Demodulat ed chip error rate as a function of the arrival angle of an undesired 55 signal.
10' ~--T""---r--...,..---"---,
10' 10· ' w
:i
'" ~
10-'
ill
s ~
..
10· 0
2
10
6
, --
--,..--
-
-,-- ---y-
-
Fig. 10. Demodulated chip error rat e as a function of the angular speed of a moving undesired user.
---,
~
'" lO- l ~ e,
10
J
10"
o
or the desired one: is cha nged . while: the D/l ratio for the same arriv al angle is one . Since an .v-clerncnt arra y has N - I degrees of freedom in its pattern. the 2-elelllent array can form only one null. In the simulation model. the arra y may only form a null so as to suppress the interfe ring SS signal with an arriva l angle different from that of the desired 55 signal. The proposed array system can carry out such a nulling operation avo iding the effe ct of the interfering 55 signal with the same arri val angle as that of the desired user because the system uses the canceller output signal. which includes no such interfering 55 signal. in updating the arra y weights as shown in Figs. :2 and 3. Fig. 5 shows the convergence property of the mean square error E Ie ( k )~ I of the proposed array system with the canceller and a con ventional array system with ADFO and without a canceller as a function of the D / I ratio for
differe nt from that
CONVENTIONAL SYSTDA (W1TIlNOCANClliIIlI
~
10' ANGllAR SPEEDldeg" elchl01
Fig. 6. Demodulated data error rate versus D/I ratio. -
PROPOSEDSI'STIM IWITH CANCEllfRl Oil • 10
10·' 10- '
01l1 X.l1
10' .......-
PROP06EO SYSTEM {WmtCANCEUfRI0/1 .0.3
10· ' 10- 1
SIN • IOdS LOCK lEVEl • 0.4 ANGl£ =90"
CONVENTIONoOl SYSlal (W1TIlNOCANCWBlIOil • 10 & OJ
SIN • IOdS lOCXlEYB. • 0.4 ANGlE .!K7'
10
ont x.u
Fig. 7. Demodulated chip error rate versus D/I ratio.
arrival angle) is changed , while the undesired signal with a different arrival angle has the same average power as that of the desired signal. In Fig. 10. the D/ 1 ratio for the different arrival angle (that is, the power ratio of the desired signal to the undesired signal with an arrival angle
404
the same arrival angle. Note that the proposed system can achieve more stable convergence than the conventional system. Figs. 6 and 7 show error rates of decoded data and chips as a function of the 0/1 ratio for the same arrival angle, respectively. In the conventional system. the data error rate is more than 10- 1 corresponding to D /1 $ 0.8 in spite of the inherent processing gain of the system because the chip error rate is considerably large. On the other hand, there is no error in data and chips of the proposed system for 0 /1 ~ 0.4. The error rate of the proposed system is large in the range 0/1 < 0.4. since mistakes during lock of the PN acquisition may occur in SSDEMO before the canceller achieves complete rejection of interference. Fig. 8 shows the chip error rate as a function of the lock .evel, and thus illustrates the robustness of the perforrnance of PN acquisition. Here. the D / I ratios for the same arrival angle and the di tlcrent one arc one. The proposed system can accomplish PN acquisition corresponding to a wide range of 0.2 $ lock level ~ O.YS. while the con ve ntion a I s y ste 111 can dot hat 0 n Iy a ro Ll nd a 10 L k level = 0.4. Fig. 9 shows the chip error rate as ~l tunct ion of the irrival angle of an undesi red 55 signal in the range 0 ~ J ~ 90° where the D / I ratio for the same arrival angle is one or two. Even in the case of the D 'I ratio == 2.0. the error rate of the conventional -v-tcrn I" l.iruer than 1a-.~ be C J U ~ can i nI e rfc r i n g SS "i g n~11 ha\ I n ~ an arr i\ aI angle the same as that of the dc- ircc] SS "lgn~t1 prevents the array from forming a proper null. Since the proposed system can cl irninate such an intcrtcring 5S signal. a nulllng operation can he improved Fig. 10 shows the eh i p error rate a" a tuner ion of the angular speed of a moving undesired user In order to Investigate adaptability for xpacc-variauon of J channel. The undesired user having arrival angles different from that of the desired user is I110veJ in the range f1 == -'+5 ± 2():J where the D / I ratio for the same arriv al angle is one and the 0/1 ratio for the di tfcrcnt arri val angle 1:-, one or () 3. In a case of the latter 0/1 ratio == o. J. the convcru ional system cannot achieve acquisition. but the proposed system can do that and track the spacc-variution for the angular speed of a moving undcxircd user < I () - 2 (degree. chip). Hence. it is noted that the proposed syxtem can achieve more stable aduptabil ity than the conventional system in spite of larger time-delays within the feedback loop updating the array and the ADF·s.
know which interfering SS signals of undesired users arrive from the same angle as that of the desired user in the canceller. This can be found by observing the output level of the correlator which uses every PN sequence assigned to the SSMA user selectively. REFERENCES
II) M. K. Simon. J. K. Omura. R. A. Scholtz. and B. K. Levitt. Spread Spectrum Conununicutions, Rockville, MD: Computer Science. 1985. 12) R. Kohno. H. lmai , M. Hatori . and S. Pasupathy. "Adaptive cancellation of interference in direct-sequence spread-spectrum multiple access systcrns ;" in Proc. IEEE Global Tclecommun. COil! /987. vol. 1. pp. 630-634. Nov. 19X7. [3\ R. Kohno and H. Imai. "On pseudo-noise sequences for direct-sequcnce QPSK spread spectrum communication system» ... Tn/II.\. IECE I apau . vol. J65-A. no. I. pp. 69-76. Jan. 1<)~2. 141 R. T. Compton. Jr .. "An adaptive array in a spread-spectrum <:0111murucuuon sy ... tcl11.·· Proc. 1f.."Ef.·. vul . 66. pp. 2~9-2<)H. Mar. 1978. 15\ - . Adapt! t·£, Ant cnnus Concctn» lind Pcrtormancc, Enulewood Chtl». NJ: Prentice-Hall. IYHX. . "[61 L. B. Milsrcm and R. A. litis. "Slgnal processing for interference rejccuon in ... prcad spectrum conunumcations ." /1-'-1:'1-.." ASSP Mug .. vol. 3. Pp. I X- J I, A Pr . IlJX6 . [71 1., 1.1 .md L. B. M rl xtcm. "Rcjccuon of narrow-hund interference in p~ -prcad-vpccrrum . . ~ -tcm-, lISI11~ transvcr.... ul filtcrs ." IEEE Trans. COn/n/UIl,. '.(11.
0
0
V.
C()NCLUSI()~
The proposed adaptive array system can improve the performance of demodulated data error rate and achieve stable PN acquisition with greater tolerance for selecting
the lock level to a correlator output: the reason is due to its ability to reject cochanncl interference in DS-SSMA. which cannot be suppressed by a conventional adaptive array. However, for proper operations. it IS necessary to
405
CO\t-JO. pp. Y25-l)2X.
~by
IlJH2.
Direction Finding in the Presence of Mutual Coupling Benjamin Friedlander, Fellow, IEEE, and Anthony J. Weiss, Senior Member, IEEE changes due to the environment around the sensor array (e.g., the effect of metal objects near an antenna array on its beam pattern), and changes in the location of the sensors (e. g., an antenna array located on the vibrating wing of an aircraft or a hydro.phon~ ~rr~y towed behind a ship). In many practical situations It IS Impossible to ,maintain array calibration to the accuracy required for the proper operation of these eigenstructure-based techniques. This results in significant degradation in system performance, sometimes to the point that these superresI. INTRODUCTION olution techniques perform no better (or worse) than convenIRECTION-find ing techniques based on eizenstructure tional direction-finding methods. methods have been discussed extensively in the literature . A pr~ctical approach to alleviating the problems introduced by since the beginning of the last decade. Computer simulations and Imprecise array calibration is to use the received signals to adjust a relatively limited number of experimental systems have ?r fine-tune the array calibration. Self-calibrating or self-coherdemonstrated that in certain cases these techniques have superior mg antenna arrays have been developed and tested by Steinberg p~rformance compared to conventional direction-finding tech[11] and others. Schultheiss et al. [51. [6] have studied in deta~ niques. the self-calibration issue in the context of passive sensor arrays In spite of the potential advantages of eigenstructure methods with imprecisely known sensor locations. Self-calibration techtheir application to real systems has been very limited. One of niques for eigenstructure-based array processing techniques seem the main reasons for this situation is the practical difficulties to have received little attention. Lo and Marple [7] discussed a associated with calibrating the data collection systern. Eigencalibration technique that requires calibrating sources 'whose structure-based direction-finding techniques such as MUSIC [81 directions are known (at least two sources are required) and require precise knowledge of the signals received bv the sensor theref~re their t~chnique is not a true self-calibrating technique. array from a standard source located at anv di;ection. The Paulraj and Kailath [1] presented a method for direction-ofcollection of the received signal vectors for ;11 possible direcar.rival (DOA) estimation by eigenstructure methods for an array tions is often called the array manifold. The performance of the with unknown sensor gains and phases. Their method does not eigenstructure based system depends strongly on the accuracy of require calibrating sources with known directions, but is limited this array manifold. to uniformly spaced linear arrays. The process of measuring the array manifold can be time In this paper we address the problem of estimating the direcconsuming and expensive. Calibrating an antenna array designed tion-of-arrival of plane waves impinging on a sensor array for two-dimensional (azimuth and elevation) direction finding w~ose elements have unknown (or imprecisely known) coupling, with the accuracy required by these superresolution techniques gains and phases. We develop an eigenstructure-based method poses numerous practical problems. The amount of memory for simultaneously estimating the DOA's and the unknown courequired for storing the array manifold once it has been meapling, gain, and phase parameters. sured may also increase the size and cost of the system. even if interpolation techniques are used to reduce the number of points II. MUTUAL COUPLING MODEL that need to be stored [10]. Ignoring for a moment the mutual coupling between the array In addition to the problem of initial array calibration. there is sensors, we first formulate the data-signal for an ideal array with the problem of maintaining array calibration. Manv factors no mutual couplingcontribute to changing the response of the sensor array over Consider N radiating sources observed by an arbitrary array time: gradual changes in the behavior of the sensor itself and of of M sensors. The signal at the output of the mth sensor can be the electronic circuitry between the sensor and the output of the described by digitizer (due to thermal effects. aging of components, etc.),
Abstract-An eigenstructure-based method for direction finding in the presence of sensor mutual coupling, gain, and phase uncertainties is presented. The method provides estimates of the directions-of-arrival (DOA) of all the radiating sources as well as calibration of the gain and pha~e o! each sensor and the ..~utu81. cQ~pling in the receiving array. Calibration sources at known locations are not required. Conditions are provided for the existence of a solution. The proposed algorithm is described in detail, and its behavior is illustrated by numerical examples.
D
Manuscript received September 29, 1988~ revised September 19. 1989. This work was supported by the Army Research Office under Contracts DAAL03-86-C-OOI8 and DAAL03-89-C-0007. sponsored by the U.S. Army Communications Electronics Command. Center for Signals Warfare. ~. Friedlander is with Signal Processing Technology. Ltd.. 703 Coastland Drive. Palo Alto. CA 94303. and the University of California. Davis, CA. A. J. Weiss is with the Department of Electrical Engineering, Tel Aviv University. Tel Aviv. Israel. IEEE Log Number 9039141.
xm(t) =
N
L
n=l
cxmSn(t-Tmn-1/;m)+um(t), - T /2 -s t
-s T /2,
m=1,2,···,M;
where {sn(I)};;=1 are the radiated signals, {um(l)}~=t are sample waveforms from additive noise processes, and T is the observation interval. The parameters {Tmn} are delays associated with the signal propagation time from the nth source to the
Reprinted from IEEE Transactions on Antennas and Propagation, ve; 39, No.3, pp 273-284, March 1991.
406
(1)
mth sensor. These parameters are of interest since they contain information about the source locations relative to the array. F inall y, the parameters a m and 1/; m are the gain and the delay associated with the mth sensor. A convenient separation of the parameters to be estimated is obtained by representing the signals using Fourier coefficients defined by
(2) where WI = 2r(/ 1 + 1)/ T, I = 1,2,"', L; and 11 is a constant. In principle the number of coefficients required to capture all the signal information is infinite. However, if we consider signals with energy concentrated in a finite spectral band, we can use only L < 00 coefficients. Moreover, in this work we are interested in narrow-band signals with spectrum concentrated around wo, with a bandwidth that is small compared to 2 7r / T and hence L = 1. Taking the Fourier coefficients of (1) and suppressing the dependence on Wo we obtain N
X m == ~ ex e-jwO!/;m. Lm
e-jwoimnS
n=l
n
+
o
o
o -----ttr----------------__ x Sensor 1
Fig. 1.
Vm :.
m=1,2,"',M;
element in the array acts independently of all the others. This assumption is often invalid in practice. Reflected radiation from one element couples to its neighbors. as do currents that propagate along the surface of the array. The output voltage of each array element is the sum of the primary voltage due to the incident radiation, plus all the contributions from various coupling sources from each of its neighbors. Hence. the actual voltage at the output of the array is given by the following modification of (4):
(3)
where S; and Vm are the Fourier coefficients of sn( t) and um ( t), respectively. Equation (3) may be expressed using vector notation as follows:
X(j) = r . A . S(j) + V(j):
j
=
1,2,"', J:
(4)
where j is the index of different (independent) samples and
X(j)
X(j) == [X1(j), X 2 ( j ) , · · · , X.W(j)]T,
=
(VI (j), V2 (j)
,... , ~w (j)]
T,
n == 1,2,"', N. To further simplify the exposition we assume that the sensors and sources are coplanar and the sources are far enough from the observing array so that the signal wavefronts are effectively planar over the array. It is easy to verify that the delays T m n are given by Tm n
(5) where c is the propagation velocity, d mn is the distance from sensor m to sensor number one (reference sensor) in the direction of the nth source, (x m: Ym) are the coordinates of the mth sensor, On is the DOA of the nth source relative to the Y axis, and the origin of the Cartesian coordinate system coincides with sensor number one-see Fig. 1. From (4) and (5), it follows that the elements of the matrix A are given by eJ(wo/c)(X m
Sin
0n+ Ym cos On)
r . A . S(j) +
vr»:
j == 1.2.···. J (7)
A. Linear Arrays
== -dmn/c,
A mn =
= C .
where C is an M x 1\1 complex matrix. Unfortunately. the matrix C tends to change with time due to environmental factors such as temperature, humidity, pressure, vibrations and nearby objects. It is therefore desirable to estimate the matrix C without interrupting the ongoing DF mission. Since DOA estimation is a complicated process even when C is perfectly known we concentrate here on estimating a first order approximation for the matrix C. as well as for r. and the DOA's.
S(j) = [Sl(j), S2(j),"', SN-(j)]T, V (j)
Problem geometry.
(6)
It also follows that { A 1n}::= 1 == 1 and that only the nth column of A depends on On' We are now ready to address the mutual coupling between the array elements. In the foregoing theory it was assumed that each
In general, the matrix C has no special structure. However, for a linear uniform array that is well balanced a banded matrix provides an excellent model. The rationale behind this model is the fact that the mutual coupling coefficients are inversely proportional to the distance between the elements. Therefore, the mutual coupling between two elements that are far enough from each other, can often be approximated as zero. Moreover, we expect that good linear uniform arrays will exhibit a banded Toeplitz mutual coupling matrix (i.e., the coupling between any two equally spaced sensors is the same).
B. Circular Arrays U sing the same set of considerations leads to the conclusion that the mutual coupling matrix for a uniform circular array consists of three bands; a center band, a band at the upper right-hand corner and a band at the lower left corner. Moreover, it is expected that a good circular uniform array will exhibit a circulant mutual coupling matrix (MCM).
407
c. Existence of a Solution The proposed method for estimating the DOA's and the MCM is based on the eigendecomposition of the sample covariance matrix of the vector of received signals. We make the standard assumptions underlying the MUSIC algorithm and other eigenstructure-based methods for direction finding. 1) 2) 3) 4)
The signals and the noise processes are stationary and ergodic over the observation period. The columns of B = C . A are linearly independent. The signals are not perfectly correlated. The noise is uncorrelated with the signals and its covariance matrix is full rank and is known except for a multiplication constant
r·
u;.
The covariance matrices of the signal, noise, and observation vectors are given by R, = E{SSH}
an2~o = E{VV H }
R x = E{XX H } = CrARsAHrHC H
+
(8)
an2~o
where (.) H represents the Hermitian transpose operation. While conditions for uniqueness are still an open research problem it is easy to derive necessary conditions for the existence of a solution. Referring to the basic equation (8) we observe that R x can be perfectly described by 2 MN - N 2 + 1 parameters. These parameters are the N + 1 different (real) eigenvalues and 2 NM - N 2 - N parameters that define the N complex eigenvectors describing the signal subspace, that satisfy N( N + 1)/2 complex orthogonality constraints. On the other hand we have r· N unknown location parameters (r = 1 for azimuth only system . r = 2 for azimuth and elevation system. etc. ), N 2 unknown parameters that define the Hermitian matrix R S' a single unknown parameter 2( M - 1) parameters associated with rand P parameters associated with C. Thus . the problem is not strictly well posed (in the sense of [8, p. 84J) unless
Proof: The proof is a straightforward extension of the proof in [4]. This theorem suggests that one should first estimate R x and use the estimates of the eigenvectors to estimate the number of signals. Once N is known reasonable estimates of {f)n}' r, and C may be obtained by minimizing the cost function
2
+ 1 ~ r· M
~
+
N
N'2
+ 1 + 2(M - 1) +
P
2N 2 + r · N + P - 2
--------2( N - 1)
':=
D .X
ESTIMATING THE
DOA's,
MCM
2)
AN+l = AN+2 =
...
[Ql(X)]IJ=X;·Oij i.L> 1,2,···,M. Lemma 2: For any N x 1 complex vector X and any M x M complex symmetric circulant matrix A we have
(9)
GAINS, PHASES, AND
= AM, =
i = 1. 2, ... , M
d , = D jj
Q,
[WI] pq =
i= 1,2,···,L
Ali'
{ X p +q _ 0,
[W2 ] pq = {
for p
l,
I
=
{ X p + q _M _ 0,
= M /2 for even
1
p~q~2
Xp _ q+ l , 0,
otherwise
0,
[W4 ] pq
+ q -s M +
otherwise
[W3 ] pq = { X M + 1+P - q '
2
408
=
where L = M /2 + 1 when M is even and L = M /2 + 1/2 when M is odd. The M x L matrix Q2(X) is the sum of the four M x L following matrices:
an
Each of the columns of B g erA is orthogonal to the matrix U = [U N + 1, UN+2'· •• , uAwl·
I vector a are given by
where the components of the L x
The proposed method is based on the eigendecomposition of the sample covariance matrix of the vector of received signals. In addition to the standard assumptions underlying the MUSIC algorithm we assume that relation (9) is satisfied. Theorem 1: Let Ai and U i' i = 1, 2, · · ·, M be the eigenvalues and corresponding eigenvectors of the matrix pencil (R x' ~o), (i.e., the solutions of R xU = A~OU), where the Ai are listed in descending order. Then, I)
= QI(X) . d
where the components of the M x 1 vector d and the M x M matrix QI(X) are given by
Thus. for an azimuth only system (i.e .. r = I) with two sources and with P = 2.. we need an array of at least five sensors. However, if the sensor gains and phases are known, or if R s is known, then a smaller number of sensors will suffice.
III.
(II)
which is the squared Euclidean norm of the matrix 0 H cr A . Here 0 stands for the estimate of the matrix U. If 0- were a perfect estimate of U (i.e., 0 = U) then the minimum value of J c( J c = 0) will be achieved for the true C, I', and {On}' When (; is an imperfect estimate U, the minimization of J c will provide estimates of C, I', and {On}' the true MCM, gain-phase parameters and DOA's. The accuracy of these estimates has been investigated by simulations. A detailed error analysis has not been carried out as yet. The proposed minimization algorithm is based on a three step procedure. First, we assume that the gain/phase and mutual coupling coefficients are (approximately) known, and we estimate {On} 1 using the principles of the standard MUSIC algorithm. Given estimates of {f)n} we then minimize J c over the gain/phase parameters. Given {f)n} and r we minimize J; over the MCM components. These minimization steps can be repeated until J; converges. Before presenting the algorithm for minimizing J; we introduced three useful lemmas. Lemma 1: For any M x 1 complex vector X and any M x M complex diagonal matrix D we have
a;.
2MN - N
N
2 H L IIO Cr a(On) 11 n=l
Jc =
1'
M and I
p
s: q
S:
I, P
+q
2:
otherwise.
= (M + 1)/2 for
odd M.
M
+
2
Lemma 3: For any M x 1 complex vector X and any M x M banded complex symmetric Toeplitz matrix A we have
Hence, we want to minimize (14) with respect to 8 under the constraint OH W = 1, where w = [1,0,0,' .. , OlT. The result of this quadratic minimization problem under linear constraints is well known and given by
where the LxI vector a is given by Q;=A
(15)
i= 1,2,···,L
u,
where Z k is the matrix
and L is the highest superdiagonal that is different from zero. The M x L matrix Q3(X) is given by the sum of the two M x L following matrices
[W.] pq [W2 ] pq
=
{X +
=
{X
p
for p
q_ l '
0,
N
z, ~ I:
n=l
+ q -s M + 1
2)
otherwise
0,
rk +
1)
1) Set the iteration counter to zero: k = O. Select initial values for the gain-phase matrix rand initial value for the MCM (i.e., C). Usually the initial values are based on some previous knowledge (e.g.. last measured values or predictions based on idealized model). 3) Use all available data vectors to compute the data covariance matrix estimate:
4)
1 ~
1
L
j=l
H
X(j)X(j) .
=IIU H C(k)r (k)a(O) \\- 2.
(13)
These peaks are associated with the N DOA' s {On} ~= I' Also note that substituting {O~k)}, so estimated, in (given by (11)) guarantees that i~ k v is minimized for given C(k) and r(k).
r,
Step 2: Estimating Gain-Phase: 1)
Fixing the DOA's and the MCM we now rmrurruze J; with respect to the gain and phase of each of the sensors. Using Lemma 1 in (11) we obtain
Jc =
where
IV
L
n=l
a(On)HrHCHOOHCra(On)
;V
=
L
a( en) II CHl)[~Ca(en)
n=l
=
~
cH {
11=
1
Qc(lllfl{;{;HQc(lll}C
( 18)
where we used the following notation:
a(On)
= ra(On)
Q2 ( n) = Q2 ( a(8,J )
Search for the N highest peaks of the spatial spectrum defined by p(k)(lJ)
In this step we hold the DOAs and sensor gain-phase fixed and find the MCM that minimizes the cost function The minimization step capitalizes on Lemma 2 if C is circulant (circular array) and on Lemma 3 if C is Toeplitz (linear array). In the following we assume that C is a complex symmetric circulant matrix.
J;
Step 1: Estimating DOA 's: 1)
(17)
Using Lemma 2 in (11) for the cost function J; we obtain
(12)
Perform eigenanalysis and construct (; according to Theorem 1.
(6).
a
r.:
2)
= -
= diag
from the vector
Step 3: Estimating the Mutual Coupling Matrix:
A. Initialization
~
I
r
( 16)
Thus, we minimized the cost function J, by holding the MCM and the DOAs fixed and searching over the space of the sensor gain-phase parameters.
otherwise.
Proof' The proof of the lemmas is based on the special properties of diagonal matrices, circulant matrices and Toeplitz matrices. (See Appendix I.) The proposed algorithm for minimizing the cost function may be described as follows.
Rx
Compute the gain-phase matrix given by (15)
p"?q"?2
p _ Q+ I ,
Q(I(n)CHDOHCQI(n).
i = 1.2.···. L.
Note that Q2(X) and L are defined in Lemma 2. Relation (18) represents (again) a quadratic minimization problem under linear constraints. The linear constraints represent the assumed model of C. (e.g.. ell = 1). Hence if the constraint equation is WITe = u then ( 19) where G is the matrix G ~
N
L
n=l
Q2(n)H{;{;HQ2(n)
and WI represents the linear constraints.
2)
Reconstruct the MCM matrix from the vector ( 19).
(20)
c given by
B. Convergence Check
o = [r 11' r 22 , .•• , rMM I T
Compute lck + I using the estimated DOA "s, sensor gain-phase and the MeM. If
{a(0n)}.
lc.~ - lc.~ + I > e (a preset threshold)
Q1( n)
= diag
409
SNR = 30 dB. SOO SNAPSIIOTS
SO 40 . ..
iii ~
g
-s
...~9.
30
::<
SNR = 30 en, SOO SNAPSHOTS
0
::>
iii
...'"-e
~
.. .
20
~
'"-e
· IS
0
·20
::> -J
!= -e e,
10
>
'
z
"
1=
'"
.
.. . -
.
•. . .. _.. .. .
1. r ··········.··....·.·.. ·T.. ···..····j·· ·········· ~
. j
,
.. ··. r····.· . r ······.··T···········T·······..··,·····..····T···········
U
z
it
0
·10
·SO
8'"
·40
·30
·3S
DlRECllOS OF ARRIVAL [DEGREESI
Fig. 2.
·23
I-
-40
Spatial spect rum of the proposed algorithm after iterati ons 1. 2. 4 .
30.
0
3S
ITERATIONS
then update the ite ration counter k = k I.
+
Fig . 3.
I and go back to step
Value of (he cost function versus iteration number.
If
SNR = 30 . SOO SNAPSHOTS
SO
stop . T he algorith m performs the iterations until J.. co nverges . Note that at each step that cos t function reduce s so that 1,~OI
> 1,(0) > '"
>
J,~ k l?:
4S
i= z;
~o
u a:
3S
w
w
O.
Hence. 1,~ kl is a convergent se ries and convergence teed .
IS
guaran-
~
JO
a: a:
ZS
a: '" 0 w ~
-c
:I:
IV .
~
NU~lERI CAL E.\A~IP LES
To illustrate the behavior of the algorithm . cons ider a circular unifo rm array of six omnidi rectional sensors separated by half :I wavelength of the actual narrow-band source signals. We used simulated signal vectors S(j) and noise vectors V(j) drawn from a complex Gaussian distribution with zero mea n and covariance mat rices a} . 1 and a; . I . respectively . We assumed that eac h sensor is sig nifica ntly coupled with his nearest neigh bo rs while the cou pling with other sensors can be ignored . T his assumption red uces the MCM to a 6 x 6 matrix with five nonzero d iagonals . The gain of the sensors was sele cted using the following relation : OJ
=
[(13, - 0 .5)aa · v'12 + 1]
i= 1. 2... · . M
where 13; is a uniformly distributed random number between zero and one. and a'; is the variance of the gai n. The phase of each of the senso rs was selected according to
'¥; = [(-y; - 0 .5)a",·
/12] ,
i= 1.2.· .. , M
ai
10
--.------ -.+..·
,
·-----. i--. --
:C·,··.·j·.··· 5
-~. __. _---_
,
i
+.·
~
l········..···j·············I.. ······· ·· ~··········· ·.., 10
IS
:W
Fig. 4 .
30
ZS
410
3S
Relative gain /phase errors versus iterati on numbe r.
\
90
..,
;:::
80
w
70 .. ': ' :' ::::
z, U
a: w c,
lI)
'"o '" w '" o
40
:J e,
JO
::>
8
20 10 •..
(
.
l .::::.:::J::.:::::.:::::t::.:::.:'.':. .
. ._
. . ..
:
.
:
.
·············[·············t·············;···
60 50
.:.
.+
·····t··············i·············;········
.... . ...•...... . ...... . . . .;_..
. . . . . . . . .:.
_.....
...\.·..··.······r···· ········i······
..........::::.:. ::::::::r:: .
.. _. .
':.
...;.-
.
3S
ITERATIONS
= 20·
.
SNR = 30 , SOO SNAPSHOTS 100
Fig. 5.
aa = 0 .2
.
ITERATIONS
dB
1 = 500 snapshots
a",
IS
o
Z
where 'Y; is a uniformly dist ributed random numbe r between zero and one and is the variance of the se nsors phase . Figs. 2-6 describe an experime nt with the follow ing parameters :
SNR = 10 log 10 ( a}/ a,;) = 30
:;;: o
ZO
Coupling coefficient error versus iteration number.
-- ---
10.1
SNR = 30. SOO S NAPSHOTS
/ I'---.. ~
0
w
'" ~ ~
2
ffi < 8
-4
o
-I--.. ~
C
·8
o
V
----
r-,
t-....
L....-
l---
~
w
~
10'
"'" 10
10
IS
20
2S
30
3S
10
IS
2S
20
~ 3S
30
Fig. 7. RMSE of the magnitude of the coupling coefficient versus SNR . The solid line depicts theoretical values comp uted by the Cramer- Rae lower bound . The point estimates are the means and 90% confidence intervals from Mon te Ca rlo experi ments for three values of the SNR . Each Monte Carlo experiment consisted of 30 runs of 500 snapshots each .
DOA errors versus iteration number.
coup ling coefficie nt = 0.2
.
to-....
SNR ldB J
ITERA TIO NS
+j .0
DOA I = -30·
10'
DOA 2 = _ 5· DOA 3
<,
1-. _ -
I Fig . 6.
<,
:--..
= 35·
>---- -
gains = 1.000 ,0 .5800,0 .8215 ,1.077,0.7970,0.9620
10 '
phases = O· , - 3 .8· , 14.6· , 34.4 • , - 3 .4• , 41 .4• Fig. 2 shows the spatial spe ctrum of the proposed procedure at the first, second , founh and 30th iterat ion. It is clear that major DOA errors are corrected . Fig . 3 show s the red uction of the cost function value duri ng the itera tions, until co nvergence is obtained . Fig . 4 shows (II I', - I', 1 / III', II> . 100 which is a measure of the relative gain /phase errors of all the sensors . I', is the gain /phase matrix at iteration i while I', is the true ga in/ phase matrix . Fig . 5 shows the relative coupling coefficient error as a function of iterations. Befor e the first iteration the err or is 100% and it reduce s to 1.9 %. Fig. 6 shows the DO A errors for the three sources as a function of the itera tio n number . To demo nstra te the statistical efficiency of the propo sed procedure we performed the following Monte Carlo experiments . The six sensor circular array described above was used, with three far-field narrow-band emitters . The gains and phases were selected as before, with UOI = 0.02 and Ucp = 2· . The DOA 's were 'YI = 0·, 'Y 2 = 120· , 'Y3 = 240· . The coupling coefficient between any two adjace nt senso rs was cc = 0.2(x + iy) where x and yare two i.i .d . random variables with uniform distribution over the interval [-0.5, 0.5] . The coupling coefficient for any nonadjacent sensors was assumed to be zero . We performed 30 experiments for each signal-to-noise ratio , for SNR = 10, 20, 30 dB. In each experiment 500 snapshots of data were collected and processed by the algo rithm. The values of the sensor gains and phases were kept cons tant throughout these simulations . For each SNR we used the results of the 30 experiments to compute the estimated root mean square error (RMSE) and the bias. Figs . 7 -11 depict the RSE' s and compares them to the corresponding Cramer- Rao lower bound. (computed as shown in Appendix II) . The se figures clearly indicate that the proposed algorithm is statisticall y efficient even for fairly low SNR 's, at least for the test case conside red here. The bias was small co mpa red to the RMSE in each case .
_.
<,
----
~
f--.
.'
10
10
15
2S
20
35
30
S NR IdBI
Fig . 8. RMSE of the phase of the coupling coefficient versus SNR . The solid line depicts theoret ical values computed by the Cramer - Rao lower bound . The point estimates are the means and 90 % confidence interval s from Monte Ca rlo expe riments for three values of the SNR. Each Monte Carlo experiment consisted of 30 runs of 500 snapsh ots each . 10 '
_.
._- _.
, F----: 10'
1-----
I
1 - - -- 1-- .... 1 - - -- - --- 1- ----
- -- _.
10.)
~-
1---.
--.....
- - -- --
--+..-
---1
~- .. -T -
10
l
10
:
.....J .. -
<;
---- --
t--...
.-:....;.....
..
._ . . .
--
---! . --
~-=:-- .: t:~:::: .I
.
...... '.---
_
IS
20
---ji
2S
30
35
S NR (dB)
Fig. 9 . RMSE of the gain of sensor 2 versus SNR . The solid line depicts theoretical values computed by the Cramer-Rae lower bound. The point estimates are the means and 90 % confidence intervals from Monte Carlo experiments for three values of the SNR. Each Monte Carlo expe riment consisted of 30 runs of 500 snapsh ots each .
41 1
sensor characteristics is essential in order to estimate the DOA's accurately. The algorithm presented here is able to calibrate the array parameters (mutual coupling. and sensor gains and phases) without prior knowledge of the array manifold, using only " signals of opportunity" and avoiding the need for deploying auxiliary sources at known locations. We found some necessary conditions for the existence of a solution. Deriving sufficient conditions for convergence to the global minimum. and obtaining a unique solution, are still open research problems .
10'
--; <;
t---..
I
-..... r---.-..
-
ApPENDIX
.-
PROOFS OF LEMMAS
10
IS
20
35
30
25
The proof of Lemma I can be obtained by direct multiplication .
S!'o'R (dB)
Fig. 10. RMSE of the phase of sensor 2 versus SNR. The solid line depicts theoretical values computed by the Cramer-Rae lower bound. The point estimates are the means and 90% confidence intervals from Monte Carlo experiments for three values of the SNR. Each Monte Carlo exper iment consisted of 30 runs of 500 snapshots each .
B. Proof of Lemma 2 By definition an M x M symmetric circulant matrix A , with
i, jth element A ij' has the following relations between its elements :
A ij
10'
r----. I
where we used the 'notation: ~
i
;;;
i
JO"
--...
- -----
(j _ i) I M
r-------
10
15
:0
15
- ---
30
L = {( M
{~+. j } -
I.
when i > i . when i =::; j .
- i,
when M is even
+ 2)/ 2,
(M + 1)/2.
35
(23)
when M is odd
(24)
and for even M. A has the form
S:W (dB)
Fig. II . RMSE of the direction of arrival of the source at O' versus SNR . The solid line depicts theoretical values computed by the Cramer-Rae lower bound. The point estimates are the means and 90 % confidence intervals from Monte Carlo experiments for three values of (he SNR . Each Monte Carlo experiment consisted of 30 runs of 500 snapshots each.
We have repeated this type of experiment with different array geometries and different initial gain and phase errors. using error values which are large compared to what one may expect to find in practice . We found the algorithm to be fairly insensitive to the initial gain and phase error values. As long as the number of snapshots was reasonably large (as in the examples above). the performance of the algorithm was very close to the Cramer- Rao lower bound . As the number of snapshots is decreased, the performance starts to deteriorate and departs from the CRB, as expected. A small sample performance analysis of this problem unavailable at this time .
V.
=
From these relations we see that a symmetric circulant matrix contains at most L distinct elements a,. a2.· .. , aL where
I
5
(21) (22)
A ij = A j l
.-
---- r----....
= A pq if (j - i) I M = (q - p) I M
and
-+-
<,
10' )
1-3
A . Proof of Lemma J
,
10·
:;:; x
I
a, a2
a2
0,
a2
a2
aL
aL_1
aL a L_ 1
A
while for odd M . -A has the form
CONCLUSION
In this work the eigenstructure approach has been used to obtain estimates of directions of arrival as well as estimates of gain. phase and mutual coupling of the observing array sensors. We have shown in previous publications [12], [13] that the basic MUSIC method does not perform well when the array properties are not known accurately . Therefore, the estimation of the
412
a\ a2 A
a2 al
aL
a,
aL
a2
aL aL a2
It is easy to see that the matrix A may be represented by the sum
A
M
L
k=-M
diag ( A • k)
where diag (A, k) is an M x M zero matrix except for the kth diagonal, which is equal to the corresponding diagonal of A. Since the elements on each of the diagonals are equal we have for even M: L-l
L
A =
diag
ak+l
k=O
L
+
-L+I
L
+
diag
a 2 L-k-1
k=L
-M+l
L
a2L+k-t
k= -L
(JM , k)
L
W2
= L
L
a2L_kdiag
k=L
k= -I
+
k=
k=L
column
(l.w. k)
w~
-1\1+1
L
k= -L
+ I • Xk +2 • . . • •
0,
Ax
(26)
diag (l,w' k) . x
=
k~O
k
•
•. ,
c.
k=O
even M
column
(xtt, 2L - k)
odd M
f
1
k=L
{:~:
+
{:~~
column column
(x t/,2 L +
column
(x;', 2 L + k)
-1\1+ 1
~
0.
:=
(WI + W 2
column (x~.
j) . a
+
(28)
+ Jt~)a = Q2(x)a.
Q.E.D.
II
(31 )
and covariance
(32) The unknown parameters () are imbedded in the covariance R. The logarithm of the pdf for 1 1 statistically independent observations can be written as
( 0)
= - J 1 In {det ( R )}
I)} .a
(x~, 2L + k - 1)} . a
(x~, 2L -
~V3
E{x} = O~
it
L
-
x( j) H R -
)=1
IX(
j)
where
(x~, 1 - K)} . a
column
odd ,'vI
Consider a complex Gaussian vector x with zero mean.
(34 )
(29)
{:t~ column(x~,k+ l)}-a
x.;
even !vI
Proof of Lemma 3
The m, nth element of the Fisher information matrix is given by
J
+ {
k - 1)
THE CRAMER-RAO BOUND FOR GAUSSIAN SIG:"JALS
and for odd M,
Ax=
column
a L]T we have
column (x ~, 2 L - k -
+
L
ApPENDIX
(27)
column (xtt, k + I)} . a
+ {M
1 - k)
(xtt, 2L - k - 1)
k= -L
L
1)
The proof of Lemma 3 goes along the same steps as the proof of Lemma 2.
where x is the right side of (27) and column (x, j) represents an M x L zero matrix except for the jth column which is equal to x. Using (25) we obtain. for even A1,
{Lf.'
(30)
Hence.
~. . . 01 T •
Xl.···. X A1 + k]
. a.
column
k= -L
a2 L + k diag ( J.H . k ) :
X \1 •
=
+
k
(xr,
-Al+1
a_ k + I d iag ( 1.\1. k)
Using the notation a = [at, a 2 , '
=
-l
L
k) . x
[0,··',0,
Ax
(xr,
column
1\-1-1
where J M is an M x M matrix of ones. Now. note that
r
k=O
k=L
M-I
. L
•
(x~, 2L + k)}
-L+l
(25)
diag ( J M, k),
L
=
a k + I diag ( J,\1' k)
k=O
-L+l
aj
column
L
L-l
+
Xk
{:~;
M-l
A =
= [[
(x~, 1 - K)} . a
L-l
WI
and for odd M:
diag ( J,w,
column
It is now easy to verify that
a -«+ 1 diag ( J A1 , k)
k= -1
+
+
(JA1 , k)
M-l
:~:
+ {
mn
=-E {
1
a2L
aomae
(35)
'
n
To evaluate (35) we use the following relations:
d (R-
I)
= - R-
I .
d (In {det ( R) }) = tr { R -
k)} . a
E{R} = R. 413
dR . R -
I •
dR}
I
(36) (37) (38)
Taking the first derivative of L(O) we obtain
oL/iJO m
=
J 1 • tr {R-
l
•
oRjoOm· R-
1
•
Using (49), we first write the partial derivative of the covariance matrix with respect to the jth DOA as
R}
.
aR / alj = A-yjPA
- J, . tr {R- l • oR/(J0m}
J-y;'Yj = 2Re
{tr {..1-y;PAHR-'AP..1~R-'}
.( -R- 1 • iJR/iJO n· R- 1 • R)}.
+tr
Taking the expectation of both sides we obtain =
'
tr {R-
1
•
(JR/dO m· R- '· OR/dOn}.
{..1-y;PAHR-I..1-YjPA~R-I}}. (52)
Observe that
-E{a 2L/oO moOn }
= 11
(51 )
we obtain
(J R / oem) / (JO n
·(R-lR - I) + R- 1 • oR/oem
1m n
(50)
tr ( A H) = con j { tr ( A ) }
The second derivative is given by 1•
'H
+ APA-Yj'
Substituting in (49) and noting that
= J l • tr{R- I • oR/oO m ' (R-1R - I)}. 0 2 L /00 mOO n = J 1 • tr { 0( R -
H
(39)
where the unit vector e I is the itb column vector of the identity A is the matrix of derivatives given by
Thus, the number of observations enters the result only as a multiplicative constant. To simplify the exposition we assume 1 1 = 1. The modification for other values of J, is straightforward.
M x M identity matrix, and
A. The FIlvt for a General Passive Array
We use the notation
For the parameter estimation problem posed earlier in this work, the covariance matrix of the data vector x can be written as R = CrARsAHrHc H + a~'/ (40)
(54)
A ~ erA
(55)
to simplify our formulas. Using (53) and (55). (51) becomes HR J'Y(Yj.=2Re{tr{Ae-€!PA / I
where A is th~ direction matrix. To simplify the derivation we assume that a - and R 5 are known. Dividing both sides of (40) by a ~ we obtain ..
1APe }
1} eTAHRj
1 +tr{Ae I eTPAHR-'Ae eTPAHR} I j }
}
= 2Re {eTPAHR-lAPe eTAHR-IAe· / } } I
where
(42)
P=Rs/a~.
+eTPAHR-lAe eTPAHR-lAe } I j j I
erA,
(44)
W~AHA,
(45)
Q~(P-I+W)-l
(46)
A g
In
(43)
and therefore we use R instead of R to derive the FIM. Moreover, we use the following notation to simplify the formulas:
We also use the relation
PWQ
= QWP = P
- Q.
(48)
B. Derivatives with Respect to DOA
{(PAHR-1AP) x (..1HR-'..1)T
(A X B) ij = A i)Bi)'
(57)
(58)
Equation (57) may be further simplified using (46) as follows:
In
= 2 Re {(p =-
Q) x (A HR- 1..1) T
+ (QAH..1) x (QAH..1{}. (59) c. Derivatives with Respect to Sensor Phase Repeating the same set of considerations leading to (52) we obtain
(60)
(49)
414
2Re
where i-y-y is the submatrix of the FIM associated with the DOA derivatives and x denotes the Hadamard product of two matrices. defined by
In the ensuing development, we make use of the notational device.
for the partial derivative of the matrix A with respect to the DOA v, of the jth sensor.
=
+(PAHR-1A) x (PAHA{}
(47) which implies that
(56)
Hence,
It is easy to verify that
J(R) = 1(R)
•
where
(61 )
E. DOA-Phase Cross Terms
Here F is a diagonal matrix containing the exponents of the sensors' phases while G is a diagonal matrix of the sensors' gains, thus (62) r == OF == FC.
F as
It is useful to define
We first write the cross-term equivalent of (69), J'Yi¢j == 2 Re
tr
{
" H A'YiPA R
the matrix of derivatives:
(
'YI
= 2Re
-(fAPAHR-1C) x (fAPAHR-1C)T}
=2Re{(C-
= 2Im{[(p-
(C-1APAHR-1C)T}
Q)AHC- H)
{-J[ (p - Q)(C-1A)H]
+J(QAHC)
{(C-IPAHR-IAPAHC-H) x (CHR-1C)T
1A(P-
-(QAHC)
x (CHR-1C)T
- ( C- IAQA HC) x (c- IAQA HC) T}
J,/aj
Q)(C-1A(] x (CHR-1A)T X
= 2 Re {tr {A"PAHR-IAPA~R-I}
·APA HC-HG-1e eTCHR-
T'
T
-
-1-1 C A.
T
J
(68)
= 2Re
{tr
{A a,.PAHR-1APA.:!RUoj
t.: = 2Rc
}
= 2Re
Substituting (68) in (69) we obtain I
j
J
(70)
2 Re {( C- le-lAPA HR-1APA HC-HG- I) x(CHR-lC)T
+
X(CHR-1C)T
+
{
"
H
AatPA R
-I
'H
APA
-I}
(78)
J ajcPj == 2Re {-tr {Cete;G-IC-IAPAHR-I
Q)( G-1C-1A) H)
"APAC-Heje;CHR-lj}
(G-1C-1AQAHc)
x(G-1C-1AQAHC)T}.
(77)
Substituting (68) and (64) one obtains
X (G-1C-1APAHR-1C) T}
-
Jat
+tr {Aa;PAHR-IAjPAHR-I}}.
(G-IC-1APAHR-lC)
= 2 Re { (G-1C-1A( P
(CHR~IA)T
X
The cross-term equivalent of (69) is given by
+tr {CejefC-tC-tAPAHR-t
t.: =
{[(P - Q)(G-1C-1A)H]
G. Gain-Phase Cross- Terms
-APAHC-HC-Ie e~CHR-I}
-Ceje3"C-1C-tAPAHR-I}}
(76)
{(PAHR-IAPAHC-HG-I) x (CHR-1A)T
+(QAHC) x (G-1C-1AQAHA)T}.
{tT {Ceje;C-lC-lAPAHR-1 )
1}
+(PAHR-1C) x (G-1C-1APAHR-1A)T}
+tr {Aa;PAHR-IA"jPAHR-I}} . (69) la'a' == 2Re
j
+tr{Ae I eTPAHR-ICe.eTO-Ic-tAPAR-ll} , j j f
Thus, repeating the considerations leading to (60) we obtain t
(75)
J-Yiaj == 2Re {tr {Ae/e;PAHR- 1
!.w . (67)
Hence AO~i == Ce,ejGFA == Ce,e1FA == CeielG
(74)
(C-1AQAHA)T}.
Substituting (53). (55). and (58). we obtain
and
==
(C-1AQAHA{}
+tr {A"PAHR-IA"JPAHR-I}}.
(66)
6 22 ( ex 2) • " . " • 6,'-'1.\,1 ( ex .\1 )}
X
(CHR-'A)T
X
The cross-terms equivalent of (69) is given by
We first define
I) •
(73)
F. DOA-Gain Cross-Terms
(65)
D. Derivatives with Respect to Gain
~ diag {Gil ( a
}
+J(PAHR-1C) X (C-1APAHR-1A)T}
J = 2Re {(fAPAHR-IAPAHf H) x (CHR-1C)T
X
JJ
J-y
+tr { - Ceje;r APA HR-1CejeJr APA HR- 1} }
-(C-1APAHR-1C)
II
+tr {jAeje;PA HR-lCe jeJC-lAPA HR- 1 }
1} J ¢icPj == 2Re {tr {Ce.eTrAPAHR-IAPAHrHe 1 , j.eTCHRj
"
(72)
J .. ==2Re{tr{-jAe.e!PAHR-IAPAHC-He.e~CHR-I}
64)
Substituting (64) in (60) we obtain
1aiaj
"H
Substituting (64) and (53), (55) we obtain
" T" . T . T -1 Aet>j==CejejFGA==jCejejrA==jCejejC A.
c
-1APA¢jR -I}
+tr {A1'iPAHR-IAcI>jPAHR-l}.
Thus (61) becomes
= 2Re
{
+tr {Ce 1·e!OIC- lAPA HR1
(71) 415
-CejeJC-lAPA HR-1j}}
1
(79)
= 2 Re {tr {AI-tPAHR-lCeje;G-lC-lAPAHR-I}
latt> = 2Re {_j(O-IC-IAPAHR-IAPAHC-fl
x)
1} +tr {A IL PAHR-IAPAHC-HG-le j .e'fCHRj
X (CHR-1C)T +j(O-IC-1APAHR-1C)
X(C-1APA HR-1C) T}
= 2Re {e'fO-1C-1APAHR-IA j IL PAHR-1Ce.j
+e~CHR-IA PAHR-IAPAHC-HO-le.}
= 21m {[ O-IC-1A(P - Q)(C-1A)H]
}
+e;eHR-1Ap.(p - Q)AHC-HO-le j
(80)
X(C-1AQAHC)T}.
p.
and
+CHR-1Ar(p - Q)AHC-HG-'}}
(81)
jr
r are real variables. Using the notation
Jp.tt>j
A~ ~ (aC/ aIL)rA
= C~C-IA
(82)
A r ~ (ac/at)rA
= CrC-1A
(83)
=.., Re {tr {A -
~
P~4 H R - lA IJ.PA H R lAPA:R -
(84)
I} }
= 2 Re {tr {jAp.PA HR-1Ce jeJC-1APA HR- f }
-jeJ~HR-IAp.PA HR-1APA He- H ej}
+ CHR-IAp.PA HR-1APA HC- H}
(85)
{Al-tPA HR-1APA fR-1}.
="'Re{tr{A
p.
(86)
+CHR-1Ap.(P - Q)AHC- H }
+cHR-1Ar(p -
+tr { AI£PAHR-IAPA~R-l} }
PA H R - lAe j
J=
(87)
+eJAHR-IAAIJ.PAHR-IAPej} .
JIJ.~
= [A II'
A 22' ••
"
[1]
(88)
Jr~ = 2Re {diag {QAHArQAHA}
J P.Q). = 2 Re {tr
{A
lAo
lA r (p
PA H R -
- Q)} } .
J~~
l-yr
lad>
Jap.
Jar
Jtt>~
J<pa
J 4Jtt>
JtiJp.
Jtt>r
Jp.~
lp.a
Jp.¢
il-t1L
ip.r
ira
J rtP
ir~
i rr
l~cr
(97)
REFERENCES
= 2 Re {diag { QA HA p. QA HA }
+ diag { AH R -
(96)
Remark: If one or more parameters are assumed known (e.g., gain/phase of a reference sensor) the corresponding columns and rows in J must be removed.
+diag {AHR-lAp.PAHR-IAP}}
- Q)} } .
l-y¢
lOla
-;
A MM]'
= 2Re {diag {PAHR-IAI1PAHR-IA}
+ diag { AH R - lAp. (P
Q) AHC- H}}.
.;
J~~
1} +tr{A IJ.PAHR-lAPe } eTAHR}
Introducing the notation diag ( A) we obtain.
(95)
}
The FIM is given by
{AIJ.PAHR-IAeje;PAHR-I}
~
}
Jrt/J = 21m {diag { -C-1AQA HA r QA He
PAHR-1A , j PAHR- 1 }
= 2 Re {eJJ:PA H R - lA
(94)
= 21m {diag {-C-lAQAHA~QAHC
J~r = 2Re {tr {A~PAHR-IArP44HR-I}
= 2Re {tr
(93)
JIJ.4> = 21m {diag{ -C-IAPAHR-IA",.PAHR-IC
+tr {A~,-P44 HR-IAPA7R-l}}.
-
{AI£PAHR-IAPA~R-I}}
= 2 Re {jere-lAPA HR-1AIJ. PA HR-1Ce j}
I}
J rr = 2 Re {tr {A;-PAHR-IArPAHR-I}
JWI}
= 2 Re {tr {A IL PA H R - lAtPj PA H R - I } +tr
Jp.f/Jj
(92)
+tr{-jA lAo PAHR-IAPAHC-He.er:CHR-l} j }
+ tr { A)J. PA H R -
+ tr
(91)
J ra = 2 Re {diag {G - I C- lA QA H A r QA H C
we obtain
J~I-t
(90)
+CHR-1AIL(p - Q)AHe-HG- 1}}
To simplify the analysis we concentrate here on a circulant matrix with only a single coupling coefficient given by
where
}
JI1Q = 2Re [diag {G-IC-lAQAHAILQAHC
H. Derivatives with Respect to Mutual Coupling Coefficient
p.e
j
= 2Re {eJG-lC-JAQAHAp.QAHCej
x(eHR-1C)T - (G-1e-1AQAHC)
C 12 =
11
(89)
fA Q-j PA H R - I }
+tr {AI£PAHR-IAPA~R-I}} 416
A. Paulraj and T. Kailath, "Direction of arrival estimation by eigenstructure methods with unknown sensor gain and phase," in Proc. IEEE ICASSP'85, Tampa, FL, 1985, pp. 640-643. [21 A. M. Bruckstein, T.-J. Shan, and T. Kailath, "The resolution of overlapping echoes, " IEEE Trans. Acoustics, Speech, Signal Processing, vol. ASSP-33, pp. 1357-1367, Dec. 1985. [3] A. J. Weiss, A. S. WHIsky" and B. C. Levy, "Eigenstructure approach for array processing with unknown intensity coefficients," IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, Oct. 1988. [4] H. Wang and M. Kaveh, "Coherent signal subspace processing for detection and estimation of angles of arrival of multiple
[5]
wideband sources," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 823-831, Aug. 1985.
Y. Rockah and P. M. Schultheiss. "Array shape calibration using sources in unknown iocations - Part I: Far field sources," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 286-299, Mar. 1987. [6] - , " Array shape calibration using sources in unknown locations-Part II: Near-field sources and estimator implementation." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP35, pp. 724-735, June 1987. [7] J. T -H. Lo and S. L. Marple, Jr., "Eigensrructure methods for array sensor localization," in Proc. IEEE ICASSP 1987. Dallas, TX, 1987, pp. 2260-2263. [8] R. O. Schmidt, '''A signal subspace approach to multiple emitter location and spectral estimation," Ph.D. dissertation, Stanford University, Stanford. California, 198 t . [9] A. J. Weiss and B. Friedlander, "Array shape calibration using sources in unknown locations - Maximum likelihood approach." IEEE Trans. Acoust., Speech, Signal Processing. vol. 37, pp. 1958-1966, Dec. 1989. [10] R. O. Schmidt, "Multilinear array manifold interpolanon;' Tech. Memo ESL-TM166J, ESL Inc., Sunnyvale. CA. Sept. 1983. [11] B. D. Steinberg, Principles of Aperture and Array System Design Including Random and Adaptive Array. New York: Wiley, 1976. [12] B. Friedlander and A. J. Weiss. "Eigenstructure methods for direction finding with sensor gain and phase uncertainty." in
Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing.
[13]
Apr. 1988. pp. 2681-2684. (Also. J. Circuits, Systems and Signal Processing, vol. 9. no. 3. pp. 271-300. 1990.) B. Friedlander. A sensitivity analysis of the MUSIC algorithm." IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. H
1740-1751, Oct. 1990.
417
Improving the Performance of a Slotted ALOHA Packet Radio Network with an Adaptive Array James Ward, Member, IEEE, and R. T. Compton, Jr., Fellow, IEEE
Abstract-The use of an adaptive antenna array is presented as a means to improve the performance of a slotted ALOHA packet radio network. An adaptive array creates a strong capture effect at a packet radio terminal by automatically. steering the rece.ive antenna pattern toward one packet and nulling other contending packets in a slot. A special code preamble and randomized arrival times within each slot allow the adaptive array to lock onto one packet in each slot. The throughp~t and delay perform~nce of a network with an adaptive array IS computed by applying the standard Markov chain analysis of slotted ALOHA [1], [2]. It is shown that throughput levels comparable to CSMA are attainable with an adaptive array without the need for stations to be able to hear each other. The performance depends primarily on the number of adaptive array nulls, the array resolution, and the length of the randomization interval within each slot.
A
I. INTRODUCTION
LOHA packet radio communi.cation syste~s are ?f int.erest because they provide a Simple way of multiplexing many users into a single radio channel. In these systems radio terminals transmit packets to each other whenever they have information to send, regardless of whether other terminals may be transmitting at the same time. Because terminals do not coordinate their transmissions, packets from different terminals frequently collide. A collision destroys all packets involved, and these packets must then be retransmitted after a random delay. Collisions limit the maximum throughput at one receiver in an ALOHA system to 18% if the system is unslotted and to 36% if it is slotted [3]. Because of these low throughputs, much effort has been devoted to finding improved packet radio protocols. One wellknown improvement is carrier sense multiple access (CSMA) [4], in which terminals listen to the channel before transmitting to determine if it is busy. If the channel is busy, transmission is delayed until the channel becomes idle. Kleinrock and Tobagi have shown that choosing the retransmission probability carefully in a CSMA system can yield high throughputs [4]. However, the usefulness of CSMA depends on whether all terminals in the network can hear one another. When this is Paper approved by the Editor for CATV of the I~EE Communications Society. Manuscript received February 10, 1990. This work was supported in part by the U.S. Army Research Office, Research Triangle Park, NC. and by the Office of Naval Research. Arlington, VA, under Contracts DAAL0389-K-0073 and NOO014-89-J-I007 with The Ohio State University Research Foundation, Columbus, OH. 1. Ward was with the ElectroScience Laboratory, The Ohio State University. He is now with M.LT. Lincoln Laboratory, Lexington, MA, 02173. R. T. Compton, Jr., is with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212. IEEE Log Number 9106306.
not the case, as in satellite or mobile communications, CSMA is less effective. In the standard slotted ALOHA analysis, it is assumed that if two or more packets arrive in the same slot, none of them is received correctly. In reality. the correct reception of a packet depends not only on whether interfering packets are present, but also on the received power of each packet. Roberts [5] first noted that if one of the packets is of much higher received power than the others, it may still be correctly received. This "power capture" effect improves the throughput and delay performance of a packet radio system. Power capture has been studied by Abramson [3] and Namislo [7] when it occurs naturally as a result of different propagation distances from transmitt~r to receiver and/or channel fading. Lee [6] considered assigning random signal levels to the stations to induce the capture effect. Also. since the received power from a given direction is proportional to the receiver antenna response in that direction. directional antennas can he used to create the capture effect at the receiver. Binder ct al. (K J have considered using directional antennas to resolve potential crosslink conflicts in a multiple satellite packet system. In their work the direction to which an antenna is steered i~ obtained a priori from a form of scheduling used to set up each communication link. Their scheduling procedure. in addition to providing direction information. also reduces the contention somewhat at the expense of increased packet delay. In this paper, we examine the use of an adaptis'C antenna array to create a capture effect and thus improve the performance of a slotted ALOHA system. An adaptive array is an antenna system that controls its own pattern in response to the signal e~vironment [9], [10]. An adaptive array can capture a packet by pointing the peak antenna response toward that packet while simultaneously forming pattern nulls on other interfering packets [11]. An adaptive array can do this automatically without requiring any a priori direction information. Thus. there is no need for prearranged scheduling in a system with an adaptive array and the delay performance should be improved. Furthermore" an adaptive array provides a much stronger capture effect than an ordinary directional antenna, because pattern nulls are placed in the directions of contending packets. We shall show that the use of an adaptive array can provide throughput and delay performance comparable to that of CSMA. Moreover, with an adaptive array there is no need for users to be able to hear each other. In Section II we describe the communication system we
shall consider. Section III gives a brief overview of adaptive
arrays. Section IV describes how an adaptive array can acquire
Reprinted from IEEE Transactions on Communications, Vol. 40, No.2, pp. 292-300, February 1992.
418
ARRAY
OUTPUT
s
Fig. 1.
A single-hop packet radio system.
the first packet to arrive in a slot while nulling subsequent packets in that slot. In Section V we calculate the throughput and delay performance of a packet system using an adaptive array. Section VI presents numerical results. and Section VII contains our conclusions.
Fig. 2.
II. THE
..
+
WEIGrlT FEEDBACK
REFERENCE SIGNAL
F(t)
An adaptive array.
CO~1~lUNICAT[ON SYSTEM ~lODEL
We consider a simple ALOHA system in which a repeater occurs.) The method used to form the antenna pattern is links a network of radio terminals. as shown in Fig. 1. In this described in Section IV below. network terminals transmit messages to each other through Now let us consider this system in more detail. We begin the repeater. We assume time is slotted and that the network in the next section by reviewing the adaptive array concepts uses a slotted ALOHA packet radio protocol. Transmissions needed. between terminals occur randomly in each time slot. Each terminal transmits a packet in a given slot whenever it has Ill. ADAPTIVE ARRAYS one to send, without regard for whether other terminals rnav An adaptive array is an antenna system that controls its own he transmittiug in that same slot. pattern, by means of feedback. while the antenna operates [9], ,\11 packets are transmitted to the central repeater. w hieh [ 12]. [13]. The signal from each element in an adaptive array retransmits them hack to the network. The repeater is assumed is multiplied by a weight and then summed to produce the to be a storc-und-Iorward repeater. It demodulates each packet array output signal. A control system adjusts the weights to and checks it for errors. If there arc no errors. the packet is maximize the signal-to-interference-plus-noise-ratio (SINR) at retransmitted on the downlink. If there arc errors. the packet is the array output. After adapting, the pattern of an adaptive discarded. The repealer downlink is on J different frequency arra y has a beam pointed at the desired signal and has nulls on than the uplink. so both the repeater and the local terminals interfering signals. In a packet radio system, the desired signal can transmit and receive at the same time. Since only the is just the first packet in each slot. The interfering signals are repeater transmits on the downlink. there is no contention on the other packets contending for channel access in that slot. the downlink. Fig. 2 shows an adaptive array with N; elements. The signal Each terminal monitors all downlink packets. By examining the address contained in each packet. a terminal determines i J (t) from element j is multiplied by a weight W j and then whether it is the intended recipient of that packet. A terminal summed to produce the array output signal .~(t). The weights retains packets addressed to itself and discards others. More- are controlled by a feedback system that minimizes the meanover. when a terminal transmits a packet of its own over the square value of the error signal i( t), which is the difference repeater, it listens for that packet on the downlink to determine between the array output /3( t) and a signal r( t) called the if the packet was successfully forwarded. If the packet is not reference signal. The reference signal is a locally generated heard on the downlink. it is assumed that the packet suffered signal that determines which received signals are retained in a collision on the uplink, and the packet is retransmitted after the array output and which are nulled. Minimizing the meansquare value of E'( t) is equivalent (for narrow-band signals) to a delay of some random number of slots. We assume the receiving antenna at the repeater is an maximizing the signal-to-interference-plus-noise ratio (SINR) adaptive array. I The purpose of the adaptive array is to aim the at the array output and causes the array to steer a beam toward repeater antenna pattern at the first packet to arrive in each slot any signal correlated with the reference signal and to null any and then to null subsequent interfering packets in that slot, to signal uncorrelated with it [9]. It may be shown [9] that the optimal (maximum SINR) prevent them from destroying the first packet. This technique array weights are given by will allow one packet to be received successfully, even when several packets arrive in the same slot. (In a conventional (1) ALOHA system, all packets arc destroyed when a collision where W is the weight vector, I The transmitting antenna at the repeater is assumed to cover all the users of the network so that each terminal can hear all downlink packets.
419
W = ['Wi. ui» I
..
,
... ' lWl\T]T Ve ,
(2)
~
is the covariance matrix, (3)
and S is the reference correlation vector,
S == E[X*'i=(t)].
(4)
In these equations, X is the signal vector, i.e., a vector containing the element signals,
X == [1; 1 ( t ) . :1: 2 ( t), . . . ~ :i;.v (t)]T . e,
(5)
E[] denotes expectation. * denotes complex conjugate, and T denotes transpose. The weights in (1) are known as the Wiener weights. A well-known method of controlling the weights in an adaptive array is the sample matrix inverse technique of Reed, Mallett, and Brennan [14]. In this technique, the element signals are sampled periodically in I and (2 (inphase and quadrature) channels and an estimate of the covariance matrix is computed from the sampled signals. If X(j) denotes the value of the signal vector X at sample time l- the sample covariance matrix is computed from /\-
~ == LX*(j)XT(j)
(6)
)=1
where K is the number of samples used. The notation ep is used to indicate that (6) is an estimate of
S ==
t:
L X*(j),-'(j)
(7)
)=1
where r(j) is sample j of the reference signal ,-.( / ). The optimal weights are then estimated by solving the system of equations
~w==s
(8)
for the weight vector. Reed et al. [ 14] have shown that this technique produces an average SINR within 3 dB of the optimal SINR if the samples X(j) are statistically independent and if the number of samples K is approximately twice the number of array elements. When several signals are incident on the array, the reference signal r( t) determines which signals are retained in the array output and which are nulled. Any signal correlated with i'( t) is retained in the array output and any signal uncorrelated with r( t) is nulled [9]. To use an adaptive array in a communication system, the main challenge is to find a way to obtain a reference signal correlated with the desired signal and uncorrelated with the interference. In Section IV we describe a method for doing this with packets. An adaptive array has two limitations that are important for this application. The first is that an array with 1'1(. elements has only N; - 1 degrees of freedom in its pattern [9]. Each null or beam maximum formed by the array requires one degree of freedom. In our case, the array needs to form a beam maximum on one packet and nulls on all other packets in a slot. Thus"
an Nt:-element array using one degree of freedom to form the required beam maximum can also form nulls on up to N = N; - 2 packets. When there are more interfering signals than the available degrees of freedom, the array will not be able to null them all [9]. Another limitation of adaptive arrays is that a given array has only a certain ability to resolve signals in space. If the arrival angles of an interfering packet and the desired packet are too close, the array cannot simultaneously null the interference and form a beam on the desired packet. In this case, the array output desired signal-to-noise ratio drops and the adaptive array may not capture the desired packet. To characterize the resolution capability of an adaptive antenna, we define the resolution width (}" to be the minimum angular separation between two signals at which the adaptive array can place a pattern maximum on one signal and null the other. The resolution width H,. is taken to be Hb/2" where fib is the bcarnwidth of the array, i.e., the angular separation between the first nulls on each side of the mainbeam. H,) depends primarily on the array aperture size but also to a lesser extent on the element patterns and the number of elements. In the analysis below, we relate the performance of the packet radio system to the number of nulls available and to the resolution capability of the array. With this background, we now describe a technique for operating an adaptive array in a packet radio system.
IV.
ACQUISITION
The main difficulty in using an adaptive array in a packet radio system is the acquisition problem, i.e., the problem of forming the beam on the first packet and nulling subsequent packets in the slot. Each packet to be received hy the array will arrive at an unknown time and from an unknown direction. The array must form its pattern on a packet very rapidly ~ in time to receive the message portion of the packet. To allow an adaptive array to do this, we add a special two-part preamble to the beginning of the packet. The first part of this preamble will be used to trigger the acquisition process, and the second part will be used to form the array pattern on the packet. Fig. 3 shows the organization of a packet. A packet will be formed by first adding an address preamble to the beginning of a fixed number of message bits, as shown in the top of Fig. 3. The address preamble will identify the destination terminal and may contain other information such as the originating terminal or a packet number. Next, the combined address and message segments will be encoded with an ('11,. k) linear block code [15], which will be used for error detection at the repeater. Finally, after encoding, an additional two-part preamble will be added to the beginning of the packet. This preamble, called the acquisition preamble, will be used to lock the array pattern on the packet. The acquisition preamble will consist of two consecutive code sequences, called Codes 1 and 2. Code I will he a 13 bit Barker code [16], which has a highly peaked aperiodic autocorrelation function as shown in Fig. 4. Code 2 will be one or more periods of a pseudonoise (PN) code [17]. The periodic autocorrelation function of such a code has a sharp
420
Accress
E=j.-------=---Message
Pream ble
packet
k bits
I
T s,- -
Fig . 6.
I --
Cn,k ) li near code
ACQU ISi tIo n
Pre amble
o=Jl-.--~ CO de I
c:
-
-
-
-
-
-
-
-
Slot width , packet width , and uncertainty interval.
n bit s
Threshold Detector
Encocec ador ess pream ol e ana mess age
Coce 2
Fig . 3
.~o
-
Packet organi zation .
Fig . 7.
13
"
Packet acquisitio n circuitry.
u,
e o
.~
~
s~ 26
13 Fig~ .
lime Shift
Aurocorrclauon tun ction <,I a 13 hit Harke r code .
Aut ocorrelation
Function
Fig. 5.
Autocorrel ation function of a PN code
peak of height .V, at zero shift (and at shifts of any multiple of the code period) and then drops to a constant value of - 1 for shifts over I bit where N, is the code period. as shown in Fig. 5. To allow the packet acquisition. the width of the slot T, will be made larger than the packet width T p by an uncertainty interval T as shown in Fig. 6. To exploit the autocorrelation properties of the preamble codes, the starting times of packet transmissions from all terminals will be randomized over the interval TIL, as in [18]. The uncertainty interval also makes the acquisition process fair (by preventing stations closest to the repeater from always acquiring the repeater first) and gives the designer control over the probability that two packets arrive at almost the same instant. The adaptive array will operate as follows. At the beginning of each slot. when the repeater is ready to acquire a new packet, the array weights will be set so the array pattern covers all users in the net. Such a pattern is easily obtained by turning one array weight on and the rest off. With one weight on, the I"
array pattern is just the pattern of the element that is turned on. This element pattern will be chosen so it covers the entire net. We call this the uniform coverage mode. In this mode, any user can access the system , To acquire an incoming packet, we use the following technique. At the array output is a filter matched to Code 1, followed by a threshold detector. and then a reference signal generation circuit. as shown in Fig. 7. Assume first that only one packet arrives during the slot. With the array in its uniform coverage mode. the incoming packet will pass through the array and into the matched filter. The output of this filter will contain a sharp peak at the end of Code I . This peak will serve as a timing spike to trigger generation of a reference signal during Code 2. The reference signal will be a signal modulated by the same PN code as in Code 2. The timing spike will start the reference signal at the proper time so it is correlated with the received packet during Code 2. The reference signal will continue only during Code 2. The array pattern will be adapted during Code 2. Because the reference signal code is synchronized with Code 2 in the packet, the array will optimize its weights for reception of the packet." At the end of Code 2, the array weights will be frozen. The array pattern will then be held fixed during the address and message portions of the packet. Now suppose two or more packets are received in the same slot. Each of these packets will cause a timing spike at the matched filter output. But only the first timing spike will trigger reference signal generation and begin array adaptation. Timing spikes due to later packets will be ignored by the system, because the acquisition circuit will be designed so that once it has been triggered, it will not trigger again in the same slot. Because the reference signal code will be aligned with Code 2 of the first packet, it will be essentially uncorrelated
421
2 The reference signal does not have to be locked in frequency or phase to the received packet for this process to work. The only requirements are that the PN codes be synchronized to within about one fourth of a code bit, and that the difference between the reference signal frequency and the received signal frequency be less than the reciprocal of the adaptation time [19), [20) .
with the second packet as long as the second packet is at least one bit later than the first. This is so because the autocorrelation function of a PN code has a very low value for shifts of 1 bit or more. (See Fig. 5.) The second packet and all later packets will therefore be regarded as interference by the adaptive array and will be nulled. At the end of Code 2, the array pattern will be optimized for receiving the first packet and will have nulls on later packets. If the second packet arrives less than one bit after the first, the first two packets will be correlated. The adaptive array will not null the second packet in this case and there will be no throughput. In this case we say that the first packet is not acquired. With the uncertainty interval TtL properly chosen, however, the probability of this event is small. The throughput analysis below takes this possibility into account. The uncertainty interval T u and the durations of Codes 1 and 2 will be chosen so that all packets in a given slot begin no later than during the Code 2 preamble of the first packet in the slot. For this reason it is possible to finish adapting the array weights at the end of Code 2 and fix the array pattern during the address and message segments. The adapted pattern at the end of Code 2 will have nulls on the interfering packets, and these will be retained for the rest of the slot. In the analysis below, we assume that the packet SNR is high enough so that if a packet is present, it is always detected by the acquisition circuitry. We also assume that the possibility of a false alarm, i.e., the triggering of a reference signal without the presence of a corresponding packet, is negligible. We assume the array acquires the first packet to arrive in each slot as long as another packet does not arrive in that slot less than one bit after the first. However, even if a packet is acquired, it may still not be successful. An acquired packet will be unsuccessful in either of t\VO cases:
slot until successful, at which time it becomes unblocked and resumes transmitting new packets. Typically, PT' > Pn so that backlogged packets are quickly cleared. At the end of each slot, the downlink transmission provides immediate feedback to the terminals regarding the success of their packets. Let .J'Y k denote the number of blocked terminals at the beginning of slot k. The number of blocked terminals at the end of the slot depends only on the number at the beginning of the slot and the events occurring during the slot. Thus, the time-varying state of the network can be described by a Markov chain, where the state represents the number of blocked terminals. At slot k, the state Xi, can vary between o and M, We shall compute the one-step transition matrix P == [Pt , ) ] and then the equilibrium probabilities of the Markov chain describing this system. In a given slot, there will be a total ofn, == 11,,, + n; packets transmitted where nn and n; are the number of new and previously backlogged packets transmitted in the slot. Given the state ..Yk == i. nil and ti; are independent Bernoulli random variables with distributions
1) when more interfering packets arrive during a slot than the number of available nulls, or 2) when another packet arrives too close in angle to the acquired packet. At the end of each slot, the array is reset into its uniform coverage mode, and the acquisition cycle starts over for the next slot. We now consider the throughput and delay performance of a packet radio repeater using an adaptive array with this acquisition technique.
Oc » (ll '.) l
~ -
[I ..,,-\.k -,.}I ( -?' - ( .\ I I- l) Pit 1-
p I.{ nn --
Pit ) .\ I -
I-I(9)
Thus, the distribution of the total number of packets per slot is
L (21l(sli)(2,·(1 - ·-;Ii). l
Qdlli) ~ Pr{nt == ll"\k == i} ==
( 11)
Let P, (I) be the probability that a packet is successful given that l packets are transm itted in the slot. The success probabilities P., (l), which depend on the adaptive array characteristics and the acquisition parameters, will be determined below. Given P.,(l), the transition probabilities PI.) may be found by enumerating the possible ways that each transition may occur.
• j < i-I. i == 2..... J\1: Not possible, since at most one backlogged packet can be cleared in a slot.
V. THROUGHPUT AND DELAY ANALYSIS
( 12)
To determine the throughput and delay performance, we apply the Markov chain analysis of a slotted ALOHA network [1], [2] to include the effects of the adaptive array and the acquisition process. We consider a finite population of M terminals transmitting to a central repeater equipped with an adaptive array. At the beginning of each slot, each terminal is either blocked or unblocked, depending on whether its previously transmitted packet was unsuccessful or successful. An unblocked terminal transmits a packet with probability Pn in a slot. Only unblocked terminals generate new packets. A blocked terminal retransmits its backlogged packet with probability Pr in each
422
• j == i - I , i == 1,···. M: 1) nn == 0, n.; ~ 1, and one backlogged packet is successful.
Pi . i -
1
==
o; (Oli) L Q,.( iii)?., (I).
(13)
[=1
• j == i 1) n.;
+ k, i ==
0, ... ,lvI, k == 0, ... , AI - i :
== k + 1, ti;
~ 0, and one packet is successful.
2) nn = k: ti; 2: 0, and none of the transmitted packets are successful. Pl,i+k
==
o; (k + Iii) L
(2,.(il,t)
For, I ~ 2, we use the uniform distribution of the transmission times to write
r.(l + k + 1)
I=U
+ (2
rt (
k Ii)
L (2,. (IIi) (1 -
Thus, from (17) and (18),
t-. (l + k)).
[=0
Pa(l)
(14)
=
I,
{
(19)
where u. == Tu/Tb is the length of the uncertainty interval in bits. Once a packet is acquired, two conditions must be satisfied for it to be successful. First, there must be no more than .N == N, - 2 additional packets transmitted in the slot, because the adaptive array can place pattern nulls in at most N directions. Second, no other packet can arrive from an angle within ()b/2 of the acquired packet arrival angle. If this happens, the adaptive array will be unable to resolve the acquired and interfering packets and there will be no throughput for the slot. The P""\l!(l) may be computed as follows. First, we have
This Markov chain analysis is similar to that of Namislo [7]. (Namislo determines the success probabilities for a fading environment by using a Monte-Carlo simulation. We will derive them directly for the adaptive array.) To compute the P, (l), we first note the distinction between acquired packets and successful packets. An acquired packet is one for which the array acquisition circuitry generates a reference signal that is not correlated with any other packets. Note that for a packet to be successful, it must first be acquired by the array. Once a packet is acquired. it is successful only if the adaptive array can form a beam on the acquired packet and place pattern nulls in the directions of the other contending packets. Given that there are l packets in a slot. \V~ characterize each packet by an arrival time t i == 1.··· / within a slot and an arrival angle HI' i = i.···.!. In accordance with the acquisition procedure in Section IV. we assume that the t are i.i.d. random variables uniformly distributed on the uncertainty interval [0. T u ] within the slot. We also assume packet arrival angles are i.i.d. random var iables (independent of the arrival times) uniformly distributed in azimuth [0. 27\] about the central repeater node. Then
(20)
since with only one packet present there are no other packets to interfere with the acquired packet. Moreover, because the adaptive array has only ~\T nulls, we set
I'
l>;.V
I
r.;
:s :s
+ 1.
(21)
To find (I) for :2 l lV + 1. recall that Ol is the arrival angle of the acquired packet and define D 1 [H t - Hb/2. 01 + 01>/2]. Then [).~ia(l)
( 15)
where PI!(l) is the probability that a packet is acquired given 1 packets arc incident. and P.. . la (I) is the probability that a packet is successful given it is acquired and l packets are present in the slot. The ~L (I) depend on the arrival times and the length of the uncertainty interval. while the !)"lfL (/) depend on the arrival angles, the resolution capability of the adaptive array, and the number of available nulls. With the preamble code structure described in Section IV, the first packet in a slot is acquired as long as all subsequent packets in that slot arrive at least one bit duration Tv later than the first packet. If the first packet is not acquired, no packets are acquired for that slot. Thus,
Fa(I) = l P r { t 2 > i. 1
1 == 1 1
(1 _ ~ )1; 1 >
+ Ti; t J > f 1 + Ti; . . . . . t I > t 1 + Tv} ( 16)
== Pr{H 2 ~ D 1 · f) 3 ~ D l · · · · .H[ ~ D l } == E o1[Pr{ t1 2 ~ D 1·03 ~ D1 ... ·.O[ ri D1IB l
= Eli,
[g
Pr{ H,
~ D1IHd]
}]
(22)
where Eel [] denotes an expectation over the random variable H1, and we have taken advantage of the independence of the arrival angles. However,
Pr{ H t
~ D1IBd =
(1 - ~~ ).
(23)
which is independ-ent of Ol. Thus, (22) becomes 2~ j SN
+ 1.
(24)
Hence, from (15), (19), and (24), the success probabilities are
where the factor of I accounts for the fact that any of the l packets transmitted can be the first packet in the slot. If only a single packet is transmitted in a slot, it is acquired, so (17)
423
1
[=0
1~
l=1
O'
P~(l) ==
{
(1 - -1 )l( 1-~ (} )l-l~ U
27T"
0;
2~l~N+1·
1> N
+1
(25)
Given that the system is in state i, the probability of a successful packet transmission is the conditional throughput S(j) , given by
0.9
M
S(j)
=L
Qt(lJj )Ps(l ).
(26)
1=1
The average number of new packets entering the system state j is
In
(27) The Markov chain described above is irreducible. Since we assumed a finite population, all states are recurrent non-null. The states are also aperiodic. Consequently, this Markov chain has a limiting distribution denoted by 11"
= [1l" (0), 1l"(I), ·· · , 1l" (M )]
j, the number
(a)
0.9
(28)
where
1l"(j) = Pr{X=
.
= j} = lim,,_oc Pr{X k + n = jlXk = i }.
~
f
(29) The steady-state probabilities are found by solving the linear system of equations [21] 11"
= 1l"P
§
(30) 10
along with the constraint that
L 1l"(j) = 1.
(31)
25
(32)
) =0
First we examine the conditional throughput S(j ) of systems with and without an adaptive array. We consider a network of 50 users. We start with an example where P71 0.002 and p; 0.2. For this case. lvl p" 0.1, which is a low traffic situation where slotted ALOHA may typicall y be used. Fig . 8(a) shows the conditional throughput S(j ) and the new packet input rate Sin(j) versus the state j . Curves for various numbers of adaptive array nulls are also shown. For these curves we have Bb = 10° and u = 62. There is a significant increase in conditional throughput as the adaptive array is added and the number of nulls is increased. Also, note that there is a fixed number of nulls above which little further improvement is gained. The stability problems of ALOHA systems have been well documented [1], [2], [22]. The finite population ALOHA model is said to be stable if there is a single intersection point of the S(j ) and Sin(j) curves and this intersection point is in a region of low delay. In Fig . 8 we have intentionally chosen Pr high enough so that the system without an adaptive array is unstable. The curves with an adaptive array are stable. Moreover, for an adaptive array with 4 nulls or more. the
=
and the average throughput is M
S(j )1l" (j ).
(33)
j=O
In the steady state, the average input rate equals the average throughput, so (34) We use Little's theorem [23] to express the average delay D experienced by a new packet as
B
B
S in
S
D= =- = = .
-10
U» \ r~
VI. RES ULTS
AI
B = Lj1l"(j ).
35
:ilJ
Fig.8. Cond itional throughput comparison . For the curves with an adapt ive array: 0. = 10", U = 62. (a) M = 50, P» = 0.002, p, = 0.2. (b) M = 50, P. = 0.006. Without the adaptive arr ay, p, = 0.1; p, = 0.2545 with the adaptive array.
Once the 1l"(j ) are found. they can be used to determine the average throughput, delay, and backlog of the system. Given 7r(j), the average number of blocked terminals B is
-
:0
(b)
j =O
=L
15
J, Ihe num ber o f blocked
;\I
S
orblocked users
(35)
We now use these results to examine the performance of a slotted ALOHA system with an adaptive array .
424
=
=
0.9
200
S.(j)
180 160
~
!=
= ;; ....
.e
.
,., " Q
-=
~
r <
;:
~
;;; 0.1 0' 0
100
Neu-no AA
10
N=6
120
~
;:
N=O--no AA
140
80
j
60
40 20
15
20
j, the number
25
30
or blocked
40
35
45
0
50
I
--=:==::;.=====-_~~~_J 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
users
New transmission probability (packets/slot)
Fig. 'I . Co ndit iona l through put for a network of 50 users with I' " as the number of adaptive array nulls is varied. 1//, 100 and /I
=
=
0 .11l 8.
value of PI! could be raised substan tially without introducing instability. Thus, it is seen that the adaptive array has a stabilizing effect on the system . In Fig. 8(b) we compare two stab le cases for fi " = O.OO(j. In eac h case we have chosen the largest retransmission probability possible for stable operation . Without the adaptive array. fi r is set to 0.1. which results in an average throughput of S = O .2~i::\ packets/slot. an average backlog of 1] = 2.0l users. and an average delay of D = (j .'l) slots/packet. The maximum possible average throughput is .\ / /1" = (U packets/slot. With an adaptive array . /1, is set to 0.2)-15. resulting in S = O.2!JV. IJ = (Ull. and D = 1l.:-:0(j slots. For such low traffic scenarios. the adaptive array provides only a slight increase in throughput but a marked improvement in the delay performance . The main advantage of using an adaptive array in an ALOHA network is the abil ity to handle much higher traffic rates and operate at a much higher throughput than is possible in a standard ALOHA system. In Fig. 9. we consider a case with P» = (J.(lli::\. so that on average . more than .\! l! » = O.V packets (new plus back logged) are transmitted per slot. We fix [I ,. = 0.2. To have a stable system. the adaptive array needs at least 5 nulls. For :V 2: ;j. the average throughput is O.~ packets/slot. This example shows how performance can be improved by increasing the adaptive array capabilities . We note that a throughput of (J.~ is comparable to typical values attainable by CSMA [24 ], and with slotted ALOHA under other capture mechanisms [5)-[7] . In general, performance improves as the number of adaptive array nulls increases or as the array bcarnwidth is reduced. Increasing the number of avai lable nulls allows more collisions to occur without reduci ng the number of successful packets. Reducing the array beamwidth allows the array to successfully null interfering packets over a larger angular region. Performance is also improved as the length of the uncertainty interval is increased. (Of course, a longer uncertainty interval requires a longer slot width and reduces the number of message bits transmitted per unit timc.) As the adaptive array capabi lities (resolu tion. number of nulls) are increased, average throughputs close to unity can be approached. The
(a)
:[ ~
= 62.
~
,
0.7 t
~
I
;
/
I
= rL ~
,e.0.6 0.5
~
~
~
-<:
0.3 f
0.1 : / 0 ''
o
/
\ N=6 ,
\
\
\
/,/
0.4 ~
I
"
/
1
0.2[.
/
/
/
I
\
/
-,
\'
N=O--no AA
! '
. 0.01
0.005
0.015
0.02
0.025
O.oJ
0.035
0,04
New transmission probability (packets/slot)
(b)
/---- - --- --- -.---=.:=-=-.=-= -===C0'l
50 ~ I - - -- ..-- - -..-.--- ----45
r
40 t ~
35 ~
I
3O ~
-g
!
[ <
I I I
I
25
::
N=O"no AA
~
I
~I
': L_==-===-_~__~_~_0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
New transmission probability (pack"!tslslot)
(c)
n.
Fig. 10. Average S. D performance . Without the adaptive array. I' , 0.1. With the adap tive array. IJ,· 0 .2. lib 10°. 11 62. (a) Average dela y versus /,,, . (b) Average throughput versus P» - (c) Average back log vers us u« .
=
=
=
=
limiting case of fh = 0°, 1L = oo(n = 0), N = M - 1 correspo nds to perfect capture where one packet is successful in eve ry slot in which at least one packet is transmitted. Fig. 10 compares the average delay, throughput, and backlog performance of systems with and without an adaptive array for the case fh = 10°, 'U = 62, and N = G. The retransmission probability Pr is 0.1 withou t the adaptive array and 0.2 with the
425
array. These curves were obtained by varying Pn and computing S, D, and B from (32)-(35). We see from Fig. 10(a) that the delay with the adaptive array is always better than without it, and the difference is greater as Pn is increased (as the input traffic is increased). Fig. 1O( b) shows the average throughput. For low traffic, both systems are stable and provide nearly the maximum possible throughput. However, the system without the adaptive array becomes unstable at relatively small Pn while the throughput with the adaptive array keeps increasing, to a maximum of near 0.83. Finally, if Pn is increased too far, the system with the adaptive array also becomes saturated and the network becomes highly backlogged. The average backlog for the two cases is shown in Fig. lO(c). Again, these curves indicate that by using an adaptive array, we can achieve acceptable delay at throughput levels that are much higher than are possible in a standard ALOHA system. VII.
[8] R. Binder, S. D. Huffman, I. Gurantz, and P. A. Vena, "Crosslink architectures for a multiple satellite system:' Proc. IEEE, vol. 75. pp. 74-82, Jan. 1987. [9] R. T. Compton. Jr., Adaptive Antennas-Concepts and Performance. Englewood Cliffs, NJ: Prentice-Hall, 19HK [10] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wilev, 1980. [II] M. Azizoglu, R:T. Compton, Jr., F. D. Garber, G. M. Huffman. and H. C. Yu, Adaptive arrays in satellite packet radio communication systems," Final Rep. 718163-1, The Ohio Stale University, Dcp. Elee. Eng., ElectroSci. Lab., Nov. 1987. [12) S. P. Applebaum, "Adaptive arrays." IEEE Trans. Antenna."; Propagat.. vol, AP-24, pp. 585 -598, Sept. 1Y76. [13] B. Widrow, P. E. Mantey, L. J. Griffiths. and B. B. Goode. "Adaptive antenna systems," Proc. IEEE, vol. 55, pp. 2143-2I5Y, Dec. 1967. [14] I. S. Reed, J. D. Mallett. and L. E. Brennan, "Rapid convergence rate in adaptive arrays," IEEE Trans. Aerospace Electron. Svst .. vol. AES-IO. pp. 853-862, Nov. 1974. [15] S. Lin and DJ. Costello, Error Control Coding: Fundamentals and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1983. [16] M. I. Skolnik, Radar Handbook. New York: McGraw-Hill, 1970. [17] S. W. Golomb, Shift Register Sequences. San Francisco, CA: HoldenDay, 1967. [18] D. A. Davis and S. A. Gronemeyer, "Performance of slotted ALOHA random access with delay capture and randomized time of arrival," IEEE Trans. Commun., vol. COM-28, pp. 703-710, May 1980. [19] D. M. DiCarlo and R. T. Compton, Jr., "Reference loop phase shift in adaptive arrays," IEEE Trans. Aerospace Electron. Syst., vol. AES-14, pp. 599-607, July 1978. [20] D. M. DiCarlo, "Reference loop phase shift in an n-element adaptive array," IEEE Trans. Aerospace Electron. Syst., vol. AES-15, pp. 576-582, July 1979. [21] E. Cinlar, Introduction to Stochastic Processes. Englewood Cliffs, NJ: Prentice-Hall, 1975. [22] D. Bertsekas and R. G. Gallager, Data Networks. Englewood Cliffs, NJ: Prentice-Hall, 1987. [23] J. D. C. Little, "A proof for the queueing formula: 1 = AW," Oper. Res., vol. 9, pp. 383-387, May 1961. [24] L. Kleinrock, Queueing Systems, Volume II: Computer Applications. New York: Wiley, 1976. U
CONCLUSION
In this paper we have shown how an adaptive antenna array may be used to improve the performance of a slotted ALOHA packet radio network. The adaptive array creates a capture effect by separating packets in angle and thereby preventing collisions at the receiver. We described how an adaptive array could be used in a slotted system and analyzed the performance of such a system. Typical performance results were presented. It was shown that this technique achieves a performance level comparable to CSMA. Unlike CSMA.. however, a slotted ALOHA system with an adaptive array does not require that all users be able to hear each other in order to attain high throughput. The performance is determined primarily by the array resolution, the number of nulls, and the length of the uncertainty interval in each slot. ACKNOWLEDGMENT
Significant contributions to this work were made by M. Azizoglu, Dr. F. D. Garber, G. M. Huffman, and Dr. H. C. Yu under a previous NASA contract [11] at The Ohio State University. The authors gratefully acknowledge their contributions. REFERENCES
[1] L. Kleinrock and S. S. Lam, "Packet switching in a multiaccess broadcast [2]
[3] [4]
[5] [6] [7]
channel: Performance evaluation," IEEE Trans. Commun. vol. COM-23, pp. 410-422, Apr. 1975. A. B. Carleial and M. E. Hellman, "Bistable behavior of ALOHA-type systems," IEEE Trans. Commun., vol. COM-23, pp. 401-410, Apr. 1975. N. Abramson, "The throughput of packet broadcasting channels," IEEE Trans. Commun., vol. COM-25, pp. 117-128, Jan. 1977. L. Kleinrock and F. A. Tobagi, "Packet switching in radio channels: Part I-Carrier sense multiple access modes and their throughput-delay characteristics," IEEE Trans. Commun., vol. COM-23, pp. 1400-1416, Dec. 1975. L. G. Roberts, "ALOHA packet system with and without slots and capture," Comput. Commun. Rev., vol. 5, no. 2, pp. 199·-204, Apr. 1975. C. C. Lee, "Random signal levels for channel access in packet broadcast networks," IEEE J. Select. Areas Commun., vol. SAC-5, pp. 1026-1034, July 1987. C. Namislo, "Analysis of mobile radio slotted ALOHA networks," IEEE Trans. Vehic. Technol., vol. VT-33, pp. 199-204, Aug. 1984.
426
Signal Acquisition and Tracking with Adaptive Arrays in the Digital Mobile Radio System IS-54 with Flat Fading Jack H. Winters, Senior Member
Abstract- This paper considers the dynamic performance of adaptive arrays in wireless communication systems. With an adaptive array, the signals received by multiple antennas are weighted and combined to suppress interference and combat desired signal fading. In these systems, the weight adaptation algorithm must acquire and track the weights even with rapid fading. Here, we consider the performance of the Least-MeanSquare (LMS) and Direct Matrix Inversion (DMI) algorithms in the North American digital mobile radio system IS-54. We show that implementation of these algorithms permits the use of coherent detection, which improves performance by 1 dB over differential detection. Results for two base station antennas with flat Rayleigh fading show that the LMS algorithm has large tracking loss for vehicle speeds above 20 mph, but the DMI algorithm can acquire and track the weights to combat desired signal fading at vehicle speeds up to 60 mph with less than 0.2 dB degradation from ideal performance with differential detection. Similarly, interference is also suppressed with performance gains over maximal ratio combining within 0.5 dB of the predicted ideal gain.
A
I. INTRODUCTION
NTENNA arrays with optimum combining reduce the effects of multipath fading of the desired signal and suppress interfering signals, thereby increasing both the performance and capacity of wireless systems. To be practical, though, the implemented combining algorithms must be able to rapidly acquire and track the desired and interfering signals. Most previous theoretical and computer simulation studies of the increase in performance and capacity with optimum combining, e.g., [1]-[6], assumed ideal tracking of the desired and interfering signals. In the computer simulation study where block-by-block adaptation was considered [7], the data rate was at least 5 orders of magnitude greater than the fading rate. Although this is appropriate for the indoor radio system studied in [7] which used kbps data rates at 900 MHz, digital mobile radio systems have a much lower data-to-fading-rate ratio. For example, in the North American digital cellular system IS54 [8] with a data rate of 24.3 ksymbols/s in the 800 MHz band, at 60 mph the data-to-fading ratio is only 300, while in the Western European GSM [8] it is around 2 000. In a previous experiment [4]-[6] that demonstrated the feasibility of optimum combining with a three-fold increase in capacity (suppression of two equal-power interferers with eight antennas), the Least-Mean-Square (LMS) algorithm tracked these Manuscript received September 8, 1992; revised October 26, 1992. The author is with AT&T Bell Laboratories, Holmdel, NJ 07733. IEEE Log Number 9211016.
signals with data-to-fading-rate ratios as low as 25. However, the tracking error loss could not be measured because of NO quantization noise. Furthermore, this experimental system had many more antennas than interferers, which is not typical of most wireless systems. Here we consider the dynamic performance of adaptive arrays in wireless communication systems. Specifically, we consider the performance of the LMS and Direct Matrix Inversion (D MI) algorithms in tracking the desired and interfering signals in the digital mobile radio system IS-54. We show that implementation of these algorithms permits the use of coherent detection, which improves performance by 1 dB over differential detection. Results for two base station antennas and flat Rayleigh fading show that the LMS algorithm has large tracking loss at speeds above 20 mph. However, the DMI algorithm can acquire and track the weights to combat desired signal fading at vehicle speeds up to 60 mph with less than 0.2 dB degradation from the ideal (perfect tracking) performance of optimum combining with differential detection. Similarly, interference is also suppressed with performance gains over maximal ratio combining within 0.5 dB of the predicted ideal gain. In Section II, we determine the performance of optimum combining with ideal signal tracking. In Section III we study the performance of the LMS and DMI algorithms for acquisition and tracking of the signals in IS-54. A summary and conclusions are presented in Section IV. II. IDEAL PERFORMANCE
A. Weight Equation Fig. 1 shows a block diagram of an M antenna element adaptive array. The complex baseband signal received by the ith element in the kth symbol interval Xi (k) is multiplied by a controllable complex weight ui, (k). The weighted signals are then summed to form the array output So (k). The output signal is subtracted from a reference signal r ( k) (described in Section III) to form an error signal E(k). Weight generation circuitry determines the weights from the received signals and the error signal. In this paper, we are interested in determining the weights that minimize the mean-square error, i.e., IE 2 (k)l. Let the weight vector w be given by
(1)
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 42, No.4, pp. 377-384, November 1983.
427
We define the received desired signal to noise ratio p as So(k)
l - - -.......- - - - . .
i == 1,···,M
ARRAY OUTPUT
(9)
the interference-to-noise ratio (INR) as i -
+
E(k)
r(k) REFERENCE SIGNAL
~ ----~
L - - -_ _......
SINR
Block diagram of an M element adaptive array.
where the superscript T denotes transpose, and the received signal vector x is given by (2)
L
Xd
+ Xn + L
(3)
Xj
j=l
where Xd, X n , and x i are the received desired signal, noise, and jth interfering signal vectors, respectively, and L is the number of interferers. Furthermore, let S d (k) and S j (k) be the desired and j th interfering signals, with
E[s~(k)] == 1
(4)
E[s](k)] = 1 for 1 < j ::; L.
(5)
Then x can be expressed as L
X
=
UdSd(k)
+ Xn + L
(6)
ujsj(k)
j=1
where Ud and Uj are the desired and j th interfering signal propagation vectors, respectively. The received signal (desired-signal-plus-interference-plusnoise) correlation matrix is given by
Rxx = E [
(Xd + Xn +
t Xi) (Xd X t Xi) T] +
*
n
+
1=1
J=1
(7) where the superscript * denotes complex conjugate and the expectation is taken with respect to the signals sd(k) and S j (k). Assuming the desired signal, noise, and interfering signals are uncorrelated, the expectation is evaluated to yield
R x x ==
uduI + a I + L ujuJ 2
(10)
L
(8)
j=l
where (J2 is the noisepower and I is the identity matrix. Note that R x x varies with the fading and that we have assumed that the fading rate is much less than the bit rate.
==
P
l+INR·L
(11)
where lLdi and Uji are the ith elements of Ud and 11,j, respectively, and the expected value now is with respect to the propagation vectors. The equation for the weights that minimize the mean-square error (and maximize the output SINR) is [9]
The received signal consists of desired signal, thermal noise, and interference and, therefore, can be expressed as
x =
1 to M, j == 1 to L
and the signal-to-noise-plus-interference ratio (SINR) as
ERROR SIGNAL
Fig. 1.
:=
- 1U,d* w == R xx
(12)
where the superscript -1 denotes the inverse of the matrix. Note that scaling of the weights by a constant does not change the output SINR. In (12), we have assumed that exists. If not, we can use x x is nonsingular so that pseudoinverse techniques [10] to solve for w. These optimumcombining weights are the same as those in [5], as shown in Appendix A.
R
R;;
B. Optimum Combiner Performance We determine the performance of ideal optimum combining in the digital mobile radio system IS-54 in the fcllowing manner. We first determine the bit error rate (BER) with the IS54 modulation technique, 1r /4-shifted differential quadrature phase shift keying (DQPSK), for ideal maximal ratio combining. With maximal ratio combining, the received signals are combined to maximize the signal-to-noise ratio at the array output, which is the optimum combining algorithm when interference is not present. Analytical results are presented for both differential detection and coherent detection, since both cases are studied in Section III. We then determine the reduction in the receive SINR required for a given BER, with optimum combining as compared to maximal ratio combining when interference is present. This gain with optimum combining is determined using analytical results with one interferer and Monte Carlo simulation with L 2: 2. Although these gains are generated only for coherent detection of binary PSK (BPSK), these results are also applicable to both coherent and differential detection of DQPSK, but at different BER's. This is because, for given receive SINR, the output SINR with optimum combining is independent of the modulation and detection technique, as can be seen from the equations in Section II-A. Thus the gain with optimum combining and BPSK at a given receive SINR will be similar to that of DQPSK at the same receive SINR-only the corresponding BER will be different. Note that IS-54 specifies a maximum BER of 2 x 10- 2 , and, therefore, our results are generated for BER's around 10- 2 .
428
With cohe rent det ection of DQP SK and maxim al ratio combining, the ave rage BER wi th flat Rayl eigh fading is approx ima tely given by --
B ER
= 2PE -
2
PE
10- 2
(13)
where (fro m [11])
PE
=T
M
(
1-
J
10- 4
2:
p)
U
1 k -k +
·t,C
a: w
M
[J::J
)T"(I+V :p)" 2
10- 6
( 14)
BER
=
1
PEh ) = e-"!
o
(15)
w here PEb ) is given by (fro m [12 ])
[~ loh/v'2) + ~(V2 - 1)" h h /V2)]
"'(' \if - I e -'ll P
....L.
-'-
10
20
30
.--J
40
Fig. 2. Average BER versus E,./ No for co herent and diff erential detectio n of DQPSK and DBP SK with M = 1 and 2 wit h flat Rayleigh fading and maxi ma l ratio combining.
12 , - - -- - , - - - - , - - - - . . - - - - - - ,
10
1 Interferer
(17)
No te that wi th differential detection of differ ent ial BP SK (DBPSK), the av erage BER is 1/2 (1 + p)-M . Fig. 2 shows the ave rage bi t erro r rate ve rsus p (SINR wit h INR - 00 dB ) for DQPSK and M 1, 2 w ith maxim al rat io combining. Results are also shown for DBPSK, w hich requir es 3 dB low er p for th e same BER with co herent detection . Note that for M = 1, w ith both DQPSK and DBPSK, differ enti al detection requires a 0.4 dB high er p for a given BER than coherent detection. For 1\1 2, differential detection of DBPSK requires a 0.7 dB high er p than co herent de tect ion , w hile differentia l det ecti on of DQPSK requires a 1.0 dB high er p than co herent detection. Differential dete ction of DQPSK requ ires a 11.2 dB SINR for a 10- 2 BER . Now , let us consider the BER wi th idea l optimum co mbining. The BER w ith optimu m co mbi ni ng and flat Rayleigh fading in the pre sen ce of noise only is given by the results above for maximal ratio co mbining . With one interferer that also ex periences flat Rayleigh fading , the BER for co herent de tect ion of BPSK is given by [1, eq. (25)]. For mu ltip le interferers wit h flat Ra yleigh fa ding, this BER ca n be determi ned by Mo nte Ca rlo si mulation as de scribed in [1] . Fig. 3 shows the gai n in dB of idea l optimum co mbining over maxim al ratio co mb ining with two antennas for one to six interferers ve rsus the interference -to- noise ratio (INR). This ga in wa s determ ined fro m the reduc tion in the requ ired SINR for a 10- 3 BER at the receiver wi th cohe rent detection of BPSK. The ga in occurs because optimum co mbining is suppressi ng interfere nce in additio n to increasing desired signa l-to-noise rat io. A 10- 3 BER was chose n beca use the
=
..L-
P (S INR with INR =-oodB) (dB)
(16) w here h is the kth orde r modified Bessel fun ction of the first ki nd, and p(I) is the probabil ity densi ty of the signal-to- noise ratio afte r maximal ratio co mbin ing and is giv en by [131
ph ) = pM(jVJ - I )!'
- - - - Differential Detection
10-10 ' -
00
pb )PEb ) d-y
- - Coherent Detection
10-8
W ith differential detection of DQPSK and maxim al ratio combining, the average BER w ith flat Ra ylei gh fadin g ca n be s how n to be given by
=
_ - - - - - -\ 2
::::=.---------1 ~ 5 6
=
5
10 INA (dB)
15
20
Fig. 3. Gain in dB of ideal optimu m combi ning over max imal ratio combining wit h two ante nnas for one to six interferers versus INR at a 10- 3 BER for co herent detection of BPSK.
11.1 dB SIN R requ ired for a 10- 3 BER w ith m axim al ratio co mbining and coh erent detection of BPSK [11], [14 ] is close to the 11.2 dB SINR requ ire d for a 10- 2 BER w ith maximal ratio co mbining and differen tial de tec tio n of DQPSK. T hus Fig. 3 also shows the gai n for a 10- 2 BER w ith diffe rent ial de tection of DQPSK. As shown in [1], the ga in does no t va ry significantly for BER ' s between 10- 2 and 10- 3 . W ith two ante nnas , optimum co mbining ca n co m pletely suppress one interferer. T hus the maximum gain w ith optim um co mbi ning and one int er ferer is 10 IOg lO (10INR/ 10 + 1) dB , w hich is approximately INR for large INR. How ever , this ga in can only be achieved w ith out desir ed sign al fading . With fading , as show n in [4)-[61, the complete suppress ion of one interferer results in the loss of one order of div er sit y aga inst
429
multipath fading, which corresponds to a 12.9 dB increase in the SINR required at a 10- 3 BER [11], [14]. Thus to achieve gain, optimum combining must trade off a partial loss in diversity improvement for partial interference suppression. The resulting gain is approximately half (in dB) the maximum gain possible without desired signal fading. With more than one interferer and two receive antennas, the gain is seen to be much lower. However, the gain is almost 1 dB even with six interferers. III. PERFORMANCE OF LMS AND DMI IN IS-54 In the digital mobile radio system IS-54, the frequency reuse factor (number of channel frequency sets) is 7. However, as shown in [6], it may be possible to reduce the frequency reuse factor to 4 (nearly doubling the system capacity) through the use of optimum combining of the signals from the two existing receive base station antennas. However, for this result in [6], we assumed ideal optimum combining, i.e., perfect tracking of the desired and interfering signals by the combining algorithm at the base station. Below, we consider the dynamic performance of optimum combining in IS-54.
A. Weight Generation The weights can be calculated by a number of techniques. Here, we will consider two techniques: the Least Mean Square (LMS) and the Direct Matrix Inversion (DMI) algorithm [9]. For digital implementation of the LMS algorithm, the weight update equation is given by
w(k
+ 1) == w(k) + J-LX*(k)E(k)
(18)
where J-L is a constant adjustment factor, x (k) is the received signal vector in the kth bit interval, and the error is given by
E(k) = r(k) - .so(k)
(19)
so(k) == w T x(k).
(20)
where
With DMI, the weights are given by [9] (21) where the estimated receive signal correlation matrix is given by K
n: = l/KL x * (j )x
T
(j )
(22)
j=1
where K is the number of samples used, and the estimated reference signal correlation vector is given by K
rXd ==
l/KL x* (j )r (j ).
(23)
j=l
Note that, as before, we have assumed that tc; is nonsingular. If not, pseudoinverse techniques can be used [10]. The LMS algorithm is the least computationally-complex weight adaptation algorithm. However, the rate of convergence
to the optimum weights depends on the eigenvalues of R x x , i.e., on the power of the desired and interfering signals [9]. Thus weaker interference will be acquired and tracked at a slower rate than the desired signal, and the desired signal will be tracked at a slower rate during a fade (when accurate tracking is most important). The OMI algorithm is the most computationally-complex algorithm because it involves matrix inversion. However, OMI has the fastest convergence, and the rate of convergence is independent of the eigenvalues of R x x , Le., signal power levels. One issue with the OMI algorithm is its modification for tracking time-varying signals. Here we consider calculating the weights at each symbol interval using one of two data weighting functions: 1) a sliding window (fixed K in (22) and (23) and 2) an exponential forgetting function on ii., and rXd' namely,
rxx(k
+ 1) == {3r x d(k) + x*(k)r(k)
(25)
where 13 is the forgetting factor. For M == 2, the OMI algorithm has about the same computational complexity as the LMS algorithm. In particular, weight calculation from the inversion of the 2 x 2 correlation matrix (21) does not even require division by the determinant, since this is only a weight scale factor that does not affect the output SINR. For larger M, since the complexity of matrix inversion grows with M 3 (versus M for LMS), DMI becomes very computation intensive. However, the matrix inversion can be avoided by using recursive techniques based on least-square estimation or Kalman filtering methods [9], which greatly reduce complexity (to the order of M 2 ) but have performance that is similar to DMI" [~l Similarly, pseudoinverse techniques [10] can be used if R x x does not exist. Therefore, our performance results for OMI should also apply to these recursive techniques. N ext, consider reference signal generation. Since this signal is used by the adaptive array to distinguish between the desired and interfering signals, it must be correlated with the desired signal and uncorrelated with any interference. Now, the digital mobile radio system IS-54 [8] uses time division multiple access (TDMA), with three user signals in each channel and each user transmitting two blocks of 162 symbols in each frame. For mobile to base transmission, each block includes a 14-symbol synchronization sequence starting at the 15th symbol. This sequence is common to all users in a given time slot (block), but is different for each of the six time slots per frame. Since base stations operate asynchronously, signals from other cells have a high probability of having different timing (since there are 972 symbols per frame) and being uncorrelated with the sequence in the desired signal. Thus as proposed in [6], for weight acquisition we will use the known 14-symbol synchronization sequence as the reference signal. DMI is used to determine the initial weights using this sequence, since accurate initial weights are required. Note that the weights must be reacquired for each block, because with a 24.3 ksymbols/s data rate and fading rates as high as 81 Hz, the
430
fading may change completely between blocks received from a given user. After weight acquis ition, the output signal consists mainl y of the desired signal, and (during proper operation) the data is detected with a bit error rate that is not more than 10- 2 to 10 - 1 . Thu s we can use the detected data as the reference signal, using eithe r the LMS or OMI algorithm for tracking.' In our simulation results shown below, we did not consider the effec t of data errors on the reference signal; i.e., the reference signal symbols were the same as the transmitted symbols. Note that since the modulati on technique is OQPSK, the error of interest is only the relative phase between adjacent symbols, rather than the error vector r (k) - So (k) in (19). Indeed , the LMS algorithm can use the phase error of each symbol, i.e., / r (k) - / so(k ), where lv is the phase of y, as the error signal. ? This results in no amplitude control of so(k ), but the amplitude is not used for OQPSK detection anyway. However, we found better trackin g with the error vector (19) and, therefore , used (19) for our results shown below. Note that with the OMI algorithm we do not have the option of using the phase error- we must use the error vector (19).
B. Results To determin e the performance of the acquisition and tracking algorithms in IS-54, we used IS-54 computer simulation programs written by S. R. Huszar and N. Seshadri . We modified the transmitter, fading simulator, and receiver program s for flat Rayleigh fading with one interferer and added our optimum combining algorithms with both coherent and differential detection. Specifically, the transmitted desired signal consisted of blocks of 162 symbols with 7r / 4-shifted OQPSK modulation. The symbols in each block were random ly generated 2-bit symbols for symbols 1- 14 and 29- 162, and a synchronization sequence for symbols 15-28. This signal, sampled at 8 times the symbol rate, was filtered by a square root cosine rolloff filter with a rolloff factor of 0.35. For the interfering signal, randomly generated symbols, independent of the desired signal symbols, were used for the data, and a synchronization sequence that is orthogonal to that to the desired signal was used for symbols 15- 28. The relative timing of the interfering and desired signals was adjustable in increments of 1/8 of the symbol duration. Independent, flat Rayleigh fading for each signal at the two receive antennas was generated by multiplying each signal by a complex Gaussian random number, which varied at the fading rate [13]. The received signals were then weighted, combin ed, and filtered by a square root cosine rolloff filter, followed by coherent or differential detection. Let us first consider the performance with OMI for acquisition and LMS for tracking with differential detection without interference. Fig. 4 shows the BER versus SINR for vehicle [ We do tracking in each block (starting from the synch roniza tion seq uence) in the forward direc tion for symbols 29 to 162, and in the reverse direct ion for symbols 14 to I. 2 Since OQPSK also has consta nt amp litude, the consrant modulus algorithm can also be used to ge nerate an error signal, i.e., e(k ) = 8 0 (1.: ) so(k )/l so(k )l , fo r the LMS algorithm, as shown in [15j . A reference signa l is, therefore, not needed , but this means that the rece iver can acquire and track an interfering signal rather than the desi red signa l. and, therefore. the algorithm cannot be used for optimum co mbi ning when interferenc e is prese nt.
INR =
lO,t
ffico
·~ d B
10,2 Ideal Maximal
Ratio Combining OOmph - Differential Detection
10.3
60 mph . Coherent
Detection
Omph . Ditter entrat Detection
Omph - Coherent Detection
10-4 '--- - - - ' - - - - - - - ' - - - - --'--------' 10 o 20 SINR (dB)
Fig. 4.
BER versus SINR for vehicle speeds of 0, 20, and 60 mph wi th OMI for acquisi tion and LMS for tracking.
speeds of 0, 20, and 60 mph at 900 MHz , corresponding to fading rates of 0, 27, and 81 Hz. Computer simulation results are shown for the BER over 178 blocks (~ 28000 symbols, which should be adequate for BER > 10- 3 ) , along with theoretical results for maxim al ratio combining (15). At o mph , the fading channel was constant over each block , but independent between blocks. Also, LMS tracking was not used at 0 mph, and thus the results show the accuracy of weight acquisition by OM!. OMI is seen to have less than I-dB implementation loss for BER 's betwe en 10- 3 and 10- 1 . At 20 and 60 mph , the tracking performance of the LMS algorithm is poor. For SINR below 14 dB, the LMS algorithm tracks so poorly that the best BER is ob tained with Jl, = 0, i.e., if LMS tracking is not used . Thi s lack of tracking causes little degradation at 20 mph, but a several dB loss in performance at 60 mph. For SINR above 14 dB, the LMS algorithm improves performance, with the best Jl, equal to 0.08. At 20 mph, the performance with the LMS algorithm is about the same as at o mph. However, at 60 mph there is a 4.2-dB implementation loss at 10- 2 BER. Thus the LMS algorithm is not satisfactory for optimum combining in IS-54. 3 Next , consider OMI for both acquisition and tracking with differenti al detect ion without interference. Fig. 5 shows the average BER versus SINR with OMI and vehicle speeds of 0 and 60 mph. For these results, we used OMI with a 14-symbol sliding window (K = 14 in (22) and (23», which gave us the best results for a 10- 2 BER at 60 mph. At this BER , OMI has a negligible increase in implementation loss at 60 mph as co mpared to 0 mph. Alth ough differential detection is typically used in mobile radio becaus e of phase track ing problems, we can also use coherent detec tion with optimum combining. Thi s is because 3 In [15] it was show n that the LMS algorithm was satisfactory for div ersit y comb ining and equal ization using the co nstant modulus algorithm for erro r signal generation in GSM , with data-to-fading -rate ratios as low as 1700. However, as mentioned before, this techn ique cannot distinguish betwe en the desir ed and interfering signals.
431
INR =
10- 1
-~dB
10- 1
Ideal Maxim al Ratio Co mbining
. .......
o mph
"
ffiCD
a:
10-2
ill
co
Ideal Maximal RallO Combinmg
10-2
lOdB
~~~2~~:~~>"
"'<>~<",-,>,.
.......'<:....
60mph • Drtterentiat Detectio n 60mpn • Co here nt Detect ion
10- 3
"
o
-----JL-_ _-...JL.-
10
,,
L.-_ _--.J
20
10
SINR (dB)
Fig. 5.
"
'", ,
Detecnon
' - --
.
"'" '..\~.>.",
Omph • Differentia l Detection Omph - Coherent
10-4
K=1 4
SINR (dB)
BER versus SINR for vehicle speeds of 0 and 60 mph with DMI for acquisition and tracking.
optimum combining requires coherent combining of the received signals, which means that the weights must track the received signal phase , and the array output signal phase should match the phase of the coherent reference signal. Thus coherent detection of the array output is possible, which, as shown in Section II, decreases the required SINR for a 10- 2 BER by 1.0 dB with ideal phase tracking ." With the LMS algorithm, however, tracking is so poor that coherent detection is worse than differential detection. On the other hand, with the DMI algorithm, there is improvement with coherent detection. Fig . 5 shows that coherent detection decreases the required SINR for a 10- 2 BER by 1 dB, resulting in performance that is 0.3 dB better than the theoretical performance of differential detection (but 0.7 dB worse than ideal coherent detection). At 60 mph, the performance degrades by an additional 0.5 dB; i.e., the performance is 0.2 dB worse than ideal differential detection (and 1.2 dB worse than ideal coherent detection). Thus the use of coherent rather than differential detection cancels most of the implementation loss of DMI at 60 mph. Finally , consider the dynamic performance of optimum combining for interference suppression. For the results shown below , the symbol timing for the desired and interfering signals was the same. Our results showed that this was the worst case since there was a sligh t improvement in performance with timing offset between the two signals (see below) . With the LMS algorithm, even at 20 mph the performance does not improve with the INR, showing that the algorithm is not accurately tracking the interferer. However, with DMI, the performance improvement with INR agrees with ideal tracking results. Fig. 6 shows the average BER versus SINR at 0 mph with one interferer with INR = - 00 , 0, 3, 6, and 10 dB . DMI with a 14-symbol sliding window and coherent detection was used as before . 4Note that this is significant in comparison to the 3.6 dB gain with optimum comb ining in IS-54 with 2 receive antennas [6], Also , it is almost half of the 2.5 dB gain needed for a frequenc y reuse factor of 3 rather than 4 (and an additional 33% capacity increase).
Fig, 6.
.....
-.-.
',
"
20
BER versus SINR with one interferer for a vehicle speed of 0 mph with OMI for acqui sition and tracking .
10-1
ldeaJMaximal Ratio Combining
~~~"-~~'a
3dB 6dB
ffi
[l)
60 mph
K=14
l OdB
10-2
10-3
10-4 '--_ _----J'--_ _- J L -_ _- J o 10
----I
20
SINR (dB)
Fig. 7.
BER versus SINR with one interferer for a vehicle speed of 60 mph with DMI for acquisition and tracking, and l\ = 14.
The requ ired SINR for a 10- 2 BER is 10.2, 9.5, 8.6, and 6.5 dB for INR = 0,3,6, and 10 dB, respectively, which is within 0.5 dB of the predicted gain shown in Fig. 3. Fig. 7 shows the average BER versus SINR at 60 mph with one interferer. Again , a 14-symbol sliding window was used since this gave the best results at a 10- 2 BER. At a 10- 2 BER these results show a gain with INR that is within 0.5 dB of the gain shown in Fig. 3. The implementation loss increases the SINR , though , resulting in poor performance at a 10- 3 BER with K = 14. However, note that the optimum window size for a given BER is determined by a tradeoff of two effects . As the window size decreases, the weights have more error due to the averaging of fewer samples, but less error caused by channel variation over the window. Our results showed that as SINR increases, the performance is improved by decreasing K.
432
10- 1
IdealMaximalRatio Combining
60 mph K=7
Ideal Ma ximal Ratio Combining
10-1
. Noise Only INA - OdB.> 6dB
ffim
10- 2
10.3
'
'~" "
1OdS -,
..'~"'" '. "
,>-,
ffi
~.l'oo.
m
""<~~>:~~.~>,.,.,
,: ~~:!~~~~~~~,
. . -,
.
' . ' .. '... ' .
......... , .
10-4
20
,',
................... ~.~~ .....
..... , ." . . ,
10- 3
" , ". . , ...
10
10-2
....
'",::,~~~ ~.~.;~;>, ''''
L....
......
.
.....
...
..
.
, "y . , . . , ... .....
" .
........
'c.
,
--.I
--'-
a
10
20
SINR (dB)
SINR (dB) Fig. 8.
60 mph ~ = 0 .675
Noise Only INA .OdB ::-.,
3dB,··-,
Fig. 9. BER versus SINR with one interferer for a vehicle speed of 60 mph with OMI for acqui sition and tracking, and exponential weighting with /3 = 0 .6 75 .
BER versus SINR with one interferer for a vehicle speed of 60 mph with OMI for acquisition and trackin g, and J( = 7.
Fig. 8 shows the performance with K = 7, which gave the best results at a 10-. 3 BER. At this BER, with interference, the improvement of optimum combining is seen to be close to that with K = 14 at a 10- 2 BER (Fig . 7). Furthermore, with K = 7 at a 10- 2 BER , the improvement with interference is similar to that with K = 14. However, with noise only , the BER for a given SINR is higher with K = 7 than with K 14, because fewer samples are averaged to determ ine the weights. Fig. 9 shows the performance of DMI with exponential weighting for {3 = 0 .675. This {3 gave the best result s for BER = 10- 2 and 10- 3 . With noise only , the BER is seen to be lower than with either K = 14 or 7, and at a 10- 2 BER the performance is close to that of ideal maximal ratio combining with coherent detection (i.e., 1.0 dB lower SINR than the curve shown for ideal maximal ratio combining with differential detection). With interference at a 10- 2 BER , the gain with optimum comb ining is close to the predicted ideal gain; i.e., the perform ance is slightly better than DMI with a sliding window and K = 14. How ever , at a 10- 3 BER with interference, the performance is slightly worse than that shown in Fig. 8 with K = 7. Thu s either the sliding window or the exponential weighting technique can be used to generate accurately the optimum combining weights, even at 60 mph . Finally , Fig. 10 show s the effect of timing offset between the desired and interfering signals. Results were generated for a 10 dB SINR at 60 mph with K = 14, as in Fig. 7. These result s show that the BER varies with timing offset by less than 12% « 0.4 dB improvement in SINR at a 10- 2 BER) , with the best performance when the interfering and desired signals are offset by approximately half the symbol dur ation .
2xlO-2 , - - - - - - , - - - - - . - - - - - , - - - - - ,
-'-'- '-'-'-'-'-'-'-'-'-'--- '- '-'-'- '-'-'-'-'-'- '- '-'-'-._ ._._. Noise Onty
10-2
=
IV. S UMMARY ANO CONCL USION S In this paper, we have studied the dynamic performance of adaptive arrays in wireless communication systems . Specifically, we studied the performance of the LMS and DMI
b:.-
.. ... ....... .. ....... .. .. .. ... ... ....... ..... .. ....... ............ . .......... ..
-
---
INA.OdB 3dB
• • •• • • . .• . . • • • . - . 6dB
a: w
m
----------------____
5x10 -3
------ ......... -
_ __ --
10dB
SINR=10dB
2xlO- 3
L....
a
---...l
0.25
--L
.....L_
_
0.50
0.75
0.875
Timing Offset (Fraction of Symbol Duration)
Fig. 10. Effect of timing offset on the BER for a vehicle speed of 60 mph 14 , and SINR 10 dB. w ith OM! for acquisition and trackin g, II:
=
=
weight adaptation algorithms in IS-54 with data to fading rates as low as 300. We showed that implementation of optimum combining allow s the use of coherent detection, which improves performance by over 1 dB as compared to differential detection . Although the performance of the LMS algorithm was not satisfactory, results showed that the DMI algorithm acquired the weights in the synchronization sequence interval and tracked the desired signa l for vehicle speeds up to 60 mph with less than 0.2 dB degradation from the ideal performance with differential detection at a 10- 2 BER. Similarly, an interfering signal was also suppressed with performance gain s over maximal ratio combining within 0.5 dB of the predicted ideal gain . Thus our results indicate that we can obtain close to the ideal performance improvement of optimum combining even in rapidly fading environments.
433
ACKNOWLEDGMENT
We gratefully acknowledge useful discussions with G. D. Golden, J. Salz, and N. Seshadri. APPENDIX
A
To relate the weight equation (12) to that of [5, Eq. (11)], we need to consider three differences between the analysis given here and in [5]. First, in [4]-[6], we considered the generation of N == L + 1 separate outputs at the receiver, each with minimum mean-square error, while here we consider only the output of the desired signal. Using the notation of [4]-[6], the channel matrix C that relates the transmitted signal vector (including the L interferers) to the received signal vector x at a given time is given in our notation by
(A-I) Thus the weight matrix W for the optimum linear combiner that generates N output signals is given by (from (12»
W == aR;;C*
(A-2)
with the vector s at the output of the combiner given by
s == W T x.
(A-3)
Note that the weight vector w of (12) is just the first column of W. Now, we can show that
R x x ==
+ CC t
(A-4)
+ CCtJ-1C*.
(A-5)
(72]
and from (A-2),
W == a[a2 ]
matrix. However, the weights can be shown to be equal (with a scalar multiple) in the limit (j2 ~ O. The change in the weight equation was done to put it the form for DMI (21).
A second difference is that in [4]-[6] we considered the zero-forcing weights, which can be obtained from (A-5) in the limit, (72 ~ 0, i.e., (A-6) Note that [CCt]-l exists only when N == M. Otherwise, the inverse becomes the pseudoinverse. Finally, the weight matrix of [5], which we will denote as W[5] , was defined as the transpose of the weight matrix given here, i.e.,
REFERENCES [1] 1. H. Winters, "Optimum combining in digital mobile radio with cochanneI interference," IEEE J. Select. Areas Commun., vol. SAC-2, July 1984. [2] 1. H. Winters, "Optimum combining for indoor radio systems with multiple users," IEEE Trans. Commun., vol. COM-35, Nov. 1987. [3] 1. H. Winters, "On the capacity of radio communication systems with diversity in a Rayleigh fading environment," IEEE J. Select. Areas . Commun., vol. SAC-5, June 1987. [4J 1. H. Winters. J. Salz, and R. D. Gitlin, "The capacity of wireless communication systems can be substantially increased by the use of antenna diversity," in Proc. 1st Int. Conf Universal Personal Commun., Sept. 1992, pp. 28-32. [5] 1. H. Winters, 1. Salz, and R. D. Gitlin, "The capacity increase of wireless communication systems with antenna diversity," in Proc. 1992 Conf. Inform. Sciences Syst., vol. II, Mar. 18-20, 1992, pp. 853-858. [6] J. H. Winters, J. Salz, and R. D. Gitlin, "Adaptive antennas for digital mobile radio," Proc. IEEE Adaptive Antenna Syst. Symposium, Melville, NY, Nov. 1992, pp. 81-87. [7] S. A. Hanna, M. EI-Tanany, and S. A. Mahmoud, "An adaptive combiner for co-channel interference reduction in multi-user indoor radio systems," in Proc. IEEE Veh. Technol. Conf., St. Louis, MO, May 19-22. 1991, pp. 222-227. [8J D. J. Goodman, "Trends in cellular and cordless communications," IEEE Commun. Mag., vol. 29, pp. 31-40, June 1991. [9] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. [10] A. Dembo and J. Salz, "On the least squares tap adjustment algorithm in adaptive digital echo cancellers," IEEE Trans. Commun., vol. 38, pp. 622-628, May 1990. [11] P. Bello and B. D. NeJin, "Predetection diversity combining with selectively fading channels," IRE Trans. Commun. Syst., vol. CS-IO, p. 32, Mar. 1962. [12J P. G. Proakis, Digital Communications. New York: McGraw-Hill, 1983, p. 175. [13] W. C. Jakes, Jr., et al., Microwave Mobile Communications. New York: Wiley, 1974. [14] J. H. Winters, "Switched diversity with feedback for DPSK mobile radio systems," IEEE Trans. Veh. Techno/., vol. VT-32, pp. 134--150, Feb. 1983. [15] T. Ohgane, "Characteristics of CMA adaptive array for selective fading compensation in digital land mobile radio communications," Electron. Commun. Japan, Part 1, vol. 74, no. 9, pp. 43-53, 1991.
(A-7) and is given in [5] as W[5]
= lim [0" 21 + ctcr1c t. 2 a --+0
(A-8)
Although (A-6) and (A-8) look similar, note that CC t (A6) is an M x M matrix, while
atc
(A-8) is an N x N
434
Effect of Fading Correlation on Adaptive Arrays in Digital Mobile Radio Jack Salz, Member, IEEE, and Jack H. Winters, Senior Member, IEEE
Abstract-In this paper, we investigate the effect of correlations among the fading signals at the antenna elements of an adaptive array in a digital wireless communication system. With an adaptive array, the signals received by multiple antennas are optimally weighted and combined to suppress interference and combat desired signal fading. Previous results for flat and frequencyselective fading assumed independent fading at each antenna. Here, we present a model of local scattering around a mobile where the received inultipath signals arrive at the base station within a given beamwidth, and derive a closed-form expression for the correlation as a function of antenna spacing. Results show that the degradation in performance with correlation in an adaptive array that combats fading and suppresses interference is only slightly larger than that for combating fading alone, i.e., with maximal ratio combining. This degradation is small even with correlation as high as 0.5.
A
I. INTRODUCTION
NTENNA arrays with optimum combining combat multipath fading of the desired signal and suppress interfering signals, thereby increasing both the performance and capacity of wireless systems. This increase is reduced, however, by correlation of the fading signals between the received antennas. Previous theoretical and computer simulation studies of optimum combining (e.g., [1 ]-[6]) assumed independent fading of the desired and interfering signals at each receive antenna. Such independence occurs if multipath reflections are uniformly distributed around receive antennas that are spaced at least a half wavelength apart. However, the signals often arrive at the receive antennas mainly from a given direction. For example, in rural or suburban mobile radio, a high base station antenna typically has a line-of-sight to within the vicinity of the mobile, with local scattering around the mobile generating signals that arrive mainly within a given range of angles or beamwidth. This problem was studied in [7], where theoretical and experimental results showed the relationship of angle of arrival and beam width with the correlation of fading between antennas. Specifically, as the angle of arrival approaches end-fire (parallel to the array) and the beamwidth decreases, the antenna spacing must be increased to reduce correlation. When this correlation is high (>0.8), because the signals at the antennas tend to fade at the same time, the diversity benefit of antenna arrays against fading (i.e., with maximal ratio combining) is significantly Manuscript received November 14, 1992; revised February 25, 1993. The authors are with AT&T Bell Laboratories, Holmdel, NJ 07733 USA. IEEE Log Number 9403803.
reduced [8]. On the other hand, because independent fading is not required for interference suppression, antenna arrays can suppress interference even with complete correlation (== 1), i.e., in line-of-sight systems without multipath. In particular, theoretical and computer simulation results [1], [3], [4], [9], [10] have shown that with M antennas, M - 1 interferers can be completely suppressed in both fading (with zero correlation) and nonfading (with complete correlation) environments. Thus, we need to understand the antenna array performance with joint fading reduction and interference suppression. In addition, the effect of correlation with frequencyselective fading, when equalization is also used, must be evaluated. This paper considers the effect of correlation of the signal fading at the antennas of an adaptive array with optimum combining to combat desired signal fading and suppress interference, and optimum linear equalization to combat frequencyselective fading. We first present a model of local scattering where the received multipath signals arrive within a given beamwidth. We derive a closed-form expression for the fading correlation with this model as a function of the angle of arrival, beamwidth, and antenna spacing. Using these theoretical results with Monte Carlo simulation, we then generate results for the effect of beamwidth (i.e., correlation) on the adaptive array performance with given antenna spacing and random angles of arrival. Results are presented for optimum combining with flat fading, as well as for frequency-selective fading, using a two-path delay spread model with joint optimum combining and linear equalization. Computer simulation results show that the degradation in performance with correlation in an adaptive array that combats fading and suppresses interference is slightly larger than that for combating fading alone, i.e., with maximal ratio combining. This degradation is small even with correlation as high as 0.5. Results for an adaptive array with either flat Rayleigh fading or frequency-selective fading show that with an antenna spacing of four wavelengths, there is little performance degradation as long as the beamwidth of the received signals is greater than 20°. Further increases in antenna spacing would reduce this beamwidth even more. In Section II, we describe optimum combining and equalization with antenna arrays and discuss how fading correlation can occur. The model and theoretical analysis of wireless systems with fading correlation is presented in Section III. In Section IV, we describe the computer simulation technique and present results on the performance degradation with correlation. A summary and conclusions are presented in Section V.
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 43, No.4, pp. 1049-1057, November 1994.
435
T
User 1
T
User 2
T
UserN
(1)
In this paper, we are interested in linear processing at the base station of the M received signals to generate an output signal that corresponds to the data from one desired signal (user 1). Specifically, we consider ideal optimum combining and linear equalization, where the M received signals are combined to minimize the mean-square error (MSE) in the output. For ideal optimum combining, we assume perfect tracking of the desired and interfering signals by the combining algorithm at the base station. For ideal linear equalization, we consider a synchronous tapped delay line with an infinite number of taps, as shown in Fig. 1. This equalizer is the optimum linear equalizer under the assumption that the desired signal spectrum is bandwidth-limited to the data rate (1IT). As shown before [1], with ideal optimum combining and linear equalization, the minimum MSE for user "1" for given C (w) is given by
T
= CT~-2 1r
j7r/T -7r/T
2
M-1
M
Fig. 2. Wireless environment where all signals from a mobile arrive at the base station within ±~ of angle d».
BACKGROUND
Fig. 1 shows a digital wireless communication system employing adaptive arrays where a base with M antennas receives signals from N users. These N users operate in the same bandwidth simultaneously and include signals destined to the base, as well as those destined to other bases, but interfering with the desired signals, as in cellular systems. Let the complex channel transfer function from user "i" to antenna "j" be denoted as Cij (w). Then, the channel vector from user "i" to the base antennas is C, (w) = [eil (w) ... CiM (w)]T, where the superscript T denotes transpose, and the M x N channel matrix between the N users and the base is given by
MSE[C]
D
D
1
II.
.+---+•
•+----+.
Fig. 1. Wireless communication system employing adaptive arrays, where a base with AI antennas receives signals from 1'1 users.
1
[I + pct(w)C(w)];l dw,
(2)
where I is an N x N identity matrix, p is the signal-to-noise ratio, and CT~ = E [Ian 12 ] , where the an'S are the complex data symbols. The superscript denotes complex conjugate transpose, and [] ~11 stands for the "11" component of the inverse of a matrix. For coherent detection of binary phase shift
keyed (BPSK) or quadrature amplitude modulated (QAM) signals, the error rate can then be upper bounded by 1-
P; ~ Ec [ exp [ -
MSE[C]/a~]] -MSE[C] ,
(3)
where Ec[ ] is the expected value with respect to the channel matrices. With multipath, the C-ij (w ),s are modeled as complex Gaussian random variables at each frequency w. The variation of Cij(W) with w depends on the delay spread model of the channel. In this paper, we examine numerically two such models: 1) flat fading, i.e., C,ij(W) = Cij for all w, where equalization is not needed, and 2) a two-path delay spread model, .. (
C1.] W
) _ -
1
Cij
+ cije 2 -
jW'T 1
,
(4)
where Ti is the time delay between the two paths for the i-th user and j and are complex Gaussian random variables, and the fading in the two paths with different time delays is independent, i.e., j is independent of CTj (but the Cij'S are not necessarily independent). Previous papers have assumed that the Cij 's are independent. Such independence occurs if multipath reflections are uniformly distributed around the receive antennas that are spaced at least a half wavelength apart (this situation is examined in detail below). However, the signals often arrive at the receive antennas mainly from a given direction. For example, in rural or suburban mobile radio, a high base station antenna typically has a line-of-sight to within the vicinity of the mobile, with local scattering around the mobile generating signals that arrive mainly within a given range of angles or beamwidth. Fig. 2 shows a typical scenario where all signals from a mobile arrive at the base station within ±~ at angle ¢. This problem was studied in [7], where theoretical and experimental results showed the relationship of angle of arrival and beamwidth with the correlation of fading between antennas. Specifically, [7] assumed that the probability density function for the angle
436
ct
Crj
ct
The overall channel matrix C can now be expressed in terms of the M-column vectors (see (1)),
of arrival of the z-th ray is given by 1r
- 2+¢
::;
(Pi
:s "2 + ¢ 1r
(5)
(9)
where n is an even integer chosen to determine the beamwidth and Q is a normalizing constant chosen to make p((Pi) a density function. The correlation of the fading between two antennas spaced D apart is then [7 J
R x x == and
R x y ==
j
' 1r
/2+(p
. --rr/2+¢
j
' 1r
/2+(p
. --rr /2+(p
cos (2JrD / A sin (cP'L - cjJ) )p( dJi )dcjJ'i
(6)
sin (2JrD / A sin (cjJi - (D) )p( tDi )d¢i
(7)
where A == w/(2Tic), C is the speed of light R x J; is the correlation between the real parts of CiJ and e,k, and R:L'!J is the correlation between the real part of CiJ and the imaginary part of Cik. Unfortunately, (6) and (7) must be evaluated numerically. Therefore, in the next section, we present a generic model, where the probability density function of (P, is assumed to be uniform,
-
~
+ (j) :s; (PI :s;
elsewhere
~
+ (/)
:l;kl\]
:t)kilJ
and seek to evaluate the 21\;] x 2l\!! correlation matrix
( 1I) Defining the 2 x 2 matrix D[L-Ji
CHANNEL MODEL
We develop a mathematical model for multipath media applicable in wireless digital communications employing antenna array processors. The model is useful for the evaluation of signal correlations among the antenna array elements which are critically important in determining ultimate system perfonnance. The degree of correlation depends on the element spacings and signal scattering angles resulting from the physical surroundings. Using the model of a uniform probability density function of. ¢'i (8), in the Appendix we derive closed-form expressions for the correlation of the fading between the j-th and k-th antennas (as compared to (6) and (7)). Note that using these expressions, (A-19) and (A-20), when ~
( 10)
(8)
This allows for the derivation of a closed-form expression for the correlation coefficient, with results that agree with the results obtained numerically using the model of [7] (see Section IV).
III.
where the column vector Ci.(w) represents the transmission characteristics from user" k" to all the antenna elements. Since each user is characterized by its own surroundings, and if the users are not on top of one another to within wavelengths, it is reasonable to assume that the columns in (9) are statistically independent. Consequently, we need only to characterize the correlation properties of a typical user, which we have already accomplished. Expressing the complex column vector Ck(w) == Xk(W) + 'lYk (w) where Xk and Yk are the real and imaginary 1\11-column vectors associated with user" k." we define the 21''1/ augmented column vector as
k xxCi - j) == [-R- xy (I:I, - J:\)
~xy(li
-]1)]
i, j == 1..... IVI
R y y ( / -.1)
( 12) where the entries are given in (A-19) and (A-20), it is easy to see that k, can be represented in terms of these block 2 x 2 matrices as follows:
12 x 2
Rk
7?k
D1T DT2
DIJ
D1
12 x 2
DT1
.«:
D2 D1
12 x 2
DII - 2
D
AJ
Di\1-1 D1\I-2
( 13)
12 x 2
where (j~ is the received signal power for the k-th user (see (A-16), with a subscript k" denoting the k t h user~(A-16) applies to a typical user). Clearly, k; is a Toeplitz matrix. 1.1.
== Jr,
IV.
RESULTS
A. Correlation and
If.
where z == 27r The implication of these results is that when reflections are allowed to arrive at the antenna array from all directions, the correlation of signals at adjacent antenna array elements is determined from J o(z) == 0 which implies that z == 27r~ ~ 2.4 or ~ ~ ;; ~ .382. This sets the minimum spacings between antenna elements yielding zero correlation.
Let us first consider the correlation as a function of the antenna spacing D / A, angle of arrival cP, and beamwidth ~. When the signal arrives from broadside (¢ == 0°), R x y == 0 for all D/ A. Thus, the envelope
correlation, R (IR x x 12 + IR x y 12 ) 1/2 is just IR x x I. R x x versus antenna spacing is shown in Fig. 3 for 6. == 180°,90°,40°,20°, 10° , and 3°, These results agree with results using the model of [7] with ~ equivalent to the 3 dB beamwidth of [7]. The figure shows that, as ~ decreases, the
437
0.5
rl
-0.5
~
0
= 90'
-0.5
.1 '---"-_ _-=---'a 2
·1 ' - - - - - - - - - ' - - - - - - - - ' : - - - - - - - - - : 4 6 2 a DIA
-1.-
--'
4
6
Of }"
Fig. 3. Correlation of the real portion of the fading versus antenna spacing for 0 = 0° .
first zero in the correlation occurs at larger antenna spacing. Specifically, the first zero-crossing occurs at D/).. ~ 30/6, with 6 in degrees . Thus, these results depend mainly on D /)../6, and show that independent fading occurs when the antenna beamwidth from two elements of the array is about the same as the beamwidth of the arriving signal. When the signal arrives from other than broadside, ¢ -:F 0°, the antenna spacing for low correlation increases and the envelope correlation is never zero for almost all values of ¢ -:F 0° and 6 < 180° (since zero envelope correlation requires that Rx x (A-27) and R x y (A-28) have zero crossings at exactly the same spacing). Fig. 4 shows R x x versus antenna spacing for the worst case of ¢ = 90°. For R x y , the peak value of the oscillations is similar to that shown in Fig. 4. Note that the correlation decreases much more slowly with antenna spacing at ¢ = 90° than at ¢ = 0°. Fig. 5 shows the antenna spacing required for the envelope correlation to remain below 0.5 as a function of ¢ and 6. The required spacing is only a few wavelengths up to very small beamwidths, unless ¢ is close to 90°. Experimental measurements of the beamwidth in mobile radio are presented and discussed in [7], [12]. These results show that, as expected, the beamwidth decreases with the antenna height. Fortunately, in most cases, antenna spacings on the order of only 10).. (several feet at 900 MHz) are required to obtain low correlation. The effect of correlation on reducing the effectiveness of antenna diversity against desired signal fading is shown in [8]. With maximal ratio combining and two antennas, small correlation « 0.3) has a negligible effect on performance, and the degradation is small unless the correlation is large (> 0.8). The effect of correlation on reducing the effectiveness of antenna arrays against interference suppression is as follows. With M antenna elements, the array has M - 1 degrees of freedom. Thus, as shown by theoretical and computer simulation results [1], [3], [4], [9], [10], an M antenna element
Fig. 4. for d>
Correlation of the real portion of the fading versus antenna spacing
= 90°.
8 , - - , - - - - - - , - - - - - - - . - - - -r - - - - - . ,
6
'" o v
~' .E
s
2
0'----'-------':------'------'------' 90
80
60
40
20
a
6 (Degrees)
Fig. 5. Antenna spacing required for the envelope correlation to remain below 0.5 as a function of
array can null out M - 1 interfering signals independent of the fading correlation (i.e., with or without fading). The only factor that changes with the environment is the required spatial separation of the interfering signals: without fading, the signals must be separated from the desired signal by the antenna beamwidth, while with fading (with 6 = 180°), the signals need only be separated by about half a wavelength. For 6 < 180°, we note the following. Spacing the receive antennas at greater than )../2 decreases the beamwidth of the array but also creates grating lobes, i.e., the antenna pattern repeats every 90° / (D / )..). Because of these grating lobes, with large antenna spacing to reduce fading correlation, interfering
438
signals outside of the antenna beam width can not alway s be suppress ed . However, bec au se of the multipath fading, mo st interfering signals within the beam width but separated by at least half a wa velength , can be suppress ed. Therefore , as before , onl y the required spatial separation of the interfering signals changes with the environment. Thus. as the signal beamwidth decreases (i.e., as the corre lation increa ses), the effectiveness of adaptive arrays to suppres s interference alone does not change, but the effectivene ss aga inst fadin g does.
10. 1 ,-----r-
- - - . . . , -. -- - - - , - --
p
ffi lD
~
--,
18dB
Flat Fading
10.2
M=N=2
B. Performan ce with Fading and Interferen ce The effec t of corr elati on on an adaptive array that jointl y suppresses interference and reduces fadi ng effec ts was detennined in the foll owing manner. For fixed D[); and the same fixed 06. for the desired and interfering signals, we use Monte Carl o sim ulation to derive 10.000 channel matri ce s C with random ¢ and fadin g and then calculate the performance averaged ove r the se matri ces (i.e.. (f and fading ). We ass ume that the user s are rand oml y located (se parated by at least half a wavelength ) and thu s ¢ is an independ ent rand om variable for eac h user with a uniform pro babil ity de ns ity function. i.e .. -
t;
< (/J
~
n
( 14 )
pIs('w lwrc .
The performance mea sur es we co nsider are the ave rage erro r rate . as well as the outage probability. i.e.. the probabilit y that the error rate exceeds a gi ven value . The error rate for a given i/J and fading was calculated as follow s. For give n (p for eac h user . D.. and D[ ); the correlation matrix RI.: / (J~ for ea ch user was calculated using ( 13). To generate C , we first ge nera te a 21\1 vec to r A. I.: for each user with eac h element Uki bei ng an inde penden t. zero -mea n Gau ssian rando m va ria ble with a variance of 1/2. Thu s.
= [al.:!
A I.:
... (lud
T
( 15)
for the k- th user. The A:-th column of C. CI.: . is then • I I:!
Ck Note that
; 2 is give n by . I /
k
(J-
I.:
:1:
(16)
(J ie
~
R~:2 = :1: where
= -HI.:.)- ,h-
[
~ .
o
= [Xl ' . . :1:2M ( ,
and eigenvalues of
o()
0
J ,\ U E
o and
.1:,
]
T I:
( 17) .
and '\ , are the eigenvectors
~, respectively. For frequency- selective t;
fading with two-path delay-spread (4) , the abov e procedure was repeated twice to obtain the cL ' s and c~) ' s. Th e MSE is then given by (2) and the err or rate by (3). We first consider the effec t of correlation with flat fadin g, two recei ve antennas, and one interferer with the same power as the desired signal. Fig . 6 exhibits the average error rate versus 06. with p = 18 dB and 27 dB , and D = 0.382'\ and 3.82'\. Note that D = 0.:382'\ corresponds to zero corre lation when the signa l arr ives uniformly from all angles
3.82
I
p ~ 27dB
10.3 L.-_--'180
-'--'---_
_
100 90
--'
-'
a
50
6 (Degrees ) Fig. 6.
Average erro r rate ve rsus
~
With lial fad ing and J [
= .Y = 2 .
(6 = HmO). At D = 0. 38:2'\, the performance is degraded slightly at 06. = 90° and become s much wor se with smaller D. . However. at D = 3 .82'\, there is little degradation until D. is 10°- 20°. Thus, increasin g the antenna spac ing by a factor of 10 decr eases the tolerable ~ by abo ut a factor of 10 as well (co rrespond ing to the decrease in antenna bearnwidth as discussed in Section IV-A) . As shown in Fig . 5. at 20°, the correlation is about 0.5 in the worst case of (p = DO°. Fig. 6 also shows that the degradation with .0. is larger with high er o, but the abo ve conclusion s are the same. Similar result s were obtained for the outage probability. Fig. 7 shows the outage probab ility vers us .0. with flat fading. N - 1 equal -power inte rferers, and M = N + I. Result s are shown for the probabil ity of excee d ing a 10- 2 error rate. with p = 17 dB. and D = 0 .382 '\ and 3.82'\ as in Fig. 6. As compared to Fig. 6, these result s sho w that for D = 0.3 82'\ correlation degrade s the performance more wh en there is an additional antenna. Additional result s we obtained for M = N + 2 and M = IV + :3 show that the degradati on with co rrelation gro ws eve n larger with more ante nnas . In Fig. 7, the IV! = :2 result s are without inte rference and thu s correspond to the performance with maximal ratio combining . The performance with D = 0.382'\ is seen to be degraded somewhat more by co rrelation when interference mu st also be suppress ed (i.e., AI = 3 and 4 versus JV! = 2 result s). However, in all cases when the spacing is increased to D = 3.82'\, the performance rem ains constant I as long as 06. is greater than about 20°, i.e.• the correlation is below 0.5. Finall y, we consider the effect of corr elation with frequency selec tive fading when jo int optimum combining and equalization is used . Fig . 8 shows the average error rate versus 6 with two-path dela y spread and 1\:1 = IV = 2. Results are for 'p = 17 dB , D = 0 .382'\ and 3.82 '\ , and dela y T = 0 , 0.7 T . and T for the de sired and interfering signals. Note that The outage probabil ity is seen in Fig. 7 [0 increase slightly with increasing JI 3 and 4 with D j-\ 3.82. but this is ju st a num erical aberration due to using only 10.000 channe l matrices in the simulations. I
~ at one point for both
439
=
=
.2 , --
--.-
-
-
--,-
-
-
-
,..--
-
-
little degradation, but higher correlation significantly decreases performance. Although our results show that this degradation incre ases with the number of antennas, these results are for a linear array, which cau ses all fading to be highly correlated when signals arrive from endfire, i.e., as if> -+ 90 0 • Since this problem can be reduc ed when M > 2 by not arranging the antennas linearly, we may be able to avoid this increase in degradation with the number of antennas. However, in all cases, increased antenn a spaci ng reduces the ~ at which degradation occurs.
-,
.1
N
o , w
II:
.05
Flat Fading M=N+l =2104 P = 17d6
~
~
'5
rf
.02
.01
~~=~;::;;:;;;S~~l2:on.= 3.82
.008 '-----'-180 150
-
- - - ' - - - - - - ' - -100 50
V. CONCLUSIONS -
In this paper, we have investigated the effect that fading correlation has on the performance of an adaptive array in a digital mobile radio system. We described a mathematical model of local scattering around the mobile where the received multipath signal s arr ive at the base station within a given beamwidth and derived a closed-form expression for the correlation as a functi on of antenna spacing. Monte Carlo simulation results sho w that the degradation in performance with correlation in an adaptive array that combats fading, suppresses interference, and equalizes frequency-selective fading is only slightly larger than that for combating fading alone, i.e., with maximal ratio combining. Th is degradation is small even with correlation as high as 0.5. Our results show that, with an antenna spacing of four wavelengths, there is little performance degradat ion as long as the beamwidth of the received signals is greater than 20 0 . Thi s tolerabl e beam width can be reduced even further by larger antenna spacing since this beamwidth is inver sely proportional to the antenna spacing.
-'
o
to (Deg rees)
Fig. 7.
.07
Outa ge probabil ity versus
.:j.
with tlat fadin g and .\f = .\'
+ 1.
, - - - - - - r - - - - , - - - - - - - , - -- --,
05
II:
w
Two-Path Delay Spread
02
lD
M=N=2
p = 17d6 .0 1
150
Fig. 8.
100 to (Degrees)
Average error rate versus
J I = .\' = 2.
.:j.
50
ApPENDIX
o
DER IVATION OF C LOSED-FORM
the error rate decreases with T , due to the diversity benefit of frequency-selective fadin g with equalization, as shown in [1]. This improvement increa ses with T until T = T and then rema ins constant since the two path s are resolvable for T 2 T . The figure also shows that there is some improvement even if only the interference has frequency-selective fading, but the best improvement occurs when both the desired and interfering signals have frequency-selective fading. A large portion of the max imum possible improvement is obtained when T O.7T . Fig. 8 shows that for D 0.382.\ , the degradation with correlation increases with frequency-selective fading. As before , however, with D 3.82.\ , the performance is not degraded until ~ is less than about 20 0 • Thu s, correlation degrades the performance of an adaptive array that combats fading , suppresses interference, and equalizes frequency-selective fading somewhat more than an array that only combats fading. Correlation up to 0.5 causes
=
=
EXPRESSIONS FOR
with two-path dela y spread and
=
Rx x
AND
u.,
The most fundamental description of a linear, quasistationary , multi path medium in wireless systems employing antenna arra ys is the impulse respon se from user " i " to array element output "j ." Such a typical impulse response can be repre sented as the superposition of a large number of impulses, (A- I) n
where the gn' s and the tn' s are the strengths and dela ys of the possible paths. Clearly, in a time varying situation, these parameters will depend on time . In a system of N users and M antenna elements, we must describe N x M such responses. Thus, if the input to the medium of a typical user is s(t )eiwot , where W o is the angular carrier frequency, the output of a typical antenna element becomes,
440
So (t ) = eiwot
L gns( t n
tn)e-iwotn .
(A-2)
We now derive the correlations among array elements for
Following the seminal work of Turin [11], the set of all in's is partitioned into L disjoint sets .il t , £ == 1 ... L. With each set A€, we associate a representative delay Tf such that i-. EAp if :,;(t - tn) ~ 8(l - Te). In other words, the differences Tf - t n are much smaller than the reciprocal bandwidth of s( t). With these approximations in mind, we rewrite (A-2) in the form 8o
L s(l - L
a single user by assuming plane waves at the array. This is
a reasonable assumption when users and antenna array are separated by many wavelengths. Suppose the reference wavefront plane coincides with element 1 (see Fig. 2). Then, the wave arriving at element 2 suffers a delay relative to the first element,
L
(l ) == el~'l)t
Tp)
£==1
.f}nC-1W·"t"
(A-3)
T
L
+ w) L
bfe"(~'
(A-4)
Sok,j(t) = eiwol
Thus, a typical baseband-equivalent frequency transfer characteristic from user "I;" to antenna element "j" can be represented in the form L
: r/,j
)Lu...
1. . f1. -
1 . . . . 'V .J.: -- 1 ... iv 'I . 1
•
Sill
¢.
I(p\
~
7f
(A-7)
and (n - l)T at element "ti," Thus, if we denote the output signals at antenna elements "k" and "j" by .';ok(t) and .';oJ(t), respectively, due to the transmission of a signal of the form, .)(t) e'r> t, located at an angle (P, we can write
(=1
. ( ) - ~1J..:j Ck.J W ~ J{ (,
C
HI
where ne is the set of integers such that I; ri f~4t· Denoting Lnr gne-iwotn == be and taking Fourier transforms of both sides of (A-3), we obtain the standard L-ray, or frequencyselective multipath description of fading channels,
So(w) == S(wn
D
== -
L
L set -
Tf)b~ktj)
f==1
(A-8)
where
(A-5)
1/
{=l
For this model to be useful, a statistical characterization of the set of M x N frequency functions (:~ ..J (w) must be provided. In our application, we shall assume that the terms in the various sums defining hi's arc random quantities and so it is reasonable to assert that the h(l' s are complex random variables. Furthermore. we assume that there are large number of terms in each sum and that each sum includes different random terms and. consequently, from the central-limit theorem. the hI" S, f == 1 ... L, ITIay be regarded as i.i.d. complex, zero-mean. Gaussian random variables. If we let wot n == f)n in the sums defining br, we write the real and imaginary parts as
1
where (/)1/ is the angle of arrival of the n-th ray. As we have already argued, the h;,(\) 's are complex i.i.d. Gaussian random variables associated with array numbers n, and therefore the sought-after correlations are determined by each h~(\) and different O"s. Thus, we seek the correlation coefficients between the following random variables: (k)
(k)
. (k)
h['
==:r p
+ I,Yr' l: ==
IJ (j) t
- ,,(.d -.L (
" (J). + 1,.tJ p •J
1.. ...J.\;1
and -·1 - ., .. 1"/ ~
where '1,(0:) == R(:"\b(n) ./(' " e
(A-6)
Now, it is reasonable to regard Hn IllOd ulo27f as i.i.d, unifonnly distributed random variables with the consequence that Ie and Ye are now independent and so lb, I is Rayleigh distributed and / be is uniform. This is then the rationale for regarding Ck] (w fin (A-5) as a complex Gaussian process in the frequency domain. For our application, the correlation among the elements of Ckj' s is of paramount importance. In order to facilitate the evaluation of these parameters, we must return to the basic definition of the br's in (A-6). We begin by considering the following geometrical model. This entails placing the users and the antenna array in a reasonable geometrical relationship, Without loss of generality assume that the antenna array is linear with M elements with identical spacing, D, between elements. We label the elements in ascending order. Users are located at arbitrary angles and distances with respect to the antenna array as depicted in Fig. 2. With each user, we associate a scattering angle of size 2~. This implies that all subpaths from the user to the antenna array are restricted to emanate from within this angle.
(A-9)
and ') L{ ?ie(0) -_ I III b(O:) P ,n - 1.-.- . ...ivl .
We note that since the en's are i.i.d. uniform, the real and imaginary parts of b~o) are independent for any n. We now calculate for any n
E[x~(»f = E[y;L\)f
=
~L
ELq;]·
(A-IO)
Tlf
It is now straightforward to calculate the four correlation coefficients
441
E [:E~k) .T~j)] = E
[y;k) y;j)]
= ~ L E [g; cos nf
[(I, - j)21r ~ sin ¢nJ J (A-II)
where the Jm's are Bessel Functions of integer order and
and
D
E[X~k)y~j)] = -E[x~j)y?)]
(A-I8)
z = 27r"I'
= ~ tE[g~Sin [(k - j)27r~ Sin¢n]] nt
(A-12)
we can integrate (A-I3) and (A-I4) and obtain the following convenient formulas for the desired correlation coefficients:
where
D
wo-
C
D
= 27rfoC
D = 21r,.
.
= Jo(z(k - J))
1\
~
+ 2 L..J
J 2m (z(k - j)) cos (2m
sin(2m~)
m=l
According to our hypothesis, there are a large number of terms in the sums indicated in (A-IO) and (A-II) and if we make the additional physically reasonable assumption that the ¢n's are dense in the range (1) - D.., 1> + A), the sums can be expressed as integrals of the form, independent of l,
(A-19)
and
~
= 2 ~ hm+l(Z(k -
.
j))sm [(2m + 1)¢]
where the _normalized R's are defined as be readily checked that
and
2mLl
+ 1)~) (2m + 1)~
sin ((2m
R=
(A-20)
R/ (J2. It can
and
as they must be for "physically consistent" considerations. (A-14)
ACKNOWLEDGMENT
where the density function of the returned strengths a 2 ( ¢) must satisfy (A-I5)
Making the reasonable assumption that this density function is a constant over the angle segments, we then obtain the relationship (72
=
~ LE[g~] ne
=
~E[lblI2],for all I.
We wish to acknowledge Professor Bar-David from the Technion for generously sharing his expert knowledge with us while he was a consultant at AT&T Bell Laboratories, during the evolution of this investigation. Also, thanks are due to our colleagues at AT&T Bell Laboratories, Noach Amitay and Jim Mazo, for listening to our frequent arguments and sharing their expertise with us.
(A-I6)
which is consistent with the definitions in (A-8). Now, by making use of the well-known series representations,
= Jo(z) + 2 L 00
cos (zcosB)
L 00
sin (z sin B) = 2
m=O
J 2m (z) cos (2mB)
m=l
J 271'1 + 1 (z) sin [(2m + 1)8] (A-I7)
442
REFERENCES [1] 1. H. Winters, J. Salz, and R. D. Gitlin, "The impact of antenna diversity on the capacity of wireless communication systems," IEEE Trans. Commun., vol. 42, no. 4, pp. 1740-1751, Apr. 1994. [2] J. H. Winters, "Signal acquisition and tracking with adaptive arrays in the digital mobile radio system IS-54 with fiat fading," IEEE Trans. Veh. Technol., vol. 42, no. 4, pp. 377-384, Nov. 1993. [3] _ _ , "Optimum combining in digital mobile radio with cochannel interference," IEEE J. Select. Areas Commun., vol. SAC-2, no. 4, pp.. 528-539, July 1984. [4] _ _ , "Optimum combining for indoor radio systems with multiple users," IEEE Trans. Commun., vol. COM-35, no. 11, pp. 1222-1230, Nov. 1987. [5] _ _ , "On the capacity of radio communication systems with diversity in a Rayleigh fading environment," IEEE J. Select. Areas Commun., vol. SAC-5, no. 5, pp. 871-878, June 1987. [6] S. A. Hanna, M. EI-Tanany, and S. A. Mahmoud, "An adaptive combiner for co-channel interference reduction in multi-user indoor radio systems," in Proc. IEEE Veh. Technol. Conf., St. Louis, MO, May 19-22, 1991, pp. 222-227.
[7] W. C. Y. Lee, "Effects on correlation between two mobile radio basestation antennas," IEEE Trans. Commun., vol. COM-21, pp. 1214-1224, Nov. 1973. [81 W. C. Jakes, Jr., et al., Microwave Mobile Communications. New York:
Wiley, 1974. [91 R. A. Monzingo and T. W Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. r 101 R. T. Compton, Jr., Adaptive Antennas. Concepts and Performance. Englewood Cliffs, NJ: Prentice-Hall. 1988. [11] G. L. Turin, "Communication through noisy, random-multipath channels," MIT Lincoln Lab., Tech. Rep. 116, May 1956. [12] W. C. Y. Lee, Mobile Communications Engineering, New York: McGraw-Hill, 1982, pp. 275-280.
443
Capacity Improvement with Base-Station Antenna Arrays in Cellular CDMA Ayman F. Naguib, Student Member, IEEE, Arogyaswami Paulraj, Fellow, IEEE, and Thomas Kailath, Fello~v, IEEE
Abstract-In this paper, the use of antenna array at base-station for cellular CDl\'IA is studied. We present a performance analysis for a multicell COMA network with an antenna array at the base-station for use in both base-station to mobile (downlink) and mobile to base-station (uplink) links. We model the effects of path loss, Rayleigh fading, log-normal shadowing, multiple access interference, and thermal noise, and show that by using an antenna array at the base-station" both in receive and transmit, we can increase system capacity several fold. Simulation results are presented to support our claims.
T
I. INTRODUCTION
HE increasing demand for mobile communication services without a corresponding increase in RF spectrum allocation motivates the need for new techniques to improve spectrum utilization. One approach for increased spectrum efficiency in digital cellular is the use of spread spectrum code-division multiple-access (CDM,,,) technology [1], [2]. Despite the high capacity offered by CDMi\ technology. the expected demand is likely to outstrip the projected capacity with the introduction of Personal Communication Networks (peN). One approach that shows real promise for substantial capacity enhancement is the use of spatial processing with cell site antenna array [3 ]-[ 11]. By using spatial processing at the cell site, we can estimate the array response vector and use optimum directional receive and transmit beams to improve system performance and increase capacity. Such improved antenna processing can be incoporated into the proposed CDMA transmission standards. The increase in system capacity by using antenna arrays in CDMA comes from reducing the amount of co-channel interference from other users within its own cell and neighboring cells. This reduced interference transforms to an increase in capacity. The currently proposed IS-95 COMA standard already incorporates a degree of spatial processing through the use of simple sectored antennas at the cell site. It employs three receive and transmit beams of width 120 0 each to cover the azimuth. Sectoring nearly triples system capacity in CDMA. While it might appear that even narManuscript received December I. 1993: revised April 10. 1994. This research was supported in part hy the SOlO/1ST Program managed by the Army Research Office under grant DAAH04-93-G-OO~9. This paper was presented in part in the 27-th Asilomar Conference on Computer. Signals. and Systems. The authors arc with the lnforrnation Systems Laboratory. Stanford University. Stanford. C A 94305 USA. IEEE Log Number 9403~06.
rower sectors might yield further capacity gains, simple planar wavefront assumptions used in sectoring are not valid for narrow beams that employ large apertures. Simple sectoring, therefore, suffers from significant losses and motivates the need for "smart antennas" that adapt to a dynamic spatial channel seen by the cell site antenna array. In this paper. we study the capacity improvement of multicell COMA cellular system with base-station antenna array for both the downlink and the uplink. As in the proposed IS-95 CDMA standard, we assume that the uplink and the downlink occupy different frequency bands. We adopt the Rayleigh fading and log-normal shadowing model in [12] to model signal level. In this model. the fast fading around the local mean has a Rayleigh distribution. Due td"_ shadowing. the local mean fluctuates around the area mean with a log-normal distribution and standard deviation a.\", which varies between 6 to 12 dB, depending on the degree of shadowing. We also assume that the received signal power falls off with distance according to a fourth power law. That is, the path loss between the user and the cell site is proportional to r:' where r is the distance between user and cell site. In the next section, we analyze system capacity in uplink, where each signal propagates through a distinct path and arrives at the base station with independent fading. In Section III. we also analyze system capacity in downlink where all signals received at the mobile from the same base station undergo the same fading. Next, in Section IV we present simulation results. Finally, Section V contains our concluding remarks.
II.
MOBILE-To-BASE LINK
We assume that the cell site alone uses a multielement antenna array to receive and transmit signals from and to the mobile. No antenna arrays are considered for the mobile due to practical difficulties in implementing such a concept. Consider a scenario where there are N users randomly distributed around each cell site at varying ranges. We assume that the receiver is code locked onto every user but does not know the direction of-arrival (DOA) of these users. Each user transmits a PN code modulated bit stream with a spreading factor (processing gain) of L. Let P be the received signal power at the cell site, let the system noise power (excluding interference from other in-
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 43, No.3, pp. 691-698, August 1994.
444
band users) be a 2 and, finally, let M be the number of antenna elements. Assuming perfect instantaneous power control, the interference from a mobile within a given mobile's cell will arrive at cell site with same power P. Since mobiles in other cells are power controlled by their cell sites, the interference power from such mobiles, when active, at the desired user's cell site is given by [I] lit
r::)
r(k»)4 IIQ'(O) 11 2 = P ( r;~) IIQ';~) 112 = P . (3~
x(t)
, Ci«(t - Tik)a"
+
-T T "
net)
j) (2)
where K is the number of interfering cells , ail is the M x 1 array response vector for signal arriving from the ik-th mobile in the k-th cell and we assume that a ~ all = 1, Cit(t) is the code used by that user, b;.(') is the bit of duration T, Til is the propagation delay , 1/;11 is a Bernoulli variable with probability of success v that models the voice activity of the same user (i.e . , a user will be talking with probability v) , and n is the thermal noise vector with zero-mean and covariance
E; {n(t)n *(T)} =
a2 - I M
=0
t
=T
(3)
t
*"
(4)
T.
_
_ Own -cell interference. constant power
_ _ Desired Signal
(I)
where is the distance from the irth user in the k-th cell to its cell site, Q' ~:) is a zero mean complex Gaussian random variable that represents the corresponding amplitude fade along that path and combines both the Rayleigh fading and log-normal shadowing effects (i.e ., IIQ' ~:) II has a Rayleigh distribution whose mean square value E;{11Q'~:)1I2} is log-normal ; i .e. , 10 loglo E;{11Q'::)11 2 } is normally distributed with zero mean and variance a~), r ~~) is the distance between the same ik-th mobile in the k-th cell and the desired user's cell site (i .e . . cell site 0), and finally Q' :~) is the corresponding amplitude fade . Note that in [1], only the effects of the log-normal shado wing is considered , Note also that since the mobile will be controlled by the cell site that has minimum attenuation (3 " ::5 I [I] . Fig . 1 shows a desired signal and interference signals from mobiles within cells and outer cells for both omnidirectional beams and directional beams . Clearly. direc tional beams reduce the interference power and boast the signal to interference-plus-noise ratio . To be able to form such beams , we need to estimate the array response vector, or the spatial signature , of the de sired user mobile . Using this estimate of the array response vector . we can form a beam towards each mobile . Assuming a narrowband signal model. the M x I output of an array of M sensors at the cell site can be written as
I
. . __.. __ Outside-cell interference . variable power
445
.. _. ... . Out side -cell inter ference, varia ble
pow ~r
_ _ Own -cell Inter ference. constant power
_ _ Desired SIgnal
x
x x
FIg . I . Interference in uplink w ith and " uhout bcam tornu ng .
Equation (3) implies that the noise is both temporally and spatially white . For the desired user . let Ow T". (' " . and b, (.) be the array response vector, the time delay. the used code, and the transmitted bits, which are assumed to be i. i.d . binary random variables taking values ± I with equal probability, respectively . The antenna outputs are correlated with the desired user's code c., to yield one sample vector per bit. Without loss of generality . as sume that T o = O. The post-correlation signal vector for the desired user's L-th bit is given by z; (l)
= \'1x(t) Co (I) dt
11
N
= So (l )0" + in2::2 h =
K
.v
+ 2:: 2::
k=1 ;, = \
where t l
= (/ -
I)T,
(2
Ii" (l )ai"
1/;;J;,(l)ah
= LT,
+
nr(t)
(5 )
and (6~
K
fjJ/) =
Var {n2}
r
It ~Th J )
JPl3jk bjk (
,12 Co (r)n(r) J/
dt,
= g {xx *}
(12)
(13)
N
I(
t_2: o == 2 ~ioHa:at"lJ2, K
11
(14)
N
= k2: 2: ~h{3~ lIa(;ah 11 2 . = I u. == ,
(15)
The probability of outage is defined as the probability of the bit error rate exceeding a certain threshold P0 required for acceptable performance. As noted in [1], with efficient modems and powerful convolutional codes, adequate performance (BER < IO-'~) is achieved with Eb / (No + (J < 7 dB. Let S be the Eh/(N(} + 1 value required to achieve the level of performance. then the outage probabil ity is 0 )
Pr (BER > Po)
=
Pr (
h
E < No + 1 0
Pr (II + f, > ~ _ (J~). S MP
III
."J'
= LJPbo(l) + L: I"
-= 2
V;t"("a(;ar,,
.V
2:-= I u;_2:== I
if; u; l., a (~ar, + a (;n T (I ) .
( 10)
The first term So (I) is due to signals from the desired user, the second term n I is due to interference from users within its own cell. the third term n~ is due to interference from users outside the cell: both are zero mean, and I1r is due to the additive thermal noise, which is normal with zero mean and variance equal to Var tnT} == La 2/M. Additionally, we assume that each user's code consists of a sequence of L i.i.d. binary random variables taking values ± I with equal probability. As noted in [15], with asynchronous transmission, random-sequence codes give approximately the same analytical results for nonrandomly chosen codes. Under this assumption and using the results in [15], we can show that the variances of nl and n2 are given by N
Var {n I}
2.
where I, and 12 are the interference-to-signal power ratios due to own cell and outer cell users respectively, and are given by
do (I) == a {; Zo(' )
k
11
~
as the generalized principal eigenvector of the matrix pair (R u • R:..:-.). Using this estimate of a". the post-correlation antenna outputs are combined via bearnforrnins to estimate the signal from the desired user. The decision variable. which is the output of the bearnforrner. is then given by
+
Ik
(9)
R~)~} = e{Z(1:(~}
K
0
L
In order to combine the array outputs to estimate the signal from the desired mobile, we need to determine the array response vector for the wavefront arriving from this user. In general, in COMA systems the number of users will far exceed the number of antennas. Therefore, subspace methods of direction-of-arrival estimation (e.g., MUSIC [13] and ESPRIT [14]) are not applicable. In [3], we showed that the array response vector of the desired mobile au can be estimated from the pre-correlation and post-correlation array covariances Rand R.. where r.r ~",-",
= So (/) +
Ik,
pend on the voice activity of the users, their array response vectors, and fading and shadowing effects. The faded energy-per-bit to interference-plus-noise densities ratio can be written as
cjl(r - T1k)CO(t) dt,
1
e;
ik,
(7) These variances are themselves random variables that de-
(8) nr(!) =
N
= LP k2: 1/;·13~'ie lIa*a, = , 2: = 1
= LP j"L;;2 1f,,,IIa:aj"ll\
(11)
446
s)
(16)
This expression gives the outage probability as a function of the random variables II and 12 • The distribution of the random variables II and 12 depends on the number of active users, their relative distances, their array response vectors, array parameters, and fading and shadowing effects. The capacity of the system in terms of maximum cell loading can be found by finding the maximum N such that for the required BER. Pout will not exceed the present threshold. To obtain Pout as a function of N, we need to specify the array (i.e., the number of sensors, spacing between them, and their arrangement) to be able to find the distribution of Ila: aik tt2 , and hence the distribution of I, and 12 • To simplify the evaluation, we use the following first order approximation. As we pointed out earlier, the effect of forming a beam towards the desired user is to reduce the effective number of interferers to those mobiles that fall within the beam formed towards the desired mobile. Since the number of those inteferers is random, we ap2 proximate this effect by replacing the lIa: a., 11 term in II and /2 by a Bernoulli random variable Xik that has a probability of success B/21r where B is the effective bandwidth
and is equal to 8 {lla~aik 11 2 } . This random variable represents the interference activity of the users, i.e., a mobile will cause interference to the desired mobile if it falls within its beam. In this case, we can write /1 and /2 as N
I
= 2.:
J
N
i., == 2
,I,. 'Yl o
K
x. = 2.: i.,
.
lo
== '2
( 17)
A..
'P/ o
N
K
L: L:
L: L:
l/;ikXI~{3T~
k=1 h==l
lV
k==1 h=l
cjJh{3~
( 18)
where cPit.: = l/;ik Xii.: is a Bernoulli random variable with probability of success v = vBI27f. The distribution of f \IQd~) Il/lla~;) I is given by [12] Pr (f
<
r) =
1
J;
roo
L",
1
exp (- U 2)
+
r-21O-2a"ulo du,
(19)
For a large number of users. the random variable /2 (interference due to K . N users) can be approximated by a Gaussian random variable with mean JJ-i N and variance a; N that depend on li, the degree of shadowing as, and the number of interfering cells K. We have evaluated the mean and variance of 12 using Monte Carlo integration considering only the first two tiers of interfering cells (i. e .. K = 18) and these were found to be given by u,
=
pacity. To be able to form such beams, the cell site needs to have an estimate of the transmit array response vector to each mobile. However, in the current standard, frequencies for the uplink and downlink differ by 45 MHz. In this case, the receive and transmit response vectors can be significantly different [16], [17]. Hence, reciprocity between uplink and downlink does not hold and the beamformer weights used for reception cannot be used for transmission. A method of performing transmission beamforming is the feedback method [18], [19], where training signals, or tones, are periodically transmitted from the cell site to all mobiles on the downlink. From the received signal information that the mobiles feedback to the cell site on uplink, it is possible to estimate the downlink spatial channel, and thus estimate the transmit array response vector. All signals received at the mobile from the same base station will have propagated over the same path, hence they will experience the same fading and path loss. Therefore, we assume that cell site transmits the same power to all mobiles controlled by that cell site. With this assumption, the power of each signal arriving at the desired mobile from the k-th cell site is given by
a~ = 0.463[' - 0.2741'2.
(21)
Also, the random variable II has a binomial distribution with parameters tN - 1. f ). Let LIS - a ~/PJl =: O. Since II, [.2, and all cPi k are independent. we can usc the results in [1] to shows that Pout
= ,vi: I
k=O
. Q
(lV - 1) l' ~ k
(1 _
f
r\'
(0 - k ~ !J.,N)
I - I..
where 1
OO
.s: i.t
e -.'"2/2 dy.
This equation gives the outage probability as a function of the number of mobiles per cell that can be supported. Note that this reduces to the result in [1] when no antenna arrays are used at the cell site. The results of evaluating (22) as a function of cell loading and beamwidth are shown in Fig. 3. Also, simulations to evaluate the accuracy of the above approximation are shown in Fig. 5 and discussed in Section IV. III. BASE-To-MoBILE
LINK
Consider now the base-to-mobile link. We assume a similar scenario as in the uplink. With antenna array at the cell site, the cell site must also beamforrn on the downlink in order to effectively increase the system ca-
447
(23)
where a t' represents the fading and shadowing experience by all signals arriving at the desired mobile from the k-th cell site. and ,.~Ol is the distance between the desired mobile from its cell site. As in [1]. we assume that the power received by the mobile from its cell site is the largest among all other signals from other cell sites (otherwise the mobile would switch to the cell site whose received power is maximum). That is. we assume that k
(22)
Ja~ ,v
Q(x) = -
p . p~
(20)
0.523~.
=
1. . . . • K.
(24)
Fig. 2 shows desired signal and interference powers seen by the desired mobile for both omni- and directional beams. Assuming N users per cell randomly distributed around each cell site at varying ranges, we can write the received signal at the mobile of interest as xo(t) =
iO~1 '1/;;" JP13"b +
f f
k = I i~ = I
jo (
[t -TT,,,
(J;,JP13k b" ( I t _
I
)
-
T
edt -
T,~
TIJa,:'a~»)
\ )
(25)
where n(t) is the background noise received by the mobile, and a ~:) is the transmit array response vector of the desired mobile as seen by the k-th cell site. All other notations remain the same as in the previous section. The mobile correlates the received signal by its code to yield
terference due to signals from its own cell site nl is zero . However, we assume here that there will be cross-correlation between those signals, which represents a worst case. Hence, similar to the uplink case, we can show that the variance of n I and n2 is given by
... .. .. . Variable Power
N
Var {nIl
1ol Il 2 = LP o illL:= 2 !/;·lIa*a 'I}
Var {n2l
= L k=2:I r, il 2:= I
10
K
(28)
0
N
!/;illla~a~k lI12,
(29)
and the energy-per-bit to interference-plus-noise densities ratio can be written as
L
,
0'-
.. .. . .. Variable Power _ _ Cons tant Power
(30)
+ G 1 + G2
Po
where G 1 and G2 are the interference-to-signal power ratios due to their own cell and outer cell signals, respectively , and are given by (31) K
-
\ x \
x
\
X
P "lIr
= s,,(/) +
III
(l)
+
II~ (/)
+
"r(l)
.v
= L(3" JPb,,(/) + 2: !/;,) ,,,a;' a::" I"
K
=~
.V
L: L:= I O',Ji,a,:a :;' + k= I 'I
"r(ll.
(26)
where 1'1 is defined as before but with (3k instead of (3'1 and "r(t)
I{
k
P"
-
IIa "* a ()tkl 112 .
(32 )
= Pr (BER > P,,) = Pr
(0'2
P"
+ G1 +
G~
>
!::.) . S
(33)
the decision variable
+
k = I il = I
The corresponding outage probability is then given by
X
Fig. 2. Interference in downlink with and without beamforming.
d;
P
.V
G~:- " ~ " LJ 1/;
= 1, (c,,(t)n(r)
dt ,
(27)
As in the uplink. the first term s.; is due to the desired signal from the cell site to the desired user, the second term n I is due to interference from the same cell site into the desired user, which is zero mean, the third term n2 is due to interference from other cell sites into the desired user. and nr is due to the additive thermal noise and it is normal with zero mean and variance equal to Var {nr} =
La 2 •
In the proposed CDMA standard 15-95, orthogonal codes (Walsh codes) are used on the downlink for all users within a cell, i.e .. in the ideal case (no multi path) there is no cross-correlation between those signals and the in-
448
Unlike the uplink case. the distribution of G~ does not yield itself to analysis (here we have only K independent fading variables, while in the uplink case we had K . N fading variables. and when N is large we were able to model 12 as Gaussian) . Therefore. we resort to simulations to estimate P"1I1 as a function of cell loading and number of sensors, from which we can obtain the system capacity (maximum cell loading) as a function of cell loading and number of sensors. The results of these simulations are shown in Figs. 6-9 and are discussed in Sec tion IV . IV. SIMULATION AND NUMERICAL RESULTS In all of our simulations and numerical results, we consider only the first two tiers of interfering cells, which means that K = 18 cells . We assume that the voice activity factor v is 0.375 . We assume that for adequate performance, the required BER is 10- 3 which corresponds to Ebl(N" + 10 ) of 7 dB . We also assume that the processing gain L is 128. Finally, we assume that a, is 8 dB . For the uplink, the outage probability was computed using (22). The results are summarized in Fig. 3. From this figure, it is shown that by using antenna array to form narrow beams towards the desired mobiles, a many-fold increase in system capacity can be obtained. For example,
t
: :.)//::
I
.Qllini
• 8c:i
., .. .. ,.
I;
.)
.. .. .eW'!'.1.20 .
. ...
"
• . . . .. . . . . .
,
:
I
'r
Bvo. 060
/
w III
if
..
10
..
10
I
J
50
100
I:
._.
. _ _ ... approximalebeam panern ..
I:
...
0.7r ··············:···, ... ... t;···
~
..
I:
0.6r ······.. ······:···"
.....I:( . I:
/
f
0.41-
. ..... ...............
·········· , {,
0.3r ·············-:·j
"-
f
I
0
..t:
e
.. .'.
O.sr · ············, ..,,·,
_--,_ _ .. .lIUll beam paIlem
,:
~0.51- · · · · · · · · · · · ·· ·:· ·1
.. .. , ... .
...~
I:
.;
~
a:
I~
BW=30
I:
0.91-·· ·······.. ··:· ,· ·/
I 150
200
r.
250
N. Number of Users per Cell
300
350
4
400
5
6
Fig . 4. Actual and approximate beam pauerns .
Fig . 3. Uplink outage probability as a function of beamwidth .
for 0 .01 outage probability, the uplink system capacity goes up from 31 users per cell for the single antenna case to about 320 users per cell if we use an arra y. such that we have beams with beamwidth of 30 ° . Also . to evaluate the accu racy of the approximation we used. we simulated the system (based on (14) - (16) ) with a cell site circular antenna array with nine elements and radiu s equal to AI:" corresponding to half-power beamw idth of 4:"° . In fact. the beamwidth was taken to be slightly more than 42 ° to account for the interference energy picked up through the side lobes of the array pattern . Fig . 4 show s the actual array pattern versus the app rox imate pattern. In Fig. 5 . we plot both the outage probabil ity computed from (22 ) and from simulations. which indicate good agreement between the simulation results and the approximation . For the downlink , results for the outage probability were obtained by simulations based on (31 )-(33) . A circular array with one, five , and seven elements and A/2 spacing was used in the simu latio n. For all other parameter values above. the histogram of Eh/(N" + U is obtained for each M and N value from 20 .000 runs. In each run , 19 Rayleigh random va riables with mean squa re value that hav e a log-normal di stribution with o, = 8 dB are generated, and the maximum of the se is taken to be that of the desired mobile's cell site . Also. we assume that the mobile is positioned on the boundary between cells . which represents a worst case situation . Some of the generated histograms of G, + G2 are shown in Figs . 6 , 7 , and 8 . The generated histograms are used to est imate the probability of outage as function of cell loading and number of sensors. These results are summarized in Fig . 9 , which also shows a many fold increase in capacity by using an tennas to form narrow beams towards the desired user. Note that as we mentioned before, if orthogonal codes are used on the downlink and in the case of no rnultipath , interference will be primarily due to outside cell interference and the corresponding cell loading N at which outage will occur will be larger.
10·
.---,---:--.----:-:0----.----.....---,---, BW~49
Degrees
: Simulations
_
EquatIon 21
(;
o c:i
a: ~
w
~
Ii:
449
10"
i
1,"'
, 1
! 10''-[---=-":-----:-'-::-----:-'----'---:-------:-------:' 190 200 210 220 230 240 250 260 !
,
I
N, Number 01Users per Cell
Fig . 5. Uplink outage probabil ity : simu lation vs . approximat ion.
3500 Number of Mobiles per Cell, N-30
3000
Number 01Array Sensors . M, l Number oj Runs. R,20000
2500
~20oo
i !
. ..
"' 1500 1000
..
500
..... .
0
Jl
o
5
. ..
"'
..
10
15
Innnnn. 20
25
Interference Power
Fig . 6. Histograms of G, -;- G2 for M
30
= I . N = 30 .
35
40
V.
4000 3500
.Number 01Mobiles psr CeU. N~130
3000
.Number 01Runs. R=20000
Number of Array Sensors. 1.1=5
2500
rr:
0 1500 1000 500
5
"n
10
nnnnn~_
15 20 25 Interference Power
Fig . 7. Histogram of G, + G, for M
35
30
40
REFERENCES
= 5. N = 130.
4 500 r---.---..,------,--~--.._-_._--..,_-___r--...,
I
Numoer of Mobiles per Cell. N=170
4000 "
Number 01Array Sensors. 1.1=7 Numoerol Runs. R=20000
3500,. 3000 -
!d 2500"
;;; cr ~
c" 2000 ~ : 500 1000 500-
20
25
Interference Powe r
30
35
40
45
FIg . H. H"tngral1l of G , + G, for JI = 7. .V = 170
I
Ou tag e Probablhty vs. Number a t Sensors
!
r M=7
50
100 150 N. Number 01MObilesper Cell
CONCLUSIONS
We have studied the capacity improvement for CDMA cellular communications systems with base-station antenna array for both uplink and downlink. The outage probability was evaluated as a function of cell loading, array parameters, fading and shadowing effects, and voice activity . Our analytical and simulation results show that there can be a substantial increase in system capacity by incorporating antenna arrays at the base-station. Our approach uses spatial processing to determine the dynamic spatial wavefront at the cell site and constructs a robust beamformer. Our model, used in this paper, does not include the effects of multi path which will be presented in a different paper.
200
Fig . 9 . Downlink outage probability vs. number of array sensors .
250
[11 K. S. Gilhousen. I. M. Jacobs . R. Padovani, A. Viterbi. L. A. Weaver. and C. Whe atly , "On the capa city of a cellular CDMA sysrem," IEEE Trans. Veh. Techn o! .. vol. 40. no . 2, pp. 303-312 . May 1991. [21 A. M. Viterbi and A. 1. Viterbi . "Erlang capac ity of a power controlled CDMA system," IEEE J. Sel ect . Areas Commun . . vol. 11. no . 6 . pp. 892 -900 . Aug . 1993. [31 B. Sua rd. A . Naguib . G. Xu . and A . Pau lraj. "Performance anal ysis of CDMA mobile: co mmunicatio n systems using antenna arrays," in Proc. 1C.~SSP ·93. vol. VI. Minneapol is. MN . pp. 153-156. April 1993 . 141 A. F . ~aguib. \_~ . Paulraj . and T. Kailath , "Capacity improvement with base-station antenna arrays in cellular CDMA . " in Proc. 27th Asil omar COil! 0 11 Signals. Systems and Computers: vol . II. Pacific Grove. GA . pp . 1437-1441 . Nov . 1993 . 151 1. C . Liberti and T . S Rappaport . " Reverse channel performance improveme nts 10 CDMA cellular com munication sys tems employing adapt ive antennas ;" in Proc. GLOBECOM ·93 . vo l. I. pp . 42 -4 7 . 1993 . 16J V. Wee rack ody , " Diversity for the direct-sequence spread spe ctrum sys tem using multiple transmit antenna s. " in Proc. ICC'93 . vol. Ill. Gene va. Switzerland , May 1993 . (7J P. Balaban and J. Salz , " Optimum divers ity combining and equalization in data transmission with application to cellular mob ile radio Part 1: Theoretical considerations," IEEE Trans . Commun .. vol. 40. no . 5. pp. 885-894. May 1992. 18\ S. C. Swales . M. A. Beach. D . J . Edwards. and J . P. McGeehn . "The performance enhancement of multibeam adaptive base station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Technoi .; vol. 39 . no . I. pp . 56-67 , Feb. 1990 . (9] J. Winters. 1. Saltz . and R. Gitlin . " T he capacity of wireless com municat ion systems can be substantially increased by the use of antenna diversity. " in Proc, Can! on Inf ormation Scien ce and Syst ems . vol. II. Princeton . NJ. pp . 853-858 . Oct. 1992. [10] S. Anderson. M. Millnen. M. Viberg . and B. Wahlberg , "An adap live array for mob ile communication systems." IEEE Trans . Veh , Techno!. . vol. 40. no . I. pp . 230-236. Feb . 1991. [II J R. Kohno. Hi-lmai. and S. Pasupathy , " Combination of an adaptive antenna array and a canceller of interference for direct-sequence spread spectrum multiple-access system." IEEE J. Select. Areas Commun., vol . 8. no . 4. pp. 675-682. May 1990 . [12] R. C . French. "The effect of fading and shadow ing on channel reuse in mobile radio ." IEEE Trans . Veh. Techno! .• vol. VT-28 , no . 8, pp. 171-181. Aug . 1979 . [131 R. O . Schmidt. "A signal sub space approach to multiple-emitter 10cal ion and spectral estimation ." Ph .D . dissertation , Stanford Uni v . . Stanford . CA 94305. 1981. [14] A. Paulraj . R. Roy. and T. Kail alh. " Estimatio n of signal parameters by rotational invariance techniques (ESPRIT)." in Pro c. of 19th Asilomar Can! on Circuits , Systems and Camp . , 1985. [15] W.- P. Yung , "Direct sequence spread-spectrum code-division-multiple access cellular systems in Rayle igh fading and log-normal shad owing channel," in Proc. ICC'91. vol . II. pp . 871-876.1991. (16) G. Xu. H. Liu, W. Vogel . H. Lin. S. Jeng, and G . Torrence, "Ex-
450
perimental studies of space-division-multiple-access schemes for spectral efficient wireless communications, " submitted to SuperCom/ ICC'94, May 1994. [17] J. H. Winters, "Signal aquisition and tracking with adaptive arrays in wireless systems," in Proc. 43rd Veh. Technol. Conf., vol. I, pp. 85-88, Nov. 1993. [18] D. Gerlach and A. Paul raj ..• Base-station transmitting antenna arrays with mobile to base feedback," in Proc. 27th Asilomar Con! on Signals, Systems and Computers, Pacific Grove. CA, pp. 1432-1436. Nov. 1993. [19] Y. Akaiwa, •. Antenna selection diversity for framed digital signal transmission in mobile radio channel," in Proc. VTC'S9, vol. L pp. 470-473, 1989.
451
Analytical Results for Capacity Improvements in CDMA Joseph C. Liberti, Jr., Student Member, IEEE, and Theodore S. Rappaport, Senior Member, IEEE
Abstract-In this paper, we examine the performance enhancements that can be achieved by employing spatial filtering in code division multiple access (CDl\'IA) cellular- radio systems. The goal is to estimate what improvements are possible using narrow-beam adaptive antenna techniques, assuming that adaptive algorithms and the associated hardware to implement these systems can be realized. Simulations and analytical results are presented which demonstrate that steerable directional antennas at the base station can dramatically improve the reverse channel performance of multicell mobile radio systems, and new analytical techniques for characterizing mobile radio systems which employ frequency reuse are described using the wedge-cell geometry of [1). We also discuss the effects of using directional antennas at the portable unit. Throughout this paper we will use phased arrays and steerable, fixed pattern antennas to approximate the performance of adaptive antennas in multipath-free environments.
by battery consumption at the portable unit, therefore there are limits on the degree to which power may be controlled. Finally, to maximize performance, all users on the forward link may be synchronized much more easily than users on the reverse link [6]. Adaptive antennas at the base station and possibly at the portable unit may mitigate these problems. In the limiting case of infinitesimal beamwidth and infinitely fast tracking ability. adaptive antennas can provide for each user a unique channel that is free from interference. All users within the system would be able to communicate at the same time using the same frequency channel. in effect providing space division multiple access (SDMA) [7]. In addition, a perfect adaptive antenna system would be able to track individual multipath components and combine them in an optimal manner to collect all of the available I. INTRODUCTION signal energy [81. In this paper. we will investigate the URRENT day mobile radio systems are becoming effects of spatial filtering by simulating a phased array and congested due to growing competiton for spectrum. by simulating antenna patterns with fixed patterns but adMany different approaches have been proposed to maxi- justable boresight angles. Furthermore. multipath is not mize data throughput while minimizing spectrum require- considered. ments for future wireless personal communications serClearly ~ the perfect adaptive antenna system described vices [2], [3]. One way to increase capacity without added above is not feasible since it requires infinitely large anspectrum is to reduce cell sizes [4]. For this reason, cell tennas (or alternatively ~ infinitely high frequencies). This sizes in emerging cellular communication systems are raises the question of what gains might be achieved using much smaller than cells used in land mobile cellular sys- reasonably sized antenna arrays which operate at UHF and tems designed previously. This, however, also leads to microwave frequencies. increased infrastructure (base station) costs. Furthermore. While both TDMA and COMA systems have been proto maximize capacity in CDMA systems, power control posed for emerging personal communication systems. is required [5]. COMA is more naturally suited to the pseudo-SDMA enThe reverse link (the link from the mobile unit to the vironment. This is because co-channel users do not have base station) presents the most difficulty in COMA cel- to be synchronized with each other in a CDMA system. lular systems for several reasons. First of all, the base As the advantages of SDMA are realized, the interference station has complete control over the relative power of all levels seen by each simultaneous CD~'lA user drop. and of the transmitted signals on the forward link; however, the bit-error performance will improve for each COMA because of different radio propagation paths between each user. On the other hand, when no SDMA is achieved, user and the base station, the transmitted power from each CDMA performance is no worse than the case where omportable unit must be dynamically controlled to prevent nidirectional antennas are used at both the base station any single user from driving the interference level too high and the portable unit. In a single cell TDMA system, users for all other users [1]. Second, transmit power is limited must be reassigned to new time slots to take any advantage of SDMA. Manuscript received September 30. 1993~ revised March 31. 1994. For interference limited asynchronous reverse channel The authors are with the Mobile and Portable Radio Research Group. CDMA over an additive white Gaussian noise (AWGN) Bradley Department of Electrical Engineering. Virginia Tech. Blacksburg. channel, operating with perfect power control with no inVA 24061. IEEE Log Number 9403205. terference from adjacent cells and with omnidirectional
C
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 43, No.3, pp. 680-690, August 1994.
452
antennas used at the basestation , the bit error rate (BER),
E1em~ IO
Pb • is approximated by [6]
(1.1)
where K is the number of users in a cell and N is the spreading factor. Q(Y) in (1.1) is the standard Q-function. the probability that y > Y when y is a zero-mean. unit variance, Gaussian distributed random variable . Equation (1.1) assumes that the signature sequences are random and that K is sufficiently large to allow the Gaussian approximation described in [6] to be applied . To il\ustrate how directive antennas can improve the reverse link in a single cell CDMA system, consider the case in which each portable unit has an omnidirectional antenna, and the base station tracks each user in the cell using a directive beam. Assume that a beam pattern, G(
ALFN_l.i QN ' I.,n
Q N. I.OUl
Fi g. 1. A generalized adapti ve antenna a rray w ith N ele me n ts . The inputs from ea ch antenna are m ixed do wn to a n intermed iate frequen c y a nd di vided int o I and Q co m po ne nts . The I and Q co m po ne nts fro m eac h antenna are tiltered by an ad aptive line ar filte r (A LF' J is the ALF correspond ing to the ith e le me nt and the jth user ). Th e I o u tputs from each ALF a re su m med to provide I,," ,. Similarl y th e Q o u tpu ts from each ALF are su m med to provid e Q"" . I ,,", and Q,,", form the sig na l wh ich is avail able to the rec ei ver.
Fig. 2 . An ideal ize d tla t-to p power pun ern With a 60 ° beam wid th a nd a - 6 dB side lob e lev e l . Th is pat tern ha s no variat ion In the /) dire ction (the ele va tio n plane) for 0 s < 11'. T hrs coo rd ina te svstem is used throughout ' th is paper.
a
Fig. I) of the base station antenna array. which is steered
to user O. is given by
f = E
K- I [
I ~I G(cf>;)P r .
] 1
( 1.2)
where
453
( 1.3)
Assuming that users are independently and identically distributed throughout the cell. the average total interference power received at the central base station may be ' W h ile this work considers adaptive a nte nnas at the ba se sta tio n . power co n tro l could be im ple me nted using a reference omnidirectional antenna at the base statio n to receive all mobile signa ls .
expressed as
I
= Pc(K -
Jro Jro R
1)
21r
ftr, cp)G(cp) dcp dr
(1.4)
where f(r, cp) is the probability density function describing the geographic distribution of users throughout the cell. Assuming that users are uniformly distributed in the cell, we have
I =
r,
(K -
211"
I)
r:!7r
Jo
G(cp) dip,
( 1.5)
The directivity of an antenna which has no variation in the () direction is [11]
21r
( 1.6)
Therefore the average total interference seen by a user in the central cell is given by
I
=
Pc(K - 1) D .
In order to develop simple bit error rate simultaneous asynchronous interference users when directive antennas are used, the bit-error-rate expression of (1. I) can
Ph ~ Q(.J3,V x CIR)
( 1.7)
expressions for limited CDMA we assume that be expressed as (1.8)
pattern shown in Fig. 2 with a side lobe level of 0.25 and a main beamwidth of 60 0 , the directivity of the antenna is 2.67 or 4.3 dB. The bit error rate with the directive antenna at the base station is 2.5 x 10- 5 , a BER improvement of two orders of magnitude. This example illustrates the possible improvements that can be achieved using adaptive antennas at the base station. In the remainder of this paper, we remove the constraint that users in adjacent cells do not inerfere with the received signal, and develop a general analysis technique which is confirmed by simulation. Section II describes analytical techniques used to detennine bit error rates in cellular CDMA systems employing adapti ve antennas. Section III presents simulations in which we compare the performance of five base station antenna configurations, three of which use adaptively steerable antennas at the base station. It is assumed that the portable units use omnidirectional antennas. We also compare the simulation results with the analytical results developed in Section II. In Section IV, the effects of adaptive antennas at the portable unit are examined using several different base station configurations. Furthermore. we demonstrate the two distinctly different effects achieved by using directive antennas at the portable unit versus using directive antennas at the base station. Finally. Section V summarizes the results of this paper.
where N is the spreading factor. and CIR is the ratio of the power of the desired signal to the total interference. In (1.8), it is assumed that M interfering users. each with a received power level of P/M. have the same effect on bit error performance as one interfering user with a received power P. This assumption is known to be inaccurate when the powers of users are widely different and when the number of users is small [12]: however. it provides first order approximation for the case of a large number of users. Using the fact that the power of the desired signal, weighted by the array pattern. is P; and using (I . 7). the bit error rate for user 0 is given by
II.
REVERSE CH:\:'-J~EL PERFOR~1A~CE \VITH ADAPTIVE ANTEN~AS AT THE BASE STATIO!\
The use of adaptive antennas at the base station receiver is a logical first step in improving capacity for several reasons. First of all . space and power constraints are not nearly as critical at the base station as they are at the portable unit. Second. the physical size of the array does not pose as much difficulty at the base station as at the portable unit. Note that adaptive antennas may also be used at the base station for directing energy in the forward channel. in which case the analysis is similar to the reverse channel case because of the perfect power control assumption. The only difference on the forward link is that interferers are (1.9) other base stations. rather than portable users. Since the transmitter and receiver typically operate in two different Thus, (1.9) holds for any single cell system with perfect frequency bands in a duplex manner, the adaptive antenpower control when base station antenna pattern which nas at the base station transmitter would be adjusted by has no variation in the (J direction. Equation (1 .9) is useful performing a transformation on the tap weights adapted in showing that the probability of error for a CDMA sys- for the receiver, and copying the new weights to the transtem is related to the beam pattern of a receiver. If we use mitting antennas [9]. This is reasonable if an assumption the idealized antenna pattern illustrated in Fig. 2 to ap- of retrodirectivity on similar frequency bands is apporproximate a realizable directive antenna pattern then it is priate. If the multipath components arriving in the reverse immediately apparent that the gain of the antenna directly channel do not have the same angles of arrival as those in contributes to the performance of a CDMA system. For the forward channel, then it is no longer appropriate to instance, if K = 250, and N = 511, with omnidirectional derive the transmitter tap weights from the received sigantennas at the base station, an average bit error rate of nal. 6.6 X 10- 3 is obtained per user. Using the flat-top beam Equation (1.9) is only valid when a single cell is con454
sidered. To consider the effects of adaptive antennas when CDMA users are simultaneously active in several adjacent cells , we must first define the geometry of the cell region . For simplicity, we consider the geometry pro posed in [I] with a sing le la yer of surrounding cell s, as illustrated in Figs . 3 and 4 . Let d i • j represent the d istance from the ith user to base j as illu strated in Fig . 3 . Let a,« represent the distance from the i th user to ba se sta tio n O. the center base statio n . Assume that path loss in dB between user i and ba se j is given by a sim ple di stance dependent path loss rela tion ship suc h that the pow er received at base station j , fro m the tran sm itter of user i . Pr . i . ]> is give n by
Pr . i . ]· =P T.I
(47fd ) (di.J A
--
drcl "
-'
-
ref
(2. 1)
Fig . 3 . Th e wedg e ce ll geo me try proposed in [I I.
where II is the path loss exponent typically ranging between 2 and 4. and d rd is a clo se-in reference distance
[u.
If we assume that perfect power co ntro l is appl ied to the i th user. and all ot he r users in ce ll i. by base j , suc h that power P, :) is rece ived as base i . then the power tran smitted by user i . Pr l • is give n by P
=P /.I
'·1
(47fd"'I)~( ~)"
(2.2)
A d""
The power receiv ed at ba se statio n 0 from user i . g ive n by
P r:l 1l
2R,:
is \
(2.3) Sub st ituting (2 .2 ) into (2 .3). the power recei ved at base user i . in adjacent ce ll. j . is g iven by
Fig . ·L Geo metry fo r detc rrruning d r , ) as a func tio n o f tiL'" the dis ta nce betwee n user i and the centra l ba se stati on and ~ . u . the ang le of the use r i rela tive to the line betwe en the ce ntral base sta tio n and base sta t ion } .
o from
Pr:I .1l = P, OJ ( dd~ ) " I .U .
jacent cell is given by (2 .4 )
To analyz e (2 .4) . we co ns ide r the geo me try sho wn in Fig . 4. From the law of cosi nes .
Sub stituting (2 .5) into (2 .4) . the po wer received at ba se o from user i is given by P r .1.0 =
r .,
2R)2 4R ),,/2 ( I + (d - d cos l{!r.O 1.0
(2.6)
1. 0
To determine the average out-of-cell interference power inc ident on the central ba se statio n . we ass ume that users are uniforml y di stributed in a typi cal adjacent cell from r = R to r = 3R and fro m l{! = to Thus , we use a modified geometry from [I) where eight equal are a cells surround the center cell . The probability density function (pdf) for the spatial di stribution of users in a single ad -
-7f/8 7f/8.
455
fe r. l{!)
=
r -R'
7f -
R < r < 3R:
(2.7) Let X represent the e xpec ted value of the inte rfe rence power from a single user in one of the adjacent cell s when om nid irectio nal base station antennas are used .
))/12 'R)2 4R (( I + ( -r - ~ cos l{! dr dl{!
(2 .8)
If it is assumed that all nine base stations control power such that Pc: j = Pc, then given a value of n, we can ex press the expected value of central cell interference power for a single adjacent cell user as X
=
{3P e
(2 .9)
where (3 =
[3R [11"18
JR J-1C18 Iir. 4R - -;: cos cp
({J)
((
(2R)2 1 + -;:
))n/2 dr dcp
VALUES OF
(2.10)
No No + Na1M 1
!=---
(K -
l)Pc
for
K»
:::
= (K - l)P c + 8K{3Pc-
e
2
0.14962
3
0.08238
4
0.05513
E[Pr~I.O
10
<
<
r
R]
r r
21r
R
= Pc
r 2 G(lP) dr dlP
Jo Jo
wR
= P,
(2.14)
D
where D is the directivity of the beam with pattern G('P) and the average received power at the base. Pr : i . O from an interfering user in the central cell is directly a function of the base station directive gain. Then the average interference power at the array port of the antenna array at the base station, as shown in Fig. 1. due to a single user in an adjacent cell is given by
E[P r : 1•O IR
< r < 3R]
1 7 i·~R 1~:~ =-2: 8 p =0
R
-
. p( ( I + (
s
7r
(
P7r) r 'P+- --~ 4 7rR-
G
2rR)2-
4R - cos
,.
(~)
)" .: dr d4P. (2.15)
(2.12)
where we have assumed that there are K users in each of the nine cells. For 11 = 4, from Table I, {3 = 0.05513, and, from (2.12), f = 0.693, implying that 31 % of the interference power received at the central base station is due to users in adjacent cells. Note that, when omnidirectional antennas are used at both the base station and the portable unit, the value of the reuse factor . f.. is determined by the cell geometry, the power control scheme . and the path loss exponent. When omnidirectional antennas are used at both the base station and the portable unit, the total interference seen on the reverse link by the central base station is the sum of the interference from users within the central cell, (K - l)Pc ' and users in adjacent cells, 8K{3Pc-
I
n
EXPONENT, n AS
1
--1 + 813
1
Loss
(2.10)
is thus given by
(2.11 )
where No is the total interference, seen by a desired user in the central cell, at the central base station on the reverse link, Nat is the total interference seen "by the desired central cell user from all users in a single adjacent cell. M I is the number of cells which are immediately adjacent to the central cell, which is always eight for the geometry considered in this paper. This reuse factor is a measure of the impact of users in adjacent cells on the performance of the link between a user in the central cell and the central base station. When power control is perfonned as described in this section, such that the power received from each mobile unit in the base station controlling that unit is P, . then (2.11) may be expressed as
(K - l)P c + 8K{3Pc
TABLE I A FUNCTION OF THE PATH DETERMINED BY
Table I lists the values of (3 for several values of n. When omnidirectional antennas are used at both the base station and the portable unit, 13 is related to the reuse factor, [, which is defined in [1], for a single layer of adjacent cells, as
f =
(3 AS
Here a special case is considered. If G( 'P) is piecewise constant over the region C2p - 1)(7r/S) < ~ < (2[1 + 1)(7r/8) for p = 0 . · . 7 . then the antenna pattern may be expressed as 7
G(lP)
')~() G"V
V(lP) =
l~
(_
lP
P1r)
(2.16)
4
where (1.17)
Substituting (2. 16) into (2. 15) . we obtain.
(2.13)
Let us assume that for the mth user in the central cell, an antenna beam from the base station with pattern, G«({J), may be formed with maximum gain in the direction of user m. It is assumed that perfect power control is applied such that all base stations controlling reverse link received power to the same level, Pc. The average interference power contributed by a single user in the central cell
456
E[P r :1 •O IR < r < 3R]
= P; -1 2:7
8 p =0
1 iit'/H 3R
Gp
4R - -;: cos (c,o)
R
)n/2
-1r/8
r
-.,
7rR-
dr de,
(
I
+
(2 R)2 r
(2.18)
The directivity of the antenna pattern described by (2.16) is
8 D =7 - -
2: c,
p=o
(2.19)
Therefore, (2 .18) may be rewritten , using (2 .10) and (2 .19) , as E[Pr:i.O I R
< r < 3R] =
PJ3 D
(b) Sectorized
(2 .20) 18
It can be shown that (2 .20) remains valid when the beam pattern, G('P) , is rotated in the 'P plane . Therefore (2 .20) is appropriate when G('P) is piecewise constant over (2p - I) (1r/8) < 'P - 'Pel < (2p + 1) (71"/8) for any angle 'Pel between -1r/8 and 1r/8 . Using (2 .20) with (1. 7), the total interference power at the array port (in Fig . 1) of the center base station receiver is given by
1=
(K - I)P, + 8KPJ3 D
.
(e) Flat- topped
For K
»
3ND K(\ + 8/3)
270
2:'0
(2 .23)
r..
111.
SIM ULATION OF
ADAPTIVE A:-
REV ERSE CHANNEL PERFORM AN CE
271"
= ----):" G('P) dip
Fig . 5 . The rive base statio n antenna patterns used in this study. These patterns arc shown for the case when the desired user is at an angle 'P = 600 from the X axi s . Shown here are (a) the omnidirectional pattern, (b) the 120 0 secton zed pattern . (c) the tlar-topped patt ern. (d) the three clement binomial pha sed array (re ferred (0 ;IS the .. ;ldaptive " pattern in this pape r i . and (c r thc binomial phased arra y pattern overlaid with a I:W o se ctorrzunon patt ern (re fe rred to as thc " aua pt;ve -se ctorized " pattern) .
The second configuration. illustrated in Fig . 5(b), used 120 secto rization at the base station . In our model. the base statio n used three sectors. one covering the region from 30 0 to 150 0 , the second covering the region from 150 0 to 270 0 • and the third covering the region from -90 0 to 30 The first sector is illustrated in Fig. 5(b) since this sector would be active when the desired user is at an angle of 60 0 • In this system. only interfering users within view of the same secto r as the desired user were included in the CIR calculation. The effective gain of this antenna is 4 .8 dB. The third simulated base station configuration, shown in Fig . 5(c) , used a "flat-topped" beam pattern similar to that shown in Fig . 2 . The main beam was 30 0 wide with uniform gain in the main lobe . Side lobes were simulated by assuming a uniform side lobe gain which was 6 dB below the main beam gain . From (1.6), the directivity of this beam is 5 .1 dB . The fourth configuration , which used a simple three element linear array, is illustrated in Fig. 5(d). This is the beam pattern formed by a binomial phased array with elements spaced a half wavelength apart . The axis of the array is in the 'P = 0 0 direction . Like a111inear arrays, this array exhibits a pattern which is symmetric about the axis of the array (the X-axis , as shown in Fig. 2) , therefore a 0
0
To explore the utility for (2.23) and to ver ify its accuracy . we considered five ba se station antenna patterns which are illustrated in Fig . 5 . These antenna patterns are assumed to be directed such that maximum ga in is in the direction of the desired mobile users . The first-base station antenna pattern is an omnidirectional pattern which models that used in traditional cellular systems . This configuration . shown in Fig . 5(a) was used as a model for standard omnidirectional systems without adaptive antennas . In order to make a fair comparison between the effects of various antenna types on bit error rate as a function of directivity, and given the fact that the simulations were performed in two dimensions only. antenna gains cited in this section are defined by (1 .6) which is restated here :
D
270
(2.22)
Equation (2 .23) relates the probability of error to the number of users per cell . the directivity of the base station antenna . and the propagation path loss exponent through the value of p. It is assumed that perfect power control is applied as described in Section 1. with all base stations controlling reverse link received power to the same level.
BA SE S TATION FOR
(e) Adap tive-Seetorized 90
I . Ph is approximated by
3ND \ K(l + 8J)} '
270
90
(2 .21 )
Substituting (2 .21) into the (1. 8). using the fact that the desired signal power at the array port is Pc' we obtain an average bit-error probability for the CDMA system em ploy ing a piecewise constant directive beam : PI> == Q (
270
(3.1)
457
•
mirror image of the main beam is also present as illustrated in Fig. 5(d). This array is not capable of adaptively nulling interfering signals; therefore we expect the perfonnance of this array to be poorer than that of a truly adaptive system. On the other hand, we did assume that the array was able to direct the one of the two main beam components in the direction of the desired user. For each desired user, the phase was computed for each element of the array and the new beam pattern was formed at the center cell base station. While the three-dimensional gain of a binomial phased array is constant at 4.3 dB regardless of scan angle, the two-dimensional gain defined by (3. 1), which is more appropriate for comparison given our assumption of users in the horizontal plane only, varies between 2.6 and 6.0 dB, depending on scan angle, with the higher gain corresponding to broadside scan angles. The pattern for the fifth simulated base station configuration, a sectorized adaptive antenna, is shown in Fig. 5(e). Beginning with the sectorizing system whose pattern is illustrated in Fig. 5(b), we added a three element linear phased array to each sector. The linear array for each sector is aligned such that the broadside direction is in the same direction as the center of the sector. This base station configuration actually uses a total of nine elements, however, only three elements are used to track any given user. For example, in Fig. 5(e), the desired user is at an 0 angle of
was calculated from Desired Signal ~
C1Ri = K - l
l:
n=O
Pi.O.O 8
Pn .O•O +
K-l
(3.2)
l: l: r..:
m= l n=O
~~ In-cell Interference
Out-of-cell
Interference
The bit error rate for the i th user in cell 0 on the reverse link was determined by first calculating the CIR for the ith user from (3.2) then using that value in (1.8), which is restated here:
Ph., = Q( ~3N x CIR,)
(3.3)
where N is the spreading factor. For each of the simulations performed in this study, a spreading factor of N = 511 was used. It was assumed that any portable unit in the nine-cell region (except for the desired user) contributed to the interference level of the desired user in the central cell. This calculation was carried out for every user in the central cell and the resulting bit error rates were averaged to obtain an average bit error rate for the cell. For instance. if there were 1700 users in the nine cells and 300 users in the central cell . then the bit error rate was determined for the 300 users in the central cell. and 2699 interfering users contributed to each CIR computation. Each base station configuration wassimulated for user densities ranging from 15 to 500 users per cell, in steps of 25. Fig. 6 shows average bit error rates resulting from the simulation for the five previously described antenna patterns for several values of path loss exponent. n . The three element linear array. whose pattern is shown in Fig. 6( d) . was able to achieve almost an order of magnitude improvement in BER despite the large backlobe , By eliminating the large back lobe. but still retaining significant side lobes. the flat-top pattern. shown in Fig. 6(c). achieves a BER which is better than two orders of magnitude less than the BER when omnidirectional antennas are used at the base station. with fewer than 200 users pe r cell. The average bit error rate alone is not a sufficient metric of system performance. Rather. the distribution of BER·s over the user population is a second-order measure which provides insight about the performance of a CDMA cellular system. Fig. 5 relates the average BER to the BER which is not exceed by 50, 90, 95, and 99 % of the users. Note that for a given bit error rate, two to four times as many users many be supported using directional antennas as for omnidirectional antennas. It is useful to note that these increases in performance were made by applying relatively modest requirements to the base-station adaptive antenna. The flat top antenna was specified to have a 30° beamwidth and a side lobe level that was only 6 dB below the main lobe. It should be noted that these bit-error-rate improve-
458
TAB LE II R ELATIONSHIPBETW EEN THE AV ERAGE BIT ERROR R ATE. AND P WHERE P,..r IS DE FINED SUCH THAT X% OF THE USERS IN THE C ENTRAL C;L~ HAVE A BIT ERROR RAT E WHICH IS L ESS THAN P,. ,. THI S IS FOR THE C ASE OF K = 200. AND A PATH Loss E XPONENT OF II = 2. N OTE THAT TH ERE IS A M UCH WIDER RA NGE OF BI T ERROR R ATES FOR THE H IGHER G AI N ANTENN AS. FOR EX AMPLE. 2 USERS OR I % OF THE USER POPULATION E XPERIENCED A BER WHICH WAS W ORSE THAN 1.5e-3 WHEN THE SECTORIZELJ ANTENNA PATTERN WAS C ONSIDERED
le
Pe.sO
Pe.90
Pe.95
Pe.99
Omn i
3.0e-2
3.1e- 2
3.2e-2
3.2e-2
3.2e -2
Sec torized
6.le -4
5.3e·4
LOe-3
l.le-3
L5e- 3
Adaptive
2.ge -3
2.6e -3
6.4e-3
7.2e -3
77e- 3
Flat- lopped
4 .0e-4
3ge-4
5.2e-4
5.6e -4
6.5e -4
Adapt ive-Se ctori zed
16e-7
6.5e-8
J Oe- 7
4.7e ·7
2.4e-6
--''----'-
--'=
Ic " r--;;:-:;--.--~--===
=
Scelonud -- Ad_pll" _ . R.... c»pprd
Ie- I
.( le.J
~Ie-
.
.<'e-J
. . . ...~ ... .
.'
100
Avg BER
Base S tation Pattern
- -, 'd'r_--:Om n
200 )00 400 Numb::r of Users p:r Ce ll ( K)
SOO
(, l D=2
Idr-----.--~-~=~
- Omn i Stdoriud Ie: _ . Ad_ptlwt
•.
F'J.I .lopp~d
Ic:. J
. .. -
~ 1C: .2
....-;. :-;.:.: ..~:: ..
te-a 1, -
'01,--- - - -,-- - -== = - Omnl le
re. I
100
"'. ,-------,-~--===
S«totirrd •• Ad_pUn . FlaHoppRt
200 JOO .too Num ber DCL' SEf'1 oer Ce ll \K l
500
(C) a=J
• • Adapthtl ,s«luflred
~ le - 2
Fig . 7. PIOls o f an al yt i ca l result s usi ng equation 2 .23 w ith two-di me nsiona l di rec ti v it ies of 1.0. 2 .67 . 3.0 . and 3 .2 f o r the o m n i . ada pt i ve. secton zed and flat -topped patt crn s, res pec tive ly .
,-
lelr----,,-----.--~--~---, IO}
nn
~OO
:'O um l"o:: l o ll...:1S rocr \ 'd l I K,
un
\ 00
I Ul
n=2 - . n=J
: 00 )00 .&00 :-
lea - -. n=4
l b) n= I
(Ill na 2
Omnt Base Station
" "" r_ ~ll~mn , ---'---== k l
..
<>:
S«tOf"lu d
Ie ·1
UJ
"d.rlll~
CD 1e -2
F1.H~pf"d
.('"
• • A d . p ll "? ,S f'( Irn"' l rd
Ie-3 ;' It . :
Ie-4
l r·1
le_5L-..Lu~'-L-:...u..,---
100
100
S u.." tEr of
t ~ 'Ilr n
(c) 0=4
; 00
JOO
rer l -elJ '" I
100
200
--"""'--....J 400 500
300
Number of Users per Cell (K)
~~
Fi g . 8. BER fo r the o rn ru and flat -topped beam sys tem s as a f u nctio n o f II .
Fig . 6 . B ER usrng ad ap t i ve anten na s at the base stuno n tor l a l 1/ = 2 . ( b l II 3. and (c) /I = ~ . These resu l ts were develo ped throug h srrnulano n by averaging the BER III every u ser," th e ce nt er cell
=
ment s are primarily due to the dire ctivity of the antenna arra y. The improvements are also dependent on the geographical distribution of interfering users . but in the case of uniformly distributed users . as noted in Section II. the improvement is approximately equivalent to increasing the carrier-to-interference ratio by the gain of the d irectional antenna. Even more drastic improvements were available when sectorization was combined with the adapti ve antenna approach . Adding the three element array to the secto rized system, as shown in Fig . 5(e) . provided a reduction in BER of th ree orders of magnitude for 200 users per cell. Fig. 6 shows results calculated result s from (2.23) for four of the antenna patterns shown in Fig . 5 . By comparing Figs. 6 and 7. it can be seen that for omnidirectional antennas. 120 0 sectorization , and the flat-topped pattern, the calcul ated bit error rates from (2.23) matches the simulation results exactly , even for a relativel y small number of users (K = 25 , 50) . For the case of the binomi al phased
arra y . the analytical results for Ph are optimistic by almost an order of magnitude when K < 200 for all values of n. For K > 350, the anal ytical result s for Pb are only smaller than the simulation results by a factor of 0 .3 or less . Unlike the omnidirectional. sectorized, and flat-topped pattern s. the binomial phased arra y did not exhibit constant two -dimensional gain as a funct ion of scan angle. Therefore . the use of the three-dimensional directive ga in as an " average " gain in (2 .23) is an approximation. By comparing Figs . 6 and 7 it may be concluded that a smaller value of average directive ga in might result in a better match between the simulated and analytical results for the binomial phased array . Nevertheless, these figures demonstrate the accuracy of (2 .23) when compared with extensive simulations . As noted in [13], use of a path loss exponent of n = 4 can result in overly optimistic estimates of system capacity and performance . The different base station antenna configurations demonstrate vary ing sensitivity to the path loss exponent , n. As illustrated in Fig . 8, the flat-topped beam system is highly sensitive to changes in the path loss
459
...,.........,.......,,.....---......,..--..-...
exponent. This is reasonable to expect since , when the CIR is large , the bit error rate is more sensitive to rela tively small changes in interference power. IV. SIMULAnON OF ADAPTIVE ANTENNAS AT THE
...,.......----..-.---,.-...,
Ie
PORTABLE UNIT TO IMPROVE REVERSE CHANNEL
Ie- ..
PERFORMANCE
100
In this section, we examine how the reverse channel is affected by using adaptive antennas at a portable transmitter. A flat-topped beam shape, as illustrated in Fig. 2 , was used to model an adaptive antenna at the portable transmitter. Since space is extremely limited on the portable unit, the gain achievable by the portable unit antenna will be considerably less than that at the base station. For this study, it was assumed that the portable unit could achieve a beamwidth of 60° with a side lobe level that was 6 dB down from the main beam. This corresponds to an antenna with a directivity of 4.3 dB . The pattern is similar to that shown in Fig. 5(c) except that the beamwidth is wider in this case. It was assumed that each portable unit was capable of perfectly aligning the boresight of its adaptive antenna with the base station associated with that portable unit. In this manner. portable units could radiate maximum energy to the desired base station. while reducing battery power proportional to the directivity of the portable antenna . Portable units with adaptive antennas were simulated for each of the five base statio n patterns described in Section III. As in Section III. average values of Ph were found by averaging the bit error rates of each user in the central cell, subjected to interference from the central cell and all immediately adjacent cells . The resulting bit error rates for these systems are shown in Fig . 9. Note that. com paring Fig. 6 and Fig . 9. the bit error rates for the reverse channel are improved when directive antennas are used at the portable unit. For omnidirectional base stations. the BER is only decreased by a small amount (20% or less) for K > 200 when steerable directive antennas are used at the portable unit. However, for highly directive base station antenna patterns such as the adaptive-sectorized pattern, the BER was decreased by an order of magnitude for K > 300 . In Fig . 10, we have defined the BER factor as the ratio of the BER with adaptive antennas at all portable units to the BER without adaptive antennas at the portable units. A small BER factor indicates that adding adaptive antennas improved the BER significantly. For example, a BER factor of 0.5 indicates that using an adaptive antenna at the mobile unit resulted in a reduction in BER of 50% compared with the case of omnidirectional antennas at the mobile unit. As shown in Fig. 10, the adaptive sectorized base station pattern improved greatly by adding adaptive antennas at the portable unit. The resulting BER for this base station configuration when using adaptive antennas at the portable unit was decreased by an order of magnitude
200
300
Numbct of Uan pct Ceq (K)
400
100 200 :JOO 0600 N.."bc, of UKr, per Cell f K )
SOO
(b) 0=3
(I) 0=2
I" ,.......-::-c,.....--
-
.....
,...-
-r--:-n
~1c. :
"::C lc-.
Ic ·J
101) 200 JOO ..00 Nllrnbcr of Ux n ptr Cell fK )
SOO
(c) 0=4
Fig. 9. BER for five different base stat ion configurations using adapti ve antennas at the portable unit for (a) n = 2, (b) n = 3, and (c) n = 4. These results were developed through simulation by averaging the BER of every user in the central cell.
-
0 ......1
o. .-
..e..,
S«toril~
- . Ad.,lIn F1at.lopped • Ad-.pli,,~Srr;'larfud
~
D.
,
., . D.J
..-r : . . ~
~
~
.
...
-~_
... .. ..... ",
*
....::,
~
I ..
...
. ....
-
~. -
D ~ "'-'- -'U." .... C. MIKI
:
"
~
4~
m
Fig. 10. BER factor, defined as the ratio of the BER with adapt ive antennas at the portable unit to the BER without adaptive antennas at the portable , for five different base station configurations using when using adaptive antennas at the portable unit. This comparison is made for n = 4.
compared with the BER when omnidirectional antennas were used at the portable unit. In general. the more directive base station configurations benefitted more from adding adaptive antennas at the portable unit. Using a 60 " beamwidth fiat-topped pattern with a -6 dB side lobe level at the portable unit, the reverse channel BER for omnidirectional base stations was only improved slightly over the case of omnidirectional antennas at the portable . For directive antennas at the base station, the improvements were more dramatic , as illustrated in Fig . 10. The relatively small improvements obtained by using adaptive antennas at the portable unit can be explained by the fact that when omnidirectional antennas are used at the mobile unit. no more than 1-0.455, or 0 .545, of the total interference power is due to users in adjacent cells (see Table III where f = 1/(1 + 8(1». When using adaptive antennas at the mobile unit, all users in the central cell will appear no different to the central base station than if they had used omnidirectional antennas. Thus, adaptive
460
TABLE
III
RATIO OF IN-CELL INTERFERENCE TO TOTAL INTERFERENCE, FUNCTION OF PATH
Loss
f,
more efficient reuse, and for more frequent reuse of signature sequences throughout a large coverage area.
AS A
EXPONENT, FOR FIVE BASE STATION ANTENNA
PATTERNS WITH OMNIDIRECTIONAL ANTENNAS AT THE PORTABLE UNIT
Base station antenna pattern
n=2
n=4
n=3
Omni
0.4535
0.6012
0.6927
Sectorized
0.4532
0.6008
0.6924
Adaptive
0.4524
0.6002
0.6920
Flat-topped
0.4534
0.6011
0.6926
Adapti ve-sectorized
0.4531
0.6007
0.6922
0.4552
0.6028
0.6939
1
1 + 8{3 (Eq.2.13) (values of
r3 from Table
2.1)
TABLE IV RATIO OF IN-CELL INTERFERENCE TO TOTAL INTERFERENCE, FUNCTION OF PATH
Loss
V. CONCLUSIONS
f,
AS A
EXPONENT, FOR FIVE BASE STATION ANTENNA
PAfTERNS WITH ADAPTIVE ANTENNAS AT THE PORTABLE UNIT. THIS DATA IS FROM THE SIMULATION DESCRIBED IN SECTION IV
Base station antenna pattern
n=2
n=3
n=4
Omni
0.6752
0.8155
0.8826
Sectorized
0.6749
0.8153
0.8824
Adaptive
0.6753
0.8152
0.8822
Flat-topped
0.6751
0.8154
0.8826
Adaptive-sectorized
0.6747
0.8150
0.8823
antennas at the portable unit will only reduce out-of-cell interference levels, Therefore. the maximum improvement in CIR. on the reverse link. that can be achieved by using adaptive antennas rather than omnidirectional antennas at the portable unit is only 3.5 dB. Table III shows several values of the reuse factor, f, defined in (2.12) as the ratio of in-cell interference to total interference. for several base station patterns when omnidirectional antennas are used at the portable unit. Similarly, Table IV shows values of f when steerable. directional antennas. with directivities of 4.3 dB, are used at the portable units. Comparing Tables III and IV. it can be concluded that the use of adaptive antennas at the base station does nothing to improve the reuse factor, f: however the use of adaptive antennas at the portable unit does allow f to be improved. When omnidirectional antennas are used at the portable unit, f is entirely determined by the cell geometry, the power control scheme, and path loss exponent, n, which is a function of propagation and not easily controlled by system designers. Using adaptive antennas at the portable unit, it is possible to tailor fto a desired value which is greater than the reuse factor obtained using omnidirectional antennas at the portable unit. Ideally, driving f to unity would allow system design to much less sensitive to the intercell propagation environment, when perfect power control is assumed. This is an important result for CDMA cellular systems because it indicates that use of adaptive antennas at the portable unit could help to allow greater capacity through
It was shown in this study that adaptive antennas, with relatively modest bandwidth requirements, and no interference nulling capability, both at the base station and at the portable, can provide large improvements in BER, as compared to omnidirectional systems. Analytical expressions which relate the average BER of a CDMA user to the antenna directivity and propagation environment were derived and used to determine capacity improvements offered by a number of antenna patterns. It was demonstrated in Section III that the linear phased array provided an order of magnitude of improvement over the omnidirectional base station. The low-gain (5.1 dB) flat-top pattern provided almost two orders of magnitude of improvement over the omnidirectional system. In addition, it was shown that up to three orders of magnitude of improvement can be achieved by adding a simple three element linear array to a three-sector base station. In terms of capacity, the results of Section III indicate that using adaptive antennas at the base station can allow the number of users to increase by a factor of 2 to 4, while maintaining an average BER of 10- 3 on the reverse link. The bit error tate on the reverse channel is further improved by adding adaptive antennas at the portable unit. Using a 4.3 dB gain antenna at the portable, the bit error rate for the directive base station configurations (but not the omnidirectional base station) was at least half of the bit error achieved without directive antennas at the portable unit. For the highly directive adaptive sectorized base station, the improvement was over an order of magnitude for user densities less than 425 users/cell when each user employed an adaptive antenna. Since the directivity of portable unit adaptive antennas is limited by the size of a handheld device, improvements achieved on the reverse channel at the portable are not as dramatic as gains achieved by adaptive antennas at the base station. In addition, cost issues may limit the application of portable unit adaptive antennas. However, the reduction in reverse channel BER may be critical in extremely high traffic environments. In addition, the portable unit is required to track the only current base station, while adaptive antennas at the base station must track every user in die cell. It should be noted, however, most importantly, Tables III and IV showed the increase in reuse efficiency which portable adaptive antennas provide. By using modest gains at the portable unit, such antennas ameliorate the loss in capacity due to intercell propagation through interference control. In short, adaptive antennas at the base station can have a major effect on bit-error-rate performance, but cannot impact the reuse factor" f. Conversely, it has been shown in this paper that adaptive antennas at the portable unit can provide no more than a 3.5 dB improvement in reverse channel CIR; however, they allow the reuse factor,
461
!,
to be altered. It should be noted, however, that the use of directional antennas at the portable unit can only result in an increase in reuse factor of approximately 1/3. It was assumed throughout this study that the adaptive algorithms and hardware could be designed to meet the specified requirements on beamwidth, side lobe level, and tracking ability. It should be noted that, unlike the arrays discussed in this paper, a properly designed adaptive array can null out interference. Conversely, tracking a large number of users with an adaptive array is nontrivial, and it was assumed that each of the base station arrays described here were able to track all of the portable units without error. The multipath channel was not considered in detail in this study; however, it will be significant in developing algorithms for successful adaptive antenna steering. Rather than tracking users, the adaptive array in a multipath environment must track the angle of arrival of multipath components in order to distinguish the maximum signal. This problem is currently under investigation. Furthermore, efforts are currently underway to develop bit error rate expressions which are accurate for small numbers of simultaneous CDMA users with non-identical power levels.
[12] R. K. Morrow and J. S. Lehnert, "Bit-to-bit error dependence in slotted OS/SSMA packet systems with random signature sequences," IEEE Trans. Commun .. vol. 37, Oct. 1989. [13] L. B. Milstein, T. S. Rappaport. and R. Barghouti, "Perfonnance evaluation for cellular CDMA," IEEE lSAC, vol. 10, May 1992. (14] B. Widrow, P. E. Mantey. L. J. Griffiths, and B. B. Goode, "Adaptive antenna systems." Proc. IEEE, vol. 55, no. 12. Dec. 1967. [15] R. Kohno, H. Irnai, M. Hatori, and S. Pasupathy. "Combination of an adaptive array antenna and a canceller of interference for directsequence spread-spectrum multiple-access system." IEEE lSAC, vol. 8, May 1990. [16] S. Anderson, M. Millnert, Mats Viberg. and Bo Wahlberg, "An adaptive array for mobile communication systems," IEEE Trans. Veil. Technol., vol. 40, Feb. 1991.
REFERENCES [1] T. S. Rappaport and L. B. Milstein, "Effects of radio propagation path loss on OS-COMA cellular frequency reuse efficiency for the reverse channel.·· IEEE Trans. Veh. Techno/ .. vol. 41. no. 3. Aug. 1992. [2] G. R. Cooper and R. W. Nettleton, "A spread-spectrum technique for high-capacity mobile communications." IEEE Trans. Veh. Technol .. vol. VT-27, Nov. 1978. [3] A. Salmasi ... An overview of advanced wireless telecommunication systems employing code division multiple access." Con! Mobile, Portable & Personal Commun., Kings College, England, Sept. 1990. [4] W. C. Y. Lee. Mobile Cellular Telecommunications Systems. New York: McGraw Hill, 1989. [5] K. S. Gilhousen et al., Han the capacity of a cellular COMA system:' IEEE Trans. Veh. Technol .. vol. 40, May 1991. [6] M. B. Pursley, "Perforrnance evaluation for phase-coded spread spectrum multiple-access communications with random signature sequences, " IEEE Trans. Commun., vol. COM-25, Aug. 1977. [7] W. A. Gardner, S. V. Schell, and P. A. Murphy, "Multiplication of cellular radio capacity by blind adaptive spatial filtering, " IEEE Con! Sel. Topics Wireless Commun. Mobile. Vancouver, B.C., Canada, Jun 1992. [8] S. C. Swales, M. A. Beach, D. J. Edwards. and J. P. McGeehan, "The performance enhancement of multibeam adaptive base-station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Technol., vol. 39, Feb. 1990. [9] R. T. Compton, Adaptive Antennas. Englewood Cliffs, NJ: Prentice Hall, 1988. [10] B. Agee, "Solving the near-far problem: Exploitation of spatial and spectral diversity in wireless personal communication networks, " in Proceedings Third Virigina Tech Symp. Wireless Personal Commun., June 1993. [11] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1981.
462
Adaptive Transmitting Antenna Arrays with Feedback Derek Gerlach and Arogyaswami Paulraj
Abstract- We address the problem of transmitting multiple cochannel signals from an antenna array to several receivers so that each receiver gets its intended signal with minimum crosstalk from the remaining signals. In addition to the usual "information" mode, we propose a "probing" mode during which probing signals received at the mobiles are fed back to the transmitter. These probing signals are used to identify an unknown propagation environment, enabling the transmitter to form the necessary transmission beampatterns.
A
I. INTRODUCTION
DAPTIVE receiving antennas have been widely applied to military and communication problems to eliminate unwanted interference or separate multiple signals. The aim of receive beamforming is to form a spatial filter that passes the desired signals and suppresses unwanted components. A receiving beamformer can observe its own output and modify its spatial filtering to improve the signal suppression/enhancement [1] . By contrast, the aim of transmit beamforming is to launch a signal into a propagation environment so that each receiver gets its desired signal without crosstalk from the signals intended for other receivers. This task is complicated by the presence of reflecting bodies of which the transmitter has no knowledge. The proposed adaptive transmit beamforming approach uses feedback of the signals received at the mobiles. This feedback makes possible the transmission of multiple signals to multiple receivers with low crosstalk, even in the presence of an unknown multipath environment. While receiving adaptive antenna arrays have been widely studied [2], [3], the transmit problem is equally important and has received little attention so far, except in [4]. In this letter, we formulate the adaptive transmit array problem and present simulations of its signal separation performance. We consider antenna arrays at the transmitter only, and the receiver has a single omnidirectional antenna. II. PROBLEM STATE~IENT AND ASSUMPTIONS
The goal of adaptive transmit antenna arrays is to send multiple cochannel signals from an antenna array through Manuscript received April 18, 1994; approved July IS, 1994. This work was supported by the Army Research Office under Grant DAAH04-93-G0029 and by ARGOSystems, Inc. under Subcontract 59613. The associate editor coordinating the review of this letter and approving it for publication was Prof. Moura. The authors are with the Information Systems Lab, Stanford University, Stanford, CA 94305 USA. IEEE Log Number 9405627.
Fig. 1. Multiple infonnation bearing signals transmitted from an array to multiple mobile receivers.
a propagation environment to several receivers so that each receiver only gets its intended signal with minimum crosstalk. Let (1)
be the d information bearing signals intended for d remote receivers. Let the antenna array consist of tri transmitting elements, and let the complex vector channel from the array to the kth receiver be given by ak:
(2) where aik is the complex channel response from the ith element to the kth receiver. The channel vector ak represents the total channel including the transmitter electronics, antenna array, and reflections.within the medium. In order to ensure that the vector channel is adequately described by a single vector, we need the following assumption:
bm p
«
BW- 1
(3)
where bm p is the maximum differential delay due to multipath in the propagation medium, and BW is the information signal bandwidth (same for all information signals). This narrowband condition is present in today' s advanced mobile phone system (AMPS), which has a 25-KHz bandwidth. Digital systems which meet (3) do not suffer from lSI.
Reprinted from IEEE Signal Processing Letters, Vol. 1, No. 10, pp 150-152, October 1994.
463
We can now define a channel matrix (4)
where A is a m x d complex matrix in which the (i, k) entry gives the complex channel gain from the ith transmitting element to the kth receiver. The channel matrix provides a complete description of all reflections and scattering in the environment. In order to transmit the d signals to the receivers, let W j be the beamforming weight vector for the information signal. Let (5)
receivers form the entries of the probing response matrix B. Next, the complex amplitude data is fed back to the transmitter. Knowing B, the base can estimate the matrix A. Let the I probing signals be
Pl(t), ... ,pl(t).
(11 )
Unlike the usual information signals, each probing signal is sent on an orthogonal channel (time, frequency, code) so that the receivers may measure the response of each probing signal. As before, each probing signal is transmitted according to its own probing vector
If we consider the array output due to only the jth signal t) with its corresponding weight vector W i: then the signal recei ved at the kth receiver will be
(12)
Sj (
(6)
where CJk represents the information signal amplitude received at the kth receiver, which was intended for the jth receiver, and * denotes conjugate transpose. If we let [C]jk = Cjk, we have
W*A=C
(8)
Diagonal elements of C are desired signal levels, and offdiagonal elements of C are crosstalk amplitudes. To ensure that each information signal is received only by its intended receiver with unit amplitude, we would like
C =1.
(9)
Since each element adds a degree of freedom, to achieve (9), we must have m 2 d. III. INCORPORATION OF FEEDBACK The channel matrix A summarizes the channel including both the antenna array and the propagation environment, and matrix A is not known to the transmitter. We therefore cannot directly find W such that "W* A = I. We propose to use feedback from the receivers to estimate A and hence W. Once A is estimated, a W that achieves (9) is
W=A+*
(13)
where bj k is the amplitude received at the kth receiver due to the jth probing signal. If we let [B]jk = bj k , we have
(7)
Equation (7) is a vector version of the familiar statement
in). reCeiver ) = (transmitter) . . x (h c anne I gaIn ( amphtude amplitude
The response at the kth receivers due to the jth probing signal is given by
(10)
where + denotes psuedoinverse. To estimate A, we introduce the concept of probing and information modes. In the probing mode, the transmitter transmits probing signals, whose responses at the receivers are measured and fed back to the transmitter from which A is estimated. During probing, transmission of the usual (information) signals is temporarily halted. Instead, the array is excited in tum by several probing signals, and each receiver measures the relative complex amplitude response of each probing signal. These complex responses measured at the
B
= V* A.
(14)
The probing signals are agreed on by the transmitter and receiver. One choice for Pj ( t) is Pj
(
t) = {
~xp( ua; t )
o
if t E
[(j -
otherwise
1)f, j f]
(15)
where We is the carrier frequency, and T is the probing signal duration. In any case. since the 1probing signals are orthogonal using a bank of 1 matched filters matched to the probing signals, each receiver can measure a column of B corrupted by measurement noise: (16)
where the entries of E are assumed to be zero mean i.i.d. Gaussian random variables. Next, each mobile digitizes the received probing signal amplitudes and feeds these back to the base on its own reverse (digital modulation and assumed error free) channel. These reverse channels are assumed to be available, and for the purposes of this discussion, they can be on different frequency channels or even a wireline channel. The spatial reuse of the reverse channel is an independent problem and has been well studied [3]. The cell site assembles 13 and then computes A. Knowing V, which are the inputs to the channel, and B, which are the noise-corrupted probing responses, the transmitter identifies A using a least squares estimate
464
(17) where + denotes the pseudoinverse operation. Once A is in hand, the transmitter can determine W using (10). Each additional probing vector provides another equation involving A. Since A has m rows" the condition to uniquely determine A is 1 2: m. Additional probing vectors will improve the accuracy of the least-squares estimate (17).
Channel reuse in an access method therefore consists of the following five steps: 1) Transmit information signals to the receivers using information weight vectors W. 2) Monitor the level of crosstalk at the receivers periodically, and halt the information transmission when the crosstalk exceeds a threshold of acceptab ility. 3) Enter the probing mode: a) Choose probing vectors V. b) Transmit the probing vectors V , and measure the response matrix B at the receivers. c) Feed back the probing response matrix B to the transmitter. d) Estimate A via A = V+*B. e) Form information signal weights according to W =
A+* .
. 'l:OP: 8prObesIWavelength
~
:s .1l
£. 10.
IV. SIMULATIONS
Simulations were carried out to evaluate the performance of an adaptive array sharing a single channel among two receivers simultaneously . Using a six-element circular array with a 15.0° beamwidth, a beampattern for each signal was created according to (17) and (l0). The propagation environment contained 20 local scanerers for each mobile placed randomly in a 250 wavelength vicinity of the mobile. Energy arrived at the receivers only via local scatterers, and no line of sight was present. The receivers moved around the transmitter in a circular path of radius 5000 wavelengths (carrier = 900 MHz) at 2.5 inilhr, maintaining a fixed angular separation. To track A, the transmitter periodically alternated between probing and information mode. Because the channel varied as the receivers
2
~ '5 o
10-' L - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - - - - ' ' - - -,, - - . l 0 .4 0 .5 0 .6 0.7 0 .8 0.9 1 1.1 1.2 1.3 1.4 Mobile Spacing in Beamwidths (=15 degrees)
Fig. 2.
4) Resume information transmission with the new choice of weight vectors. 5) Go to step 2. The frequency with which it will be necessary to enter the probing mode will be determined by the receivers' speeds and the propagation medium complexity . If two mobiles sharing a channel approach each other, then the channel matrix will become singular. Since this method is not designed to accommodate singular channels, the transmitter should hand off one receiver to a new channel. To accommodate the probing signals in a TDMA system, a portion of each slot should be devoted to probing. If the mobile motion, and hence, the required probing rate is slower, probing could occur every n th slot.
.
Bottom .: 16 _ p robesiw aYe l e ng~
Outage probability versus mobile spacing.
moved, the interference was least immediately after each probe and worst just before the next probe. Two probing rates of 8 and 16 probes/wavelength were used, and each real entry of A was specified with 4.2 bits, for a net feedback rate of 1379 and 2753 bps, respectively. Fig. 2 shows the probability that the channel's SINR was below 7.3 dB for various mobile spacings. The 7.3 dB threshold is a BER of 10- 3 for B~SK. As the mobile spacing increased, the channel quality increased because the two channel vectors were less parallel. V. CONCLUSION
We have proposed an adaptive transmit antenna array that uses feedback to achieve low signal crosstalk at the intended receivers . Simulations show that at low mobile speeds (2.5 mi/hr), adequate signal separation requires feedback data rates in the thousands of kilobits per second, making the approach most applicable for static of slow-moving receivers. Methods of reducing the feedback rates are needed.
465
REFERENCES [1] B. Widrow and S. Stem. Adaptive Signal Process ing . Englewood Cliffs. NJ: Prentice-Hall. 1985. [2] A. Naguib and A. Paulraj, "Performance of COMA cellular networks with base-station antenna arrays," in Proc. Int. Zur ich Seminar Digital Commun. (Zurich. Switzerland ), Mar. 1994. [3) B. Sublett. R. Gooch, and S. Goldberg, "Separation and bearing estimation of co-channel signals," in Proc. MILCOM '89, May 1989, pp. 629--Q34. (4) O. Gerlach and A. Paulraj, "Spectrum reuse using transmitting antenna arrays with feedback," in Proc. Int. Conf Acoust.. Speech, Signal Processing (Adelaide, Australia), Apr. 1994, pp. 97-100.
Adaptive Antennas for Third Generation DS-CDMA Cellular Systems George V. Tsoulos, Mark A. Beach, Simon C. Swales Centre for Communications Research University of Bristol Bristol, UK Fax: +44 117 9255265, Tel: +44 117 9287740 e-mail: [email protected]
Abstract: This paper considers the perfonnance of a DS-CDMA system employing adaptive antenna technology at the base station site for both an Umbrella and a Micro-cell in a hierarchical cell structure. The possible advantages and problems from such a deployment are discussed. By exploiting the capabilities of Ray Tracing to provide the complex channel impulse response, a new adaptive antenna simulation model is presented along with some initial results for the perfonnance of well known adaptive algorithms in a multiple interference scenario. These provide insight into how the adaptive antenna operates when used in conjunction with DSCDMA and illustrate the potential benefits. Finally, propagation measurements are provided in order to validate some of the claimed capabilities. 1.
INTROD UCTION
Figure 1: Hiera rchical cell structure concept.
The need for mobile radio systems with increased spectrum efficiency is paramount in the drive towards third generation systems [1]. Currently favoured solutions in today's systems include the deployment of smaller cells as well as fixed sector, or multi-beam antennas, at the base stat ion (BS) site. In terms of modulation schemes and access techniques, application of spread spectrum modulation with Code Division multiple access (COMA) and especi ally Direct Sequence (OS) COMA, look to be amongst the favoured approaches. Recognising that the ambitious requirements of UMTS & FPLMTS can not be fulfilled with the known cellular architectures (macro, micro, pico cells) led to the conception of the idea of a hierarchical cell structure [2 - 3]. The key issue for this type of cell architecture is to apply multiple cell layers to each service area , with the size of each layered cell tailored to match the required traffic demand and environmental constraints (Fig. 1). In essence , microcells will provide the basic radio coverage but they will be overlaid with Umbrella cells to maintain the ubiquitous and continuous coverage required. Especially for the OS-COMA system, this mixed cell technique gives answers to situations where a possible performance degradation may occur, e.g. fast moving users requ iring handover, or black spots in coverage .
Advanced antenna techniques, such as adaptive antennas, is an area which seems to gather momentum recently [4 - 7], as another possible way to increase the efficiency of a given system. Adaptive antennas, based on the spatial filtering at the base station, separate the spectrally and temporally overlapping signals from multiple mobile units. This can be exploited in many ways such as: - Support a mixed architecture. - Comb at the near-far effect. - Support higher data rates. - Combine all the available received energy, (multip ath). In the following section , a brief discussion will be presented on the application of adaptive antennas in an Umbrella cell. The conclusions are taken from an earlier publication [5], but include some additional propagation measurements to support previous claims. The remaining sections focus on the use of adaptive antennas in a microcellul ar environment operating with OSCOMA. This work includes the development of a detailed Ray Tracing based simulation model and the pre sentation of some initial results .
Reprinted from Proceedings of 45th Vehicular Technology Conference, Vol. 1, pp. 45-49, July 1995.
466
gree of spati al selectivity that can be applied by the antenna system, i.e. whether to form a single narrow beam or adopt an optimum combining approach. In a large cell application, the use of an A DA based approach for a beamformer, would po tentially be more desirable since the ADA of the signals has a relatively narrow angular spread [9]. In a microcellular environment, th e angul ar spread of the signal from a single user is much greater , (figures 6b , 6c), due to the lower height of th e BS antenna and the close proximity of the scattering objects. Also, the ADA of the signa ls will change rapidly, with the do minant direction not always towards the desired user , as in the large cells case . Therefore , in the microc ellular case , the optimum combining approach see ms to be more flexible, providing increased capacity, as it will be shown in the following sections.
2. A N ADAPTI VE BASE STATIO N A NTENNA FOR THE UMBRELLA CELL OF A MIXED CELL STRUCTURE The potential advantages offered by employing an adaptive ante nna at an U mb rella BS site with a OSCOMA system, can be summarised as follows :
• • • • •
Mitigation of the near-far eff ect. Capacity enhancement. More efficient handover. "In-f ill" cove rage for the dead-spot s. A bility to support high data rates.
These wer e discussed in greater detail in an earlier publi cati on [5], although in orde r to support the last claim, some propagation measurements have been carried out. The measurements were performed with a Fast Fouri er Transform (FFT) D ual Channel Sounder at 1.823 GH z [8]. The RMS del ay spread was calculated using a 10
900 "0
750
'" 600 '" .s e-, '"
a; 0
Ul
.:
f:
450
..J
II:
300
distance (meters) Figur e 2: Wid eb and measurement s
dB power window on each measured impulse response profi le. The results are shown in figure 2, while figure 3 shows the map of the area where the mea sur ements were per formed . For the umbrella cell base sta tion which was at the roof of a building with approximat e height 50m, two ante nnas wer e used : on e omnidirectional end-fed dipole (identical to the mobil e ante nna) and one directional shro uded yagi with 15dBd gain. From the above figur e can be see n that th e RMS delay spread is much less for the case of the directiona l antenna, with a reduction which can be up to 1/5. The reduced de lay spread results in less int er symb ol interferenc e and, therefor e, provides the pos sibility of suppo rting higher bit rate services .
Figur e 3: Map of the area un der investigation
4. SIMULATION MODEL T he simulation model can be separated into two basic block s: a) The block which generates the impulse response of the channel under investigation. This is done with th e help of a Ray Tracing simulatio n tool developed by the Un iversity of Bristol [10]. Th e input parameters include the number of reflections and diffractions, th e tr ansmitted power, antenna radiation patterns, etc. Th e result ant output file includes the time delay, the angle of arrival and the power of each received ray. h) The block which simulates th e adaptive antenna array, illustra ted in th e next figur e 4.
3. AN ADAPTI VE BASE STATION A NTENNA FO R SM ALL CELLS The angle of arrival (ADA) of the radio signal, along with its multipath compon ents. dir ectly affect s the de -
467
-
,
X,
(k)
where N is the total number of antenna elements. The desired, or reference signal, roCk) is simply the PN sequence from on e user, (i.e. no data modulation is considered at the moment) , and the error signal is defined as the difference between the array output and the desired signal e(k) = y(k) - roCk). This model for the adaptive antenna offers the capability of selecting one from several adaptive processing algorithms, such as the LMS , NLMS, RLS, SQRLS and the OMI, [11 - 13].
x,(kJ
y (k) Array Output
J---r---~
An/enna Array I >-I....,+-T-+-+-~ (N elements)
Adap tive Co ntrol Pro cessor
5.
Reference '. (k)
The aim for the simulations is to investigate the performance of the adaptive algorithms on an environment basis and to provide insight into the mechanism followed by the adaptive antenna, when operating in conjunction with OS-COMA. Parameters used in the simulations include: averaging over 15 runs, 8 antenna elements with half wavelength spacing, 1023 chips M-sequence with 1.25 MHz chipping rate, step for the LMS and NLMS algorithms 0.01 and a value of 1 for the forgetting factor for the RLS and the SQRLS algorithms. From figure 5 can be seen that, as it was expected, the recursive least squares algorithms, converge very fast, (within around 50 samples, while neither of the LMS - NLMS have reached the same level even after ten times that time) . The RLS and the SQRLS algorithms have very similar behaviour, with the SQRLS giving the best output and being more robust. The choice of an adaptive algorithm must be made on the basis that the algorithm must be able to rapidly acquire and track the signals in a variety of mobile scenarios. Therefore the obvious choice is either of the RLS - SQRLS algorithms. In the following simulations the RLS algorithm is used.
Figure 4: Adaptive Antenna Arr ay
x.(k) is the sample of the total received signal at the nth element at instant t = kT, where T is the sampling interval, as well as being the chip duration of the PN sequence. x.(k) consists of the desired and interfering OS-COMA signals and random noise, and it can be expressed as:
x.(k) =
2: 2: hm,e Jkd(. -I )sin (~,)rm(k M
R
m=l r=1
t,)
SIMULATION RESULTS
+ N(k) (1),
where h"" and r",(k) are the elements of the vectors of the impuls e response and the OS-COMA signal from the mth user respectively:
h, = [hi'll' hm2," ' , hm,," ' , hmR]T, r m = [rm(k), rm(k - t l ) , " ' , rm(k - t,),· .. , rm(k - tRW. rm(k) = dm(k)· PNm(k) . ei~m, with dm(k) the binary data and 'Pm the carrier phase of user m. N(k) represents the
random Gaussian thermal noise. M is the total number of users, R is the total number of rays, d is the interelement distance, {J, and t, are the angle of arrival and the delay of each ray r respectively and [ ]T denotes the transpose. Although the total received signal at the n th antenna element is calculated by considering the interelement phase shift for each incoming ray , (n - 1)kd sine {J,), depending on the environment under investigation, it can also be calculated directly from the ray tracing to ol. The output from the adaptive array in vector notation is: y(k) = wT(k)x(k), where w(k) and x(k) are the weight and element vectors respectively. Using (1), this gives :
0 r---;--~-,",-.,..--"'---c----,-----,---,----, - 10
1.. .• _ tll-~==="'-l==='4-=~======"'"'-1 _ ~_.
.. •
100
:-. . _.
.:..
200
:_..
samples
J . _.
300
~
__
;.. _
400
Mea n weight for the second antenna eleme nt
Figure 5: Mean weight convergence for different algorithms, with 16 users.
468
soo
SINR_IN
-+-
;
~ a:
z
0;
.... _
1------.----;.--
- -.-
than steering it towards the first desired ray, because there is much more interference around the first ray which would be accommodated by the main lobe and hence would decrease the output SINR.
"_ ' . . .. .. •, . . ..
,
.. __
- ._
,~ _
__ _
,
,~
.
If better output SINR than the one depicted in figure 6a , is needed, then an increase in the antenna elements would offer great improvement, as it is depicted in figure 7.
_..
.
·30 ' - - - - - - '- - - - ' - - - -........- - - - ' - - - - 4 12 8 20 24 Users
( a)
18 . - - - - - . - - - - - - - - - , , . - - - - - - - - , - - - - - . ,
16
a:
14
;!; 12 CIl
10
8
·30
0
Angle
61!!-- - - - ' - - - - - " " ' - - - - - - - - - - l 20 16 12 4 8 No of elements
•
Ifl)ut SINA ... ·19.61 LIB
30
Figur e 7: Output SINR for the RLS algorithm and 16 users as a function of the number of a nte nna elemen ts,
(h i
Simulations showed that the influence of the thermal noise (modelled as White Random Gaussian noise), to the adaptive antenna performance is negligible. For example , for a microcellular environment with 16 users and the RLS or the SQRLS algorithm, there is a reduction of less than OAdB in the output SINR. This maximum reduction corresponds to the rather worst case situation of an input SNR of 3dB . The above behaviour can be explained on the ground that the influence of th ermal noise in a system can be neglected when traffic in the system is close to its capacity limit, because then interference power becomes a dominant factor for determining communication quality and channel capacity. This obviously is even stronger for the case of OSCOMA .
-10
·60
30
30
60
90
I.e)
Figure 6: (a) Output SINR for the RLS algorithm as a function of the number of users. (b) & (c) Produced radiation patterns for 8 and 24 users respectively.
6.
The results depicted in figure 6 show that the array is capable of adapting to the given user scenario even with as many as 24 users . It has to be mentioned here that the SINR values shown in figure 6a are the mean values after convergence. By comparing the results depicted in figures 6b and 6c, the concept of the " smart" antenna is revealed: Although the array should direct its main lobe towards the ray with the maximum incoming power, its first sidelobe towards th e next ray with the next maximum power and so on, it doesn 't do so for the case of figure 6c. The reason for this behaviour is that the criterion used by the adaptive algorithm is the optimum SINR. This is going to be achieved by steering the main lobe towards the seco nd desired ray rather
DISCUSSION
The advantage from using an adaptive antenna with a OS-COMA system is two-fold: First , the output SINR is greatly improved, which corresponds to an improvement on the capacity of OSCOMA, which can be substantial. Second, the produced radiation pattern has a directionalit y which varies according to the environment under invest igation. For an umbrella cell scenario, due to the small number of signals and their very narrow angular spread, the produced radiation pattern can be very directional, which can be exploited in a number of ways as it was described in [5]. Even for the microcells, where the number of users is great and the angular distribution
469
of the incoming signals very wide, the produced radiation pattern is going to be better than an omnidirectional pattern (even slightly). Obviously, the pattern oriented analysis for the benefits achieved with an adaptive antenna, (discussed in [5]), can not be applied for the case of microcells.
REFERENCES
[1] IBC Common Functional Specification, "Mobile Communications: General Aspects and Evolution", Specification RACE D731, Issue D, Dec. 1993. [2] Hakan Eriksson et aI, "Multiple Access Options for Cellular Based Personal Communications", 43rd VTC, Secaucus, New Jersey, USA, May 18 - 20 1993, pp. 957-962. [3] S. Chia, uThe Universal Mobile Telecommunications System", IEEE Communications Magazine, pp 54-62, December 1992. [4] J.S.Winters, "Signal acquisition and tracking with adaptive arrays in digital mobile radio system IS54 with flat fading", IEEE Transactions on VI: Vol. VT-42, No.4, November 1993, pp. 377-384. [5] G.V.Tsoulos, M.A.Beach, S.C.Swales, "Application ofAdaptive Antenna Technology to Third Generation Mixed Cell Radio Architectures", 44th VTC, June 8-10 1994, Stockholm, Sweden, pp. 615-619. [6] G.V.Tsoulos, M.A.Beach, S.C.Swales, "Adaptive Antennas for Third Generation Cellular Systems", 9th ICAP, 4 - 7 April 1995, Eindhoven, the Netherlands. [7] Race Tsunami Project, "Requirements for Adaptive Antennas for UMTS", R2108/ART/WP2.1/DS/I/ 004/bl, 22 April 1994. [8] M.A.Beach, S.Chard, J.Cheung, T.Martin and T.Wiltshire, "Description ofthe advanced handover experiment", PLATON R2007, 1993. [9] S.C.Swales and M.A.Beach, Direction Finding in the Cellular Land Mobile Radio Environment", lEE Fifth International Conference on Radio Receivers & Associated Systems, RRAS90, University of Cambridge, England, 23rd - 27th July 1990, pp.192-196. [10] G.E.Athanasiadou, A.R.Nix, J.P.McGeehan, "A Ray Tracing Algorithm for Microcellular Wideband Modelling", 45th VTC, Chicago, USA, July 1995. [11] Adaptive Filter Theory, S.Haykin, 2nd edition, Prentice Hall 1991. [12] Introduction to Adaptive Arrays, R.Monzingo, T.Miller, John Wiley, 1980. [13] Advanced Digital Signal Processing, J.Proakis et al, Macmillan Publications, 1992.
In a system like the DS-CDMA, the optimisation process must be repeated cyclically for each desired user. This can be done either in parallel with the help of a bank of beamformers or with one time shared beamformer. Considering as an example, the case of a channel which is sampled every 1ms, the following can be mentioned: • For the case of an umbrella cell with 10 users, the time available to the Beamformer to optimise its response for each user in a serial mode, corresponds to 125 samples for a 1.25MHz PN sequence. This means that if fast algorithms are used, the use of one Beamformer in a serial mode, can be possible for this kind of cell structures. • For the case of a microcell with 24 users, the samples available for convergence when one Beamformer is used, are limited to 52. This obviously indicates the need for a bank of Beamformers and parallel beamforming. 7.
CONCLUSIONS
Work presented in this paper discussed the application of adaptive antennas in a third generation DS-· CDMA mixed cell architecture system, at both the umbrella and the microcell base stations. It was shown that an adaptive antenna can be used in order to enhance the performance of a DS-CDMA system. In the microcellular environment, simulation results were presented which employed a Ray Tracing tool to provide the radio channel characteristics. Work currently under way is investigating the performance of an adaptive antenna in different cellular environments with moving users. Also, different forms of adaptive antennas are considered as a function of the environment they are operating, in an attempt to provide a unified approach for all the different environments. ACKNOWLEDGEMENTS
George V. Tsoulos wishes to thank the Centre for Communications Research (University of Bristol) for his postgraduate bursary. The authors would like to thank Professor J.P .McGeehan for his continuous encouragement and the provision of laboratory facilities. Also, the authors would like to thank C.M.Simmonds for the propagation measurements and the postprocessing of the results. Finally, many thanks to G .E.Athansiadou for her help with the Ray Tracing and M.P.Fitton for his help with the field trials.
470
The Spectrum Efficiency of a Base Station Antenna Array System for Spatially Selective Transmission Per Zetterberg, Student Member, IEEE, and ·Bjorn Ottersten, Member, IEEE Abstract- In this paper we investigate the spectrum efficiency gain using transmitting antenna arrays at the base stations of a mobile cellular network. The proposed system estimates the angular positions of the mobiles from the received data, and allows multiple mobiles to be allocated to the same channel within a cell. This is possible by applying a transmit scheme which directs nulls against co-channel users within the cell. It is shown that multiple mobiles per cell is an efficient way of increasing capacity in comparison with reduced channel reuse distance and narrow beams (without directed nulls). The effect of the spatial spread angle of the locally scattered rays in the vicinity of the mobile is also investigated.
U
1.
INTRODUCTION
SING antenna arrays, at the base stations, to perform spatially selective reception and transmission is a newly proposed way of increasing the capacity of a cellular network [2], [13], [15]. In [13] and [15], the reduction of the channel reuse factor is investigated as a means of increasing capacity. The analysis in [13] assumes that ideal sectorized beams are formed in the direction of the mobile, thereby reducing the probability of co-channel interference. In [15] the antenna array outputs are linearly combined to produce the least mean square error at the output. Since the weights are updated at the fading rate, not only is co-channel interference suppressed but the fading is also mitigated. For base to mobile communication reuse of weights adapted during reception is proposed in [15]. However, this requires a system which uses time duplex division (TDD); that is, contiguous timeslots are allocated for mobile to base and base to mobile communications (at the same frequency). Since outdoor systems such as TACS, GSM, DCS-1800 and IS-54 use different frequency bands for receive and transmit [12], the base to mobile scheme described in [15] cannot be applied in any of these systems. Since it is desirable to increase capacity also in the base to mobile link, we here investigate a transmit scheme which does not rely on reuse of weights adapted during reception. The technique here is based on array response and directional information. In the proposed scheme, the angular positions of the mobiles are estimated during reception and then used to calculate the transmit weights for array transmission. The problem of angle estimation is not addressed in this paper, and we refer to [71, Manuscript received May 25, 1994; revised February 8, 1995. This work was supported in part by the Swedish National Board for Industrial and Technical Development (NUTEK). The authors are with the Royal Institute of Technology, S-100 44 Stockholm, Sweden. IEEE Log Number 9413245.
[9], [10] and [14] where algorithms for solving this task may be found. In order to calculate the transmit weights, the antenna transfer function, at the transmit frequency, is assumed known. In this paper we introduce a new approach for increasing capacity. While [13] and [15] explore reduced reuse distances we here also investigate the reuse of channels within the cells (with unchanged distance between co-channel cells). This permits the use of a simple dynamic channel allocation scheme which avoids major interferers from getting close (in angle) to the desired mobile. Transmit weights are chosen such that a main beam is pointed at the desired mobile with nulls in the direction of co-channel interferers within the cell, but not outside the cell. It is possible to direct nulls against cochannel users in other cells also. However, the implementation of such a scheme in the downlink of a TDMA system might be difficult due to synchronization problems. In this paper, a comparison between reduced cluster sizes and multiple mobiles per channel is also made. Our results indicate that the latter technique is more effective. However, it should be kept in mind that the transmit scheme directs nulls only against co-channel users within the cell. In the analysis, the spread angle of the locally scattered rays in the vicinity of the mobile is a crucial factor. We find that it is possible to increase the capacity between two and twelve times using up to 20 antenna elements. The capacity is largely dependent on the spread angle of the locally scattered rays in vicinity of the mobile and on the number of antennas at the base stations. The paper is organized as follows: Section II explains the cellular network, the base station antenna array transmission system and the propagation modeling. The weight selection algorithm used in base to mobile communication is derived in Section III-A. In Section III-B, the channel allocation scheme is presented and finally, Monte Carlo studies are used in Section IV-B to determine the spectrum efficiency gain. II.
PRELIMINARIES
A. The Cellular Network
The coverage area of the mobile radio system is assumed to be divided into a network of hexagons [8], where each hexagonal cell is covered by a base station site. The channel reuse factor (cluster size) will be denoted with C, the cell radius with R, and the channel reuse distance with D. The parameters C, Rand D are related through D == J3CR. Thus, a large set of channels, C, implies a large distance
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 44, No.3, pp. 651-660, August 1995.
471
Fig. 2.
Illustration of the transmission system.
Fig. 1. The Cellular Network with channel reuse factor four, C = 4.
between co-channel cells, D. Increase in C means decreased interference but also a decrease in the number of channels available in the cells . In this paper, the hexagonal cells are divided into three 1200 sector subcells . Each of these subcells uses a fixed third of the channels available in the hexagonal cell. The subcells are covered by 1200 base station antenna arrays. The sectorization reduces the number of interfering cells in the first tier of interferers from six to two. The concept is illustrated in Fig. I for the case C = 4. Interferers in the second tier and further away will be neglected .
B. The Base Station Antenna Array Transmission System
• • •
x (t ) = W ' (9 )s(t) x ,(t)
•
•
•
!xm(t j
/.
( >120 \ .,:
0
The base station transmission system is based on four Fig. 3. The Uniform Linear Array (ULA) and the spatial multiplexer. algorithmic blocks as depicted in Fig. 2. One of the building blocks is the direction finder. This algorithm estimates the mobile angular positions 4' and their angular spreads r; (to where c denotes complex conjugate. The resulting m dimenbe defined) of the mobiles in the subcell from the received sional vector, x(t), is the input to the m antenna elements. The data D . As mentioned in the introduction, multiple mobiles implementation of the transformation can be done in analog or will be allocated to the same channel within the subcell . digital hardware and at different intermediate frequency bands. The channel allocator determines which mobiles should be The exact interpretation of the messages S k (t) depends on this allocated to the same channel. This algorithm uses only the implementation, but the analysis of this paper is independent angular positions of the mobiles 4' as input. The channel of this. allocation is represented by the n c x d matrix e, where n c is the number of channels within each subcell and d is the number of C. The Antenna Array Configuration simultaneous mobiles on the channel within the subcell . Each The antenna array of the base station is assumed to be linear contains the angular positions of the mobiles on a row in with uniformly spaced antenna elements. This form of antenna corresponding channel. The elements of an arbitrary row of configuration is known as a Uniform Linear Array (ULA) . The will be denoted by the 1 x d vector (). The corresponding individual elements of the array are ideal sectorized antennas angular spreads will be denoted by a , with a sector of 1200 • The active sectors of the antennas The weight selector calculates matrices of weights to be are positioned towards the broadside of the array (see Fig. used in transmission. One matrix is calculated for each chan3) and the spacing between the antenna elements , is set to nel. The angular positions , (), and angular spreads a of A/ v'3 where A is the wavelength of the carrier wave. The the mobiles allocated on the channel in the cell are used number of antennas in the configuration, m, is an important in this calculation. In order to simultaneously transmit d parameter of the system. In Fig. 3 the polar coordinates (T, a ) different messages, {s( t) , .. . , Sd(t)} , to d different mobiles, are introduced. The elements of the 1 x d dimensional vector the messages are spatially multiplexed. This operation can be of mobile positions, 0, are given in terms of the angle a. represented by the multiplication of the m x d dimensional matrix, W C ((), u), with the d dimensional vector s(t) = D. Propagation Modeling [Sl (t ), " ' , Sd(t )V i.e. In this section, we define the channel model between the x(t) = W C(O, u)s(t) (I) antenna elements of the array and a receiver at the position
e
e
472
(T, a). The transfer function consists of three factors: path loss, shadowing, and fast fading. The path loss and the shadowing are common to all the antenna elements. The path loss is modeled as (1/ T ) T where ry is the path loss exponent. The shadowing is modeled by a factor L which has log-normal distribution [5]. The standard deviation of 10 log L is denoted as adB (the mean is zero). The fading gain and the phase of the m antenna elements of the array are stacked into a vector denoted v(a~ a). The vector v(a, 0") is a random vector with a distribution depending on a and 0" (where a is to be defined). When the receiver and transmitter are located in different cells the fading of the antenna elements is assumed to be fully correlated, or equivalently the local scatterers in the vicinity of the mobile have negligible radius (implies that 0" == 0°) in comparison with the distance between the base station and the receiver [6]. Mathematically we model this as v( a, 0°) == {response of a single ray}
== Fa(a)
Fig. 4.
Illustration of local scattering.
where
CJ
. 27r . ( )) = F [ 1, exp (-J J3sm a , " ' ,
~ sin (a))]T
~ak E~
(2)
where F is the common fading of the antenna elements and a(a) models the phase differences of the antenna elements due to propagation path differences. The complex random variable F has a Rayleigh distributed amplitude and is uniformly distributed in phase [0, 27r]. The Rayleigh distribution is normalized such that E{IFI 2 } == 1. The function a(o:) in (2) is assumed to be known, although in practice a(a) may have to be obtained by calibration. When the receiver is in the same cell as the base station, the fading in the antenna elements is not assumed fully correlated. The model used in this situation is discussed in more detail in the next section.
E{v ( a, a)} == E {ej ip k } E{a( a
==
21r sin E N ( J3
(o ), (j )
O.
(7)
u.;(a, a) == E{v(a, 0") V * ( a, a)}
E{ (teXPj;k) a(a + 6a k))
(t
(3)
21r sin (n + 6ak) v'3
+ ~a k ) }
The second equality follows since ip k is uniformly distributed [0, 21r] and thus E{ e j rpk } == O. The covariance matrix is derived next By definition
=
The phase shifts, 'Pk, of the rays are assumed to be uniformly distributed [0, 21r], and the angular perturbations, ~ak are assumed to be distributed
(6)
N(O, a)
where .6.ak is given in degrees. This means that the spatial spread of the energy which is received by the mobile has approximately normal shape with standard deviation a. The normal distribution has previously been used in the propagation study [1]. As will be seen in the simulations, a is a critical parameter for the system. The parameter CJ which is related to a through (5) represents the angular spread in terms of the beamwidth of the array. Since the beamwidth of a linear array increases with a, CJ decreases with a. We will refer to a as the physical spread and CJ as the spread in terms of the beamwidth. Since the number of rays, N, is assumed large, it is natural to assume that the entries of the vector v (a, a) are jointly normally distributed. The following expression is obtained for the mean
E. Modeling of Local Scattering In this section, the model of the fast varying factor of the transfer function between the antenna elements of the array and a mobile receiver in the cell is presented. Consider the situation when the signal received by the mobile is built up by N locally scattered rays in the vicinity of the mobile, as depicted in Fig. 4. Assume that each of these rays has an individual stochastic phase
(5)
2700 cos (a )a.
Equation (4) means that (21r/J3) sin (a + ~ak) when treated as a random variable is normally distributed. Equation (4) can be closely approximated as
clef
. exp ( - j (m - 1)
==
J37r 2
eXP j ; k)a(a + 6 a k
))*}
(8)
where ()* denotes complex conjugate transpose. Since the terms of v( a, o ) are independent and equally distributed the following expression is obtained
(4)
473
message divided by the sum of the squared amplitude of the interferers, i.e.: (12)
where (13)
Fig. 5. The geometry involved in reception at the mobile (k
Conditioning on {27f/V3(sin(a and using (4) yields
+ 6.ak) -
= 2. d = 3) .
sin(o))
The variable P is the squared amplitude and is essentially the power of the signals. The subscripts d, hi and ci denote "desired," "host interference " and "co-channel interference" respectively. The Outage Probability (OP) will be used as a measure of the quality of the link between the base station and the mobile. That is the probability that the SIR is below a certain threshold g, or formally
= v}
OP = Pr {SIR
< g}.
(16)
This is a reasonable measure since it is known that most digital modulation schemes perform well above some threshold in SIR. The phase shifts, ipk, in (3) are of course frequency dependent. However, within the bandwidth of the messages, Sn(t), they can be considered to be constant. F. Outage Probability Consider the reception at the kth mobile on an arbitrary channel. Let 0 be the vector of angular positions of the mobiles at the desired base station, and let 0 1 and 0 2 be the corresponding position vectors at the interfering base stations . Assume further that the position of the kth mobile is (r, Ok) seen from the desired and (T1 ' (1) , (T2 ' (2) from the interfering arrays. The situation is depicted in Fig. 5. By the assumptions made earlier, the signal , u(t) , received by the kth mobile is given by
G. The Distribution of the Mobiles
To analyze the capacity of the system, some assumptions on the positions of the mobiles must be made. Clearly, if all the mobiles in the cell have the same angular position it will be impossible to host more than one mobile per channel and cell. A natural assumption is that the users are uniformly distributed with respect to area. The resulting angular distribution is plotted in Fig. 6. To simplify the analysis the uniform area distribution will not be used, instead we will assume that a has a density function such that (27f / V3) sin (a) is uniformly distributed [-7f, 7f] , or equivalently, a has the probability density function
f (v) = Q
27f V3 cos (v), 3600 3
(17)
This distribution is also plotted in Fig. 6. III. Two ALGORITHMS
(11)
A. Selection of Weights where W n (0, u) is the nth column of the weighting matrix W(O, u) and Sn(t) and s~(t) are the messages transmitted by the desired and interfering base stations respectively. A natural assessment of the instantaneous Signal to Interference Ratio (SIR) of the kth mobile is the squared amplitude of the desired
In this section, we design the algorithm or function which calculates the weighting matrices used in transmission. The algorithm uses only the angular positions of the mobiles in the subcell, 0 = [0 1 , • . • , Od], and the angular spreads of those mobiles a = [0"1, . . . ,lid] as input. Note that the assumptions
474
and L n is the log normal fading at the nth mobile. Assume that the angle Q of a beam which hits a co-channel user in a neighboring cell is distributed according to (17). Assume further that the distance between the base station and this mobile is equal to the distance between co-channel cells D (Section II-A) . It follows that, the mean square of the interference generated by the base station at this mobile is given by
- .............
0 .01
0.007 ~
g 0.006
.~
'"
:5 0.005
-
et 0.004
- - Assumed
~
~
A rea umfcrm
E{ ~: Filw'k(O, u)a(Qi)1
2
0 .003 0.002 0 .001
}
= ciw'k(O, u) (l:~:Oooo fQ(v)a(v)a*(v) dV) Wk(O , u)
1
I
OL-.-----'---:':-----'----=---~-___:
- 60
Fig. 6.
- 40
-20
0
Angle in degrees
20
40
60
= ciw'k( O, U)Wk((}, a )
TIlustration of the angular distribution of the mobiles.
where
described in this section are for the purpose of designing an algorithm. The assumptions in the analysis will differ somewhat. The objective of the weight selection algorithm is to provide weighting vectors, Wk(O, a), such that the Outage Probability (OP) is the lowest possible. We reason as follows: The transmission of the kth message Sk (t) will yield contributions to the signal received by 3d different mobiles . These mobiles are: the desired (kth) mobile, the other d - I mobiles in the cell which use the same channel, and 2d mobiles in the first tier of interferers. Choose W k (0, a) such that the quotient of the mean power of the desired contribution to the undesired contributions is maximized, thus specifying Wk(O, a) to the degree of a scalar factor. This scalar factor is chosen such that the nominal gain in the direction of the kth mobile is one. Assume that all mobiles in the cell allocated to the channel are one half cell radius from the base station, that is r = R /2. Consequently, from (8) and (13) the expectation of the squared amplitude of the desired contribution can be written as
E{ (~) -r L 1w'k (O, a )v(th , iTk )12}
ci = E{L i}E{lFiI2} /(3R2C p /2.
(23)
Using E{L h } = E{L n} = E{L i} and E{lF i I2 } = 1 the following criterion is now obtained
where
M1
= (3Cr / 22-r Rvv (fh , !.Tk)
M 2 = (3C )-r/ 22-r
L d
n=l.n#k
Rvv(Bn , !.Tn ) + 2 dI.
(25)
(26)
The maximum of the criterion function (24), is given by the largest solution f.L to the generalized eigenvalue problem (27)
where it should be noted that M 1 and M 2 are positive definite matrices and e is an eigenvector corresponding to u: The vector Wk (O , u ) is given by
= Ckw'k(O, u )E{v(lh , !.Tk )v*(Bk, !.Tk )}Wk ((} , u )
= CkW'k (O, u )Rvv(Bk , !.Tk)Wk(O, u )
(22)
Wk (O , o )
(18)
1
= a* (B)k e e.
(28)
(Numerical algorithms for solving the generalized eigenvalue problem can be found in [3].)
where (19)
The expectation of the squared amplitude of Sk(t ) at the nth mobile (n E {I " .. , k - 1, k + 1, · ·· , d} ) in the cell is similarly given by
E{ (~) -r Lnlw'k (O, u)v(Bn,
!.T n
)12 }
= cnw'k ((}, u)Rvv(Bn , !.Tn)Wk((}, u)
(20)
where (21 )
B. The Channel Allocator
The channel allocator allocates d mobiles, on each of the n c channels available in the cell, using the information of
the angular positions of the mobiles . The objective is to keep the mobiles operating on the same channel well separated in angle. The channel allocation is represented by the (n c , d) dimensional matrix e. Each of the n c rows of this matrix corresponds to one of the n c channels available in the cell. The d entries of a row contain the angular positions of the mobiles allocated on the channel. The channel allocator works as follows : Assume that the positions of the ti ; x d mobiles are given by i[J = {(P1 " " ,
475
cPn
c
x d.
The
e
matrix is then assigned according to
e 1, k =
(37) (29)
e
This means that is filled column wise, starting from the upper left corner and finishing at the lower right comer. If the distribution of the rows of is the same, the interference will be independent of the channel number. This is accomplished by randomly permuting the rows of such that every possible permutation has equal probability. Furthermore, the columns of are circulated to remove dependence. The distribution of the positions, 0, of the mobiles on an arbitrary channel is deri ved and presented in [17]. In particular it is shown that
v ()k ~
e
e
-
-
2
(1)27r ~
27i"
d(k - 1),
as n;
-+ 00
v
(J3
when
)
1 - d)
when
d k < -
2
+1
d k> - 2
+1
B( a) = diag ([1, e- j a,
(30)
... , e-j(m-l)ii])
Qi E Uniform[O, 27i"]
-
'h
21r = y'3 sin (fh)
(31)
and ~ denotes convergence in second mean norm. It is also shown in [17] that
81
E Uniform [0, 27i"]
1) Calculate the matrices Rvv(Ok, ak) using (5) and (10).
(38-39) and q is given by (40). 4) Calculate rl and r2 using the cellular geometry and the position of the mobile (r, a) (See Fig. 5). 5) Simulate 11(0, 0- 1 ) using the formula v(O, 0- 1 ) R~G2(O, 0"1)(, where ( is an m dimensional vector of independent complex elements with real and imaginary parts independently normally distributed N(O, 1/ /2). 6) The remaining is straightforward use of (34)-(42).
A. Simulation of the Instantaneous SIR
ao
(33)
then the SIR can be simulated using the equations (34)
(42)
2) Calculate a square root factorization of Rvv (0, 0"1) i.e. determine R~G2(O, 0'1) such that R~~2(O, a1)R:~2(o, 0'1) = Rvv(O, 0- 1). 3) Calculate WI (9, u), ... ,Wd(O, u) where 'IJ is given by
IV. RESULTS
cos (a)
(41)
The most important steps when using (34)-(42) are:
i.e. 81 is uniformly distributed [0, 27i"]. In a practical application the reallocation should be made as often as possible. The computational demands seem to be very small. The most computationally demanding task is the sorting of the d x n c mobile positions.
a:=---
(39)
diag (A) denotes a diagonal matrix with elements given by A and sin -1 (.) is the inverse of sinf-). Proof' See Appendix B.
(32)
In order to estimate the OP for a mobile, a large number of realizations of the instantaneous SIR is drawn. The following theorem provides a computationally simple formula to obtain these realizations. Theorem 1: If the number of channels per subcell, n c , is large and the angular spread in terms of the beamwidth a (see Section II-E) is the same at all mobiles (seen from a desired base) i.e.,
(38)
(40)
where
where
(V3 a:(k - 1))
(h = sin- 1 a:(k -
e
(Ok -
sin- 1
Note that steps 1-4 only have to be performed one time whereas steps 5 and 6 should be repeated for every new sample of SIR. The method used to determine the OP of the system is to draw a large number of independent realizations of the instantaneous SIR and count the fraction of times that the SIR exceeds the threshold g. In order to draw these realizations we must specify a number of user parameters. In the next section the influence of the parameters m. d, C, and ao on the OP will be investigated. (Remember m is the number of antenna elements in the arrays, d is the number of mobiles per channel in each cell, C is the channel reuse factor, and 0"0 is the width of the local scatterers). The remaining parameters are kept constant at the following values: The position of (R/2, 0) and the angular the mobile is given by (r, a) spread in terms of the beamwidth is the same at all mobiles. The rationale for the assumptions on r, a and a will be discussed in Section IV -C. The threshold used in the OP calculation (16) is 9 = 9 dB (simulations using 9 == 3 dB gave qualitatively the same results). The standard deviation of the log normal fading is a ae = 6 dB. The path loss exponent used is , = 4. The number of channels in each cell n; is large i.e. ti; = 00. (Simulations have indicated that this assumption can be considered to be valid when n c > 100). The number
476
= a
,
0.09
I
~
,
~O .04
a'5
l
/
x: C=4
,+
+.. C-7 -
/
//
//
I
0.03 __ _ B.ef.e rence
-
.J -
//
(J :::::::
'0
x- - -
K
10
11
" 10 ~ ~
c
..., _
E
E
-
..J
,'"
'2 ~
_
0°
~
ro 12
"
.4"
a
~ 14
I
0- - / /
,I I
,)to -
,'"
-
-><'
I
I
0.02
-- - ,,I
.:+::x: :'.t
~
0.0 1
:; :: :: ::
o
1
1.5
I
2
2.5
3
3.5
4
Spectru m Efficiency (E)
=
=
4.5
5
=
=
=
=
=
B. The Spectrum Efficiency of the System
As promised in the introduction of this report the capacity gain which can be obtained with the proposed system will be investigated. As a starting point for this analysis the case m = 1, d = 1, C = 4 will be used as a reference. This can be thought of as the system of today, with which the new system is compared. The capacity of the system can be enhanced either by increasing the number of mobiles per channel and cell, d, or by decreasing the channel reuse factor C. To make a fair comparison of these two alternatives the (relative) spectrum efficiency factor E, defined as
d C
678
Spect rum Etticency (E)
Fig. 8. The spectrum efficiency when C
samples of the instantaneous SIR used to estimate each OP is 40000. This yields a theoretical standard deviation in the OP estimate of ~0.08 % when the true outage probability is 2.5%.
E =4-
3
5.5
Fig. 7. The OP for the case In 7 and 0"0 1.6° . (The values of d corresponding to the points in the plot are; C 1: d 1, C 4: d 1. . . .. .5, C 7: d 2. . . . . 5)\abovedisplayskip lOpt\belowdisplayskip IOpl.
=
0
x
~
I
I
"
0 : C::;l
"3sr16
I
'
X
'0
I
I
:i .05 10.
x0.8
/J '"
18
I
I
I
0 06
1.6°
I
I
I I
0 .07
a ;:::;;:
20
I
I ,
0.08
ofL
I
I
(43)
will be used. Thus, for our reference system E = 1 and the OP has been estimated to be 2.65%. Example 1: In order to determine the maximum spectrum efficiency with seven antenna elements m = 7, the OP is estimated for all combinations of d E {I,· · · , 7} and C E {I, 4, 7}. The result when ao = 1.6° is shown in Fig. 7. From this figure it is concluded that the maximum value of spectrum efficiency which can be obtained without having the outage probability exceed the reference 2.65% is E = 4. This value is obtained when C = 4 and d = 4. Example 2: The same procedure as in Example 1 is repeated here for all m E {l , .. ·, 20} and ao E {0° , 004° , 0.8° ,1.6° , . . . ,4. 8°} . In these simulations, the channel reuse factor four, (C = 4), was found to be an optimizing value in all the cases. Thus, the maximum spectrum efficiency is equal to the maximum number of mobiles per channel in each cell for the case C = 4. In some cases, the channel reuse factor one was also optimal. In Fig. 8, the minimum number of antennas required to obtain an OP
12
= 4 (0" is given at = 0). Q
less than 2.65% has been plotted as a function of spectrum efficiency. The eight different curves correspond to the eight different values of ao. It is noted that the number of antennas required increases rapidly as ao increases. In particular it is seen that in order to multiply the capacity with five, six antenna elements are needed when a = 0° whereas 16 are required when ao = 4.0°. From Example 2 we deduce that the performance of the proposed system degrades with increased angular spread. This is in contrast to the reception technique [11] which improves with angular spread. The reason is that as the angular spread increases, the wavefront becomes less planar, which can provide improved performance through diversity gain. Furthermore, when receiving with an antenna array (as in [II)) it is possible to estimate the channel transfer function between the mobiles and the antenna elements, while during transmission (as in this paper) this is impossible due to lack of feedback. C. Influence of the Mobile Position
In Theorem I and in Example I and 2, it is assumed that the spread in terms of the beamwidth of the array, a- , is the same at all angles 0: (recall Fig. 3) and independent of the distance between the mobile and the array. In this section we show how the results can be interpreted in the physically more reasonable case of a spread which is inversely proportional to the distance between the array and the mobile r, and independent of the angle 0:. Thus, assume that the physical spread a is given by a
R = -aD 2r
(44)
where ao is the spread at r = R /2. Assume further that the weight selection algorithm (Section III-A) still uses the spread a = ao/cos (0:), when it calculates the correlation matrices, independently of the actual value of a . It is possible to show that Theorem I still applies except that v(O, ao) should be replaced by v(O , ao cos (o:) /r ) where 0: is the angular position of the mobile. The results of Example 1 and Example 2 still apply in the point (r, 0: ) = (R /2, 0). However under
477
is such that a main beam is pointed at the desired mobile with nulls in the direction of co-channel mobiles within the cell, but not outside of it. For this system, multiple mobiles per cell increases capacity more than reduced reuse distance. It is found that in order to increase capacity d times ~ 1.7d antennas -are needed when 0'0 = 0.8° whereas ~ 3.2d antennas are needed when 0'0 = 4°. (The approximations are valid for
0.8
m ~ 20 .)
-0.8 -0.6 -OA -0.2
0.2
OA
0.6
0.8
ApPENDIX A
Umts ; cell rad ii.
LEMMAS TO THEOREM 1
(a)
From the definition of B(ii) in (41) we immediately obtain
1,-------r--,---.--~"""7l"~_...-,._____,r___r-_,
B-1(ii)
0.8 ~
B(ii
OAL
(46)
+~)
=
(47)
B(~)B(ii).
Using (5), (10) and (41) we obtain
0.2
Rvv(0:', O'o/cos (0:'))
O L----l. _-L_-"- _L.-~:::.....-......L --'-------'------L -...J
-0.8 -0.6
-OA
-0.2
0.2
OA
0.6
0.8
= B ( ~ sin (0:')) R,JV(O , 0'0) . B* (~ sin (0:')). (48)
Units = cell rad ii.
(b)
Fig. 9. a) III = 1.6 0 •
0'0
(45)
and
0.6r
-1
= B*(ii) = B( -ii)
= i , d = 4, C = 4, 0'0 = 1.Go , b) = 9. d = 1. C = 1, 111
Let us define
ic; (ii, 0'0)
as
Rvv(ii , 0'0) = B(ii)Rvv(O, O'o)B*(ii) .
the assumptions here, the proposed system will most likely have a lower OP than the reference system in most of celL Because when 0:' and/or r increase, the spread in terms of the beamwidth, ir, decreases and this improves the SIR of the proposed system. To test validity of this conjecture the OP has been estimated in a large number of positions (r , 0:') for a large number of cases. In Fig. 9 below, the result of this validation process in the cases m = 7, d = 4, C = 4, 0'0 = 1.6° (Fig. 9a) and m = 9, d = 1, C = 1, 0'0 = 1.6° (Fig. 9b) is displayed. The plots illustrate a subcell with the base station in the point (0 ,0). The symbol "+" indicates that the proposed system has a lower OP than the reference system, and " 0" indicates the reverse. In the case m = 7, d = 4, C = 4, 0'0 = 1.6° the proposed system is better than the reference system in approximately 85% of the area while the corresponding number in the case m = 9, d = 1, C = 1, 0'0 = 1.6° is 75%. The results in other cases are similar. To overcome the problem with the area close to the base, reuse partitioning [4] could be used. V . CONCLUSIONS
In this paper, we have investigated the spectrum efficiency gain of a base station antenna array system for base to mobile communication. The propagation model includes path loss, shadowing and fast fading. The effect of the spread angle of the locally scattered rays in the vicinity of the mobile is also taken into account. Also, we introduce a new approach for increasing capacity in which multiple mobiles are allocated on the same channel within a celL The proposed transmit pattern
(49)
From (48) and (49) we obtain - (27f Rvv(O:' , O'o/cos(B)) = Rvv .j3sin(O:') ,O'o ) .
(50)
Using (47) and (49) we arrive at
Rvv(ii + ~ii , Define Wk(iJ ,
0'0)
_ -
0'0)
= B(~ii)Rvv(ii , O'o)B*(tlii) .
(51)
as
Wk((), 0'0)
= arg
max
x*Af1x -
x· B(8k )a (O) = l x * M
2X
(52)
where (53)
Af2 = (3C)'/22'Y
L d
Rvv(Bn ,
0'0)
+ 2dI .
(54)
n=l,n¥k
From (24)-(26), (50) and (52)-(54) we obtain Wk( (),
O'o/cos (Bd, . . . , O'o/ cos (B D ))
=Wk (~ sin (Bd,···, ~ sin (B D), 0'0) '
(55)
Since B(ii) and Rvv(ii , 0'0) are periodic with periodicity 27f (from (41) and (51» we obtain from (52)-(55) that
Wk(fJ , 0'0) =
wk(B 1 + k 127f , "
where k1 , ' . . ,k d are integers . Theorem 2:
478
', BD
+ k D27f , 0'0)
(56)
Proof: Using (51) and (52)-(54) we obtain that wk(6 1 + Ll, ... ,Od + 6. ao) := Wk(O + Ll, 0"0) is given by
Similarly we obtain,
7
iiJ,,/iJ + ~, 0"0) =
W~((Ji, (Jo/cos (ei),· .. ,ao/cos (eb))a(a i )
x* B(~)MIB*(6.)X
== w~ (6, u)B( (01
arg max X· B(I:>.)M2 B · (I:>.)x. (58) x { x * B(6.)B(()k)a(O) == 1.
-
Wk((J
0"0)
y*Af2~
= argm;x y* B(fh)a(O) = 1
{ where y == B*(6.)x.
(59)
w~((Ji,
D B
ao/ cos (81 ) , ... , aO / cos (()D) )v( ()1 \ ao/ cos (()1 ))
== w~((O, 82
-
81 , "
' ,
. V(Ol, ao/cos (B l ))
Bd
-
e
1 ],
ao)B( -HI) (61)
where (62)
Since v(B1 , ao/cos(e 1 ) ) is multivariate normally distributed with mean zero and covariance R vv (01 , ao/cos (81 ) ) we obtain from (46), (50) and (51) that B( -01)V(01' O"o/cos ((;II))
N(o, B( -B1)Rvv(iJ 1 , O"o)B*( -8 1 ) ) == N(o, ~Jv(O, 0'0)). E
(63)
Applying (56) and (63) to (61) yields w~((J,
ao/cos(e 1 ) , · · · ~O"o/COS((}D))V(Ol, ao/cos(Ol)) d ist _
*
-
== Wn([O, (02 -
-
--
el)27'i~"" (()d - ( 1 )27r L
ao)v(O, 0'0) (64)
where d~t denotes that the distributions of the left and right hand side are equal. Using the property (30), (55) and (56) on (64) yields w~((},
where
ao/cos (()1),··· ,{Jo/cos (()V))V(f)l, {Jo/COS (01)) dist == W n* ((}v ,(J'V) V (0 ~ (To ) (65)
lJ is given by (38-39) and
(67)
ACKNOWLEDGMENT
1
The proof here is based on the definitions and results of Appendix A. Without loss of generality the proof is given for mobile number one, that is k == 1 in (12)-(15). Applying (55) and (57) (with L\ = -2n/V3sin(t11)) to w~((J, ao/cos(e 1 ) ,. · · ~ao/cos(eD))V(()l, O"o/COS(Ol)) yields w~ ((J,
(01),···,ao/cos (eb))a(ai)
where al is uniformly distributed [0, 2n], {} is given by (38-39) and (f is given by (40). Applying (65) and (67) to (12)-(15) with (33) yields the desired result. 0
(60)
ApPENDIX
(JO/COS
~t w~(O, u)B(ai)a(O)
From (52) and (59) it is obvious that
PROOF OF THEOREM
(66)
=
1L& +~,
Oi)21r )a(O)
where g~ 27r / V3 sin (Oi) and Oi == 27r / J3sin ( Qi)' Since 01 is independent of Qi and uniformly distributed [0, 27r] from (32), the argument [01 - Qi]21r, will also be uniformly distributed [0, 27r] and independent of ()~. Thus using (66)
Introducing y == B* (6.)x yields
~
-
if is given by (40).
Special thanks to Dr. T. Trump, Dr. U. Forsen, and Dr. M. Almgren for valuable comments and discussions.
REFERENCES [1] F. Adachi. M. T. Feeney, A. G. Williamson, and J. D. Parsons, "Crosscorrelation between the envelopes of 900 Mhz signals received at a mobile radio base station site," lEE Proc., vol. 133, pt. F, no. 6, Oct. 1986 . [2] S. Anderson. M. Millnert, M. Viberg, and B. Wahlberg, "An adaptive array for mobile communication systems," IEEE Trans. Veh. Technol., vol. 40, pp. 230-236, Feb. 1991. [3] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore: Johns Hopkins Press, 1983. [4] S. W. Halpern, "Reuse partitioning in cellular systems," Proc. Veh. Technol. Con! VTC-85, 1985, pp. 322-327. [5] W. C. Jakes, Ed., Microwave Mobile Communication. New York: Wiley, 1974, pp. 79-131. [61 W. C. Y. Lee, Mobile Communications Design Fundamentals. New York: Wiley, 1993, pp. 157-159. [7] 1. Li and R. T. Compton, Jr., "Maximum likelihood angle estimation for signals with known waveforms," IEEE Trans. Signal Processing, vol. 41, pp. 2850-2862, Sept. 1993. [8] V. H. MacDonald, "The cellular concept," Bell Syst. Tech. J., vol. 58. pp. 15-41. Jan. 1979. [9] S. J. Orfanidis, Optimum Signal Processing, An Introduction. Singapore: McGraw-Hill, 1990. [10] B. Ottersten, M. Viberg, and T. Kailath, "Analysis of subspace fitting and ML techniques for parameter estimation from sensor array data," IEEE Trans. Signal Processing, vol. 40, no. 3, pp. 590-600, March 1992. [11] J. Salz and J. H. Winters, "Effect of fading correlation on adaptive arrays in digital wireless communications," in Proc. Centre International de Conferences de Geneve (ICC-93), Geneva. Switzerland. [12] R. Steele, Ed., Mobile Radio Communications. London: Pentech Press, 1992, pp. 82. [13] S. C. Swales, M. A. Beach, D. J. Edwards and J. P. McGeehan, "The performance enhancement of multibeam adaptive base station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Technol., vol. 39, pp. 56-67, Feb. 1990. [14] T. Trump, Maximum likelihood estimation of nominal DOA and angle spread using an array of sensors, Tech. Rep., (IR-S3-SB-9422), Access: see [17] below. [15] J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE Trans. On Veh. Tech. , vol. 33. pp.
144-155, 1984.
[16] _ _ , "On the capacity of radio communication systems with diversity in a Rayleigh fading environment," IEEE Selected Areas Commun., vol. SAC-5, no. 5, pp. 871-878, June 1987.
479
[17] P. Zetterberg, "The Spectrum Efficiency of a Basestation Antenna Array System for Spatially Selective Transmission," Report Version, (IR-S3-SB-9403), available by Mosaic: Document URL: http://www2.e.kth.se/s3/signaVINDEX.html or by anonymous ftp to: elixir.e.kth.se directory/pub/signal/reports, [18] P. Zetterberg and B. Otters ten, "Experiments Using an Antenna Array in a Mobile Communications Environment, " Proc. 7th SP Workshop on Statistical Signal & Array Processing, 1994 (IR-S3-SB-9412) Access: See [17] above.
480
Capacity Enhancement and BER in a Combined SDMA/TDMA System Josef Fuhl and Andreas F. Molisch Institut fur Nachrichtentechnik und Hochfrequenztechnik, Technische Universitat Wien, Vienna Gusshausstrasse 25/389, A-I049 Wien, Austria Phone: (+43) 1 58801 3546; Fax: (+43) 1 587 05 83; email: [email protected] Many papers have been published on SFU, see e.g. [1],
Abstract - This paper considers the performance of a TDMA system employing smart antennas at the base station. Two adaptation schemes are analyzed - the switched beam approach and an adaptive array based on an adaptation algorithm to maximize the Signal-toInterference and Noise ratio. For an SDMA system the switched beam approach performs worse than the adaptive array. Adaptive arrays based on gradient-vector estimation (e.g. LMS) are not suitable for mobile radio. The class of Least Squares (LS, RLS, SQRLS) algorithms shows satisfactory performance. For a linear array with 8 elements a minimum angular separation of 100 between two users is sufficient for the adaptive array to achieve as good performance as a system serving one user per traffic channel.
[2], [3], [4]. They show that for a single user the bit error
rate (BER) can be decreased by pointing the "main beam" of the antenna towards the current location of the user. This contribution is devoted to a true SDMA scenario. We consider the canonic situation that two users are served on the same traffic channel, consequently the capacity of such a system will nearly be doubled. OUI simulations are based on a channel model including directions of arrival and on the air interface of the 2nd generation standards GSM and DCS1800. We show how the BER is changed by adding the second user, as a function of Signal-to-Noise Ratio (SNR) and the number of antenna elements. The paper is organized as follows: Section 2 discusses the channel model used for the simulations. Section 3 addresses the simulation setup for the whole system. In Section 4 we take a look at the performance of different adaptation schemes. Section 5 gives the simulation results for various parameter combinations. Section 6 concludes this work.
1. INTRODUCTION The growing number of users of cellular communication systems necessitates measures to increase the performance of such systems, i.e. their coverage and capacity. Currently, there is a considerable interest in adaptive base station (BS) antenna arrays for 2nd and 3rd generation mobile communication systems. A possible 2-step implementation procedure for smart antennas may be as follows:
II. CHANNEL MODEL
• Spatial Filtering at the Uplink and at the Downlink
(SFU-SFD): Smart antennas are used both at the uplink ( mobile station (MS) transmits, BS receives) and at the downlink (BS transmits, MS receives). Only one user is served in one traffic channel. The aim is increased coverage and decreased interference for cochannel cells.
• Space Division Multiple Access (SDMA): With the use
of adaptive directional antennas and additional hardand software at the BS, users in different angular positions can be served in the same frequency band and in the same timeslot, i.e. on the same traffic channel. The data intended for each user are separately processed in base-band in such a way as to give the user-specific antenna pattern. The signals are added (linearly superposed) and modulated onto the RFcarrier, which is radiated from the antenna. This approach leads directly to increased spectral efficiency of the system. However, it can be added to an existing 2nd generation mobile communication system only if there are also changes and redefinitions in the switching and signaling system. The concept of a cell in its traditional sense has has to be redefined.
\Ve utilize a channel model including directions of arrival (DOA) and fading [5], [6], [7], and [8] (Fig.I). It is suitable" for both rural and urban areas. Fading correlation at the receive array is automatically included by the model. Like all the references above, our considerations are restricted to a two-dimensional channel model (i.e. the horizontal plane), but as mentioned in [8] this does not impose severe restrictions for mobile radio applications. Many scatterers in the vicinity of the mobile combine to one fading signal, spread out in angle over several degrees dependent on the distance of the mobile from the BS. By this model we extend the concept of DOA to a nominal DOA associated with an angular spread in contrast to the widely used discrete DOAs. In order to model the propagation physically we partition the propagation area into two different regions [8], [7]: (1) Regions without scatterers; (2) Circular regions where the scatterers are located. The motivation for choosing circular regions where the scatterers are located lies in measurements conducted by [5] and [6]. The radius R of these regions is about 100A - 200"\, where A denotes the wavelength [6]. This models can be easily generalized as shown in [8]. The overall impulse response for this channel at the location of the m-th antenna element r-« = [x m , Ym, Zm]T
Reprinted from Proceedings of the 46th Vehicular Technology Conference, Vol. 3, pp. 1481-1485, 1996.
481
---
CDF( Q',)
" I
I
~
~
~~2R
Scatterer
""
1 ::~===--~ ", ,
0.7
\
I
.'
, "
0 .5
' ••..
": 1
I ; ··
., ...
0 .3
,
0 .2
BS
,, ,.
r
. . ..
0 .4
\ \ I
I I
. , .. ..
0 .6
\
.... . . . j ••.
r. .
0.1 - :
, ,r
Fig.! Chann el model
h(rm ,
T,
cell radius
t, cp)
L
= 2: h,(rm , T, t, cp), 1=1
- ' - : r2
(1)
where L is the number of scattering points, T is the delay (relative time), t is the absolute time, and cp is the azimuth angle. The quantity ht(rm , T, t, cp) denotes the impulse response of the l-th path at the location r m of the m-th antenna element. The impulse response is different for every location of an antenna element. A. Angular Spread
We define the angular spread a s as the angle under which the diameter of the scattering region is seen at the BS (Fig. I). It is given by
as = 2 arctan
(~),
(2)
where R is the radius of the scatterer circle and r m is the distance between BS and the mobile. We assume that the users are uniformly distributed within the cell area. The Cumulative Distribution FUnction (CDF) F(CLs,k) of the angular spread CLs,k of the incoming signals from user k, 1 ::s k -s K, where K denotes the number of users, seen at the BS can be calculated to [9)
(3) where r l (rz) are the distances from the BS to the nearest (farest) location of the user and 0'0 given by CLo =
= r l = 100,), a n d the oute r = 1000,),; - : r 2 = 5000,),; and
Fig.2 CDF of the angular spre a d for R
(1 ::s m -s M) is given by
2 arctan
(~)
(4)
is the angular spread associated with a user located at the cell fringe. Fig.2 gives the CDF of the angular spread with the outer cell radius rz as a parameter.
482
r2
as parameter. - - :
=20000,),.
r2
III. SIMULATION MODEL We use a mod el with a protocol closely related to t he 2nd generation standards GSM and DCS 1800. The only difference is that two or mor e users are served at t he same traffic channel (i.e. at th e same frequency and time slot ). Therefore each user 's mobile has to be assigned a unique training sequence, each of which must com ply with the GSM (DCS 1800) specifications. This 26-bit midam ble originally int ended for estim ation of t he impulse response of th e channel (equalizer t raining) is now used for user separation and identificati on also . We assume perfect time alignment of the received sequ ences. We consider a narrowband channel where transmission suffers from flat-fading only. As parameters for th e chan nel we assumed L 20 scattering points per user. All th ese points are randomly located within th e scattering circle. The radius of the scattering circle has been set to T\ = R = 100>' and th e outer cell radius to rz = 5000>'.
=
IV . ADAPTATION SCHEMES Two different adaptation schemes are investigated in this paper , th e switched beam approach and an adaptive array based on Least Squares (LS) adaptation. Sw itched beam uses a set of P (P 2: K) different beam positions (Fig .3) to separate t he users. The output signals from the different positions are demodulated and th e reference sequence (t raining sequen ce or user identifier) is compared to th e training sequences used . Different criteria like minimum Bit-Error-Rate (BER) , maximum received power, or a combination of both can be utilized to determine the best suited pattern . The signal containing th e training sequence which is closest to the desired sequence (in terms of the abovementioned criteria) is taken .for reception of the specific user .
2
~"~ ':_~w ,(n)
~...... j -.:.:.~-.......
wz(n) ~~: ---=-~---4C
~,..;-i
w3(n) .
-...-oGG--../
Fig.3 Switched beam
(a) 1 ~., .1. . . ._ _~~_ _.... L-I
Y
I
,< .. .:
I
2~_ ' -h--~-+- L-I
Y
I
I
I I
I 1
, :. :
3 ~_:-++r---01~-+-+--__""
L-I
I
1-
,
"
.......
"
""S..:.
Training
'
.
,.' \
.... ....
.. ,
.,.
... Co: :.
~
,
'0
,
~
,
::.:f! U HU! U Uf~~;HJ~;~: i l;U:
..
.:.
,,',
_I
5
...
"
\
Switch for Tracking
I
,
.., t ::::::.': :, ': ::: 10'
M~,., :-++h-_~H+-' y
....
'1
.10
Steps
10
'5
(b)
Fig.4 Adaptive array
-
-
.
~
20
25
Steps
Fig.5 Averaged mean power e2 of t he erro r for t he co nsidered adap-
Adaptive A rrays are based on maximization of the
Signal-to-Noise and Interference (SNIR) ratio upon a known sequence [10] (FigA). Algorit hms based on gradient-vector estimation (LMS) and/or Least Squares (LS), (LS, RLS, SQ RLS and its non- deterministic counterpart , the time- space domain Wiener filter) may be utilized. Fig.5 a shows t he averaged mea n power of the error for the LMS. T he numbe r of anten na elements is M = 8, their spac ing d = )../2. The step size J-l LM 5 of the LMSalgorithm was set to the ha lf of the maximum ste p size to obtain convergence. The Signal-to-Noise Ratio (SNR) was set to 20dB. Since th e LMS does not converge with in the used 26-bit traini ng sequence, we used it repeatedly. The slow convergence of the LMS-algorithm is remarkable . T his is due to t he eigenvalue sp read of the covariance matrix of the antenna outputs. For smart antenna applicatio ns t he corre lat ion mat rix is usua lly ill-conditioned (i.e. its eigenvalues are widely spread) , therefore t he convergence speed of t he LMS is low. Fig .5b shows the averaged mea n power of t he error for the RLS, SQRLS , and LS al-
tation sche mes. (a) LMS; (b) - - : LS, - : RLS, and 0 : SQ RLS . No te the di fferent scale on t he x - a xis .
gorithm. T he forgetting factors of bot h RLS and SQRLS were set to 1. They show that t he RLS and the SQRLS perform in the sa me way for infinite pr ecision arit hmetic . For finite precision arit hmetic the SQ RLS is preferable [11]. T hese graphs illustr ate that LMS is not suitable for a mobi le radio environment with short t rai ning sequences [12], [4], but LS (Wiene r filter) or (SQ)RLS algorith ms are well suited . V. SIMU LATION RESULTS A. Single User
Fig .6 shows the BER for t he considered adaptation st rategies for various SNRs wit h t he number of anten na elements M as parameter . The curve for M = 1 agrees well with t heory (BER 1/(2SNRlin), for SNRlin > 10, where SNRlin is the SNR on a linear scale) . Switched beam performs better tha n the adaptive array. T his may be att ributed to the non- st ationar y chan nel. T he nomi nal DOA of t he impi nging signa ls is the only quantity being st ationary . This gives adva ntages for DOA-based approaches for this specific scena rio.
483
=
1=: f- .__.
-
-
-- ..
-.
..
-
.-
~=1 -'- :;>... ~
~~
--
F '
f=.
-
~
~
.V
--
M-2 -
~-
""'-..;;
-............. ~
<,
............
~~ ,
M 5
_. -.- -
-
--- .
~
..
- ..
-- -2
-
0• •
...-
,M-a/
----. . .
-
c=
--
~
-
--
..............
~
,.. ........
._--
~;
....
..~ .""::a...
." , · · · · · · · ·_· ·~ ·· ·· · · · · · ·~· · · · · · ·· · · · ·j···· · · · · · · · · M= 1····' ········ . . . . ..~ ;
~~~ . ......... .
...........}
i ···········j············i·····- ,
( ...........•-
:
~
;
~
~ ,
;
_j ~
.
.
.
_ ._.-._- "- .._._----_._- -_._---_._.4
6
8
10 12 SNR/dB
14
16
18
20
cr:
UJ
to
Fig.6 BER versus SNR for a single-user system. - - : Switched beam, - : LS; 0 : RLS, SQRLS. A linear array with d = )J2 is used. M is the number of antenna elements .
·········:- ········M=S·· ~ "':::"; - - _ · · ·; · · · · · · · · · · r;;;~·8
... - - -: -_ ._ - _ ~ .._
i.._._
~
The simplest SDMA-scenario are two users within the same traffic channel (K = 2). With a conventional detector without any equalization the BER for one user is 0.25 for a Signal-to-Noise Ratio (SNR) of SNR = 00 [9]. Fig.7 shows the BER curves for one of the two users served in the same traffic channel. If two users are close in space they can hardly be separated by a conventional SDMAapproach . Even if they could be separated at the uplink, downlink transmission to the mobiles without causing significant interference to at least one of the users is not possible. These close-by users could only be served on the same traffic channel if more sophisticated signal processing at the mobile is used [13]. So, if the two users are too close in angle, i.e. their angle difference reaches a given threshold value fP'h' a "handover" of one user to a different frequency or timeslot has to be performed. This threshold, of course, influences the capacity: the higher the threshold, the lower the capacity. For RLS, SQRLS, Wiener filter, and the Least Squares approach a threshold of fP'h = 10° with M = 8 antenna elements is sufficient to obtain nearly the same performance as for the single-user case. Although Fig.5 suggests otherwise, we found that LS and (RLS, SQRLS) perform equally well BER-wise. The figures demonstrate that the switched beam approach performs worse compared to LS-adaptation. Switched beam uses a given set of beams and selects the best of them in accordance to a chosen criterion . Since this approach cannot place the maximum in the "best", i.e. desired, direction and a concomitant zero in the unwanted angular section, interference is always higher than for the adaptive array approach. This leads to the increase in BER especially for high-SNR conditions . For one specific scenari o Fig.8 shows the antenna pat-
484
••_
:::::::::::r:::::·::::r::::::::::l::::::::::::L~::::::::: r:: ::::::::t :::: ::_ .. ...........: : ~
............-.
B. Two User
. ..- ·· · · Tmfu54lc l~~~;j
~
:::::·::::·>· ·::::+:::· :::::::::::: :::·::~:::::: :::F:::::< :::
2
~-
~
.
4
6
8
:
-
10 12 SNR/dB
14
16
18
20
14
16
18
20
(a)
2
4
6
B
10 12 SNR/dB (b)
Fig.7: BER versus SNR for a two-user system. - - : Switched beam, - : LS; 0 : RLS, SQRLS. (a) Threshold
(3] I. R. Carden, M. Barrett. " Adaptive Antennas for Second and Third Generation Mobile Systems." RA CE Mobile Telecommunications Workshop , May 17-19 , 1994, Amsterda m, pp. 728-732.
8/deg.
.------0---
User 1 30
90
1
User 2
-30
\---+-----l--4---+---,~_+-_t'___+_-_t___1
(4) G . V. Tsoulos, M. A. Beach, S. C . Swales. " Adaptive Ante nnas for Third Generation DS-CDMA Cellular Syste ms ." P ro c. VTC'95 , Chicago, Illinois , USA, July, 25-28 , pp.45-49. (5] P. Eggers . " Einfluf der Geliindestreuung auf die Strahlun gseigenschaft en von Basisstationsantennen fiir Mobilfunks yst eme ", Nachrichtentech., Elektron., Berlin 45 (1995) 4, pp . 57-62.
-90
Fi g .S An ten n a patterns t o separat e th e t wo users d erived by t he us e of LS a d a p t a t io n . - : P a t ter n t o serve us e r 1, - - : Pattern t o se r ve us e r 2 . T he antenna is a lin ear a rray with M
=
8 elements spa ced
by d = >.. / 2.
te rns serving the two users separa tely, derived by L5 ada ptati on . The adapta tion algorit hm places a maximum into the directio n of th e wan ted signals and nulls t he interferin g ones, indeed. VI. CONC LUSIONS T he performance of two adapt ation schemes for smart ante nnas for mobile communications is ana lyzed . T he swit ched beam ap proach performs worse com pared to an adapt ive array, especially at high-SNR conditions. For the ada pt ive arr ay only the class of L5 adaptation algorit hms (LS , RL5 , and SQRLS ) shows sa tisfactory perform an ce. The convergence speed of t he class of gradient-vector estimation (like t he famous L:\1S ) is too low for application to mobile radio problems. Ad ap tive array ap proaches show surprisingly low sensitiv ity to close- by int erference. T his means that only small angular threshold valu es for handovers ar e necessary , which results in a larger ca pacity increase th an for th e switched beam approach. VII. ACK NOWL EDGMENT Support of this work by the Aust rian PTT is gratefully acknowledged . T he views expressed in t his pa per are t hose of the a utho rs and do not necessa rily reflect the views within the Austrian PTT . We t ha nk Prof. Bonek for sti mula ting discussions. VIII . REFERENCES (1] J . H. Winters . " Signal Acquisition and Tr ackin g with Ada p tiveArr ays in t he Digital Mobile Radio System IS54 with Flat Fading." IEEE Tr ans . Vehicular Technology, vol. 42, no. 4, November 1993, pp . 377- 384. (2] O. Munoz-Medina , J . Fernandez-Rubio. " Adaptive Antennas in Mobile- Satellite Communica tions." RA CE Mobile Telecom munications Workshop , May 17-1 9, 1994, Amsterda m , pp . 754-759.
485
[6] W. C.- Y. Lee. " Effects on Correla tion Between Two Mobile Radi o Base- Station Ant ennas ." IEEE Trans . on Communications, vol. C OM-2 1, no. 11, November 1973 , pp . 1241-1221 . (7) O . Norklit, J. Bach And ers en. " Mobile Radio Environments and Adaptive Ar rays" . Proc. 4th Personal, Indoor a nd Mobile Radio Conference, PIMRC '94, The Hagu e, Netherlands, pp. 72.5-728.. (8] J . J. Blan z, P. W . Ba ier and P . Jung. " A Fle xibly Configurable Statistical Chann el Mod el for Mobile Radio Systems with Direct ional Diversity" . Proc. 2. ITG- Fachtagun g Mobile Kommunikation '9.5 , Neu-Ulm, Sept ember , 26-28, 1995, pp . 93- 100. (9) J . Fuhl , E. Bonek. "Adaptation Schemes for Smart Antennas - A Comparison Bas ed on a Channel Model Including Directi ons-Of-A rri val" , C OST 231 T D(96), Belfor t , France, 24-26 January, 1996. (10] S. Haykin. Adaptive Filte r Theor y (Second Edition). P rentice-Hall , Inc. . Englewood Cliffs (New J ers ey), 1991. (11] F . M. Hsu. " Square Root Kalman Filtering for HighSpee d Data Received over Fading Dispersive HF Chan nels" , IEEE Trans . on Inf. Theory, vol. IT- 28, no . 5, September 1982, pp . 753-763. [12J J . Fuhl , A. F. Molisch , " Space Domain Equalisation for Second and Third Generation Mobile Radio Systems" , P ro c. 2. ITG-Fachtagung Mobile Kommunikation, Neu Ulm, Sept embe r 1995, 26- 28, pp . 85- 92. (13] S. W . Wales. "Technique for cochan nel interference suppression in T DMA mobile radio systems" , lEE P ro c.Commun ., Vol. 142, No . 2, Aprill 1995, pp . 106-114.
Performance of Wireless CDMA with M-ary Orthogonal Modulation and Cell Site Antenna Arrays Ayman F. Naguib, Member, IEEE, and Arogyaswami Paulraj, Fellow, IEEE
Abstract- In this paper, an antenna array-based base station receiver structure for wireless direct-sequence code-division multiple-access (DS/CDMA) with i\l-ary orthogonal modulation is proposed. The base station uses an antenna array BeamformerRAKE structure with noncoherent equal gain combining. The receiver consists of a "front end" beamsteering processor feeding a conventional noncoherent RAKE combiner. The performance of the proposed receiver with closed loop power control in multipath fading channels is evaluated. Expressions for the system uncoded bit-error probability (BEP) as a function of number of users, number of antennas, and angle spread are derived for different power control scenarios. The system capacity in terms of number of users that can be supported for a given uncoded BEP is also evaluated. Analysis results show a performance improvement in terms of system capacity due to the use of antenna arrays and the associated signal processing at the base station. In particular, analysis results show an increase in system capacity that is proportional to the number of antennas. They also show an additional performance improvement due to space diversity gain provided by the array for nonzero angle spreads.
D
I. INTRODUCTION
IRECT-SEQlJENCE code-division multiple-access CDS/CDMA) is an emerging technology for civilian wireless communications. CDMA offers improved performance in terms of capacity or coverage area over frequency-division multiple-access (FDMA) or time-division multiple-access (TDMA) based cellular networks [1]-[4J. One approach to further increase the system performance is the use of spatial processing with base station antenna arrays 15j-(8J. By using spatial processing at cell sites, \ve can use optimum directional receive and transmit beams for each user to improve coverage or increase capacity. The increase in system performance by using antenna arrays in CDMA comes from reduction of cochannel interference from own cell and neighboring cells. In general, this reduction in cochannel interference can be used to improve other system performance measures such as coverage area, transmitted mobile power. and system capacity. In [9]. we proposed a base station antenna array Bearnformer-RzvKf receiver that exploits the spatial structure Manuscript received May 1, 1995; revised November 22, 1995. This paper was presented in part at the IEEE ICC'95 Conference, Seattle, WA, June 1995. A. F. Naguib was with the Information Systems Laboratory, Department of Electrical Engineering. Stanford University, Stanford, CA 94305 USA. He is now with the Information Sciences Research Center, AT&T Laboratories, Murry Hill. NJ 07974 USA (email: [email protected]). A. Paulraj rs with the Information Systems Laboratory, Department of Elecrrical Engineermg. Stanford University, Stanford, CA 94305 USA. Publisher Item Idenufier S 0733-87 J 6(96)05239-0.
in the multipath received signal in addition to the time diversity to provide a more efficient combining of paths. This receiver incorporates a "front-end" heamsteering processor feeding a conventional coherent RAKE combiner, which assumes know ledge of the amplitudes and phases of path gains. This requires the transmission of a pilot signal to obtain good amplitude and phase estimates. However, transmitting a pilot in each user' s reverse link signal, whose power is greater than the data-modulated portion of the signal, reduces efficiency to less than 500/0. Instead, either differential phase shift keying (DPSK) which does not require phase coherence or M -ary orthogonal modulation with noncoherent reception should be used. For M > 8, where M is the number of orthogonal signals, orthogonal modulation is better than DPSK llO], [l l ]. Analysis results for CDMA communications systems employing DPSK are reported in [12]-[15]. The analysis in [15] also assumes a base station antenna array receiver structure. Analysis for DS/CDMA with M -ary orthogonal modulation but without antenna arrays has appeared in r16]-[20]. The use of very low rate orthogonal codes in spread spectrum multiple access is considered in [16]. The analysis in l17j is done for the additive white Gaussian noise (AWGN) channel. However, the analysis in [18] models the multiple-access interference (MAl) as Gaussian noise and considers a rnultipath Rayleigh-fading channel [19]. The analysis in [201 is done for a general multipath energy distribution. In this paper, we propose an antenna array-based base station receiver structure for DS/CDMA with M -ary orthogonal modulation and noncoherent RAKE combining and study its performance. The receiver consists of an antenna array followed by a bank of Walsh correlators. The base-band received signal vector and the post-correlation signal vector are used to estimate the channel vector for each path. The output of the correlators is then fed to an optimum bearnformer followed by a noncoherent RAKE combiner. The output of the RAKE combiner is then used to estimate the transmitted data. Assuming that the channel parameters are almost constant over several symbol periods, the output of the RAKE is also fed back to the channel vector estimation algorithm to determine the winning post-can-elation signal vector which corresponds to the actual transmitted Walsh symbol. We may note that, although the analysis results in this paper drew upon some of the analysis results in [19] and [20], they are different in the sense that they are done for a base station receiver with an antenna array and include the effect of closed loop power control.
Reprinted from IEEE Journal on Selected Areas in Communications, Vol. 14, No.9, pp. 1770-1783, December 1996.
486
I-Channel Short Code
(ltl)
Convolutional
Encoder
and Repetition M -ary Orthogonal Walsh Modulator W(t)
Long Code
Mask
c;Ct)
User
.LongCode Fig. l .
dQ(t) Q-Channel Short Code
Mobile transmitter block diagram.
This paper is organized as follows. In Section II, we present the channel and received signal model. In Section III, we describe the antenna array receiver structure. System analysis and derivation of the signal and decision variables statistics are given in Section IV. The probability of error analysis is given in Section V. Numerical and simulation results are presented in Section VI. Finally, Section VII contains our conclusions and remarks.
and the I or Q channel PN code as (I) ( ) = c, ( t) a (I) t ( ) cit
and
(Q) ( t ) -__ t:.l (1, - ) a.'( Q) (. t).
C1
To simplify our analysis, the PN codes c;I)(t) and c~Q)(t) are represented by [20], [22]
c~I)(t) ==
L
c~~Jp(t - kTc )
(2)
c;Q)(t) ==
L
c~;)p(L - kTc )
(3)
00
oc
ll. SIGNAL AND CHANNEL MODEL
The mobile transmitter block diagram [21J is shown in Fig. 1. The binary data at the output of the interleaver are grouped into groups of J = log2 ~1 bits. Each group is mapped into one of lvJ orthogonal Walsh sequences \tV( t). The resulting signal is then spread using the user's long PN code C; (t). The signal is further multiplied in both I and Q channels by the short PN codes (](T) (f.) and 0,( Q) (t),
respectively. The PN modulated Q channel signal is delayed by half a chip period rT~_:/2. The two spread signals are upconverted to radio frequency for transmission. The power of the transmitted signal is adjusted according to both the open and closed loop power control mechanisms (see, e.g., [21]). Then we can write the signal transmitted by the ith mobile as .s i ( t)
:=-
1/)~
IP:(vv (h) ( t) C
+ l-i/l h ) (t
1(
r=-x
where c~~,: and c)~) are assumed to be independent and
identically distributed (i.i.d.) random variables taking values ±1 with equal probability, and p(O is the chip pulse shape, which can be any time-limited waveform. Here we assume that p(t) is rectangular although our results can be easily extended for any time-limited waveform. To simplify our analysis, we assume that we have a constant deterministic power-delay profile and that the log-normal slow fading is the same for all multipath components. Therefore, we can write the complex lowpass equivalent representation of the vector multipath fading channel from the ith user to the base station antenna array as {23] L,
t ) a( I) ( t ) cos (wc t)
hi(t, T) == ~L 6(1 t=o
To)Ci (t -- To)a (Q)
x (t - T()) si Jl ( wet))
0 < t
S
T'w
( 1)
where P, is the transmitted power per symbol per dimension, 'I ~lJ is the symbol period, W r is the carrier angular frequency, To == T c/2 is the time offset between the I and Q channels, and finally 'l/;i is a Bernoulli random variable that models the voice activity of the ith user (we assume that a user will be on with probability v and will be off with probability 1 - v). vV( h) (t) is the hth orthogonal Walsh function, h == 1, ... , IvI. Let the processing gain be G ::::: T w /1~_ For simplicity of notation, we shall denote the product of the user's PN code
Tl,l)al,l
(4)
where L, is the number of multipath component for the zth user, S, is the log-normal shadowing experienced by the same user, TI,l is the time delay of the l multipath component. and al,'l is the K x 1 channel vector of the base station antenna array to signals in the lth path from the ith user, where K is the number of antennas. Without loss of generality, throughout this paper we shall assume that we have a uniform 1inear array (ULA) of omnidirectional sensors in each sector at the base station. Each of the resolvable multipath components (i.e., those separated by more than T; from one another) will actually be themselves a linear combination of several
487
L
parallel demodulators
1- '
t ·····--·----·.. - ··.········.···--·--····-···.,
l--
1
Downconverter 10-.... and LPF
-44
Walsh
H ,
·
•• •
Walsh
:
~equence
Corre lators
~equence
Correlator s
1-
..
~
L parallel demodulators ,( ········--·· ·· ·--··---- ···-·--------- ······- -1
...
rlH Walsh ~quence ~ Downconverter fand LPF
~
, ~ .................
+{ : I
1 ~
~
pre-correlation
signal vector X
Fig. 2. Basc
•••
II""
I I
Walsh:equence CorreIators
~
, ,,I , ,,I
j
.
4
stat ion recei ver
Mvary Decoder,
•
(M)
data
Measure Frame Error Rate(FER) i+-
.
Threshold
P-- Closed Loop
post-correlation Z signal vector
'. t
Deinterleaver, and Viterbi Decoder
~
Power Control Algorithm Up/Dow n
WeightVector Estimation
weight vectors
Z' ••• , WL
Wi' W
I Select Index, I ofMaximum
select post-correlation signal vector
Power Control Command
I
block diagram.
unresolvable paths of varying amplitudes and phases which arrive at the base station within angle spread ±.Ll./,i at angle BI " with respect to the array. The elements of al,; are zero mean complex Gaussian correlated random variables , each having an autocorrelation function Ju(21f!dT), where fd is maximum Doppler shift, and if 1= k otherwise.
(5)
This implies that the channel vectors of two differen t paths are independent. The matrix RI,i is a K x K Hermitian Toeplitz matrix that describes the correlation between the elements of the channel vector of the lth multipath component ar , which is a function of the angle spread .Ll././, the mean angle of arrival &/.,' the waveleng th A, and the spacing between sensors d. If we assume a uniform angle of arnval over the angle spread, then it can be shown that [24], [23], the real and imaginary parts of R I " (m , 71,) are given by
= Jo(r(m,n)) + 2 L 00
Re{R/,,(m, n)}
••
ET
l.. . .. ..... . . .. .. .. .. . .. . . ....... . ... .. .__•••••• J
.~
•••
Optimum Beamfonning and Incoherent I-RAKE Combining
......•
,
ZJ
ZJ
JIl"".
Correlators
(1)
Jl
•••
\7
K
Optimum Beamforming and Incoherent I-RAKE Combining
1• • • ••• • • • • •• •• _. - • • • • • • • • • •• • • • • - . .... . .. .....
• •
L.--
.
JII"'.
1=1
hl+l (r(m, 71,)) sin((21 + 1)B1,i )
/=0
x sinc«2l + l).Ll.l ,,)
(7)
and r(m,n) = 21fd ·Im - nl /'\, We assume that the channel parameters vary slowly as compared to the symbo l duration T", so that they are constant over several symbol durations, Therefore, after downconverting to baseband , we can write the K x 1 complex baseband received signal vector for the ith user as
= jSiPi'lfJ; L L,
x ;(t )
c~h)(t -
Tl ")ejh 'al ,,
(8)
1=1
where
¢l "
=
WeTl ,.
and c~") ( t) is defined as
Let N be the number of cochannel mobiles. The total received signa l at the cell site is the sum of all users' signals plus noise and is given by
h/(r(m,n))
x cos(2lBL / ) sinc(2l6i,/)
= 2L 00
Im{ R/,;(m,n)}
L x.;(t) + net) . N
(6)
488
x(t) ::-:
.=1
(10)
· IA = ._
_._
...................
.-
z l /J(m)
~I.l
::
~.:..:.:::..::::.:..:':..:.~.:..:..~:.:':. :'.::.~': ._....._.....::.:::.:::~::::..~::.:=1 =====::' z1.1 (m) :
_..-.
r·--······__·········_--
•
( M)
..............._.._..t:..
1 •
••
II-- ---;= ::j::::= :> z~/m) (I)
•
••
cit) I !
-========(===> <7~m) 1
wM
.
:
;! •
.
i::::::c:cc:::::::c:::~:::=: ::::::::cc=:==:::::::::::::::=::::::::: :::::c_::::c:c:c~c:c-:: ::::::::::: cc=:: ::::c :::c -:::~
r...tl-*'i;
: _
l
; •
;
:
... .... .... ......._ _
Fig. 3.
_
_
_
_
_
_._
_
--~-- ---, ...,>
: .::..7 - - - .. -. -
Z(1)( m) LI (M)
ZLI ( m)
Corrclators,
The vector n (t) = n c(t) + jn,,(t) is the K x 1 additive Gaussian noise vector with zero mean and covariance
where (J" ~ is the noise variance per antenna. III.
RECEIVER MOD EL
The block diagram of the base station antenna array receiver is shown in Fig. 2. It has a " Beamfo rmer-RAKE" structure where several multipath components are tracked in hoth time and space. After down-converting to baseband , the outputs of the LPF are fed into a bank of M Walsh correlators shown in Fig. 3. Assuming that the hth Walsh symbol was transmitted, where li = 1, · · , I'v! , the pre-correlation and post-correlation K x 1 signal vectors x U ) and y ~~ ) are used to estimate the channel vector a l. i and the corresponding K x I optimum hcamforming weight vector W I ., from the pre-corr elat ion and post -correlation array covariances R n and R yv .1-, using the code filtering approach derived in [91 and [23] . The deta ils of this algorithm and how R x ," and R zz,I" are tracked with the channel variation can be found in [23] and [25J. For each rnultipath component we have M different postcorrelation signal vectors y ~ ':) . h = 1, · · · . .~rf . The vectors
y;-';) arc fed to an opt imum bcamforrn er.
L , bearnforrners for the hth Walsh function wi.,Yi~: ) are then fed into an incoherent RAKE combiner. The output of the incoherent RAKE com bine r z; h) is the decis ion variable for the hth Walsh function. The beamformer and the incoherent RAKE combiner for the hth Walsh function are shown in Fig. 4 . In order to update the post-correlation array covariance R y y ,l , t (that will be used in e stimating a ., and WI., ), the
Th e outputs of the
489
receiver needs the post-correlation vector y;~ ) corresponding to the true transmitted Walsh sym bol ~V (h J(t ) . However, at this stage the receiver has no prior knowledge of which post correlation vector y;~ ) is the right one. Here, the receiver relies on the inherent correlation of the multi path vector channel and the assumption that the channel remains almost constant over several sym bol periods. In this case, the receiver uses a delayed updat e of R vv.i ., (and hence dela yed estimation of the channel vector and the optimum beamforming weight vector). Thi_s is done by using the decision on the CUITe nt Walsh symbol It to
yi':)
selec t the post-correlation vector to update R y y . l " and obtain the optimum weight vector WI ,,' This weight vector WI " will be used for beamforrning for the next symbol. Th e de cision variables z~ \), . . . . z}'·1) at the output of the incoherent RAKE are then fed to an M -ary decoder, deinterleaver. and Viterbi convolutional decoder. Without loss of
Time Align
~"
't J
•• •
w..
•• •
2,/
Time Align 't
I
L
//
Z (n) 1
Fig. 4. Optimum beamfonning and incoherenr RAKE.
generality let us assume that the first user is the desired user and let Tk,1 be the time delay of the kth tracked multipath which is assumed to be estimated perfectly and k == 1, ... , L 1 . Then, we can write the post-correlation signal vector yL~) [or
transmitted
( 18)
the kth tracked multipath component for the first user as
y~~l ==
== ==
1 T11.l
f1T1 y
(n ) d k ,1
(11.)
k
.
l+Tw
u(n) k,l
x(t)cin)*(t - Tk,l)dt
TJ...,l
(n). If n
+ uk, 1 ,
==
h
(14)
= 2/TwS 1 Pl ej ¢ /.; ·l a k ,1 ==
m(n) k,l
( 12) (13)
if n i=- h
U k,1 '
d~~{
and
l'T
+ sen) -+- n(n) k)l' k,l
(15) ( 16)
di',Li is the desired signal vector, m~~{ is the MAl signal
vector, sk~i is the self-interference (SI) signal vector due to
other multipath components of the 1st user. and n~~l is due to the AWGN. Let At == JSiP(l/J'I.' Also, for the kth tracked multipath let the optimum beamfonning weights determined using the previously estimated Walsh symbol be W k,l' For an equal gain combining incoherent RAKE, the nLh decision variable of the 1st user corresponding to the nth Walsh symbol is given by [10] n == 1,' .. ,A1.
However, for the data, a symbol-by-symbol M-ary decoder is used [11]. Both approaches yield exactly the same decisions for the M -ary symbol and both are optimal (i.e., a maximum likelihood rule) for an AWGN channel. Since the IvIAl is not necessarily Gaussian, this decision rule is actually not optimal. However, when the number of cochanrrel users is large, the MAl can be modeled as Gaussian noise and therefore this decision rule can be used. 'I'he primary reason for using the syrnbol-by-symbol approach for the data is to provide improved performance with error-correcting codes by using soft decision decoding [10], [1 1]. Note also that we cannot use the output after the convolutional decoding and deinterleaving to select the postcorrelation signal vector, The reason for this is that we will have to wait for a decision to be made on the current symbol and convolutionally encode and interleave again, By the time this process is over (which is at least twice the time of one frame of bits), the channel would have changed and the estimated channel vector and the channel vector of the new symbol will be quite different. This will lead to a ctegraclation in the beamformer output SINR.
(17)
IV. SIGNAL STATISTICS In order to derive the uncoded bit error probability (BEP), we need to derive the statistics of the decision variables 1 1 ' thediff .21(1 I, Zl(2) , " ' , Zl(M)', F'irst, we wi' 1 exarnme l erent terms
Now, to select which post-correlation signal vector y~~) should be used in estimating the post-correlation array covariance Ryy,l,'t, a hard decision is made on which Walsh symbol was
490
'
zi~l i.e., the multiple access interference signal vector due to other cochannel users m~ll{, self-interference signal vector in
due to the user own rnultiparh components vector
(n) uk,}'
S~11{, I
and noise
A. Noise Analysis
The noise term
nk
H
' T~" j (nJ· Uk,l = p;=;-.
j vTu· T~.l
f (11,)./1 \nk,l
_
-
h were
For
-
is given by
{
l -rT,L'
~(n)*(
(
n t. )c1
n(n),II k,l '
~
k,1
1
k,l'
A
Ll ""'
[(I(n),II + I(n),QQ) k.l,l.l
1 ~
1=1 l:f:.k
t
~
Tk,l)(
ll
+ nk,l (n).QQ) + "(_n~n),IQ + n(n),QI) } k,l k,l
(n);!! n(n),QQ n(n),IQ
ok,!
-
and
n(n),QI k,l
+ /(n).QI)] J~/.I + J/.( _ I(n).lq k, i ) \1 k, 1 ,I, 1 e al .i
( 19)
where
(20)
k.l,l,l
( n ) ,II I(n),(.JQ I(n),TQ rk,l,l,'i' k.l,l,'I. . k,l,l,l '
an
d
r(n),QI k,Ll;l.
(30)
d fi d are e ne as
are defined as
we have
Also, we can write the MAl due to other users' signals as
: : : "'.44.£ L N
L,
~
[(I(n).TI k.l,l,1
+ I(n)'~Q) k,l,l,z
i=2 l=l
+ .J.: (_z(n),l9 + I(n),Ql)] e j (/) , k,l,l,l. k,l.l,·L
.
a
i,
.
(37)
Let
. rand om vector . sh ow t h at n (n)i' II.lS a Gaussian We can easily k with zero mean and covariance
,I
f 1. Similarly, we can show ')
r=-oo
that n~~l'QQ, n~~LrIQ. and n~~rQI are all uncorrelated zero mean Gaussian random vectors with the same covariance. Hence n~~{ is a zero mean Gaussian random vector with covariance 2a~ I.
where q.~~ == W~h)c~I1~' It follows from (2) and (3) that
T ~' -00,'" 00' is an i.i.d. binary random sequence taking values ± 1 with equal probability. Hence, it follows that (n ) I I ' U' . I(n),! T I k,1:'l,i IS zero mean. sing (38) , we can rewnte k,l,l./ as
q(I),
B. Self and Multiple-Access Interference Analysis The self interference due to other multipath components is given by
(n) ,I I _ -
I k . 1 ,L i
1
1
G-l
IT' '""" Z::
V
.L u;
b=O
j,'TI.:'l +(b+l)T
c
( 1)
Q1,b
.
TI... l
+bTc
. (
Pt
_
b'T. _ .i.e
Tk,l
)
00
X
(28)
L1
::::
JCPl 1 A 1 "l(n) L..J k,l,l,l e . al.l
'L..J "'
(f) P.t ( q'i,'T'
-
I
r'7'c -
it Tl,i. ) (1/
r=-r")('J
(29)
1=1 lfk
(39)
491
where {3k,1.l,l == rl,i - rk,l modulo-T, and Rp(s) is the partial auto correlation of the chip waveform defined as
Rp(s)
=
is
p(t)p(t
+ t; -
o~
s)dt,
s
~ I~.
J
q;;-l Rp(fJk,Ll,t) + q~,~: Rp(T
c -
f3k.l,l,-L).
Using the results in [22]. we can show that {Fb}b=O,. are independent random variables and hence T
\ ar
{1(n),II} __ 2 T k,1,l,'t - '3 c·
(41) ,C-1
· ·1 I 1 h th t l(n),II j(n),QQ l(n),IQ d S irm ar y, we can a so s ow a k),lJ7.' k,lJ,i , k,l,l;i ,an
lk~/'l~I are all zero mean uncorrelated random variables with the same variance given by (42). n Remark: In deriving the variance of /,/ : , \ve used the assumption that the chip pulse shape is rectangular, In reality the channel is bandlimited (due to low pass filtering following the down converter) and the received signal cannot be a square wave. Under the condition that the same amount of energy is received regardless of the channel used, the received signal through a bandlimited channel will have a higher peak value, resulting in a higher level of M.A.I interference due to larger fluctuations. In [191 and [22], it was shown that if the bandlimited channel has ideal low pass filter characteristics with a bandwidth B = liTe, then, we would have
lk
SImilarly. covariance of
The total interference vector
21~()~I.
R(n) uu,k,l -
a
Consider the pre-correlation and post-correlation signal vectors x(t) and yk~{. With the assumption that the MAl 1S spatially white, the optimum bcamforming weights can be shown to be (48)
where ( is some arbitrary constant (that does not change the beamformer output SINR). For simplicity of the analysis, we
J
set ( == 1/ a k,l ah,1 . Define the beamformer output for the kth multipath component of the first user w*h,l yen) as U(n) k,l k,l (n ) 2A 1 -V/'--T I I ]¢~. .1 Uk,l:::= w ak.1 e
==
*
(n)
+ ,V *k, 1 Uk,(11.)1 .
for n
:=
h, (49)
for t: =1= h.
Wk,l U k. 1,
(50)
where \ak,l! == Jak,lak,l. We can easily show that V~:~)
=-
k u~ti is a zero mean complex Gaussian random variable with variance (J2. For simplicity of notation, let L 1 == L. Then,
W
1
the decision variables for the first user are
From [10], and conditioned on Al and al,l, l ~ 1, ... 1 L, vre n 2 can show that for == h, ) has a noncentral X distribution with 2L degrees of freedom and noncentrality parameter
(44)
zi
i~~l == m~~i + s~~{
(52)
The noncentrality parameter E is the symbol energy. For :f. h; n ) has a X2 distribution with 2£ degrees of freedom. Therefore. we can write the conditional probability density function (pdf) of as
n
zi
zfn)
n==h
(45)
a} is
(47)
C. Decision Statistics
n
is modeled as a zero mean complex Gaussian random vector with covariance I~1~1 == R{i~1:1i1~{*}. Although, this assumpnon does not always hold [or CD M.A. analysts, it was shown in [22] that it is valid for large G. Moreover, simulation results presented later in Section VI show that for large N . L, if we assume that the angles of arrival of the multipath components are uniformly distributed over the sector, the total interference vector i~~{ will be spatially white. In this case
where
21
(43)
ni~l will then be
Var{ n~~l} ==
covariance of the interference-plus-noise vector
(42)
I..
v ar {I(n),I I} --- Tc k,l,l.t
= 2. The
Ui~~l) is then given by
(40)
For rectangular pulses, R p (s) == s. For asynchronous networks, a reasonable assumption is that (3k,l,I,t are independent and uniformly distributed over [0, Tc ) . Let
rl ==
C
given by
ni=h
(53)
where I L (.) is the modified Bessel function of the Lth order.
(46) and C is a constant equal to two for a bandlimited channel and ~ for a rectangular pulse shape. For the remainder of our analysis we will assume the case of a bandlimited channel, i.e.,
r(.) is the Gamma function, and IS
==
e (72'
(54)
We may recognize Is as the symbol energy to interferenceplus-noise ratio.
492
Remark: In this analysis. we used the assumption that the channel vector remains constant over t\VO symbol periods. Also, we assumed that ak', 1 is estimated perfectly. In reality the channel is time varying and the array covariances are estimated using few samples. This will lead to errors in the estimated channel vector and hence a reduction in the symbol energy IS' In [25], we studied these losses as a function of the maximum Doppler shift f d and angle spread ~. These results show that, with optimum forgetting factor for recursively estimating the array covariances, the loss in [8 is less than a few tenths of a dB. Therefore, the analysis results obtained here can be regarded as an upper bound on the system performance.
A. Low Doppler Frequency
At "low Doppler frequencies and high diversity orders, fast closed loop power control can eliminate most of the channel variation due to multipath fast fading. In the case of ideal power control, 18 is a fixed quantity and is given by '1s ==
!J!\t ("fs
)
1 - P; _ 1 - P (~(2) 2,
==
== 1 -
==
1-
PROBABILITY OF ERROR ANALYSIS
In this section we derive the uncoded BEP with hard decision. As we mentioned earlier. with forward error correction (FEe) and interleaving, which is not considered in this paper, symbol by symbol soft decision decoding yields better performance. To derive the probability of error, without loss of generality, let us assume that It. == 1, i.e., the first Walsh symbol ~"l(t) is transmitted. Then the probability of symbol error is
<
(1) -y(3) Z 1 ',(..1
<
(1) .,.
21
,
-;yU'vl) <
, .... 1
(55) (56)
z(l») 1
. -
1 (
a2
(CXJ
Jo
[1 - e-z/a2 ~ ~ (;2 y]
(57)
and (2)
p (z 1
I
(1) _ -
IZ1
-y
~)
_ -
zz:
,.: .I f 0
," z ~ 2) (x)
(58)
dx L-l
z 1 - c- 12~1(Z) a Z:: T! 02
l
(59)
I=()
Finally, the corresponding uncoded BEP
})1J( T..;)
z
--
a21s
) L:;l f
The symbol error probability and the corresponding uncoded BEP derived above are conditional probabilities and are functions of /Or 8' which is itself a function of the channel vectors al,l,···,aL,1, shadowing and path loss 51, and the first user transmitted power Pl. Also note that because of power control (both open-loop and closed loop), PI, 51, and al,l, ... ,aL,i are generally dependent variables. The dependency among
these variables is in general a function of the maximum Doppler shift f a of the first user. A reasonable assumption is that the combination of open-loop and closed-loop power control is perfect in eliminating the slow fading due to shadowing and path loss [26]-[28]. Based on the mobile speed (or fd), we consider the following different cases.
_~ l L - 1 (,2 {ijJ'S Z
rt
-?
a-
>
)
dz . '">
(62)
However, due to the delay in the control loop, finite step size by which the mobile can increase or decrease its power. and errors on the downlink, power control cannot be ideal (see, e.g., (28]). Therefore, the symbol error probability obtained above needs to be averaged over the probability density function of 1t;, which is not known in general. However, an approximation to the uncoded BEP can be obtained as follows. Here, we can use the approach in [29] and [20] to get an approximation for the uncoded BEP. First, let C; denote the coefficient of variation of 1 s' defined as
c. ==
JVar{1'8 } ---e-{- } - .
(63)
Is
In [29] and [201, it is shown that for a low Cv , (less than O.~ [20)), a reasonable approximation of the symbol error probability is
P,'c' : : :; ~FAd7$) + ~PM(;s + J3(7,) 3
6
t-
~PMhs
-
v'3(7~)
(64)
Vv here is and a I are is the mean and variance of 1 ~. We can see that (J 'Y also represents the power control error. Then, the corresponding uncoded BEP is
" Pb
is given by (60)
;'\;/-1
l=O
fOG [fl(Z[2) < z I zP) = z)]M-l fz;I)(Z I 'Ys)dz
./0
(61)
where i is symbol energy to interference-plus-noise ratio per path per antenna. Also, the density function of z;n) for n == h given in (53) becomes an unconditional density. Therefore, the syrnbol error probability is P,f\".l
Y.
1 .L . K
2J - 1 " =: -J--PjH.
2 - 1
(65)
B. High Doppler Frequency
For high Doppler frequency and/or long loop delay, the fading statistics of the received signal after power control remain the same as those of the multipath fast fading with only perfect average power control. In this case, we have L
" 18 == 1- '~
'.9
lat,ll 2 .
(66)
l=1
The distribution of depends on the angle spread 6. through al,}- We consider the following three cases.
1) Small Angle Spread: For zero (or relatively small) angIe spread ~, the channel vector of the lth multipath component can be expressed as
(67)
493
where aZ,1 is a zero mean complex Gaussian random variable and for aULA Yr.i is a Vandermonde vector [30] given by Vi,l
==
where J..K
1rk=IT
1=1 L-:j=k
[1
e-rrrsinf:Jz,lD/A ... e-J7f~1Il6J,1 D(K-l)/-\]T,
rs
has a X2 distribution
with 2L degrees of freedom, i.e., £-1
' (ry) = f 'Ys, (ryK)L(L -
(II
ZL-l
"2) =
(J2L(1
Zl ) \
+ iK)L(L -
(69)
1)!
" e-z/((T~(l+l'K)). (70)
2) Large Angle Spread: For large angle spread, theLelements C
becomes uncorrelated and hence 2::£=1 lal,112 is a sum of K L i.i.d. random variables having a X2 distribution with two degrees of freedom. Therefore, '"Ys is distributed as a X2 random variable with 2K L degrees of freedom
of
al)
f-y,(-r)
=
/_
,KL-l
(71 )
C'Y)KL(KL-l),e-"'"
The corresponding unconditional probability density function of zln). for h
= ti is (see the
Appendix)
where
R == I
1
(0- 2 (1 + "y))KL-l X
((!(-l)L) l
(~)(K-l)L-l(_l
] + l'
1 + l'
)1.
(73)
3) Other Values of Angle Spread: For other values of ~, we can easily show (see the Appendix) that the syrnbol energy to interference-plus-noise ratio is L
/8 ==
K
L L ilil u
ZLI
2
(74)
l=J i=l
where 1111···'ULK are i.i.d. zero mean complex Gaussian random variables and i i i == l' )..li, where {Al,i} 1.=l,"', K are the eigenvalues of R Z,l , the covariance matrix of the first mobile's lth multipath component. Also. if the is
'8
=::
Let {i'h.h-:l. L,1.=1· K be equal to {il,}1.=l-- KD· \ve assume that the ilt' s are distinct (this is true angles of arrival are sufficiently different). Then, distributed as [101 (75)
KL {eL (2( '1=1
1
7ri
where
and
(]'
1+
_ )) * g(z)
"L .
t.::
g(z)
Hence, it can be shown that the corresponding unconditional pdf of z~n) for h == ii is (see the Appendix)
f
k=l, ... ,LK
~
Z/ (U2 ( 1+ i 1»)
fzH(Z)
e-,/Ci R-) 1)! -
_
(76)
and the corresponding unconditional pdf of z~n) for h == ti is (see the Appendix)
(68)
In this case, we can show that
,k ik - i1.
* denotes
= u 2 ( L _~)(; _
2)! e-
}
c o2 /
(77)
(78)
the convolution operation.
zi
Using the unconditional density of ) for h = n in (70), (72), and (77), the average symbol error probability is given by "
PM=::l-l
oc [
z u '2 l_e- /
n
1 z Lf!(u
£-1 l=O
2)
l ] M-l
fz~dz)dz
(79) and the corresponding average uncoded BEP is given by (65).
VI. NUMERICAL AND SIMULATION RES\.}Ll'S
First, we study the accuracy of the approximation that the MAl signal vector can be assumed to be a spatially white complex Gaussian random vector. The base station receiver in Fig. 2 was simulated. In our simulation, we assumed that the processing gain G == 256, L == 4,1'1 = 40,1\-1 == 64 and LJ == 0.375. We also assumed ideal power control. 'VJ.le assumed that the base station has three sectors, each with a five element ULA as shown in Fig. 5. The angles of arrival {8k ,2} were assumed to be uniform over [0,120°] (i.e., uniform over the sector). The angJe spreads {~kJ} were assumed uniform over [0,60°]. The results of 10000 post-correlation signal vectors were used to estimate the statistics of the Mi\I signal vector. Figs. 6 and 7 show the empirical PDP of both the I and Q component of the MAL at the first antenna. From both figures we can see the validity of the Gaussian approximation. Also, the covariance matrix of the MAl vector ft~~J.k.,l was estimated and is shown at the bottom of the next page. The
Frobenius norm [311 of the error IleilF == IIRS~:'k)l - IIIF was estimated as 0.0058 which also shows that the MAl can be assumed to be spatially white. Next, we study the system uncoded BEP. The closed loop power control was simulated according to the model in [28] and 1271. In this simulation, we assumed that the mobile can increase/decrease its transmitted power by 0.5 dB at a time and that the power control command was sent every 6Tu.We assumed that the loop delay IS also 6Tw and that the return channel error rate is 0.05. Figs. 8 and 9 show the power controlled signal level distribution versus the distribution of the simulated multipath Rayleigh fading at the RAKE output for f d == 5 and 100 Hz. From these two figures we can see that closed loop power control eliminates most of the channel variations at low Doppler frequencies, while at high Doppler
494
0.08 - ' _ .. ,.__._ -
-
..- ..- --- . ,.-
-
IE:] Simuinuo ns - "om. ',,'m,
0 07
-
-
, --
-
--,---,
I
~
0.06
----------"---1
'::':--
0.05
8
is e:
j
0 04 0.03 0.02
d
• F Ig. .)
008
000 0.05
•
•
0.01 0.00
Si rnulario n scenario
i'
I 007
•
•
-:._=-.....,. ".-
: ~ Siruulatiunv
Nurnm.l
,-
!
_." ---'-.._ -- '''''- - '- '- --,.-
f
0.9 0.8
~ E
\
r-
-e
V1
i
..
.s c
·10
o
M- -
Muttipath
0.4 0.3
- . - - . -+----
I
'//----j-
-
0.0
-
-t-
--+1--.. -
-
Ii
-·--I1---·-_.-·-t- //U l
-Tf'f --
'
-
-
20
i
-4
! /
___- ---1 ·2
i
+-!
f-
- -..,.....--1 I
----t--.
- --t-- - -
I-
i· //
0.1
f -channel: first ante nna inte rfe rence di strib ution .
/ /' I
f~~::' _ I~l-/ .-_ . / --- 1i_I
0.6 - ' -
0.2
10
11- Power-Controlled Signa l Lev el I) (
0.7
~ 0.5
§.
Fig. 6.
10
{I -c ha nnel' first ante nna Inte rfere nce di stnbution .
FI!! 7 1.0
·20
o
10
·? O
F i ll l l\ ~
'5 0 04 ~ 00.03
'--_La...,=
J-i--. : t -+--i Ll L o
2
Norm ah zed Si gnal Level (dB)
frequenc ies the rece ived signal after powe r contro l has the same distribution as that of the s imulated multi path fa st fad ing. For more de tails on the pow er control performance of the above receiver, the reader is referred [0 [28] . For the ideal power control case , the probability of e rror was computed using (60) and (62) . For the power control case with f d == .'i Hz. the approxim ation in (64) was used . The resulting j)b is plotted for L == 4 and I( =- 1, 1\' ..: :1 , and K '- fl in Fig. 10. Not e thai this is independent of the ang le spread ~ since with Id = .s Hz, the lading is slo w enough to be tracked by the pow er co ntrol loop for all va lues of ~ (see 12R]) . If we assume that the req uired uncod ed BEP is :s: 10- 3 . then fo r K == 1 the ma ximum number of allowa ble user s per sec tor N m a x is 29 for ideal pow er co ntro l and 21\ for po wer co ntro l with 1<1 = j Hz. For K = 3. these numbers go up to 85 and
10043 0.0062 - 0.0029i 002(;:1 - O.()O()..j i 0.0212 - {l.0003i 0.0204 - 0.01 1:1i
0.00(j2 ~ 0 0029i 1.0039 - OOEm + O.OOO{j i 0.0090 - 0 0009, 0.0102 -+ 0 0027,
Fig. 8. ing :
Power-controlled signal distribution versus simulated muluparh lad Hz. f, and L -1 .
J. / = '.
= ',.
=
82, respe ctively . This shows the improved performance due to beamforming . For the high f d case, as menti oned ear lier. the distribution of "I" and henc e the distribution of z ~,, ) for n = h, depends on the angle spread ~ . Fig. 11 shows the pdf of z ~ ") for different values of ll. and with ideal power contro l. From thi s figure we can see that the higher the angle spread is, the closer to the ideal power contro l case the pdf bec om es. Thi s can be explained as follo ws. At zero angle spread, the received signa l in any multipath component will experience the sam e fad ing at all ante nna s and the antenna array will not pro vide any space
002(;:1 + (1.0004i - 0.0198 - O.O()()O{ 10050 - II.OO:I!) - 0.00 11i 0.0 111 f- 0.01 l Si
495
0.0212 + O.0003·i 0.0090 + 0.0009i - 0.0039 + 0.00 1 Ji 1.0055 -O .015G - O.OOJ <Ji
0.0204 + 0.011:~ I 0.0102 - o.OO:n·i 0.01 11 - Of111 5i -0 .{H5(j + 0.00 19, J .0U27
1
1.0
0.9
H-
0 .8
.~
:i:
0.6
]
0.5 I - - -
.
~
~ 0 .4 ~
0.3
I
/ :/
0 .7
V1
I
Power -Controlled Signal Level -- Multipath Fast Fading .
-
/
.... ..
- ---
0 .2
j
--'
;g, gv 0.04
-
~ _.
l:' ;;; 0 .03
c;"
1--I
I I
.E ~
I
...
002 f---I-.;.....j.++-----'~f_----....L
.c
£
-
0.01 f--/~i++_---+~~-_+----+_--___j
I.: .
r/
l?"
0
4
Decisrcn Vari able
· . 9 . Power-con trolled si . ltd ro signa I d IStrib n u ti on versus sirnu a e mu It'ipa tb fad FIg ing: I« 100 Hz. J{ 5 . and L 4.
=
.. _... _ . ... _. .
l "
-
"
=
_. . . I
.-
_.
'-
I.'
it'
10 "
--
_.
--_··t -tl --l H 50
-
..
----
100
g
~
PI,
for ]'./
=
!j
= 11
at high
l«.,
= 0.0 84
~
'2 0."
10.
3
I--1-- -
p
)
=E'"
' ....tY
-
._ -
;:f
- --· f· - -- "..
.J.j
10. 5
rr1-;
--
40
20
~_.
_.
~-
80
60
200
100
120
i===
An e Screed t>
a· [
140
180
180
.. .
.-
---6-
-
-- ..
~
-o- K = l
I --0- KK =3 - 5
H-+-
Ideal Power Contro l
~
.. __.
...
1\
I---
- .-. _..
--
I I
I
/ /
1 0~
Power ControlJd = 5 Hz
150
7 .-
0..
1==
-
V /
- - 0 ~~
~
--_. _.
~
10-2
"3
1, C,
_..
a;
200
N . Number of Users
N, Number of Users
Fig to.
11.
W
-1- ----\
/
-#--r:~ K = 3, C,=0.077 ;? J--&-- K=5, C,=O.077
_.
~ --:-
~
K
..
f
..-J.'j
/
~l
-
--_.
PDF of : 1( " ; for
t==r=
-
-;::ft.
_.
._ - +-_.f--.
-' .
'>I
._-
Fine-. 11.
' 1( 1)
_rI:. ~
----_..
.-
IdealPower Control Fading, t. ~ D· 0 Fading, t. ~ 3 l-adin g, t. ; (fJ.
}-----A-~"'-'--...:+
Nor ma lized Signal Level (dB)
=
- -
}--- ---+-+- \ - -
"=>
. .•
·2
·4
0.05
I--
u,
. ..
,1
If
0.1
-
. ....
)
1--1
0 .0
-:
Fig . 12.
Hz and closed loop power control.
diversity for this multi path component. As the angle spread increases, the signal fading at different antennas becomes more and more uncorrelated which leads to less variations in the RAKE output. Figs. 12-14 show the uncoded BEP for 6. == 0° ,6. == 3°, and 6. - 600 for a high maximum Doppler frequency fd (high enough such that the statistics of the received signal after power control remain the same as those of the multi path fast fading). For 6. == 0° we can see that for Pb == 10- 3 the maximum number of allowable users reduces to (compared to the perfect power control case) 10, 29, and 55 for K == 1,3, and 5, respectively, which corresponds to a 65% reduction in system capacity. This capaci ty reduction is due to the multipath fading which was not eliminated by the closed loop power control. With a single antenna, the statistics of "Ys does not depend on the angle spread . Therefore , the maximum number of allowable mobiles for J{ -'= I is the same at ten mobiles per cell for any value of angle spread . However, for angle spread 6. = 3° this number goes up to 44 and 90 mobiles
Ph
for high /" and ~
~~
10' \
.--.
tJ
..n ~
..
-
1/ _.__ .
/
10"
I - ;---f-.
20
_.
I
'"
-_.
I
40
~
--. 60
80
..
-0- K =l ~--0- K =3 ---6- K -- 5
. ;.......-.
7'
_.
.--
"7
,/ ... -
..
I-----
r.--
.»:
..-
~
10"
. _---
power control.
J
H J. 0."
= 0" . and
100
120
Angle Spread t>-- 3" . -
140
160
180
200
N, Number of Users
Fig. 13.
Ph
for high / ,/ and 2l.
= 3° , and powe r contr ol.
for K == 3 and J( "..- 5, respectively. This is due to the additional space diversity gain provided by the array . For
496
_.
.-
-~
--
-
g
10-2
'"
.-
T-'---
==t --:-.- - =--= -_. r--P-- --
-
-- - f
,
.-
_." - -
/
J
-/I---j- -
-0-
-0-
--- -
-
-&-
--j- .- - -_..,- j ---~ t-::::::-:t:j
I
20
60
40
80
- r
.... --- ~-
• . _;---t.
_ .
~
- _.
C - -. _
'-
f--
(
_.
1 0~
./
-_. .
100
- --.-
-. - ..
,../'
~
---'
120
140
-
K=1 K=3 K=5
Antle SpreadIl. -
-
~ -
~. L
1aO
180
200
sen ted an approximatio n fo r the uncoded BEP as a function of the mea n symbol energy to interference-plus-noise ratio and the power co ntrol error. Fo r the case of high maximum Dopp ler shift, we derived exact expressions for the uncoded BEP for different cases of angle spread. In all cases, an improvement in sys tem performance pro portion al to the number of sensors is observed. Addi tiona l improvemen t is obtained due to space diversity gai n at hig h angle spreads . APPENDIX
To derive the uncon dition al pro bability density funct ions in (70), (72) , and (77), first we reca ll the chara cteristic function rep rese ntation of the condi tional density in (53) [10]
N. Number of Users
Fog 14.
P" for high
TABLE [ P ERCENT RED UCTION I" C APACITY AT
38 11 1
187
TABLE II
K 1 3 5
P" :::
50% 50% 50%
PFRCENT REDUCT ION IN CAPACITY AT
29
85 142
. , <,,) Z
exP{sz- ')"~2s/ (~2S+1)}
( 2
'l
fTs + l )~
Then the unconditional de nsity of z\n ) is given by [0- 2
r-OR HIGII
i\ ::: 10- '2
Reduction at 6 ::: 0° .::l. ::: 3°
K
_ _I_f
f - r ( I ')" ) - 27rJ.
j" and --'- ::: GOo. and pow er co ntrol
50% 34.6% 26.5%
Pb
.::l. ::: GO n 50% 19.6% 13.8%
::: IU -.\ FOR H IGH
65.5% 48.8% 36.9%
(80)
t"
tel
Reduction at PI, ::: 10- 2 .::l. ::: 0° .::l. ::: 3° 6 ::: GOo
65% 65.5% 65.5%
.
d«.
65.5% 31.8% 21.0%
where F-y, (.) is the characteris tic functio n of I s' Then I) For small angle spread, 1-1,(,) is given by (69) and
~ ::: GO° , these number goe~ even higher to 58 and I 10, which is. agai n, due to the space diversity gai n provided at large angle spreads. T he res ults of Figs. 12- 14 are summa rized in Tables I and II. Th ese two tables give the % reduction in capaci ty at 10- 2 and ?b ::: 10- 3 relative to the case with ideal power control, respe ctively. We make the following observation. For a fixed angle spread, increasing the number of antennas will provide more space diversity gain . Similarly , for a fixed number of ante nnas. the large r the angle spread, the larger the space diversity gain provided by the array will
Fb :::
2) For large angle spread.
l- ,h)
is given by (7 1) and
be.
VII. CONCLUS ION In this pape r, we proposed an antenna array-based base station receiver structure for DS/C DMA wireless systems with M -ary orthogonal mod ulation and studied its performance . We deve loped [he vector rnultipa th channel and received signal models. The average uncoded BEP was eva luated as a function of the number of users . number of antennas. and angle spread for different power control scenarios. For the case of low maximum Doppler shift , we pre-
Now we have "
(~2s + 1 ) (K-l )L
Fc(s ) = ( ~ 2( 1
+ ~I )S + 1)l\L
( [( - I ) L
L 1=0
497
(84)
where
1
1
Ri == (0- 2 (1 + 1))KL-lI! x
~((j2:; l ds
_I
l)(K-l)L! s=
1
- (a 2(1 + i))KL-l x
-1
(0"2(1+1'))
((!{ - 1)£) l
(~)(K-l)L-l(_l 1+1 1+1
)l
(85)
It follows that (91)
where
3) For other values of angle spread, first we need to derive
(92)
the density function for Is itself. We have L
rys
= ;y ~ lal,]
(87)
2
1 .
1=1
Let aZ,l == R~:J~2UI where Ri,l is the K x K covariance matrix of al,l and til is a K x 1 zero mean complex Gaussian random vector with covariance matrix I. Since R'l,l
is Hermitian, we can rewrite
RZ,l
as (88)
where U 1 is an orthonormal matrix and A z is a diagonal matrix of the eigenvalues of R l ,1. Let Al diag{AllA12' ··AlK}. Then we can write lal,ll~~ as
lal,l12 == u;UlA I Viul == iii AlUl
L K
==
2
Ali lil'li 1
(89)
i=l
where lit == Urlll is also a zero mean complex Gaussian random vector with covariance I. Then, we can rewri te
'5 as
(90)
where itt = 1'Al i and, therefore, Is has the density function in (75). It follows that
REFERENCES
[1] D. L. Schilling, "Wireless communications going mto the 21st century," IEEE Tram; Veh. Technol., vol. 43, no. 3, pp. 645-652, Aug. 1994. [2] A. M. Vitcrbi and A 1. Viterbi, "Erlang capacity of a power controlled CDMA system." IEEE 1. Select. Areas Commun .. vol. 11, no. 6, pp. 892-900, Aug. 1993. l3j D. L. Schilling, "Broadband spread spectrum multiple access for personal cellular commurucauons." in Proc VTC' 93, May 1993. pp. 819-822. [4] K. S. Gilhousen, L M. Jacobs, R. Padovam, A. Viterbi, L. A. Weaver, and C. Wheatly, "On the capacity of a cellular CDMA system." fEEE Trans Veh. Technol., vol. 40, no. 2, pp. 303-312, May 1991. [5J S. Simanapalli, "Adaptive array methods for mobile commumcanons." in Proc. VTC'94, Stockholm, Sweden, June 1994. [6] A. F. Naguib, A. Paulraj, and T Kailath, "Capacity improvement with base-station antenna arrays In cellular CDMA," IEEE Trans Veh. Technol., vol. VT-43, no. 3, pp. 691-698, Aug. 1994. [71 1. C. Liberti and T. S. Rappaport. "Analytical results for capacity improvement in CDMA," IEEE Trans Veh. Technol, vol. 43, no. 3, pp. 680-690, Aug. 1994. [~] S. C. Swales, M. A. Beach, D. J. Edwards, and J. P. McGeehn, "The performance enhancement of multibeam adaptive base station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Technol.. vol. 39, no. 1, pp. 56-67, Feb. 1990. [9] A. Naguib and A. Paulraj, "Performance of CDMA cellular networks with base-station antenna arrays," in Prot' International Zurich Seminar on Digital Communications, Zurich. Switzerland. Mar. 1994, pp. 87-100. [10] J. G. Proakis. Digua! Communications, 2nd ed. New York: McGraw Hill, 1989. [11] A. J. Viterbi, Principles oj" Spread Spectrum Multiple Access Communications. Reading, MA: Addison-Wesley, 1995 r121 G. Turin, "The effects of multipath and fading un the performance of direct-sequence CDMA systems," JEEEJ Select Areas Commun., vol SAC-2, pp. 597-603, July 1984. [13] M. Kavehrad and B. Ramamurthi, "Direct-sequence spread spectrum with DPSK modulation and diversity for indoor wireless communications." IEEE Trans. Commun., vol. CO~1-35, no. 2, pp. 224--236. Feb 1987. [14] M. M. 1. Wang and L. B. MIlstein, "Predetection diversity for CDMA indoor radio communications." in Virginia Tech Symposium on \Jv'lreless Personal Commun , Blackhurg, VA, June! 992 pp 13.1-13 10 [l 51 1\. F. Naguib and A. Paulraj, "Effect of multipath and base-station antenna arrays on uplink capacity of cellular CnlV1A," in Proc GLOBECO.M'94, San Fransisco, CA, 1994, pp. 395-3<19.
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\ \6) 1\.], Vnerr», "Very low rate convolutional code for maximum theoretical
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vol 40. no 1, pp. 461-464, Mar. 1992. \301 S. U. Pillai, Array Signal Processing. New York: Springer Verlag,
pseudonoise sequences," IEEE 1. Select. Areas Commun., vol. 10, no. 4, pp. 770-781, May 1992.
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[311 G. H. Golub and C. F. V. Loan, Matrix' Computations, 2nd ed.
i23] A. F. Naguib, "Adaptive antennas fur COMA wireless networks," Ph.D. dissertation. Stanford University, Stanford, CA, 1996.
more, MO: John Hopkins University Press, 1989.
499
Balti-
Abstract
Base station antenna arrays are a promising method for providing large capacity increases in cellular mobile radio systems. This article considers channel-modeling issues, receiver structures, and algorithms, and looks at the potential capacity gains that can be achieved.
Smart Antenna Arrays for COMA Systems JOHN S. THOMPSON, PETER M. GRANT, AND BERNARD MULGREW
ireless communications have become a significant area of growth within the last few years. There are a diverse range of products and services currently on the market, but cellular or personal communications services (ECS) radio networks probably have the highest public profile. These services provide highly mobile, widely accessible two-way voice and data communications links [1]. In general, the most complex and expensive part of the radio path for these systems is the base station. As a result, manufacturers have been designing networks that have high efficiency in terms of the bandwidth occupied and the number of users per base station [2]. This trend has been at the expense of highpower transmitters and receivers which employ very computationally expensive signal processing techniques. The second generation of cellular telephones, based on digital signaling and time and frequency division multiple access. have recently been introduced round the world. Typical examples include the European Global System for Mobile Telecommunications (GSM) and the North American IS-54 access protocols [3]. However, there is also considerable interest in code division multiple access (COMA) techniques for cellular systems [4]. The IS-95 standard for CD MA cellular systems was published in 1992; there is also interest in CDMA systems for the U.S. 1.9 GHz pes bands [5] and the European thirdgeneration universal mobile telephone system (UMTS) [6]. Whatever the relative merits of a given cellular system. it seems that considerable system capacity gains are availahle from exploiting the different spatial locations of cellular users [71. There are a number of methods to achieve this, from simple sectorization schemes [8] to complex adaptive antenna array techniques [9]. This article will consider antenna arrays for the mobile-to-base-station or reverse link of a COMA cellular system. It begins with an introduction to CDMA communications systems and also addresses the general topic of antenna array receivers. Channel modeling is then discussed, because this will influence the design of CDMA receivers. The specific form of receiver algorithms will then be discussed, and some performance comparisons are provided. Finally, the most important question for implementing antenna array systems is what capacity gains are achievable. Some simple analysis is presented to provide an initial answer.
of ways to achieve this, but this article will focus on directsequence spread spectrum techniques, which are used in IS95-based systems. The reverse links for all users within a CDMA system can be conducted over the same radio frequency (RF) bandwidth so that complete frequency reuse for that link is obtained throughout all cells [4]. To distinguish one user's transmission from another, each mobile modulates the voice data symbols by a pseudo-noise (PN) code. Each symbol is composed of W binary "chips which have a much shorter period than that of the original data symbols, so the signal bandwidth is considerably increased. The generic form of the reverse link for a binary phase shift keying (BPSK) spread spectrum system, using noncoherent detection at the receiver is shown in Fig. la. A PN code c(t)~ such as that shown in Fig. lb, is used to modulate the baseband data x(t) and the resulting signal transmitted to the base station. The receiver employs noncoherent quadrature demodulation to recover the signal amplitude and phase. This signal is correlated with the PN code to provide a delayed estimate of the transmitted data i(t - tei)' where td denotes the time delay. A typical PN code auto-correlation function observed at the correlator output is shown in Fig. l c: it takes on significant values only within one chip of the code arrival time. The reverse link of a CDMA system such as that specified by IS-95 has a number of essential characteristics for effective multiple access communication. A detailed introduction to spread spectrum and CDMA techniques can be found in [11-13], but here only points relevant to this discourse will be addressed.
Direct-Sequence CDMA
Modulation Srheme - The reverse link of an IS-95 system employs 64-ary orthogonal data modulation, transmitted using offset quadrature phase shift keying (QPSK) (11]. This article will be concerned with assessing general trends rather than providing specific results for different cellular systems. So, for simplicity, a system employing differential phase shift keying (OPSK) modulation will be considered [141.
C
D MA techniques are based on spread spectrum communications, which were originally developed for military applications. A simple definition of a spread spectrum signal is that its transmission bandwidth is much wider than the bandwidth of the original signal [10]. There are a number
Spread Spectrum Bandwidth - The chip rate of the spread spectrum signal is an important parameter, and is inversely proportional to the chip period t.: A number of different chip rates have been proposed for such systems, but a chip period of approximately 800 ns (chip rate 1.25 MHz) will be assumed for this article. Such a system is often called narrowband COMA, because the baseband bandwidth is much smaller than the RF carrier frequency, which is usually at least 900 MHz for cellular systems.
Reprinted from IEEE Personal Communications, Vol. 3, No.5, pp. 16-25, October 1996.
500
Multipath Diversity - In urban are as. mult ipath
pr opag ation is common . wh ereb y th e rec e iver o bse rves a lar ge number of co pi es of the t ransmitt ed signa l, e ac h with a d iffe ren t time del ay, The noise-like autocorr elat ion function of a PN code. shown in Fig. Ic , means th at the co rrelato r receiver can r e s ol ve mult ipa t h compo nent s whic h a re spac e d by I c h ip p e r iod u p to t he symbo l pe riod. Th is pr o vid es a fo rm of mu ltipa th diver sity, whic h ca n be exploi ted by using a RA KE receiver at the o utput of th e code co rrelat ors [15].
i
I
II I
I
Chip period
I
!
H
~
Asynchronous Operation - T he rever se link of
a C O MA syste m is usu ally async hro no us, in th e se nse that the arrival times for ea ch user's code a re different. This means th at th e rece iver for e ac h us er will observ e int erfer en ce fr om a ll o the r users in the syste m, since th e tr an smitted co des will not be orth og on al. Hen ce. the n umber of users that can be si m ulta ne o usly acco mmodated in one cell is inte rference-lim ited .
,
I
1 -1 1
Symbol period b)
-1 1
.
.' , ~PI;t~i'1L .'. " . . , - ',· ·0 \: ",' '.' .' Delay -+i ~ Chip period c)
• Figure 1. a) Th e gene ral form of a direct-sequ en ce spread spectru m system; b) a typ ical PN code; c) a typical auto-correlati on f unction for a PN code ,
Power Control - A co roll a ry to the above is that powe r co n-
tr ol is esse n tial o n th e re verse li nk. to min imi ze mu lti p le access inte rfe re nce . Othe rwise. mobile tr ansm itt e rs fa r awa y fro m th e ce ll's base sta tio n will be swa mped by interfe re nc e ge ne ra te d by users closer to the re ceiver. If a ll signals ar rive with the same power. the rece iver's tole rance to COMA interfe re nce is pr op ortio nal to the pro cessing gain [ 121. W = t .t t.: where t, is the symbol pe riod , T he obvio us way to inc rease the capacity of a C O MA syste m is to redu ce th e leve ls of mu lt ipl e a cce ss inte rfere nc e , This may be ac hieve d by d ire ctl y ca nce ling the inte rfe re nce [ 16] o r by e mployi ng a mult i-u ser rece iver which sirnu lta neo usly dem odul ates a ll users [ 17]. In th is a rt icle. an approac h based on antenna arr ay rece iver s will be co nside red.
A
-i
.
Why Use an Antenna Array?
n a n te n n a arrav con si st s o f .\1 ide n t ica l! a nt e n na receive rs. whos e o pe ra tio n a nd urrun g IS usu ally co ntrolle d by on e ce nt ra l pr ocessor. The ge o me try of th e antenna loc at ion s ca n va ry widely, but the most commu n con figurations are to pla ce th e ante nnas aro und a circle (circular ar ray ) o r a lo ng a stra ight line (line ar array) . Antenna a r ray s have fre que ntly been pr op osed fo r the o pe ratio n of radar and co mm unica tio ns systems in a milita ry con text [18] : it is pos sible to perform direction findin g ta sks a nd to null o ut e ne my inte rfe re rs . How e ver, in th e co ntext of c ivilia n ce llu la r sys te ms, the ai m of the a nte n na a rray recei ver is pu rel y to pro vide acceptable error pe rf orma nce an d hen ce maxim ize the signal-to- inte rference and noise ra tio (SIN R) for each use r in the syste m. An an ten na a rray co ntain ing ,\4 c le me nts can pn'vide a me an pow er gain o f 1\1 ove r whi te noi se. but suppression of inte rferenc e from o the r ce llular user s is de pende nt o n the form of the received data. A goo d model for the rece ived data in a power -con troll e d CO MA system is a stro ng de si red signal corrupted by a la rge nu mb er of sma ll cr oss-correl at ion inte rference te rms. which arrive with a uniform di str ibution fro m thr ouuhou i th e ce ll." T he arr ay can null o ut M - I inte rfe re rs. but fo-r a CO MA system thi s is unlikely to signif icant ly improve the received SIN R
[9]3 because of the very lar ge number of int erferen ce component s. In ge neral , a bett er meth od ol og y is to es timate th e fo r m of th e received signa l a nd det ermine the mat ch ed filte r sol ut io n [19] . T his fo r m of rec e iver ca n explo it an y sp a tial d iversity present, whi le suppress ing th e mean level of C OMA int e rfe re nce by a fac to r pr op ortion a l to M, Ass u ming tha t the an tenna array provides significa ntly improved SI NR levels a t the base statio n rece iver, the numbe r of channel e rro rs. me asure d by the bit erro r rat io ( BE R) , will reduce, This provides the ce llu la r o pe ra to r with so me deg re es of freedo m whic h may be used for the following pu rposes [9. 20. 211: • To inc rease the num ber of ac t ive use rs fo r a give n BER quality threshold • T o improve the BER perform an ce fo r a give n numb er o f use rs within a cell • T o reduce the SINR required at each ante nna to achieve a ta rge t BER. thus reducing the transmit powe r requ ired by the mobile hand set for the rever se link • T o increase the range of th e base sta tio n rece iver and thus the cell size • T o permit a less stringe nt for m of rev erse link power control while maintaining acce pta ble BER performan ce While antenna arrays provide man y advantages , these mu st be offset against the cost and co mplexity of the ir impleme nta tio n. The re are a numbe r of po ints whi ch mu st be taken into acco unt her e [22, 23J: • The hardwar e/softw ar e re q uire men ts incre ase as .\1 de mo dulators ar e require d for eac h user. • The M receiver s mu st be acc urately synchronized in time to prov ide effec tive per for mance. • Th e co mp uta tio na l co m p lex ity of array p roc essi ng a lgo rit hms ca n be very larg e . • T he array size will be co nstrained by the availab le spa ce for a ba se st a tio n . Usua lly, th e spac ing of a nte nna e le me nts var ies fro m one -half to ten s of RF ca rrie r wavelen gth s. Th is assump tion permits general results 10 be obtained [or system capacity. H owever, the validity of this assum pt ion will depend 0 11 the layout of the cell fo r practical situations,
!
,;An exception is if a mobile undergoes a power control error lind transII//{S
This is not strictly necessary: orthogonally po larized clemen ts. fo r (~tam pie. could be used instead . I
501
multipath components which cannot be separated in time [26, Ch . 2] . The resolvable channel taps are assumed to be uncorrelated as each tap arises due to contributions from different multipath scatterers. The exact distributions of the signal envelopes are a func tion of the signal bandwidth but , for a 0.Q1 2 4 6 8 10 0 "2 . 4 . 6 8 narrowband CDMA system , two distribuExcess time delay (microsec) Excesstime delay (microsec) tions are often proposed. If a dominant line b) a) of sight (LOS) path exists between the transmitter and rece iver, the first received signal • Figure 2. a) A typical channel impulse response for an urban area; b) the discrete component will follow a Rician distribution. tap channel model. However, in urban areas, thi s is often not true : each channel tap consists of a number of independent multi path scatterers with the same probability distribution . Applying the central limit theo• Practical antenna arrays may be adversely affected by chanrem. the received signal envelope statistics can be assumed to nel model ing errors, calibration errors , phase drift , and noise which is correlated between antennas. follow the Rayleigh distribution. As the mobile moves, the sigWith these points in mind , this article will move on to connal strength regularly changes by 20-30 dB: the phenomenon sider channel modeling aspects of antenna arrays . This will of a sudden loss of signal power in this context is often called motivate a discussion on the likely form of a CDMA antenna fading . Over a longer period of time, the average (averaged over array base station receiver. the Rician or Rayleigh fading) received signal power levels vary according to shadowing effects, which have been found to follow a log-normal distribution . The standard deviations quoted for this distribution vary between -l-12 dB according n order to correctly specify the structure of an antenna , array base station receiver, it is important to understand to the type of environment encountered [27]. In addition. the average received power varies inversely with the transmitterthe characteristics of the radio channels that are likely to occur. There are many different types of channel model receiver distance R, raised to the power n. Again, the value of appropriate to different radio systems and scenarios, but here n varies widely, but for urban areas its value is often approximated as 4 [26. Ch . 2J. In this article, it will be assumed that a channel model will be developed for a large CDMA macropower control can adequately co mpensa te for these effects cell operating in an urban environment. For simplicity, the but that it is unable to compensate for Rayleigh "fast" fadin g: case of a single antenna receiver will be initially considered: the results will be generalized for an antenna array receiver in this is somewhat pessimistic compared to the quoted results the next section . for IS-95 power control systems [11]. One of the most important methods for characterizing a The time variation of each channel tap depends on the radio channel is to det ermine its impulse response . This promotion of the mobile . As the mobile moves through spatial locations with different field strengths, each multipath comp ovides an indication of the severity of multipath propagation, nent of the received signal is subject to a Doppler shift in frewhich occurs due to multiple copies of the signal arriving at the receiver with different amplitudes and time delays. In quency. For a CDMA signal not subject to data modulation, calculating the power spectrum of each channel tap shows the dense urban areas there are many buildings and obstacles that distribution of Doppler frequencies for the constituent multigive rise to multipath propagation, and the range of times of paths. The maximum Doppler frequency Vm is proportional to a rrival can be significant. A typical impulse response for an urban area , drawn from the Europe an COST-207 channel the vehicle speed v, according to the equation models [24], is shown in part Fig. 2a. (1 ) The correlation function of a typical CDMA code takes where A. is the carrier wavel ength . There are two d ifferent significant values within :± 1 chip of the time of arrival of the forms of rnultipath scattering, acco rd ing to the exce ss time code. This mean s that a CDMA correlator receiver is able to resolve multi path components of the signal which are spaced delay of the given channel tap [13]: in time by 1 chip up to the symbol period . As a result, the channel impul se response measured by a CDMA receiver is Small Excess Time Delays - The channel tap may be modoften modeled by discrete channel taps [25], spaced in time by eled as the accumulation of multipath components received 1 PN code chip . A typical discrete tap model is shown in Fig. from scatterers close to the mobile . This gives rise to the clas2b. Each channel tap may be charact erized by the following parameters: the stati stical distribution of S(v) the received signal envelopes and phases, the temporal variation of each tap and the spatial variation of each tap . The symbol period for an 15-95 type system is quite high (approx . 100 us) compared to the impulse response duration (typically a few us) , so it is commonly assumed that the number of significant channel taps K « W. It is often assumed that each channel tap is wide -sense-stationary for the mobile moving over short distances, up to a few tens of carrier wavelengths. This means that the signal var iation is due • Figure 3. The classical Doppler model: a) Geometry of multipath scattering; purely to phase changes in a set of independent b) the associated Doppler power sp ectrum for the channel tap. " '"
Channel Modeling Considerations
502
Reflector
/
sica l Doppler power spectrum of the rece ived multi path co m po ne nts. which is illustrated in Fig. 3. The equation for the Doppler power spec tru m S(v) is given by [26. Ch. 2]
a
S(v)=
l - (. ~)'
Ivl~
v",
vm
---.'"
\//.
------.:.'--.. ..
.
rays
··- ,~obile -......
(2)
,
'<, ' ::. .
.
• Figure 4. a ) The geometry fo r scattering with large excess time delays; b) the Gaussian -distributed Doppler power spec trum .
tr ibuted over [0. 21t ] radi an s. a nd the Doppler frequen cie s v, a re di str ibuted to g ive th e a p p ro p r ia te Doppler s pe c t r u m sho wn in Fig. 3 or 4. Equati on (3 ) makes explicit th e fa ct th at eac h channel tap is a continuou s function of time. a ltho ug h th e PN co de au to- co rre lat io n functi on means th at it may onl y be o bse rve d o nc e p er tr an smitt ed sy m b o l. Equ ati on (3) ass u mes th at K « W so th at t he re is no significa nt int er-sym bo l interference . T his mod el ca n easi ly be exte nded to the case of a n an te nna a rray. using the narrowb an d C O MA ass u mp tio n fro m the second sec tio n. Th e a rrav will be co nside red to lie in the horizo n tal p lan e . wi th m aximum le ng t h a nd bread t h of seve ral ca rr ie r wav e le ngt hs. In t his case . each rnultipath arrives a t all the a rray e le me nts a t the sa me tim e . but wi th an a ppro p ria te p hase sh ift a t each a nte nna . The fo rm of the ph ase s hifts is specified by the M x I a rray stee ri ng vecto r a(e l. whic h may be tho ug ht of as the ar ray res po nse to a uni t im pulse a rrivi ng fro m beari ng 8 . T he c ha nne l model now co nsists of a n ,VI x 1 vector for each cha nne l tap : the kt h channel tap vecto r for the nth symbo l may be writte n as [32]:
The insta nta neous va ria tio n o f signal power in space for a c ha nne l ta p depends o n the a ng les of ar riva l of th e mu lt ipath co m po ne n ts. The dist ributio n of mult ip ath e ne rgy with a ng le has been si m ula te d using seve ra l different method s in the lite ra t u re . Le e [29] u se d a mode l base d o n a cosi ne fu nc tion ra ise d to a h igh p owe r to rep rese n t the ang ula r wi d th (or ang le sp rea d): a G aussian dis t rib u tio n of rnult ip ath e ne rg y with a ng le has also b ee n use d [30] . However. o ne of the simp les t models is du e to Sa lz a nd Win te rs [31]. who use a uni fo rm d istributi on of e nergy with angle - th is mo del will be use d fo r the rem a inder of th is article . T he model is sho wn in Fig . 5a . which s ho ws th e geo me try of the mode l for ne ar-in sca tte ri ng . Th e a ng u la r wid t h 2.J de pe n ds o n t he sca tteri ng circle rad ius r an d the d ista nce to the bas e statio n R. T he center bearing 80 is simply th at o f th e mobile : fo r scatte ri ng wit h la rge exce ss time del ays. a sim ilar geo me try applies replacing t he circle o f sc atterers with th e refle ct or givi n g ri se to th at co mpo ne nt.
x ( k .nt
=
(3)
1=1
wh er e Qk denotes th e nu m be r of multipaths that make up the kth t ap , d (n) denot e s th e nth tr an sm itted symbol , and {Aki. Vi. QJ;}denote t he a m plitude . Doppler fr equ ency a nd phase of th e ith m ult ip ath . In thi s art icle it will be as sumed th at for a fix ed value o f k the ampl itudes A ki a rc all equ al. th e ph ases
lx, i k ,n t I + i]: - 1)1, ), X ~ (/.. .IU, + ( k Q,
=I
d (II JA)'i exp(j2 n:I'[ IlIJ
,=1
= J (1l )Akq (k .lU, + i]:
n o rde r to mod el a mult ip a th cha nnel. it will be ass u med tha t each cha n ne l tap co ns ists of a num be r of indepen dent co m po ne nts. Ass u me th at th e firs t chip of th e firs t cha nnel tap for th e firs t tr an smitt ed sym bo l begi ns to arrive a t th e re ce iver a t time t = O. Then t he kt h c ha nnel ta p fo r the n t h symbo l will be detect ed at time lit , + (k - I )te . whe re I, a nd t, are th e symbo l a nd chi p periods. respectively. This ta p. whe n measured at th e o utp ut o f th e desi red PN code cor relator for a sing le re ce ive r. will be de noted as x l( k. lll s + (k - 1)le ) a nd may be expanded as:
,
+ ( k - \) 1I" )
I
Xw
Modeling the Received Signal
x ,(k .IIt, +( k- ! )I e )
b)
a)
does not provide a pl au sibl e geo me tric mod el fo r this type of sca tt eri ng . Inste ad. mult ip ath e ne rgy is more like ly to have a narrow Doppl er sp rea d. having arise n from re flection s off isolated ob st acl es suc h as buildings or hills. One such retlecti on ma y be m odel ed as havin g a G aussian di stri b ut e d power spe ctr u m [24 ]. as s ho wn in Fi g. 4 . The be arin g of th e rnult ip ath component m ay b e d et ermin ed by drawin g a n e llipse - so me times ca lled the "e llipse o f Kassini " - wh ose m ajor axis is t h e mu lt ip a t h len gth a nd whose foci a rc th e tra ns m itte r a n d receiver [28 ]. as shown in Fig. 4 . A typ ica l cha nne l tap with a lar ge time de lay ma y co mp rise seve ral suc h refle ctio ns.
2>").] ex p{j(2 n: v , [nr, + ( k - 1)l e ] +
......
S(v)
Reflector locus ellipse
Large Excess Time Delays - T he classical Doppler mod el
= d( lI)
..
...........................
whe re v is the D oppler freque ncy an d a is a sca ling factor.
Q,
---
-
-'-
\)r, l.. ...
ik .n s, + Ik -I )1, ljT
(k - 1)1(' I + $; )a(8, )
(4)
\)1, )e xp(<j>k l
wh ere 8 i denotes th e beari ng of th e ith multipath . xm(k . nt; + (k - \ )t e ) is the c ha n ne l t ap o u tp u t a t the mth a n te n na. Ak de notes th e signa l a mp litude. a nd xT de no tes th e vec to r tr an spose o pe ra tio n. The firs t e nt ry of q is spe cified to be a po sitive re al num be r. so th e term exp( 't',) represents the ca rr ier phase of the kth multipath a t th e first ante nna at tim e t. T he s ta tis tics of a give n c ha n ne l ta p vec to r x(k . I) a re of inte res t. T he p hase of eac h e ntry of x(k. I) is un iformly distribu te d ove r [0, Zn], so th e mean vec to r o f x(k . r) is the zero Extent of mu ltipath / scattering
Power
i
I i i
I
i ii
" ~i;~I~'ofscatterers ..
.:'..>. . ':....;.. a) • .. ·i...'.
Bearing . b) '.
,-:, '.
• Figure 5. a) The phy sical geometry f or the Salzlwinters m odel; b) the uniform distribution of multipatk ene rgy with angle.
503
vector. The second order moments of x(k, t) are specified by its M x M mean covariance matrix \Ilk, which is defined as: \}It
= E[x(k,t)x
== Ai
r
H
(k,t)]
9 0 +
2L\ J90-~
multichannel must also be subject to data demodulation to determine d(n).
A1aximal Ratio Combining - If the interference observed on (<1»/ a H (
each separate multichannel is assumed to be uncorrelated, maximal ratio combining is the method which maximizes the SINR of the combined signals. If the interference has the same standard deviation signal for all multichannels, this method scales each signal by the complex scalar Al exp l-jq»}. Practical methods for estimating the carrier phases {
(5)
where E[] denotes the statistical expectation operator. The integral term arises from the fact that the distribution of multipath energy is uniform over the bearings [90 - 6, 80 + L\]. The leading diagonal entries of this matrix specify the mean power levels of the channel tap at each antenna element; the other entries specify the cross-correlation between the channel tap signals at two different antennas. In practice, measurements of x(k, t) are corrupted by three sources of interference: • Background noise • Auto-correlation interference from all the other taps of the given user's channel • Cross-correlation interference from other users operating over the same bandwidth in the CDMA system In a cellular system, the effect of the third is usually much worse than that of the first or second, and it is the limiting factor on CDMA capacity. Neglecting the first and second, the measured data vector y(nt s + (k - 1 )t c ) ' of size M x l , from the antenna array at time Ills + (k - 1)tc may be written simply as the sum x(k, Ills + (k - l)lc) + Tl(nts + (k - l)t c ) . The 1\1 x 1 vector l1(t) represents the total multiple-access interference measured across the array. Initially, it will be assumed that T\(t) consists of spatially and temporally white Gaussian noise of zero mean and variance cr2.
Noncoherent Combining - If the receiver employs DPSK detection, the carrier phase reference is simply the data sample obtained for the previous symbol. The magnitude of the previous data sample also provides an estimate of the amplitude AI, so the multichannels may be combined as follows: " L d(n)= I9t{:(i,n)z*(i,n-l)}
(6)
i=l
whered(n) is the estimate of the current transmitted symbol,
z" indicates the complex conjugate operation and 9\ denotes
the real part of a complex number. As the combiner weights z*(i. n - 1) are noisy, the SINR of the combined signals tends to be poorer than for maximal ratio combining. For currently used dual-diversity antennas (L = 2), the loss in SINR at the system operating point is typically less than 1 dB: however, as L increases the losses become greater [34, p. 302].
Wiener Filtering - The three techniques described so far are
Signal-Combining Methods
I
t is clear from the previous section that the antenna array has to process a number of copies of the desired signal, each of which is corrupted by undesirable interference. If there are K significant channel taps observed at the A1 elements of the array, there are K x .1.\1 separate data samples to be considered when making a decision on each transmitted symbol. This situation requires what is called a "multichannel" receiver, equivalent to receiving the information over K x il1 separate narrowband flat-fading channels. The best approach to this problem is to weight each channel appropriately and combine them together, before making a data decision. In this section, methods for combining an arbitrary number of channels will be considered; in the next section, specific receiver structures will be described. The main difficulty in designing such a receiver is to decide how to scale each data sample before combining. Consider L multichannels, denoted as z(l, n), which are to be combined. The nth sample for the lth multichannel is of the form d(n) Al exp{j
based on maximizing the signal power at the combiner output. It is also possible to use a Wiener filter. which attempts to suppress interference and maximize the SINR at the combiner output. The performance of this technique is likely to be better than that for maximal ratio combining when the interfer.ence is correlated between multichannels. This technique is discussed in more detail later.
Selection Diversity - If the receiver has to process a number
of multichannels simultaneously, one method is simply to choose the multichannel which is presumed to have the largest signal power. This approach is quite simple, while permitting some performance improvement over single antenna receivers. However, this method does not provide the optimum improvement in SINR that can be obtained. The chosen
504
Multichannel combiners for other modulation schemes are described in (34. Ch. 7].
Antenna Array Receiver Structures
T
he purpose of the antenna array receiver is to estimate the transmitted data sequence d(n), based on the interference-corrupted measurements vv«, + (k - 1)tc ) of the K channel taps. There are two approaches that one might consider for combining the data samples, as described below.
1D RAKE Filter - In a single antenna CDMA receiver, noncoherent combining is often used to combine the K channel taps, a method which is normally called ~'RAKE filtering" [15]. A simple approach to designing an antenna array receiver is to employ a RAKE filter to combine all the K x M entries of the vectors {y(nt s + (k - l)tc ) } . For DPSK modulation, this is performed using Eq. (6) with L = K x M. However, as pointed out above, noncoherent combining of a lot of multichannels can give rise to large losses in SINR compared to maximal ratio combining. 2D RAKE Filter - A more effective and compact approach to dealing with the channel tap vectors {y(nt s + (k - 1 )t(')} is to
apply a separate spatial filter to each tap vector in turn. The receiver can exploit any structure that might be present, such as the directions of arrival of the multipath components, their Doppler frequencies, and so on. This permits the receiver to perform coherent combining of the tap vector elements,
!::".; -: ~ ... . ... . .
improving performance over the l D RAKE filter. Denoting the kth filter as hh K complex outputs a re generated for the nth sym bo l by the ve cto r inn er products { h l!y ( n t, + (k - l )t c )} ' wh ere denotes th e complex conjugate transpose operatio n. This means th at only K outputs ne ed to be co mbi ne d using the DPSK demodulation method o f Eq. (6). The appro ach ha s been called th e " 2D RAKE filter" because the receiv er o pe ra tes two sepa rate se ts of combiners in ti me and spa ce. It is sh own in F ig. 6: the receiver pick s out th e lar gest chan ne l taps and selects appropriate spatial filters in e ach ca se. The outputs fr om t he spati al filter banks are then combined in a conventional RAKE filter, ready for decision making. Base stations ar e fre quently split into three sectors, to provide 120 cove rage, which in this ca se corresponds to th e bearings [30°, 150°] as shown.
hI!
0
Filter
ban ks
• Figure 6. The 2-D RAKE filter combiner operating in a 1200 coveragesector.
There are a number o f method s to determine th e form o f the spatial filter hk. Many o f the se techniqu es employ th e M x M est imated covariance matrix Rk o f th e sig na l y (nt , + (k I )tc )' which is de fined as: I v H R k = N"LY lnt,+ (k -l )tc )y (nt, +(k- I)t/) .
(7)
1l= 1
The not ation N indi ca te s the number o f co nsecu tive symbols used for averaging. For the chose n filte r to opera te e ffec tively, the form of the array response vecto r q(k, nt , + (k - I )tc) sho uld not change significa ntly over this time. In th is a rt icle . it will be ass u med th a t pow e r co nt ro l ensur es that the SINR of eac h channel ta p is large eno ug h for eac h vector q(k, nt, + (k - I )tc ) to be correc tly identified witho u t th e transm itter e m p loyi ng a n in itial trai n ing se q ue nce . Th is ta sk ma y be performed by blind cha nne l ide n tifica t io n techniques. such as: Beamspace Transformation - This technique applies J fixe d spa tial filters, denoted as length M vecto rs Wj , to the dat a [23]. By measurjqg the average power at the outp uts, given by the pr oduct wfRkw), the receiver ma y act accordingly. A sim p le approach is to apply selecti on diversity by choosing the filt er with the largest power output to pick out the desired signal. Bearing Estimation Techniques - As the vector q(k, nt s + (k - I )t() is co mpos ed of a number of ste e ring vec to rs, it is possible to apply bearing estimation techniques to th e cova riance matri x Rk to pick out the major direct ional compon ents. as proposed in [36]. There a re a number o f well-known h igh-re solut ion techniques such as ESPRIT and M USI C; however , the se a lgor ithms perform poorly in th e pr esence of highl y co rr e la te d mult ip ath signal s, whi ch fre q ue nt ly occ u r in th e urban communic a ti on c ha n ne ls bei ng co nsi d e re d he re . A si m p le r approa ch is to use th e conv ention al beam forme r (C BF) den sity spect ru m, which is de fined as:
The beamspace and be aring e sti ma tio n techniqu es po int o ne o r more narrow be am s a t the in coming sign als from the mobile. Thi s c h o ice is o p t ima l o n ly if each ch annel t ap appears as a point so urce. If multipath scatte ring o f th e signa l gives rise to a significa nt a ngula r width of the signal , th e per fo r ma nc e will degr ad e . H owever , the eigenfilt er meth od a lways gives weights that maximiz e the co mbine r signa l power . so it sho u ld perform bett er in t he p resence o f sig ni fica n t angula r width. If the signa l power of a given tap is no t sign ifica nt ly lar ger th an the inte rfe rence power, t he above techniqu es are like ly to inco rrectly pick o ut an inte rfe re nce instead of th e de sir ed tap vec tor. Howe ver. give n th at C D MA inte rfe re nce ge ne ra lly co m p r is es the co n t r ib u tio ns o f a large number of users. it se e ms unl ikel y th at any technique co uld co rr ectly pick o ut the c ha n ne l t ap wi t h suffi c ie n t SIN R fo r t he purp o se s o f da ta de cis io n making. Having noted thi s po int , the proce ed ing sectio n will now lo ok in more d et a il a t th e o p e r a t io n o f th e se blind techniques.
Algorithm Performance n o r d e r to provide a n in iti al comp arison of th e se a lgo, rithms , a very simple cla ssic al Doppler one -tap channel
model will be con sid ered .' The rec eiver cont ain s an eighte le me n t uniform line ar array (ULA ), who se e le me n ts -a re space d by o ne half o f th e ca rr ie r wa ve le ng t h .> Th e cha n ne l tap ang ula r width 20 (de fined in Fig. 5) was varied fro m O- fiO° a nd a rrive d fr om th e a rray broadsi de . be ar in g 90° (i .e ., perpendicular to th e a rray, as sho wn in Fig. 6) . Results from [311 sugges t th at fo r thi s be aring th e signa l co rre la tio n betwe en e le me n ts fa lls most rapidly wi th th e a ng u la r width 20. The sym bo l rat e o f the C D M A signa l wa s 10 ksymbol s/s, and th e maximum Dopple r fre que ncy was se t to II HZ,6 50 H z. or 200 H z. Th e maximum ac hievable S IN R level , d e fin e d as A f/cr2, ~ as set to 14 dB , and N = 50 sna ps ho ts we re used to es ti ma te R I . T he beam space me thod e mployed eig h t o rthogona l spa tia l
(8) Agai n, the steeri ng ve cto r for the largest value of P(S) may be se lected - thi s approach is quite similar to jitter diversity [37]. Eigenfilter Techniques - In o r d e r t o identify the ve cto r q (k , nt, + (k - 1)tc ) ' it is po ssible to calculate the eigenvalu e dec omposition of Rk • Provided the SINR is large, th e M xl e ige nvecto r "1 corresponding to the largest eigenvalue of Rk provides an estimate of q(k, nt ; + (k - l )t c) ' This is th e statistical method of princip al co mpo ne nt analysis and was used in
[38].
505
4 Although these results are for a single rap. they provide some insight into the performance of the algorithms when they are applied 10 the individual channel taps of a multipath cha nnel.
This ant enn a spacing was chose n to amid problems with the CBF bearing estim ation approa ch. Spat ial aliasing effects can occur if the ant enn a spacing is chosen to be much larger than half the carrier wavelength.
5
6 l n the case of 0 Hz Doppler, for each realization of the fad ing channel, the channel tap vector do es not change over the time of observation.
filters : the bearings were chosen as those which provided the poorest SINR for angular width 2.1 = 0°. As the antenna array provides a gain of 9 dB, the SINR at one antenna is 5 dB . Results from [34, p. 302] suggest that applying the 1D RAKE filter to a simple nonfading 8-multichannel scenario with each multichannel having S[NR of 5 dB results in a loss of 2.5-3 dB (i.e., an SINR of 11-11.5 dB at the 10 RAKE fil-
ter output) . This provides a baseline to compare the other algorithms. The average SINR at the output of the beamformers selected by the three algorithms has been. determined from Monte Carlo simulations using 10,000 trials in all cases . The desired signal was generated according to Eq . (4) with Qk = 100. The results for no Doppler effects (0 Hz) are shown in Fig. 7a , with a horizontal line showing the maximum atta inable S[NR in all cases. The results demonstrate th at the eigenfilter method consistently performs well and achieves 16 output SINR values close to the optimum. By contrast, the Eigen <> beamspace and CBF algorithms degrade as the angular width (BF + 2.1 increases, because the anguJ.ar width of the signal energy 15 Beamspace 0 becomes wider than the main lobe of the spatial filters {a(8)} - for large values of 2.1, the performance is not much better eo 14 ....'f ....¢ .... 'O..... Q ..... <>.....o ....¢ . ....<>.....O-.... O .....<>... :s + than for the 1D RAKE technique . In the case of the + 0::: beamspace technique , selecting and combining the two or z + + Vi 13 three filters with the largest power levels may improve perfor+ ~ + mance . 0 <, 0 + 0 + 0 ::J In practice, the mobile will often be in motion, giving rise + 12 0 + 0 0 to a positive Doppler frequenc y. The performance of the 0 0 three algorithms has also been mea sured for Doppler fre 0 0 0 11 quencies of 50 and 200 Hz in Figs . 7b and 7c. Assuming the carrier frequency 'IS 900 MHz, this corresponds to vehicle 10 speeds of roughly 30 and 120 mph (the difference between a 10 20 30 40 50 60 0 a) car moving in an urban area and so me o ne communicating Angular spread of scatterers (deg) from a train) . Otherwise , the simulation conditions are unchanged fr...nn before. 16 As the maximum Doppler frequency increases, the perforEigen <> CBF + mance of all three techniques degrade for large multi path Beamspace 0 15 scattering angles. This occurs because the form of the signal vector q/\k, nt, + (k - 1)tc ) can vary significantly as the phases eo 14 ....4"....~ · ....O..·..O....O·....o..·..~ ....·~ · · ..· ~· · ..O··.. ·~· ..· of the c.onstituent multipaths change over the sampling epoch . :s + . It is interesting to note that the performance of the eigenfilter 0::: z + technique degrades significantly for 200 Hz Doppler. much Vi 13 ''''':- . + .. more so than the CBF or beamspace techniques. However. + ::J + 0 Co . .! 0 + the eigenfilter approach still provides the best absolute S[NR. 0 0 + + ::J 12 Reducing the number of snapshots N to form R, may improve o + 0 0 the SINR performance since the signal vector is more likely 0 0 0 0 0 11 to be stationary over the sampling epoch ; however, this increases the vulnerability of the algorithms to the effect of 10 't:..-_-'-_ _..L-_--'-_ _. . l - _ - - 1 _ - - - I background noise. 10 20 30 40 50 60 o The SINR comparison shown here between techniques Angular spread of scatterers (deg) b) based on beam-steering (i.e., the beamspace and CBF methi ods) and adaptive arrays (i .e ., the eigenfilter approach ) i 16 r---.---,---~--.,---......-,.---, broadly correspond to previous published results [20]. HowevI: Eigen o er, no account has yet been taken here of the statistical distri CBF + Ii bution of the signal at the beamformer output, which will .15· Beamspace 0 " significantly affect the error performance of the receiver. CalI culating the eigenvalues of 'f'k defined in Eq . (5) indicates the '. l eo 14 amplitudes of the independent Ra yleigh processes present in a given channel tap x(k, t) which can be tracked to obtain spa, ~ tial diversity gain . As the probability of several independent I VI <> 13 o o + Rayleigh processes simultane ously entering a large fade is + <> o o <> o much less than that for one Rayleigh process, the receiver is 0 + 0 " 12 0 + 0 much more likely to correctly estimate the transmitted data + + 0 0 sequence. The cumulative distribution function (CDF) indi+ + ,11 0 0 cates the likelihood of the received signal fading below a o o given threshold: th is has been measured for the three algo rithms in another Monte Carlo simulation. The simulat ion conditions are the same as for Fig . 7, except that the scattering width has been fixed at 20° and the maximum Doppler frequency is a Hz . The CDFs for the algorithms have been • Figure 7. The average SINR performance vs. angular width 2.1 calculated from 100,000 trials, and the results are shown in of the beamspace, CBF and eigenfilter algorithms for maximum Fig . 8. The theoretical CDF for x(k, t) has been calculated by Doppler frequencies ofa) 0 Hz; b) 50 Hz (30 mph); C) 200 Hz substituting the eigenvalues of 't'k into the probability density (120 mph). equation [34, Eq. (7.5.26)].
.e-
... ...
I ~
II
•
~:'!i!~J~~J~~~~,~\~~f~~ti'
506
The CDF fo r the e ige nfilte r is close to the optimum achievable, exce pt at low SINRs, where the performance of th e alg orithm de grades sligh tly du e to th e effect of no ise . Inte rest ingl y, both the CBF and beamsp ace a p proaches a re ab le to e xplo it the s pati a l d ivers it y of t he wi d e a ngle m ult ip ath scatt eri n g, a nd th e curves are o n ly slightly worse th an for the eigenfilter. As a result, th e error performan ce of three algo rit hms may be q ui te s imi la r, any d ifference bein g du e to backgro u nd no ise effec ts ra the r tha n the CDF of th e filter o utputs. As a contrast, th e CDFs a re a lso sho wn for 0° sca tte ri ng a nd for a single fixed sp a tia l filt er pl aced at th e so u rce be a rin g . In th e former cas e, the c ha n ne l ta p vec to r e n tr ies a re co m ple te ly correlated. There is no spa tia l d iversity ob tai nab le for thi s case, and the filter o u tp ut follow s a Raylei gh dis tr ibution. In the latter situation . th e filte r cannot be alte re d wh en a large fa de o ccurs at the so urc e b e arin g; inde ed, it is a we llkn own re sult of multi vari at e s ta tis tics th at th e filt er o u tput will aga in follow a Rayleigh distribution [39] . For both curves. large fades are much more likely th an fo r the eig enfilter. C BF or beamspace algorithms: th e rec ei ver error performance will be poor unl ess the SINR is very high o r o the r mult ipath co m ponents are ava ilab le to pr ovid e d iver sity. In ge ne ra l, the co rrela tio n be twee n a n te n nas re du ce s as the a ngula r width 2a inc reases o r, in th e cas e of the e ige nf ilter method. the a nte n na spaci ng is increased. As th e mobile to -b a se -st ation d ist an c e R in cr e as e s, th e val ue of 2a wi ll generally reduce fo r a fixe d val ue o f the sca tte ri ng rad iu s r ( Fig. 5) . The availa ble spatia l di ver sity will reduce acco rd ingly. so t he ra nge e x te nsio n offe red by a n te n na a rra ys ma y be affec ted by this facto r [20] . In cre as ing the o r der of di ve rs it y pro d u ce s d im in is hing re turns in terms o f the e rro r pe rforman ce of the receiver. If the decision va ria b le C D F is me asure d for a re ce ive r co mb ining sev e ral ch a n ne l taps. the im proveme nt in th e C D F d ue to inc reasin g the a ngula r width 2a o f ea ch ta p will no t be as d ram a tic as tha t s ho wn in F ig. 8 . H ow e ver. so me pe rform an ce ga in will sti ll be o b ta ined . Th e m ain d rawbac k of increas ing the va lue of 26 is that th e co rrel a tio n between the rever se a nd fo rwa rd link c ha n n e l ve c to rs re d uces [91. If it is in te nded to use a n a n te n n a a rr a y to tr an smit to the mobile s o n th e for ward link . using the reverse link we ight s for retransmission o n t h e fo rwa r d link may n ot b e ver y effe ctive . Altern ati ve a p pro ac hes su ch as transmissi on di versity [20] . tim e di vision dupl ex reception/transmission. o r tr an smi ssion feedb ack tr ain ing [40] ma y be more ap pro p ria te. A nothe r so u rce of de gr ad ation in pr act ice is th e effect of mul tipl e acc ess inte rfe re nce. U n ti l now , th e algo ri thms whic h ha ve be en used attempt to m aximi ze th e s igna l pow er a t th e o u tputs of th e co mbi ne rs fo r th e in coming cha nnel taps. T his a p p roach maximizes the SINR wh en th e in te r fe re nce is spa ti all y a nd temporally wh it e : how e ver. if th e C DMA inte rfe re nce d o e s not fo llow th is dis trib u tio n . t he per for m a nce of th ese algo ri thms will degra d e furthe r. In o rd e r to ove rcome t h is probl em , th e Wi en er fil te ri ng o r opt imu m co m b in ing techniqu es may be use d to su p p ress th e in te rfe re nce and max imise th e co m bi ne r's SI NR [9]. The Wi en er filte r sol ut ion fo r hk may be expres se d as: hk = wh ere
R;'rk rk is th e
(9)
cro ss -cor r el at ion o f the de si r ed d at a with the N sym bo ls. The ma in difficul ty he re is to determine the desired sig na l. It is possible to feedback previous deci sions: however. th ere will be lo sse s in per formance whe n de cis ion e rro rs o cc ur. T o minimize the effects of s uc h error s , i t is po ssibl e t o empl o y p er iodic tr a in in g sequences, a ltho ugh thi s reduce s the e fficie ncy o f th e rev erse link tr an smi ssions.
y(nt s
+ (k - 1)tc ) ove r
f
:~ .
:g.
0.1
1lI . .
. .0
o .
:~
.
Eigen <> (BF +
0.01
Beamspace 0 Theory - -
o deg
. : O~OOl
Fixed •
.
L.--......._--'-_---'-_---L._---'-_-'-_--'----'
-20
-15
-10
-5
0 5 SINR (dB)
10
15
20
• Figure 8. The cum ulative distribution f unctions for the beamspace. CBF, and eigenfilter algorithm s, and fo r a fixed beamfonner (rfixed ") with a scattering width of 20°. Tire theoretical distribution s fo r scattering width s f or 20° ("Th eory") and (f' ("0 deg"] are also shown.
In o rde r to avo id tr ain in g se que nces a b lind method for ma ximizin g the SI NR, based o n th e eige n fi lte r method. ha s bee n pr op osed [38]. Both pr e- a nd post-correl at ion da ta at th e re ceive r a r e u sed to es ti mate th e in te rfe re n ce cova r ia nce ma t rix 12k' It is s ub tra cte d fro m Rk in o rder to im prove th e es ti matio n of the und erlyin g sig na l ve ctor q(k. t ). T he largest eige nvec t o r u 1 is o b tained fro m th e modi f ie d cova ria nce 1 ma trix. a nd the spa tia l filte r is give n by the e q u atio n Q k- u 1. whe re Q- 1 denotes th e ma tr ix inve rse o pe ra tio n .
Performance of CDMA Antenna Arrays
T
he performan ce of a nte nna a rr ay ba se sta tio ns has been a nalyzed in a number of publications. including [36, 4 1]. Tw o papers have a na lyzed base sta tio n sc he mes spec ific to CDMA syste ms [19, -1 2]. M ore re cently, analys is has be en pr e sented for IS- 95 M- ary m odulation systems [43]. In thi s secti o n. the perform ance of a C D M A system b as ed o n th e e ige n filte r method a nd a DPSK RAKE filter will be present ed. usin g the me an BER of a give n user as a qu al ity mea sure . A si ng le -ce ll sys te m will b e co nsi d e r e d with a numb er of active users o pe ra ting ove r the sa me RF bandwidth . Th e fo llo wi ng ass u m p t io ns h a ve been m a d e a bo u t t he C DMA syste m : • Eac h us er is ass u me d to obse rve a K = 4 t ap c ha n ne l , acco rdi ng to th e im pu lse resp onse in Fig. 2. The normaliz ed c ha nne l tap pow e r lev e ls {Sj} become 0, - 3. - 6, a nd - 9 dB , respectively . T he chan ne l is " slowly fa ding" with m aximum D oppler freque ncy 0 H z for eac h cha nne l ta p. T he re ce iver is assumed to be a b le to perfe c tl y tr ac k th e d esir ed user' s cha nnel. • Eac h cha nne l tap is co rru pted by C D MA inte rf e re nc e fr om a ll o the r users. Ass um ing tha t th e me an power o f a ll use rs is the sa me . the PN co de filt e r for the desired user will suppr ess th e power level of each o the r user by a factor of 3Wj2 for rectangular pulse sha ping [44] . • The CDMA in te rfe re nce is as sumed to ha ve a uniform di stributi on o ve r the range of be arings [30°, 150°]. The normalized o utp ut power y from the desired user's spa tial filt er h for one interferer at be aring 8 is sim ply hHa(8)a(8)H h/(hHh). It is important to determine the s ta t ist ica l moments of y; for exa m ple, the mean y is given by:
507
' M = l 'M=2 :- - ·
0.0001,
',M ;,,4 ·····.,
" M = S' -
100
·
120
• Figure 9. The BER results plotted for different numbers ofusers and antenna array sizes.
y= ~J5lt6/6 (h Ha($)a($)" h dcl»/(hHh) 2lt
For each antenna size and scattering width, the largest value of y has been selected to provide a pessimistic capacity estimate . The performance of adaptive array filters can be difficult to determine analytically. but here the power suppression level y ha s been modeled as Gaussian using the central limit theorem. Its first two moments for P active users are v (P - 1) and yl (P - 1), where yl is the covariance of y for one interferer. As h always represents the matched filter for the given channel tap xik: z), h has the zero vector as its mean vector and its mean covariance matrix is '¥ k/Af. The following steps were then used to calculate the BER performance in each case : • The mean SINR at the spatial filter output for the Jth tap is given by the equation WSi p= K (11) I (2/3)il. k = l s k
where Sk denotes the normalized power of the kth channel tap and the processing gain W = 128. • The eigenvalues of the matrix '¥k ind icate the number of independent Rayleigh processes at each channel tap. If there are a total of Z significant Rayleigh components for all channel taps. denote their amplitudes as {aJ. • Assuming that the add itive interference is Gaussian distributed, zero mean. and uncorrelated between channel taps, a closed form expression for the BER may be obtained. The output of each channel tap at each antenna follows a Rayleigh distribution . The BER for a given user may be obtained by simple modification of a result from [34, Ch . 7] BER= 1
K-IK -I-m
-,-L L 2- K - 1 m=O 11=0
(
2K-I) n
-l
Z
1
LIp
L- "=1 1I,, I+l/"
)m+1
Z
a
TII=IG,,-l/ i
Conclusions
T
(10)
nt
the results demonstrate a significant performance improvement for a given number of users, by increasing the size of the antenna array. If a mean BER of 10- 3 is taken as the threshold of acceptable performance, the capacity according to this measure increases from six users for one antenna to 63 for eight antenna elements. For cellular systems with a uniform distribution of users throughout the network, an additional noise term of approximately 50 percent of the single-ceil CDMA interference is present. This effect will reduce the capacity by approximately 1/3; however, it can be compensated for by including error correction coding and data interleaving [11] . In addition, the system capacity may be doubled by using voice activity detection, which only permits mobile transmissions when the user is speaking; typically , one person speaks during 40 percent of a telephone conversation.
(12)
,>'"
Equation (12) may be integrated over the distribution of gamma to determine the final result for each number of users . In order to compare the performance of different antenna array sizes, Eq. (12) has been evaluated for M = 1,2,4, and 8. In each case, the maximum value of y for a scattering width 2t1 = 0.1 has been calculated, assuming the source bearing is in the range [30°, 150°]. The results are shown for the desired user's BER vs. total number of users P in Fig. 9. In this case,
508
his article has provided an introduction to the subject of antenna arrays for narrowband CDMA base station receivers. A number of points have been discu ssed, and are summarized below : • The topic of antenna arrays has been introduced, noting that they can reduce cellular inte rfe re nce levels and improve capacity. Results in this article suggest that employing M antennas can multiply the reverse link capacity by a factor of roughly M . However, this requires additional base station hardware and software. • Channel modeling aspects have been de scribed: in urban areas, several channel taps are often resolvable . Each channel tap arises geometrically through Classical Doppler or Gaussian Doppler models and has an angular width related to the distance and width of the scattering. • The channel taps observed at antenna arrays may be modeled as the summation of array steering vectors. In urban areas, it is common for each vector entry to fade according to the Rayleigh distribution. • The 2D RAKE filter appears to be a promising approach to handling a CDMA antenna array receiver. Several algorithms have been compared. with the eigenfilter approach performing consistently well. As the angular width of a channel tap increases, the receiver is able to exploit more spatial diversity. • All the algorithms degrade in the presence of high Doppler frequency signals, particularly when the tap's angular width is large. In these cases, the receiver may need to employ short data lengths for channel estimation. • The BER performance of an antenna array receiver has been estimated and significant capacity increases demonstrated. In general, the angular distribution of interference will affect the ability of a channel tap's spatial filter to suppress interference. The location of a mobile is particularly critical in determining the interference suppression levels . There are a number of points which remain to be addressed. An important issue for antenna arrays is the scattering width of multipath components. Values have been estimated for narrowband systems (e .g., [30]), but no results appear to exist in the literature for frequency-selective CDMA systems. Efficient implementations of channel-identification algorithms are required, and more work is needed on their performance in realistic CDMA channels. The interaction of the reverse and forward links is important in practical systems, particularly to ensure that the forward link can handle the increased traffic that antenna arrays can offer on the reverse link.
Acknowledgments This work was sponsored by EPSRC, an MOD CASE sp?nsorship, and Nortel Technology. The aut~ors would also like to thank the anonymous reviewers for their helpful comments on this article.
References
[1] J. E. Padgett, C. G. Gunther, and T. Hattori, "Overview of Wireless Personal Communications," IEEE Commun. Mag., vol. 33, no. 1, Jan. 1995, pp. 28-42. [2] D. C. Cox, "Wireless Personal Commu.nications: What Is It?" IEEE Pers. Commun., vol. 2, no. 2, Apr. 1995, pp. 20-35. [3] D. D. Falconer, F. Adachi, and B. Gudmundson, "Time Division Multiple Access Methods for Wireless Personal Communications," IEEE Commun. Mag., vol. 33, no. 1, Jan. 1995, pp. 5 0 - 5 7 . . . [4] A. J. Viterbi, "The Orthogonal-Random Waveform Dichotomy for Digital Mobile Communication," IEEE Pets. Commun., vol. 1, no. 1, 1st qtr., 1994, pp. 18-24 [5] C. I. Cook, "Development of Air Interface Standards for PCS," IEEE Pets. Commun., vol. 1, no. 4, 4th qtr., pp. 30-34. [6] P. G. Andermo and L.M. Ewerbring, "A CDMA-Based Radio Access Design for UMTS," IEEE Pers. Commun., vol. 2, no. 1, Feb. 1995, pp. 48-53. [7] M. Barrett and R. Arnott, "Adaptive Antennas for Mobile Communications," lEE Elect. and Commun. Eng. J., vol. 5, no. 4, Aug. 1994, pp. 203-14. [8] G. K. Chan, "Effects of Sectorization on the Spectrum Efficiency of Cellular Radio Systems," IEEE Trans. Vehie. Tech., vol. 41, no. 3, Aug. 1992, pp. 217-25. . . [9] J. H. Winters, J. Salz, and R. D. Gitlin, "The Impact of Antenna Diversity on the Capacity of Wireless Communication Systems," IEEE Trans. Commun., vol. 42, nos. 2, 3, and 4, Feb./Mar./Apr. 1994, pp. 1740-50. (10] J. L. Massey, "Informatron Theory Aspects of Spread Spectrum ~om munications," Proe. 3rd IEEE lnt'l, Symp. Spread Spectrum Techniques and Apps. (lSSSTA), Oulu, Finland, July 1994, pp. 16-21. [11] R. Padovani, "Reverse Link Performance of IS-95 Based Cellular Systems, IEEE Pers. Commun., vol. 1, no. 3, 3rd qtr., pp. 28-34. [12] M. K. Simon et ei., Spread Spectrum Communications Handbook (Revised Ed.), New York: McGraw-Hili, 1994. [13] W. C. Y. Lee, "Overview of Cellular COMA," IEEE Trans. Vehie. Tech., vol. 40, no. 2, May 1991, pp. 291-302. [14J S. Haykin, Digital Communications, New York: John Wiley, 1988. . [15] R. Price and P. E. Green, "A Communications Technique for Multipath Channels," Proe. IRE, vol. 2, Mar. 1958, pp. 555-70. [16] R. Kohno, "Spatial and Temporal Filtering for Co-Channel Interference in COMA," Proe. 3rd /SSSTA, Oulu, Finland, July 1994, pp. 51-60, (17] S. Verdu, "Adaptive Multiuser Detection," Proe. 3rd ISSSTA, Oulu, Fin. . land, July 1994, pp. 43-50. [18] B. D. Van Veen and K. M. Buckley, "Beamfor minq: A Versatile Approach to Spatial Filtering," IEEE ASSP, Apr. 1988, pp. 4-24. . [19] A. F. Naquib, A. Paulraj, and T. Kailath, "Capacity lmprovem~nt With Base-Station Antenna Arrays in Cellular COMA," IEEE Trans. ven«: Tech., vol. 43, no. 3, Aug. 1994, pp. 691-8. [20] J. H. Winters, "The Diversity Gain of Transmit Diversity in Wireless Systems with Rayleigh Fading," Proc. ICC '94, New Orleans, LA, May 1994, pp. 1121-25. [21] A. F. Naguib and A. Paulraj, "Performance Enhancement and Trade-offs of Smart Antennas in COMA Cellular Networks," IEEE Vehic. Tech. Conf. (VTC), Chicago, IL, July 1995, pp. 40-44. [22] S. Haykin et al., "Some Aspects of Array Signal Processing," lEE Proc., pt. F, vol. 139, no. 1, Feb. 1992, pp. 1-26. . [23] J. E. Hudson, Adaptive Array Principles, Stevenage, U.K.: Peter Peregnnus, 1981. [24] Commission of the European Communities, "Digital Land Mobile Radio Communications: COST-207 Final Report," Ch. 2, 1988. [25] G. l. Turin et al., "A Statistical Model of Urban Multipath Propagation," IEEE Trans. Vehic. Tech., vol. 21, no. 1, Feb. 1972, pp. 1-9. [26] R. Steele (ed.). Mobile Radio Communications, London: Pentech Press, 1992. [27] J. D. Parsons, The Mobile Radio Propagation Channel, London: Pentech Press, 1992. [28] A. S. Bajwa and J. D. Parsons, "Small-Area Characterisatron of UHF Urban and Suburban Mobile Radio Propagation," lEE Proc., pt. F, vol. 129, no. 2, Apr. 1982, pp. 102-9. [29] W. C. Y. Lee, "Effects on Correlation between Two Mobile Radio BaseStation Antennas," IEEE Trans. Commun., vol. 21, no. 11, Nov. 1973, pp. 1214-23. II
[30] F. Adachi et aI., "Correlation between the Envelopes of 900 MHz Signals Received at a Mobile Radio Base Station Site," lEE Proe., pt. F, vol. 133, no. 6, Oct. 1986, pp. 506-12. . [31] J. Salz and J. H. Winters, "Effect of Fading Correlation on Adaptive Arrays in Digital Wireless Cornrnurucations." Proc. ICC '93, Geneva, Switzerland, May 1993, pp. 1768-74. [32] G. Raleigh et al., "Characterisation of Fast Fading Vector Channels for Multi-Antenna Communications Systems," Proe. 28th IEEE AS/MOLAR Conf., Pacific Grove, CA, Nov. 1994, pp. 853-57. [33] D. G. Brennan, "Linear Diversity Combining Techniques," Proe. IRE, June 1959, pp. 1075-1102. [34] J. G. Proakis, Digital Communications, New York: McGraw-Hili, 1989. [35] A. F. Naguib and A. Paulraj, "Performance of CDMA Cellular Networks with Base-Station Antenna Arrays, Proe. tnt'i. Zurich Seminar on Digital Commun., 1994. (36] S. Anderson et a/. , "An Adaptive Array for Mobile Communications Systems," IEEE Trans. Vehic. Tech., vol. 40, no. 1, Feb. 1991, pp. 230-36. (37) C. Farsakh and J. A. Nossek, "Application of Space Division Multiple Acc~ss to Mobile Radio," Proe. 5th IEEE lnt'l. Symp. Persona/Indoor and Mobile Communications (PIMRC), The Hague, Holland, Sept. 1994, pp. 1736-39. [38] B. Suard, et al., "Performance of COMA Mobile Communication Systems Using Antenna Arrays," Proc. IEEE tnt'l. Cant. Acoustics, Speech and Signa/ Processing (lCASSP), Minneapolis, MN, Apr. 1993, pp. IV 153-56. [39) R. J. Muirhead, Aspects of Multivariate Statistical Theory, New York: Wiley, 1982. . [40) D. Gerlach and A. Paulraj, "Adaptive Transmitting Antenna Arrays With Feedback," IEEE Sig. Processing Letts., vol, 1, no. 10, Oct. 1994, pp. 50-52. [41] S. C. Swales et aJ., "The Performance Enhancement ~f Mult.ibeam Adaptive Base Station Antennas for Cellular Land Mobile Radio Systerns," IEEE Trans. Vehie. Tech., vol. 39, no. 1, Feb. 1990, pp. 56-67 .. (42] J. C. Liberti and T. S. Rappaport, "Analytical Results for Capacity Improvements in COMA," IEEE Trans. Vehie. Tech., vol. 43, no, 3, Aug. . 1994, pp. 680-90. [43] A. F. Naguib and A. Paulraj, "Performance of OS/COMA With M-ary Orthogonal Modulation Cell Site Antenna Arrays," Proe. ICC '95, Seattle, WA, June 1995, pp. 697-702. (44] R. Meidan, R. Kohno, and L. B. Milstein, "Spread Spectrum Access Methods for Wireless Communications," fEEE Commun. Mag., vol. 33, no. 1, Jan. 1995, pp. 58-67, Jan. 1995. II
509
Efficient Direction and Polarization Estimation with a COLD Array Jian Li, Petre Stoica, and Dunmin Zheng Abstract-This paper considers angle and polarization estimation by means of a cocentered orthogonal loop and dipole (COLD) array. We show that by using the COLD array, the performance of both angle and polarization estimation can be greatly improved, as compared to using a crossed dipole array. We present an asymptotically statistically efficient method of direction estimation (MODE) algorithm that can be used with the COLD array for both angle and polarization estimation of correlated (including coherent) or uncorrelated incident signals. Numerical examples are given to show the better estimation performance of the MODE algorithm than that of the multiple signalclassification(MUSIC) and the noise subspace-fitting (NSF) algorithms.
W
I.
INTRODUCTION
HEN array signal processing algorithms are devised to estimate incident signal parameters with uniformly polarized or diversely polarized arrays, it is important to take advantage of array geometries and receiving properties of antenna elements. Although many algorithms have been developed for array signal processing recently, the characteristics of specific antenna sensors are only beginning to attract more attention. Previous work on angle and polarization estimation using crossed dipoles [1], [2] and orthogonal dipoles and loops [3] are examples of using specific antenna sensors to estimate the angles and polarizations of incident narrowband electromagnetic plane waves. In this paper, we study the advantages of an arbitrary linear array that consists of cocentered orthogonal loop and dipole (COLD) pairs. By using the COLD array, the performance of both angle and polarization estimation can be improved significantly, as compared to using a cocentered crossed-dipole (CCD) array. We consider the case where all incident narrowband electromagnetic (EM) plane waves are completely polarized. A completely polarized EM wave is a limiting case of a more general type of EM wave, viz. a partially polarized EM wave. The state of polarization of a partially polarized EM wave is a function of time, while a completely polarized wave has a fixed state of polarization (see [4] and the references therein). Manuscript received February 10, 1995; revised October 9, 1995. J. Li and D. Zheng were supported in part by the NSF Grant MIP-9308302. P. Stoica was supported in part by Goran Gustafsson Foundation, the Swedish Research Council for Engineering Sciences (TFR), and the Swedish National Board for Technical Development (NUTEK). J. Li and D. Zheng are with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA. P. Stoica is with the Systems and Control Group, Department of Technology, Uppsala University, P.O. Box 27, S-751 03 Uppsala, Sweden. Publisher Item Identifier S 0018-926X(96)02638-5.
We present an asymptotically statistically efficient signal subspace-based method of direction estimation (MODE) algorithm [5], [6] for both angle and polarization estimation. Since the MODE algorithm is a signal subspace-based approach, it is asymptotically statistically efficient for both correlated (including coherent) and uncorrelated incident signals. We show with numerical examples that the estimation performance of MODE is better, especially for highly correlated or coherent signals, than that of multiple signal classification (MUSIC) and noise subspace fitting (NSF) [7]. (We remark that the signal subspace eigenvector based MODE algorithm and the NSF algorithm are asymptotically statistically equivalent whenever the signals are noncoherent. For coherent signals. MODE remains asymptotically statistically efficient. whereas NSF is no longer asymptotically statistically efficient [8]. This observation suggests that when the correlation coefficient is close/very close to one, the NSF may need a much larger number of data samples than MODE to converge to the asymptotics, and hence. for a given finite N, MODE is likely to perform better than NSF in such a case of highly correlated signals.) In Section It we define the array geometry, describe the properties of the COLD array and formulate the problem of interest. In Section III, we describe how the MODE algorithm can be used to estimate the angles and states of polarization of incident signals with a COLD array. In Section IV, the asymptotic statistical performance of MODE is given. In Section V. several numerical examples are presented to compare the MODE algorithm with the NIUSIC and the NSF for both angle and polarization estimation. The Cramer-Rao bound (CRB) for the COLD array is also compared with that for the CCO array. Finally. Section VI gives our conclusions. II. COLD
ARRAY AND PROBLEM FORMULATION
Consider a 2L-element linear array consisting of L COLD pairs, as shown in Fig. I. The signal received from each antenna sensor is to be processed separately for direction and polarization estimation. The lth COLD pair, l == 1.2,···,L, has its center on the y-axis at an arbitrary y == 81• For the lth COLD pair, the dipole parallel to the z axis is referred to as the z-axis dipole and the loop parallel to the x-y plane as the x-y plane loop. Assume K (with K ~ L) narrowband plane waves impinge on the array from angular directions described by () and
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 44, No.4, pp 539-547, April 1996.
510
where A o is the wavelength of the signal. The effecti ve heights of the short dipoles and small loop s are given by [Ill
z
hsd
= Lsd sin ¢
and h oi
27r A s l
(6) .
(7)
= - j - - slll ¢
>'0
respe ctivel y. Including the time and space phase factors in (4), we find that an incoming signal characterized by (() , ¢, ,. TI , E) produces a signal vector in the COLD pair centered at y ~I as follows :
Y
=
(8)
x
Fig. I.
A linear COLD array.
where Ii
wave with an arbitrary elliptical-electromagnetic polarization [9] . Assume that the electric field of an incoming signal ha s transverse components
where unit vectors e O. e 'D' and -e r • in that order. form a right -hand coordinate syste m for the incoming signals and E: and E
= E cos -y.
= E sin ~/C) '1
.
JlCOLD = [1'~:i~~~;)II]
(9)
( 10)
where (11)
( 12)
(2)
(3 )
where E denotes the amplitude of the incident signal. The --y and fJ can be used to compute (t and (3. which are the ellipticity and orient ation angles of the polari zation ellipse . respecti vely . "! is alw ays in the range 0 :S 'Y :S "if / 2 and '1 is in the range - 7r :S TI < 7r . ex and ri ca n also be used to compute '"'I and T) [Il. [I OJ . We assume that each dipole in the array is a short dipole (i.e.. the length of the dipole is equ al to or less than one-tenth of a wavelen gth ) with the same length L'd and each loop is a small loop (i.e.• the perimeter of the loop is equal to or less than three tenth s of a wavelength ) with the same area Asi ' Thu s, the output voltages from each dipole and loop are proportional to the electric-field components parallel to dipole and loop. respectively . An incoming signal described by arbitrary electri c-field components Eo and Ed> ca n be written as E = E[ (cos ; )e e + (sin;ej 'l)e o ].
sin dJ cos, ] sin ¢ sin , C)I,
27l'.-1.,1 Ao
L sd
An advantage of the COLD array is that its antenna elements are not sensitive to the azimuth angle () of the signal becau se both the loop s and dipoles have the same si n (b field pattern . as may be seen from (9) . Hence. the incoming signal described by (4) is independent of () . We assume that the antennas and the inci de nt signals are co planar. i.e.. ¢ = 90° . Thu s. (9) become s
(I)
£ (J E
j
COLD = [ _
Note that Fe and \ /~!> repre sent the comple x voltages induced at the loop and dipole outputs by a sig nal with a unit elect ric field parallel to the loop s and dipoles. respecti vely. Let s(t ) = E so(t )Ve cos -y. The ZI(t ) in (8) can be rewritten as Z(t )
= US(t )(ll
( 13)
1 ] = [ \'; F" ta n; e j'1
(14 )
where U
We ass ume that K signals. spec ified by incident angles Ih , l: = 1, 2. · · · . K. are incident on the arra y. In addition. we ass ume a thennal noise voltage vecto r 01 (t ) is present at each output vector z/(t ). The o /(t ) are assumed to be zero-mean cir cul arly-symmet ric complex-Gaussia n random processes that are statistically independent of eac h other and to have covariance matrix a 2 I, where I denotes the identity matrix . Under these assumptions. the total output vector recei ved by the COLD pair centered at y = 51 is given by
(4)
Let us define the spatial phase factor
(5) 1 For a narrowband BPSK (binary phase -shift keyed) signal. for example. so (t ) e}[-"ol + ..·(t }] . where ~'o is the carrier frequency and .;.(t) is [he modulating phase.
=
511
K
ZI(t ) =
L UkSk(t )qlk + o /(t ) ,
l = L 2. · · · . L
(15)
k=l
where U k and qlk are given by (14 ) and (5). respectively, with subscript k added to each angular quantity. Furthe r. Sk(t) = Eksok(t )Vo cos ; /c. where Eks odt.) denotes the kth narrowband signal. The incident signals may or may not be correlated (including completely correlated. i.e.. coherent) with each other.
Let z(t),s(t), and n(t) be column vectors containing the received signals, incident signals, and noise, respectively, i.e.,
z(t) ==
[:~:~g],
s(t) ==
ZL(t)
[:~~g],
n(t) ==
[:~m].
(16)
nL(t)
8l\-(t)
The received signal vector has the form
z(t) == As(t) where
A
+ n(t)
(17)
is a 2L x K matrix
A ==
[<11
<12
== [A ® I]U
lJll
lJ12
l}IK
fJ21
q22
(]2K
fJLl
and
u
==
[UI o
(24)
We assume that ic is known (if ic is unknown, it can be estimated from the data as described, for example, in [13]). Note that if no components of the signal vector s( t) are fully == K (provided N > K). correlated to one another then Further, the A in (23) is a diagonal matrix with diagonal elements ~ 1 2: ~2 2: ... 2: ~ K' which are the largest eigenvalues of it, and
tc
tc
(25)
( 18)
with 0 representing the Kronecker product
A==
subspace eigenvectors of it that correspond to the ic largest eigenvalues of it, with ic == Inin[N, rank(S)]. Here, S is the source covariance matrix
lJL2
where
( 19)
qLK
0 ]
(20)
(J2
=
~- Ai =,2£ -1 K..., [tr(R) - 1:. Ai]. .
t=K+l
where
pi == I -
where (. )If denotes the complex conjugate transpose and it denotes the estimate of the following array covariance matrix
R == E[z(t)zH (t)].
(22)
It has been shown in [5], [12], and [8] that an asymptotically (for large N) statistically efficient estimator of the angles f) == [0 1 , ()2, " ' , O](]T and the polarization parameters r == [rl' rz, ... r K]T can be obtained by minimizing the fOllowing function 'J
f(B, r) = Tr[PitsA,- A-lE~]
±
(23)
"'here the symbol P stands for the orthogonal projector onto the null space of AH, and the columns in Es are the signal
512
(26)
PA~.~~I
+ P(ATH'~:I)V
(27) (28)
v-
III. ANGLE AND POLARIZATION ESTIMATION USING MODE
(21)
1
and
with The MODE [5], [6] and (in a related form) the weighted subspace fitting (WSF) [12] algorithms were derived for angle estimation with uniformly-polarized arrays. We present below how to use the signal subspace-based MODE algorithm with the COLD array for both angle and polarization estimation. Let
t=
We show below that we can concentrate out r first, and hence, reduce the dimension of the parameter space over which we need to search to minimize (23). It is shown in the Appendix that
UK
We assume that the element signals are sampled at ~V distinct times tn. ti == 1.2.···.;.V. The random noise vectors n(t n ) at different sample times are assumed to be independent of each other. The problem of interest herein is to determine the azimuth arrival angles Ok and the states of polarization de .cribed by C:k. "k)~ or (C~k .... :h:). J-; := 1.2.···. K from the measurements z(t n ) . ti == 1.2.···. N,
1 __ 2£ - K
v» == Thus, minimizing
.
VI [
o
(29)
VK
- r* ] [ 1k ,
f (() ~ r)
!(().r) == Tr[(-PA~I
0 ]
k ==
1~
2..... K.
(30)
in (23) is equivalent to minimizing .)
+ P(AtH~I)V)EsA-A-1E~].
(31)
Let
w == {"vH[(AH A) -1 0
I]V} -1
(32)
be formed from some consistent estimates of () and r. Since
P(AtH®I)vE s
= O(ljJN),
{VH[(A H A)-l ~ I]V}-l
can
be replaced by W without affecting the asymptotics of the MODE estimator [5], [6]. Then we have
(33)
where
(34)
and " 2
f2((}, r) == Tr[(At H ~ I)VWV H (At ~ I)EsA A-1E~]. (35)
The MODE estimates {{), r} are obtained by minimizing i.e.,
j,
{{} ~ r} == arg lllin [11 (B) + f2( B, r)].
(36)
A h ==
odd columns of (At H @ I)
(37)
A u ==
even columns of (At H 0 I).
(38)
8,r
To summarize, we have the following MODE algorithm for angle and polarization estimation: Step 1) Obtain initial estimates of () and r (see the discussions below). Step 2) Determine {) by minimizing II (0) + .f3(()) as shown in (49) with W in (32) formed from the initial estimates obtained in Step 1). Step 3) Calculate by using the {) obtained in Step 2) in (47). Step 4) Determine the ~ and r, from r with
Let
r
and
,k~ ==
Let V h and V v be the following K x K -diagonal matrices (39)
tan
_1(!f-kVd>V8!)
r,k = -arg ( _r~9}
and
V u == I.
(40)
Then
(A t H ® I)V Thus,
12 (fr r)
==
Ah V h
+ Au V».
" 2
(41 )
+Tr [
A--lE~ s; Vh W]
v;H A hH"Esj\ ~
v
2
~ I"H AE s AvVvW ]
(42)
IV.
Since V h and V v are diagonal matrices, (42) can be written in the following matrix form
h(fJ. r) =
[v~
e T ] Q(fi)
[Veh
1
(43)
where (44), as shown at the bottom of the page, with (~) denoting the Hadmard-Schur matrix product (i.e., the elementwise multiplication) (45)
and
e == [1
(46)
Note that the polarization parameters are contained only in v h. By setting 8f2/8vh == 0, we obtain
Vh == -Ql 1(B)Q2(())e.
(47)
Using (47) in (43) gives
f3(()) == e T [Q3(()) - Q~ (8)Q 1l(B)Q2(8)]e
and using
e in
8
== 1,2.···.K.
(51)
(49)
STATISTICAL PERFORMANCE ANALYSIS
We present below the asymptotic (for large ;.V) statistical performance of MODE for both direction and polarization estimation with the COLD array. Before we present the analysis results, however, we first describe the method we use to describe the accuracy of the polarization estimates. For reasons discussed in [I), we define the polarization-estimation error to be the spherical distance between the two points M and JI on the Poincare sphere that represent the actual state of polarization (",(",) and the estimated state of polarization ('"Y. 'f)), respectively. Let ( be the angular distance between AI and .:.v[. Then [11
cos ( == cos 2, cos 21'
+ sin L--Y sin 21' cos( r} -
,))
(52)
where ( is always in the range 0 S ( ~ tt, Applying the first-order approximation to the left side of (52) yields ( k2
(48)
which is a concentrated function depending only on f). The MODE-estimates {{), r} can be obtained by
{) == arg min [11 (f)) + 13(8)]
k
For signals that are not highly correlated or coherent with each other, the initial estimates of 0 and r in Step I may be obtained by using MUSIC [14], which requires a onedimensional search over the parameter space. For highlycorrelated or coherent signals, the initial estimate of () may be determined by setting W == I and minimizing 11(8) + 13(B). as shown in (49). The initial estimate of r can be calculated by using the initial estimate of B in (47). The initial estimates obtained by using MUSIC for noncoherent signals or MODE with W == I are known to be consistent [6], [151·
in (35) can be rewritten as
I2(B~ r) == Tr[V~ Af:EsA
(50)
== 4(""t» -
+
. 2'2 ,k ) 2 SIll \ I k )(" 1Jk -
iT} k )? -.
(53)
The asymptotic variances of the polarization estimates are obtained with (53) and the accuracy results on :y and fry given below.
Let
(54)
(47) to obtain r.
" ?
(A~Es~: A-IE~ A v) 8 (A~EsA A-lE~ AI') 8 513
W W
T T
]
~ [Ql(()) Qlf(O)
(44)
It follows from [5], [8] that the asymptotic (for large N) statistical distribution of f is Gaussian, with mean T and covariance
MUSIC WIth COLD : NSF with COLD :
matrix equal to the corresponding stochastic Cramer-Rao bound (C R B ). The ij th element of CRB - 1 is given by [C R B-l j,.j
= ~ Re[tr{AfPiA iSAHR-l AS}]
CRB for COLD :
(55)
where A i = 8Aj8Ti with Ti denoting the i th element of T.
o o
V. N UMERICAL RESULTS
+ Lsd sin ~!ej " sin (} cos L sd 5111 . (/)" Sill Ie in
_ [LSd cos y cos (} -
¢] .
o
8
10
MUSIC Wll h COLD:
,e)"
(\Ve remark that if the antennas and the incident signals are
not coplanar, we will need two-dimensional CCD or COLD arrays for angle and polarization estimation, which is the case not considered herein. For this case, however, the COLD array will not always perform better than the CCD array.) First, we present two examples that illustrate how the angle separation between the two incident signals affects both the direction and polarization estimates. We begin with the case of two signals with identical circular polarizations (al = a 2 45°). Fig. 2 shows the root-mean-squared errors (RMSE' s) of the estimates of the first signal as a function of angle ,eparation t::.B when two correlated signals with correlation coefficient 0.99 arrive at the array from angles Bl t::.B /2 and B2 = t::.B /2. We note that MODE performs better than MUSIC and NSF. Further, MODE achieves the best possible unbiased performance, i.e. , the corresponding CRB , as the angle separation increases. Because the signals arrive from angles near the broadside of the arrays, the CRB' s for the COLD and CCD arrays are similar. This case corresponds to
=
=-
18
20
22
o
NSF w,th COLD : MODE wrth COLD : CRB for COLD:
o
.
o
o
,
(57)
.
12 14 16 Angle Separation (deg)
o
10 ' rf ---r--~-~----~---r--~--~-,
=
L sd cos I cos B] . ' [ - Lsd Sill
o
. o
.o
o
18
20
(56)
In the following examples, the antennas and the incident 90 0 , for both signals are assumed to be coplanar, i.e., ¢ 0 the COLD array and CCD array. For ¢ = 90 , (56) becomes
=
o
(a)
10
J.LCCD
o
.
We present below several examples show ing the performance of using the MODE algorithm with the COLD array and comparing the asymptotic statistical-performance analysis results with the Monte-Carlo simulation results . We compare MODE with MUSIC and NSF for both angle and polarization estimation. The simulation results were obtained by using 50 Monte-Carlo simulations. In the examples, we assume that there are K = 2 incident signals and both signals are assumed to have the same amplitude Ek, such that We Eki = W Ekl = 1, k = 1. 2. Hence, the signal-to-noise ratio (SNR) used in the simulations is -10 log 10 a 2 dB . The array is assumed to have L = 8 COLD pairs that are uniformly spaced with the spacing between two adjacent COLD pair s equal to a half wavelength. We also compare the est imation performance of using the COLD array with that of using a CCD array with the same array geometry. The CCD array consists of crossed yand z-axes dipoles . The counterpart of (9) for the CCD array can be written as f.LC C D -
°
MODE Wllh COLD : !l
,
12 14 16 Angle Separanon (
i
22
(b)
Fig. 2. Root-mean- squared errors (RMSE's) of estimates versus ~8 for the -~0/2 . O ~ +~0/2. n 1 O'! '+50 . first of the two signals when 0\ J~ 0° . correlation coefficient 0.99. X 400. and SNR 10 dB (the CRB 's for the CCD array nearly coincide with those for the COLD array ): (a) Direction estimates and (b) polarization estimates .
}\ =
=
=
=
=
=
= = =
small incident angles, which make J.LC CD similar to Jl C OLD as may be seen by comparing ( 10) and (57) . In Fig . 3, we consider the case where the signals with identical horizontal polarizations (al = a 2 = 0 0 • /31 = /32 = 0 0 ) arrive from angles away from the broadside of the array . In this case, the CCD array is outperformed by the COLD array. This result occurs because the signal outputs at the yaxis dipoles are attenuated by a factor of cos B for the CCD array [see (56)] . For the COLD array, however, the signal outputs at both the dipoles and the loops are independent of the incident angle B [see (10)]. Note also that the RMSE's of the angle estimates first decrease and then increase even as the angle separation increases. Th is result occurs because the incident angle of the second signal approaches 90 0 for very large ti.B and the RMSE's of angle estimates are approximately proportional to 9 for large ti.B [15]. We note again that MODE gives better performance than MUSIC and NSF and achieves the CRB as t::.B increases. We have also found that
514
co;2
MUSIC with COLD:
MUSIC WithCOLD : NSF WithCOLD: MODE with COLD:
°
NSF with COLD : MODE with COLD : CRB for CCD: CRB lor COLD:
g10
°•
CRBfor CCD: CRB for COLD:
1
~
'" .§ Ql
-;;
°
.-
+
+
°
_ 0
°
-
w 510°
_ - 0
ti
o ~
°
°
°
°
°
10
20
30
40
50
60
70
°
w
10 "
10.2
10
8
6
°
"0
'" ~
10 .2
°
12
14 18 16 20 Angle Separation (deg)
22
24
0
26
Potanza tion Separation (deg)
BO
90
(a )
( a) 10'
10'
r
MUSIC with COLD: NSF with COLD:
MUSIC with COLD :
°
NSF with COLD :
•
MODE with COLD :
J'i l
CRB forCCD : CRB lor COLD :
, I
;ij
~
'" Ql
"
1ii
§ -;;
w c
~
" "0 Q.
[ ° •
"0
°
~ 10' ~ [ 10"
10
°
°
o
°
°
1
~ 10' •~ r r
Q.
8
10
12
14 16 18 20 Angle Separation (deg)
22
24
o
0
,
10
I
20
Fig. 3. Root-mean-squared errors (RMSE's) of estimates versus .3.8 for the second of the two signa ls when 11 1 :)0° . (12 50° + .3.8, cq 02 0° .1 [ 12 0°, correlation coefficient 0.99, S 400. and SNR to dB: (a) Direction estima tes and (b) polarization estimates.
= =
0
0
0
•
..
II:
•
II:
..
30
, 40
50
60
70
80
0
I
Polanzation Separat ion (deg)
1
90
(b )
(b )
=
o
- - - - -- --- - - - --- --- -- - - -~--
II:
26
j 1
f- ---°
w
1J 6
1
;ij
+
I
Q.
CRB 10r CCD: CRB lor COLD:
~102
!I ~ 10' ~
°
MODE with COLD:
=
=
=
=
=
=
Fig. 4. Root-mean-squared errors (RMSE 's) of estimates versus ~o for the seco nd of the two signals when 81 50° .82 70° . o 1 = 45° - .3.0 , and 0 '2 = 45°, h = 12 0° , correlation coefficient 0.99, S = 400. and SNR = \0 dB: (a) Direction estimates and (b) polarization estimates .
=
=
=
=
the CRE's for COLD array are much lower than those for ization estimates. Fig. 5 shows the RMSE' s of the direction and polarization estima tes as a function of the correlation CCD array. especially when () approaches 90° . 0 Second, we consider how the polarization separation af- coefficient. The two signals arrive from (}1 = _6 and (}2 =6° , 0 0 fects the estimator performance. Consider the case where and we have a1 = a2 = 45 and f31 = {32 = 0 . We note that two incident signals with a correlation coefficient 0.99 arrive for highly correlated or coherent signals. the performance of from angles (}1 = 50° and (}2 = 70 0 • We assume that MODE is much better than that of MUSIC and NSF, Fourth, we consider the performance of the estimators as a the corresponding ellipticity angles are 0: 1 = 45 0 - 60: 0 function of SNR. In Fig. 6. we consider the case where (}1 == and a2 = 45 and the orientation angles are {31 = {32 = 0 0° . The polarization separa tion between the two polarization 50 , (}2 = 70 0 • 0:1 = 0:2 = 0 0 • {31 = f32 = 00 , correlation states is 260: , Fig, 4 shows the RMSE' s of the direction and coefficient = 0.99, and N = 400 . In this case, the advantage polarization estimates as a functio n of 6a for the second of MODE over MUSIC and NSF is increasingly eviden t at low signal. Note that MODE performs better than MUSIC and SNR. We note again that the COLD array gives much lower NSF since the signals are highly correlated, Since the second CRE than the CCD array, especially for angle estimation. Fifth, we consider the performance of the MODE estimator signal arrives from a large angle away from the broadside of the array, the performance of the COLD array is again much as a function of the number of snapshots N . We consider the better than that of the CCD array. case where (}1 = 50° , fiz = 70 0 , a1 = 0:2 == 45 0 , {31 == ;32 == Third, we consider how the correlatio n coefficient between 00 , correlation coefficient = 0.99. and SNR = 10 dB. It can the two incident signals affects both the direction and polar- be seen from Fig. 7, that although MODE is an asymptotically
515
10' MUSIC wllh COLD :
MUSIC with COLD : NSF with COLD :
0
MODE with COLD :
-
MODE with COLD:
Il C;;10 '
CAB lor COLD :
•
:g. ~
CABlarCCD: CAB tor COLD :
~
W
o
c:: 10°
~
(
o
~
0 .2
0 .3
0.4
0 .5
<5
.
*
0.1
0.6
Correlation C oefficient
o
NSF Wllh COLD:
0 .7
0 .8
(;
w
en
~
10 .
1
,)
0.9
0
2
(a)
6
!
10
SNA (dB)
, 12
.
I
14
16
I
18
20
(a)
'1
10' .----.,---~--,----.---~-~--~-.,.._-.,___, MUSIC With CCD: NSF With COLD :
MUSIC with COLD :
o
NSF Wllh COLD :
-
MO DE w,th COLD : CAB lor COLD :
o
MODE With COLD :
I
CAB larCCD: CAB lar COLD :
-j 1
J 0 .1
0.2
0 .3
0.4
0.5
0 .6
Co rrelation Cceruceot
07
0 .8
10 0
0.9
2
10
SNA (dB)
12
14
16
ta
20
(b)
(b)
Fig. 5. Root-mean -squared errors (RMSE' s) of estimates versus Source-correlation coefficient for the first of the two signals when a 8( 6 a • 82 6 • 0'( = 0'2 ~ 5 a . J( .h O? . .\" 400, and SNR = 10 dB (the CRB's for the CCD array nearly coincide with those for the COLD array): (a) Direction estimate s and (b) polarization estimate s.
Fig. 6. Root-mean-squared errors (RMSE's) of estimates versus SNR for 50 a . liz 70 a • n 1 n2 O" . the second of the two sisnals when /II il J2 O" , correlation coefficient = 0.99, and V 400 : (a ) Direction estimates and (b) polarization estimate s.
=-
=
=
= =
=
(for large N) statistically efficient estimator, MODE estimates achieve the CRB even for very moderate N . MODE again performs better than MUSIC and NSF . Finally, we remark that although we make the Gaussiannoise assumption in the problem formulation, the Gaussiannoise assumption is not critical to the algorithms. As a matter of fact, the asymptotic distributional properties of all of the algorithms are most likely to remain the same , even in the non-Gaussian noise case. However, of course, in the latter case, MODE is no longer asymptotically statistically efficient and the CRB given in (55) is no longer the true eRB matrix, but rather the asymptotic theoretical performance (TP) of MODE. Fig . 8 shows an example when the Gaussian noise assumption is violated. Fig. 8 is the same as Fig . 7, except that the noise is the contaminated Gaussian noise [16]. More specifically, the probability density function of n( t) has Ute form (1 - €)N(O , a 2 I ) + €N( O,9a 2 I), where N(a, B )
= =
=
=
=
=
=
denotes the Gaussian probability density function with mean a and covariance matrix B . The SNR here is defined as -10 loglO(a 2(1 + 810)] dB. We used 10 = 0.7 in the example. Note that MODE can also achieve its asymptotic theoretical performance for moderately large N for this case and MODE has the best performance. VI. CONCLUSION We have presented a cocentered orthogonal loop and dipole (COLD) array. We have shown that with the COLD array , both angle and polarization estimation performance can be significantly improved as compared to using a similar crossed dipole array. We have also described how to use the asymptotically statistically efficient MODE algorithm for both angle and polarization estimation with the COLD array . Numerical examples have been given to show that MODE gives more accurate angle and polarization estimates than MUSIC and NSF, especially for highly -correlated or coherent signals.
516
MUSIC wIth COLD: NSF wIth COLD:
MUSIC w;th COLD: NSF with COLD : MODE w ith COLD:
o
MODE w;th COLD: CRB for CCO:
TP for MODE wrth CCD:
CitO' • :8.
CRB for COLD:
o
o
TP for MO DE wIth COLD:
" '"; ;" E
+
o
~
o
o
o
o
ur
o
o
c
o
Q10°
o
o
0
a
o
1:3
o
o
i!! i5 "0 w
::; ., a: 10
Vl
100
50
150
20 0
N
250
300
350
40 0
.50
100
50
150
200
(a)
tOJ ~_ ~_~_
j
CRB forCC O: CRB lor COLD:
Fit:..
t.
" v ........ -
II I\,...'. ....
0
I
r~'-tu.'"'.I. l..u
200
....l IUI.:J
o
N
o
250
=
o
350
300
-
[
1
a
'.
"
VI C :'> lI I I1CJl t: :'>
=
=
VlIU
vt:r~U:-i I
_,
lu r
Lue
= , >2 =
= 10 dB : ra)
(58)
-
[V~(:~~I) ][B 01 0
, .
,':,. 'c' -
-0 -
-
')
o
_
-0-- -
-
_ 0::)
---
II
!
-
..
0 ..
tb ) . E . s ) 01. estimates . ..ig. lI . xoor-mean-squareu errors (KMS versus .\ " t'or the sec ond of the two signals in the pre sence of contaminated Gaussian noise 0 50 and FI ~ 70 0 " I 1 \ '2 ~5 ° i l h 0° . when Ii, correlation coefficient = 0.99 . and SNR 10 dB : (a) Direction estimates and (h) polari zation estimates.
=
=
=
=
=
=
= 2L -
Thus ~ has full column rank 2(L - K ) + K dim[N(AH )]
= 2L -
=
K. Since
K
(61)
it follows that the columns of ~ span the entire null space
N (A H). Hence
(59 )
.
This result shows that the columns of .6. belong to Moreover
(B lI B ) ® I
o
10 L_ _"'--_--'-_ _'":-_=_-::-::-:---,:-:::-_~::____:_;:;:_-_;; 450 0 100 150 400 200 250 300 350 50
. 50
400
BlI ® I ] VH (A t :2lI) .
:::.HA = [(BHA ZI)U ] _O
= [
~_---,
11:.
vf!
H
,
~
o
We first show that the columns of the matrix .6. span the Uk = 0, we obtain null space of A H , Since B H A = 0 and
.6. ti =
~_~_ _~_-.,.-
TP for MODE wit h COLD'
Proof of (27J: Let B be an L x ( L - K ) matrix whose columns span the null space of A H . Also, let a (2 L - K ) x 2L matrix .6. H be defined as H _
_
TP lor MODE wIth ceo:
ApPENDIX
~
450
N
\l!RL,L,. "'}
=
o
second of the two signals when H, 50 0 and H ~ 70°. n ~5 ° . 11 1~ 0° , correlation coefficient 0.99 . and SNR Direction estimates and (b) polarization estimates.
=
400
MODE wrth CO LD:
i
150
100
350
MUSIC w,th COLD : NSF with COLD:
1
o
MODE with COLD:
o
300
(a)
MUSIC wrtn COLD: NSF wIth COLD:
cr - _ _ o
250
N
pi = P.:> =
.6.(D.H .6. )-1 D. H
= (B 0 I)[(BHB ) -
I
® I](B H ® I )
+ (AtH @ I)V {VH[(A H Af1 V
N(A H).
H
@
I]V} - l
(A t @ I ).
(62)
Since PBt~I = P:L3;I' we conclude that (27) must hold true.
(A tH ® I )V ]
REFERENCES
0 ] VH [(AHAf l @I]V . (60) 517
[Ij J. Li and R. T. Compton, Jr. , " Angle and polari zation estimation using ESPRIT with a polarization sensitive array," IEEE Trans. Anten/las Propagat ., vol. 39, pp. 1376-1383, Sept. 1991.
[2] Y. Hua, HA pencil-MUSIC algorithm for finding two-dimentional angles and polarizations using crossed dipoles," IEEE Trans. Antennas Propagat., vol. 41, pp. 37~376, Mar. 1993. [3] J. Li, "Direction and polarization estimation using arrays with small loops and short dipoles," IEEE Trans. Antennas Propagat., vol. 41, pp. 379-487, Mar. 1993. [4] J. Li and P. Stoica, "Efficient parameter estimation of partially polarized electromagnetic waves," IEEE Trans. Signal Processing, vol. 42, pp. 31 14-3 J25. Nov. 1994. (~; P. Stoica and K. C. Sharman, "Maximum likelihood methods for direction-of-arrival estimation." IEEE Trans. Acoust.. Speech, Signal Processing. vol. 38. pp. 1132-1143, July 1990. (6] _ _ , "Novel eigenanalysis method for direction estimation," in lEE Proc.. Pt. F, vol. 137, Feb. 1990, pp. 19-26. [7] A. Swindlehurst and M. Viberg, "Subspace fitting with diversely polarized antenna arrays," IEEE Trans. Antennas Propagat., vol. 41, pp. 1687-1694, Dec. 1993. [8] B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and large sample ML techniques for parameter estimation and detection in array processing," in Radar Array Processing, ch. 4, S. Haykin, 1. Litva. and T. J. Shepherd. Eds. New York: Springer-Verlag, 1993.
[9! C. A. Balanis, Antenna Theorv-i-Analvsis and Design.
[101
[11] [12]
l13~ [14] [15] [161
518
New York: Harper & Row, 1982. G. A. Deschamps, "Geometrical representation of the polarization of a plane electromagnetic wave." in Proc. IRE. May 1951. vol. 39, pp. 540-544. R. C. Johnson and H. Jasik. Antenna Engineering Handbook. New York: McGraw-Hill, 1984. M. Viberg and B. Ottersten. "Sensor array processing based on subspace fitting," IEEE Trans. Acoust., Speech, Signal Processing, vol. 39. pp. 1110-1121. Mav 1991. M. Wax and T. "Kailath. "Detection of signals bv information theoretic criteria." IEEE Trans. Acoust.. Speech. Sig..n. al Processing, vol. ASSP-33. pp. 387-392. Apr. 1985. E. Ferrara. Jr. and T. Parks, "Direction tinding with an array of antennas having diverse polarizations," IEEE Trans. Antennas Propagat., vol. AP-31. pp. 231-236. Mar. 1983. P. Stoica and A. Nehorai, ·'MUSIC. maximum likelihood. and Cramer-Rae bound:' IEEE Trans. Acoust., Speech. Signal Processing, vol. 37. pp. 720-741. May 1989. P. 1. Huber. Robust Statistics. New York: Wiley. 1981.
Upper Bounds on the Bit-Error Rate of Optimum Combining in Wireless Systems Jack H. Winters, Fellow, IEEE, and Jack Salz, Member, IEEE Abstract- This paper presents upper bounds on the bit-error rate (BER) of optimum combining in wireless systems with multiple cochannel interferers in a Rayleigh fading environment. We present closed-form expressions for the upper bound on the bit-error rate with optimum combining, for any number of antennas and interferers, with coherent detection of BPSK and QAM signals, and differential detection of DPSK. We also present bounds on the performance gain of optimum combining over maximal ratio combining. These bounds are asymptotically tight with decreasing BER, and results show that the asymptotic gain is within 2 dB of the gain as determined by computer simulation for a variety of cases at a lO-J BER. The closed-form expressions for the bound permit rapid calculation of the improvement with optimum combining for any number of interferers and antennas, as compared with the CPU hours previously required by Monte Carlo simulation. Thus these bounds allow calculation of the performance of optimum combining under a variety of conditions where it was not possible previously, including analysis of the outage probability with shadow fading and the combined effect of adaptive arrays and dynamic channel assignment in mobile radio systems. Index Terms- Bit-error rate, optimum combining, Rayleigh fading, smart antennas.
A
1.
INTRODUCTION
NTENNA arrays with optimum combining combat multipath fading of the desired signal and suppress interfering signals, thereby increasing both the performance and capacity of wireless systems. With optimum combining, the received signals are weighted and combined to maximize the signal-tointerference-plus-noise ratio (SINR) at the receiver. Optimum combining yields superior performance over maximal ratio combining, whereby the signals are combined to maximize signal-to-noise ratio, in interference-limited systems. However, while with maximal ratio combining the bit-error rate can be expressed in closed form [1], with optimum combining a closed-form expression is available only with one interferer [2], [3]. With multiple interferers, Monte Carlo simulation has been used [3]-[5], but this requires on the order of CPU hours even with just a few interferers. Thus the improvement of optimum combining has only been studied for a few simple Paper approved by N. C. Beaulieu, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received September 21, 1993; revised November 28, 1996. This paper was presented in part at the 1994 IEEE Vehicular Technology Conference, Stockholm. Sweden, June 8-10, 1994. J. H. Winters is with AT&T Labs-Research, Red Bank, NJ 07701 USA. J. Salz, retired, was with AT&T Labs-Research, Crawford Hill Laboratory, Holmdel, NJ 07733 USA. Publisher Item Identifier S 0090-6778(98)09388-X.
y
User~
Fig. 1. Block diagram of an Jl-element adaptive array.
cases, and detailed comparisons (e.g., in terms of outage probability) have not been done. In [6], we showed that, with ]VI antenna elements, the received signals can be combined to eliminate L (L < M) interferers in the output signal while obtaining an M - L diversity improvement, i.e., the performance of maximal ratio combining with ]\II- L antennas and no interference. However, this "zero-forcing" solution gives far lower output SINR than optimum combining in most cases of interest and cannot be used when L 2: Ail. In this paper we present a closed-form expression for the upper bound on the bit-error rate (BER) with optimum combining in wireless systems. We assume flat fading across the channel and independent Rayleigh fading of the desired and interfering signals at each antenna. 1 Equations are presented for the upper bound on the BER for coherent detection of quadrature amplitude modulated (QAM) and binary phase-shift-keyed (BPSK) signals, and for differential detection of differential phase-shift-keyed (DPSK) signals. From these equations, a lower bound on the improvement of optimum combining over maximal ratio combining is derived. In Section II we derive the upper bound on the BER. In Section III we compare the upper bound to Monte Carlo simulation results. A summary and conclusions are presented in Section IV. II. UPPER BOUND DERIVATION
Fig. 1 shows a block diagram of an M -element adaptive array. The complex baseband signal received by the ith antenna element in the kth symbol interval Xi (k) is multiplied by a controllable complex weight ui, and the weighted signals are summed to form the array output signal So (k). I As shown in [7], the gain of optimum combining is not significantly degraded with fading correlation up to about 0.5. Thus our bounds, based on independent fading, are reasonably accurate and useful even in environments with fading correlation up to this level.
Reprinted from IEEE Transactions on Communications, Vol. 46, No. 12, pp. 1619-1624, December 1998.
519
With optimum combining, the weights are chosen to maximize the output SINR, which also minimizes the mean-square error (MSE), which is given by [8] MSE == (1 + U~R~~1£d)-l
(1)
where Rnn is the received interference-plus-noise correlation matrix given by
n.; =
0"
2
1+
L L
(2)
Uj1£}
j=l
CJ2 is the noise power, I is the identity matrix, Ud and
are the desired and j th interfering signal propagation vectors, respectively, and the superscript denotes complex conjugate transpose. Here we have assumed the same average received power for the desired signal at each antenna (that is, microdiversity rather than macrodiversity) and that the noise and interfering signals are uncorrelated, and without loss of generality, have normalized the received signal power, averaged over the fading, to 1. Note that the MSE varies at the fading rate. For coherent detection of BPSK or QAM, the HER is bounded by [9]
1.£j
t
r. ::; e(1/er;) E [e( -l/MSE)] = e((1/IT~)-l) E [e-U;,R;;-,:Ud]
(3)
the bound. Also, note that with only noise at the receiver, An = (1~, where O'~ is the variance of the noise normalized to the received desired signal power, and from (4) and (5) ., ((1~)~I 1 (6) Pe < -2- = = 2p AI where p is the received SINR, while the actual BER is 1/2(1 + p)}VI [1]. Thus even without interference, the bound differs from the actual BER, and this difference increases as the received SINR decreases. Let us consider the case of interference only. In this case, IRnnl, which is giyen by (2), may also be expressed as
IRnn I = IQ t QI
L
±DI Dm1D~Dm2 ... DtI D 1n M
(7)
where Q = (D 1 , · · · , D~I), D·, U = ((U1).m···(1£L)m)T, (Uj )In is the mth element of 1£j, the sum is extended over all M! permutations of the Il.;' s, D rn , is the ith element of the permutation of the D 111 's, the "+" sign is assigned for even permutations (i.e., an even number of swapping of DnJ.'s in the permutation), and the "-" sign for odd permutations. Now
"2 L
t E[D·m,D,u]
= L...-J
(8)
aj
j=l
where O'J is the average power of the .ith interferer normalized to the desired signal power, and
= L O'f· L
E[D!nDnD;"D.rn]
where now the expected value is taken over the fading parameters of the desired and interfering signals, and O"~ is the variance of the BPSK or QAM symbol levels (e.g., O'~ == 1 and 2 for BPSK and quaternary phase-shift keying (QPSK), respectively). For differential detection of DPSK, assuming Gaussian noise and interference.? the BER is given by [1] 1 [ e-udt B-1 Ud] . P = -E nn e 2
=
(9)
j=l
Similarly, from (7), it can be shown that
(4)
Thus the BER expression for both cases differs only by a constant, and we will now consider the term E[e-u~R;~Ud]. As shown in the Appendix, this term can be upper-bounded by (5)
where IRnnl denotes the determinant of Rnn, and An is the nth eigenvalue of Rnn. Since (5) is the key inequality in our bound (and is the only inequality we use in determining the bound for differential detection of DPSK), let us examine its accuracy. The bound is tight if An ~ 1, and since the An's are proportional to the interference signal powers, the bound is tight for large received SINR, i.e., low BER's. Although for all cases (1 + (l/An»-l < 1 and thus BER < 0.5, for An > 1 the BER as given by the bound may exceed 0.5. Thus with small received SINR, occasionally BER's greater than 0.5 may be averaged into the average BER, reducing the tightness of
where the sum is over all sets of positive integers ik and lk that exist such that M ~ ... > i 2 > iI, with Ek iklk ~ M. For example, when M = 5, there are 6 sets of {ik' lk} such that Ek iklk ~ M (see Table I). All sets are of the form {iI, II}, e.g., {i 1 = 3, II = I} for 3 ·1 < 5, except for the set {i 1 = 2, 11 = 1, i 2 = 3, 12 = I} for 2 . 1 + 3 . 1 = 5. Q~}VI) is an integer coefficient corresponding to the qth set with M antennas. Note that a~/)';l) is obtained by summing the coefficients (±1' s) for similar terms in E[ IQ t QI]. a.~Nf) can be determined as shown below. Since E~=l CJ; 1/ p, and a~lvI) 1 when iklk 0, (10) can also be expressed as
2 Since the stronger the interference, the more that optimum combining suppresses it, with the Gaussian assumption we overestimate the probability of strong interference. Note that this is consistent with the derivation of an upper bound on the BER.
520
E[ IQtQI]
Ek
=
=
= p-Al
[1 + L a.~AI) (t(p.aJ)i q
=
1
)
h
J=1
-(t,(pa;)i2)'2.. -]
(11)
TABLE I
VALUES OF
FOR
II
i2
AI
= 2 TO
5
V ALUES OF
TABLE II
0'
q
M
i}
I}
1
-1
6
2 2
2 3
1 1
-3 +2
1 2 3 1 2
2
1 2 1 1
-6 +3 +8 -6
1 2 1 1 1 1
-10 +15 +20 -20 -30 +24
i
2
2
3
4
I
2
3 4 2
2 3 2
4 5
/2
1
3
2 3 3 4
5
7
2 2
1 2 3
6 7 2 2 2 3
1 2 1 1 1 1 1 1 1 1
2
2
3
3
4
where now M 2: ... > ';'2 > '£1 > 1. To determine the a~ll'..t), s, first note that if a~) 1, "', L, then L~=l
aJk
== Lo?", and
E[IQtQIJ = (L M +
t,
(J''J.
' .J'
(11) becomes
IhLM-k+l)iT2M
(12)
where the 13k's and the a~Af),s can be seen to be closely related. From [6], P; == 0 for L < M, and thus the {3k's are the coefficients of the N/th-order polynomial in L, L(L - l)(L - 2)··· (L - M + 1). This result is not only useful when all interferers have equal power, but also serves as a consistency check on our calculated values of Q.~AI). (1\1) Q.q
.
were generated USIng a computer Th e va Iues 0 f program to examine every permutation in (7) for given M. The number of each type of iI, ll, i 2 , l2, ... term was calculated to determine Q.~lYf). Tables I and II list these values for M == 2-7. Note that only i 1 and II terms exist for M < 4 and i? and
l2 terms also exist for 5 ~ M Values for ;;~l\j) for higher M can also be easily calculated. However, since the amount of computer time to generate the values of a.~j\1) increases exponentially with M, our program could only generate these 0
p-M
[1 + l:= Q.~JvI) (t(PiT JY1) q
i2
12
1 1
3 4
3
1 1
4 3
1 1
4. 5
1
AND
7
a(M) q
-15 +45 -15 +40 +40 -90 +144 -120 -120 +90
-21 +105 -105 +70 +280 -210 +504 -840 +720 -420 +630 -504 -420 +210
and from (4), the upper bound on the BER with differential detection of DPSK is given by
r. < ~ p- M
[1 + l:= Q.~J\I) (t(PiTJ)i 1) q
II
J=1
o
(t,(PiTjYi2 ) 12 o' oj
0
(14)
For the case of noise with L interferers, consider the noise as an infinite number of weak interferers with total power equal to the noise. That is, let
values in a reasonable amount of computer time for up to M == 10 (where a hundred CPU hours on a SPARCstation20 would be required). From (3), the upper bound on the BER with coherent detection of BPSK or QAM is now given by
e(l/O"~)-l)
AI = 6
1
1 1 1
2
FOR
1
6 2 2
5
Fe :S
~l\t{)
a(M)
M
5
n~lV/)
2 aj
a;
== K - L'
j
== L + 1, ... , K,
(15)
II
J=l
.(t,(piTJ)i y2 .. oj 2
(13)
for i k > 1. Therefore, with noise, the BER bound is the same as in (13) and (14), but with p including the noise. In this case, if we define the received desired signal-to-noise ratio a;;2 and the jth interferer signal-to-noise ratio as as d
521
r
f
j
= aJj a~, then (14)
becomes [similarly for (13 )]
10 r - - - - - - - - - - -- - - - - - ,
...........-
8
_.---_ ---....
__-----
_----8::: M=5
f l=10dB
Coherent Detection of BPSK L=1
2;M
Since is the bound with maximal ratio combining , the tenn in the brackets is the improvement of optimum combining over maximal ratio comb ining based on the BER bound. Defining the gain of optimum combining as the reduction in the required p for a given BER, from (17), this gain in decibels is given by Gain (dB) 10
=- M
10' \ .5
Fig. 2. Gain versus BER for coherent detec tion of BPSK-compari son of analytical result s to the asym ptotic gain.
log10
12 , - - - - - - - - - - - - - - --, 10
- - Theoretical Results •••• Simula tion Results Asymptotic Ga in M=2
This gain is therefore independent of the desired signal power (because the bound is asymptotically tight as p ---+ 00 ). However, this is the gain of the BER bound with optimum combining over the BER bound with maximal ratio combining. Since the required p for a given BER with maximal ratio combining is less than the bound , the true gain may differ from (18) and to obtain a bound on the gain, the gain in (18) must be reduced accordingly . For example, with differential detection of DPSK, to obtain a bound the gain given in (18) is reduced by the factor (pj( l + p))IIJ . Note that as p ---+ 00 , this factor reduces to one and the gain approaches ( 18) . Thus we will refer to (18) as the asymptotic gain.
III.
COMPARISON TO E XACT THEORY AND SIMULATION
In this section, we compare the bound to theoretical results for L = 1 and simulation results for L ~ 2. Fig. 2 compares theoretical results (from [1]-[3]) for the gain to the asymptotic gain (18) versus BER with coherent detection of BPSK. Results are generated for M = 2 and 5, and I' 1 = 3 and 10 dB. In all cases the gain monotonically decreases to the asymptotic gain as the BER decreases . The gain approaches the asymptotic gain more slowly with decreasing BER for larger M and also, at low BER's, the accuracy of the asymptotic gain decreases with higher f l . Thus the accuracy of the asymptotic gain decreases as the p required for a given BER with optimum combining decreases, as predicted by the approximation in Section II. Fig. 3 compares theoretical and Monte Carlo simulation [5] results for the gain to the asymptotic gain with M = 2 and L = 1, 2, and 6. Results are plotted versus f j , where all L interferers have equal power , for coherent detection of BPSK
L=2
5
10
f j (dB)
15
20
Fig. 3. Gain with .\1 = 2 for I. 2, and 6 equal-powe r interferers versus signal-to-noise ratio of each interfe rer-s-co mparisc n of analytica l and Monte Carlo simulation res ults with coherent dete ction of BPSK [5] to the asymptotic gain.
at a 10- 3 BER,3 In all cases, the asymptot ic gain has the same shape as the gain and is within 1.7 dB for L = 1, 1.0' dB for L = 2, and 0.4 dB for L = 6. Since optimum combining gives the largest gain when the interference power is concentrated in one interferer and the least gain when the interference power is equally divided among many interferers, L = 1 and L = 6 represent the best and worst cases for the gain in an interference-limited cellular system. Thus from the results in Fig. 3, we would expect the asymptotic gain to be within 0.4-1 .7 dB of the actual gain for all cases in cellular systems with M = 2. 3Th is BER was used bec ause the result s in [5] were obtained for this BER. As shown in [5], the gain does not change sign ificantly for BER 's between 10- 2 and 10- 3 , the range of interest in most mobile radio sys tems.
522
of cases at a 10- 3 BER. These cases include interference scenarios that cover the range of worst to best cases for the gain of optimum combining in cellular systems with M == 2. The bound is most accurate with differential detection of DPSK and high SINR, corresponding to low BER and a few antennas. Because of the 2-dB accuracy, the bound is most useful where the optimum combining improvement is the largest, which is the case of most interest. The closedform expression for the bound permits rapid calculation of the improvement with optimum combining for any number of interferers and antennas, as compared with the CPU hours previously required by Monte Carlo simulation. These bounds allow calculation of the performance of optimum combining under a variety of conditions where it was not possible previously, including analysis of the outage probability with shadow fading and the combined effect of adaptive arrays and dynamic channel assignment in mobile radio systems,
6 -----------------., • • •• Simulation Results Asymptotic Gain
. .. .. ... . . ..... . ..... . . . ..•.. BER=10· 3
r j U=1,L)=3dB
'---------"""'
4
2
0'----.-..-----------"--------6 3 2
5
4
7
M
Fig. 4. Gain versus AI with two and six equal power interferers-comparison of Monte Carlo simulation results with coherent detection of BPSK [3] to the asymptotic gain.
ApPENDIX
Diagonalizing Now, consider the lower bound on the gain obtained from the BER bound (17), as compared to the asymptotic gain. Without interference, differential detection of DPSK with maximal ratio combining and All == 2 requires fJ ~ 13.3 dB (theoretically [10]) for a 10- 3 BER, while the BER bound (17) gives p ~ 13.5 dB. Thus the lower bound on the gain (from (17)) at a 10- 3 BER is 0.2 dB less than the asymptotic gain for any interference scenario-in particular, the lower bound on the gain is 0.2 dB less than the results shown in Fig. 3. Similarly, coherent detection of BPSK with maximal ratio combining and 1\;1 == 2 requires p ~ 11.1 dB for a 10- 3 BER, while the BER bound (13) gives 15.0 dB. Thus the bound is most accurate with differential detection of DPSK and low BER's. Fig. 4 compares Monte Carlo simulation results [3] for the gain to the asymptotic gain for L == 2 and 6. Results are plotted versus !vI with r j == 3 dB for all interferers and coherent detection of BPSK at a 10- 3 BER. Again the asymptotic gain has the same shape as the simulation results. The cases include both many more interferers than antennas and many more antennas than interferers, but in all cases the asymptotic gain is within 1.8 dB of simulation results.
Rnn by a unitary transformation W, we obtain (19)
where diag (.) denotes an M x !VI matrix with nonzero elements only on the diagonal, or
R n- n1 and
1 tRnn Ud
'Ud
-
,,/,t I.fI
di:lc1g (\/\ 1-1
t I}/,t - U d I.fI
di.rag (\Al-1
\ -1),1/ AI If),
A
\ -1),,/, " . /\;'1 If/Ud,
(20) (21)
Let (22) Then
tR- 1 u d nnU,d and
-
AI
j') An
~ ICn ... ~
n=l
(23)
E[e-U~R~~Ud] = E [exp(_~ Ic~2)] =E
IV, CONCLUSIONS In this paper we have presented upper bounds on the biterror rate (BER) of optimum combining in wireless systems with multiple cochannel interferers in a Rayleigh fading environment. We presented closed-form expressions for the upper bound on the bit-error rate with optimum combining, for any number of antennas and interferers, with coherent detection of BPSK and QAM signals, and differential detection of DPSK. We also presented bounds on the performance gain of optimum combining over maximal ratio combining and showed that these bounds are asymptotically tight with decreasing BER. Results showed that the asymptotic gain is within 2 dB of the gain as determined by computer simulation for a variety
."
[IT exp (J~~2) ].
(24)
Since with independent, Rayleigh fading at each antenna, the elements of U,d are independent and identically distributed (i.i.d.) complex Gaussian random variables, the elements of C are also i.i.d. complex Gaussian random variables with .the same mean and variance. Furthermore, the An's are independent of the c.,' s. Thus we can average over the desired and interfering signal vectors separately, i.e.,
523
E
[IT exp (_1~~2) ] E [IT E [exp(J~n~2) ]]. =
A
Cn
(25)
Since the en's are complex Gaussian random variables with zero mean and unit variance
[1]
(26)
[2]
E en [exp
IcnI2)] = 1 +1 } (- ~
REFERENCES
n
and
[3] [4]
Since the
An'S
1 1+
E[e-u~R;:~Ud] ~ E lR..nl
1
An
and, therefore,
where
[5]
are nonnegative
A
< An
(28)
[g An] =
denotes the determinant of
[6]
[7]
E A [ lR..nl]
R..n.
(29)
[8] [9] [l 0]
524
w. C. Jakes Jr. et al., Microwave Mobile Communications. New York: Wiley, 1974. V. M. Bogachev and I. G. Kiselev, "Optimum combining of signals in space-diversity reception," Telecommun. Radio Eng., vol. 34/35, no. 10, pp. 83, Oct. 1980. J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE J. Select. Areas Commun., vol. SAC-2, no. 4, July 1984. _ _ , "Optimum combining for indoor radio systems with multiple users," IEEE Trans. Commun., vol. COM-35, no. 11, Nov. 1987. _ _ , "Signal acquisition and tracking with adaptive arrays in the digital mobile radio system IS-54 with flat fading," IEEE Trans. Veh. Technol., Nov. 1993. J. H. Winters, 1. Salz, and R. D. Gitlin, "The impact of antenna diversity on the capacity of wireless communication systems," IEEE Trans. Commun., Apr. 1994. J. Salz and J. H. Winters, "Effect of fading correlation on adaptive arrays in digital wireless communications," IEEE Trans. Veh. Technol., vol. 43, pp. 1049-1057, Nov. 1994. R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. G. 1. Foschini and J. Salz, "Digital communications over fading radio channels," Bell Syst. Tech. J., vol. 62, pp. 429-456, Feb .. 1983: J. H. Winters, "Switched diversity with feedback for f)PSK mobile radio systems," IEEE Trans. Veh. Technol., vol. VT-32, pp. 134-150, Feb. 1983.
The Range Increase of Adaptive Versus Phased Arrays in Mobile Radio Systems Jack H. Winters, Fellow, IEEE, and Michael J. Gans Abstract-In this paper, we compare the increase in range with multiple-antenna base stations using adaptive array combining to that of phased array combining. With adaptive arrays, the received signals at the antennas are combined to maximize signalto-interference-plus-noise ratio (SINR) rather than only form a directed beam. Although more complex to implement, adaptive arrays have the advantage of higher diversity gain and antenna gain that is not limited by the scattering angle of the multipath at the mobile. Here, we use computer simulation to illustrate these advantages for range increase in both narrow-band and spreadspectrum mobile radio systems. For example, our results show that for a 3° scattering angle (typical in urban areas), a 100element array base station can increase the range 2.8 and 5.5-fold with a phased array and an adaptive array, respectively. Also, for this scattering angle, the range increase of a phased array with 100 elements can be achieved by an adaptive array with only ten elements. Index Terms-Adaptive arrays, mobile communications, multipath channels, phased arrays.
M
1.
INTRODUCTION
ULTIPLE antennas at the base station can provide increased received signal gain and, thus, range in mobile radio systems. Two approaches for combining the received signals are the phased array, which creates an antenna beam directed at the mobile, and the adaptive array, which maximizes signal-to-interference-plus-noise ratio (SINR). Here, we compare the range increase of phased arrays to that of the more complex adaptive array technique for both narrow-band and spread-spectrum systems. Previous papers have studied the increase in gain with phased arrays [1]-[6]. With phased arrays, the signals received by each antenna are weighted and combined to create a beam in the direction of the mobile. The same performance can also be achieved by sectorized antennas, whereby a different antenna is used to form each beam. As the number of antennas increases, the received signal gain (range) increases proportionally to the number of antennas, but only until the beamwidth of the array is equal to that of the angle of multipath scattering around the mobile. Beyond that point, the increased gain of more antennas is reduced by the loss of power from scatterers outside the beamwidth. The range can even be reduced with narrower beamwidths because the resulting reduction in delay spread can cause a loss of diversity
Manuscript received September 19, 1994; revised July 19, 1998. J. H. Winters is with AT&T Labs-Research, Red Bank, NJ 07701 USA. M. J. Gans is with Lucent Bell Labs, Holmdel, NJ 07733 USA. Publisher Item Identifier S 0018-9545(99)01067-1.
gain in systems using equalization, e.g., in spread-spectrum systems using a RAKE receiver. This limitation in range increase can be overcome by the use of adaptive arrays [5]-[9]. With adaptive arrays, the signals received by each antenna are weighted and combined to maximize the output SINR. Although the most widely studied advantage of adaptive arrays is interference suppression [7J-[ 10], maximizing SINR also forms an antenna pattern matched to the wavefront (which is not a plane wave for nonzero scattering angle) and therefore provides a range increase that is not limited by the scattering angle. In addition, adaptive arrays can provide higher diversity gain than phased arrays, since all the receive antennas can be used for diversity combining. Thus, for a given number of antennas. adaptive arrays can provide greater range, or require fewer antennas to achieve a given range. In this paper, we describe the limitations of phased arrays for range increase and describe how these limitations can be overcome using adaptive arrays.' We use computer simulation to illustrate our results for the range increase in both narrowband and spread-spectrum mobile radio systems. For example, our results show that for a 30 scattering angle, a l Otl-elcmcnt array base station can increase the range 2.8 and 5.5- fold with a phased array and an adaptive array, respectively. Also. for this scattering angle, the range increase of a phased array with 100 elements can be achieved by an adaptive array with only ten elements. In Section II, we discuss the theoretical performance of phased and adaptive arrays. We present a mobile radio system model and illustrate the performance results by computer simulation in Section III. II.
DESCRIPTION OF PHASED AND ADAPTIVE ARRAYS
A. Phased Array
Fig. 1 shows a block diagram of a phased array with omnidirectional elements linearly spaced at >"/2, where X is the signal wavelength. The signals received by the antennas are weighted and combined to form a beam at angle ¢, i.e., the signal at the i th antenna is phase shifted by IT (i - 1) sin (P ~ 't == 1,··· .Ad.
For the mobile radio base station, the antenna beam should be narrow in elevation and the antenna characteristics should be independent of azimuth. A narrow elevation angle can be I Note that we consider range increase as a convenient way to express the effect of gain increase, and it also corresponds to a decrease in required number of base stations to cover a given area.
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 48, No.2, pp. 353-362, March 1999.
525
signals should also be weighted by the voltage gain in the given direction to maximize signal-to-noise ratio (SNR) in the array output. These weighted signals are summed to generate the array output, with the output SNR for a beam with direction ¢ given by
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created by using a vertical array of antenna elements for each horizontal element. The azimuth dependence can be reduced by placing the linear elements on four panels in a square, as shown in Fig. 2(a) [11]. However, a cylindrical array, as shown in Fig. 2(b), is usually used to create azimuth independence. Each antenna element is typically spaced at A/2, since smaller spacing reduces gain by creating a wider beamwidth with increased mutual coupling, while wider spacing can also reduce gain by decreasing the beamwidth and creating grating lobes, i.e., gain in directions other than the desired angle-ofarrival. The effect of antenna spacing on mutual coupling is studied in Appendix A. To create a beam in a given direction, the signals from the antenna elements are cophased, based on a plane wave arrival. Since to reduce mutual coupling between elements, each element should have higher gain in the direction pointing away from the center of the cylinder (see Appendix A), the
i,
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is the complex received signal voltage at antenna
¢) is the expected (based on antenna location) antenna
voltage gain and phase (relative to the other antennas) for a signal arriving from angle rjJ, and the superscript * denotes complex conjugate. The weights can be implemented at radio frequency (RF) by different cable lengths for the fixed phase offsets and fixed attenuators for the amplitude weighting. The weighted signals for each beam are then combined, with a separate combiner and signal for each. beam. For each mobile radio user, the receiver then selects the beam output with the largest power to use for signal demodulation. However, this technique can require a large amount of hardware, including amplifiers, with large !VI, but the complexity can be reduced somewhat by combining only a portion of the antenna outputs-the signals from the antennas with the largest gain in a given direction-for each beam. Alternatively, the signal from each antenna can be brought to baseband and analog-digital (AID) converted, with the combining done in software. Although this method is similar to adaptive array processing, with the phased array the combining software needs to determine only one parameter, the angle-of-arrival ¢ (which changes slowly with time), for each mobile radio user. The same performance as the phased array can be achieved by using sectorized antennas, i.e., separate antennas for each beam, as is currently done at many mobile radio base stations. However, to create uniform coverage using sectorized antennas or phased arrays with predetermined (fixed) beams, overlapping beams should be used. (This is also useful for obtaining diversity-see below.) This doubles the number of antennas (with sectorized antennas) or the combining hardware (with phased arrays with fixed beams) without increasing the gain. Arrays increase the range by providing additional received signal gain due to two factors-antenna gain and diversity gain. With an M -element phased array and a point source, the antenna gain is lVI, neglecting mutual coupling (see Appendix A). The range increase is the gain raised to the inverse of the propagation loss exponent '"'(, typically a fourth power loss. Thus, with a point source, the range increase due to the antenna gain of an .1\11 -element array is MIlT. However, signal scattering around the mobile means that the signal received at the base station cannot always be considered as coming from a point source. As shown in Fig. 3, with scattering the signal arrives from a range of angles, called the scattering angle. Typically, the mobile signal is scattered mainly by objects within 1000 ft of the mobile,
526
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Base Station Fig. 3. Mobile radio environment with scattering around the mobile. where all signals from a mobile arrive within a scattering angle II
Fig. 4.
but this distance can vary widely, e.g.. with reflections off mountains 112]. Furthermore, this scattering angle increases with decreasing base-station height. Measured results for rural areas with 130-ft antenna heights show scattering angles of only a few tenths of a degree, while suburban and urban areas have much larger scattering angles r 131. Measured results in urban areas of Tokyo, Japan. for ranges up to 7 km [141, show a 3° scattering angle at a 50-m antenna height increasing to 360° at a l-m height (as on the mobile). In addition, digital mobile radio systems in North America (IS-136) and Europe (GSM) are designed to handle delay spreads up to 41 and 16 ItS, respectively, which, with an 8-mi cell radius. correspond to scattering angles of 52° and 21 0, respectively. Also, these scattering angles are for 900-MHz mobile radio systems, while at 2 GHz the range is reduced by about 509c (from the Hata model [15], for an antenna height of 50 m at the base station and 1 m at the mobile, medium-small city, and 8-mi cell radius), corresponding to a two-fold scattering angle increase. We expect that microcells will have even larger scattering angles because of the lower antenna height. Here. we do not consider what the likely distribution of scattering angles will be for any given system, but show results obtained for a wide range of scattering angles. Since receive signal power is lost when the beamwidth, which is approximately 360° /1\,1 (for a cylindrical array), is less than the scattering angle, the signal gain will be less than J.\;1 in the phased array with large enough 1\1. For example, for a uniform distribution of power within a scattering angle of a degrees, the maximum signal gain is given by an array with "Ai! == 360/ Q elements. Additional elements increase the antenna gain, but the power lost outside the beam reduces the signal gain by the same amount (under the uniform power distribution assumption). Thus, with phased arrays the signal gain, and the corresponding range increase, is limited. The other factor for receive signal gain is the diversity gain. Multipath fading results in a higher average output SNR required to achieve a given average receiver performance (e.g.,
Cylindrical array using of angle diversity.
BER in digital systems) than without fading. The fading in the output signal can be reduced by using multiple receive antennas and combining the received signals. We define diversity gain as the improvement in link margin beyond the factor of AI for array gain. For example, for a 10- 2 BER averaged over Rayleigh fading with coherent detection of PSK, a 9.5dB higher average output SNR is required than without fading. Two antennas provide up to a 5.4-dB diversity gain, while 3, 4, and 6 antennas provide up to 6.8,7.6. and 8.3 dB, respectively, with maximal ratio combining. Thus, six antennas can provide within 1.2 dB of the maximum diversity gain (i.e., the 9.5-dB gain achieved when the fading is eliminated). However, to achieve the full diversity gain, the fading at the antennas must be nearly independent. This requires that the spacing between antennas is at least the distance such that the beamwidth of an antenna with this aperture is approximately the scattering angle. For example, a spacing of lO-20A is used for the typical scattering angle of a few degrees [12], [14], [16]. For a cylindrical phased array, such an antenna spacing between elements is impractical and would create numerous grating lobes without providing the antenna gain commensurate with the diameter of the array (or providing diversity gain). However, when the beamwidth of the array is comparable to the scattering angle (i.e., the total array aperture size corresponds to a beamwidth given by the scattering angle), different beams can cover part of the same scattering angle and thereby angle diversity can be used [4], [13], as shown in Fig. 4. For the square array, another set of flat arrays could be spaced lO-20A apart on each side to provide diversity, as shown in Fig. 5. Note that this is not practical with cylindrical arrays, as the arrays would partially block each other. Similarly, to provide diversity with sectorized antennas, a separate set of antennas can be spaced lO-20A apart (as is used today) with overlapping sectors to provide more uniform coverage over all azimuth angles. In all cases, though, diversity gain requires additional hardware. To minimize the added cost, usually only dual diversity with selection combining is considered. Note that for the example case of a 10- 2 BER,
527
1\11 -element adaptive array is given by
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Square array using space diversity.
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selection diversity with two antennas provides only about 3.9 dB of the maximum-possible 9.S-dB diversity gain (which is also I.? dB less than maximal ratio combining with two antennas). Frequency-selective fading due to delay spread can also be used to provide diversity by using equalization [9] in narrow-band systems, or a RAKE receiver in spread-spectrum systems [17]. In this case, the diversity gain of additional antennas is reduced. For example, a three-finger RAKE is used in the IS-95 CDMA system (three fingers on the downlink, but four fingers on the uplink). With received signal energy uniformly distributed over three code symbol periods (2.4 J-Ls), maximal ratio combining of the three fingers provides three-fold diversity, ora 6.8-dB diversity gain at a 10- 2 BER, and dual antenna diversity provides up to 1.5 dB (the overall combining is equivalent to six-branch maximal ratio combining) of the remaining 2.7-dB maximum diversity gain. Note, however, that, compared to a narrow-band receiver, one finger of this CDMA receiver is 4.8 dB lower in signal power, i.e., the RAKE receiver does not give any increase in average SNR (antenna gain). Finally, note that beamwidths smaller than the scattering angle can reduce the delay spread, and therefore the diversity gain, in systems with phased arrays.
B. Adaptive Array With an adaptive array, the received signals are combined to maximize the output SINR. Thus, the array can null interference in narrow-band systems/ (as discussed below), but here we consider only the increase in range due to higher antenna gain. Without interference, the output SNR of an 2 For spread-spectrum systems, nulling of all strong interferers is generally not possible since the number of interferers is typically much greater than the number of antennas.
Although (2) is simpler than the SNR equation for the phased array (1), the adaptive array is more complex to implement because the weights are not fixed, but depend on the received signals. Thus, variable gains and phase shifters are needed for each signal on every antenna. These can be implemented in hardware at RF or IF, or in software at baseband. For the software implementation, the signals from each antenna can also be digitized using block processing. Another complication is the need to acquire and track the weights. As compared to the phased array where the beam or the weights only need to track the angle of the mobile, the adaptive array weights must track the rapid fading of the signal. Algorithms to generate the weights include the constant modulus algorithm (CMA) [18], least-meansquared (LMS) algorithm [19], and the direct matrix inversion (DMI) algorithm [19]. It should be noted, though, that when interference is not a concern, i.e., when range increase is the issue as in this paper, simpler techniques may be possible for determining the weights. With the adapti ve array, though, the array pattern is matched to the multi path wavefront. That is, there is no antenna gain limitation due to multipath scattering angle, as with phased arrays, and an Ail-fold diversity gain can also be obtained. Achieving this diversity gain requires adequate antenna spacing however. With a base-station array oriented broadside to a small angle, a degrees, of scatterers around the mobile and with power arriving uniformly at the base from within n~, the magnitude of the correlation coefficient between two array elements spaced x wavelengths apart is approximately [see also [14], which approximates the envelope correlation Pc(x) by the square of the complex phasor correlation I p(:t) 12 ]
( )I ~ Ipx
sin( 1r 2ax /180) . (1r 2ax /180)
(3)
Thus, an antenna spacing of (360°/1ra) (>../2) is required for independent fading at each antenna, but spacings of about half of this still give low-enough fading correlation «0.7) that nearly the full diversity gain can be achieved. However, even with a spacing of (360° /(7fa))(>"/4), the required array size can be too large. For example, a 3° scattering angle requires a 10-ft antenna spacing at 900 MHz, and, thus, in particular, a 100-element cylindrical array would require a 330-ft diameter. However, since only a few-fold diversity is needed to obtain most of the maximum diversity gain, an array with a diameter of a few times the required antenna spacing (20-30 ft in the above example) should obtain almost all the maximum-possible diversity gain. Finally, we note that, although not studied in this paper, the adaptive array can also suppress interference. With the narrow beams of large arrays, the number of interferers is greatly reduced in both narrow-band and spread-spectrum systems. Since an M -element array can eliminate N interferers with an M - N diversity gain, large arrays can eliminate any significant
528
interference with little loss of di versity or antenna gain. Thus, these arrays can not only greatly increase the range when there is little interference, but they can also be used for future expansion by permitting the capacity to be greatly increased without increasing the number of base stations.
III.
RES ULTS
A. Model To verify and illustrate the above concl usions, we used Monte Carlo simulation with the following model (see Fig. 3). We considered transmission from a mobile to a base station. The multipath model consisted of 20 scatterers uniformly distributed in a circular area of radius T around the mobile. These scatterers had equal transmitted power, with a fourth law power loss from each scatterer to the base station. The phase of each multipath reflection at each antenna was determined from the path length. Recei ved power variation due to shadow fading was not considered. The base-station array was a cylindrical array of J\;1 equally spaced cardioid antennas [20], with each antenna pointing out from the center of the array, and one element at 0°. The mobile was at 90°. Note that for AJ == 2, the mobile at 90° results in equal gain from the two antennas, while with a mobile at 0° only one antenna has nonzero gain. Thus, for ~\1 == 2, the results depend strongly on the angle of the mobile (i.e., dual diversity at 90° versus no diversity at 0°). However. for l\lI 2 -1, the effect of angle is negligible, and therefore this angle was fixed at 90°. We considered spacings between elements of A/'2 or greater, and therefore neglected the effect of mutual coupling (see Appendix A). With the phased array, the weights were set to generate a beam that was pointed directly at the mobile. From (A-8) and (A-IO), these weights are given by
s; (!.l00) = /2 cos { ~ [Sill(21f(i . e- j ( 2 7r r / ).,) sin (2;"1 ( l -
1) II'v!) - l]}
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and the SNR is then given by (1). With the adaptive array, the weights are 8;ec i == 1.···.M and the SNR is given by (2). We consider coherent detection of phase-shift-keyed (PSK) signals, for which the BER is given by 1
'
BER == ~ erfc( jS/1V).
(5)
We used Monte Carlo simulation to determine the BER averaged over 10000 cases. Note that the BER depends on the ratio of transmit power to receive noise power. This ratio was adjusted to obtain a 10- 2 average BER for the baseline case of an omnidirectional transmit antenna with the mobile at a given range and scattering radius. With this ratio and the scattering angle fixed, we generated results for the 1\1[element phased and adaptive arrays, increasing the range until the BER exceeded 10- 2 , thus giving the range increase. All the following results for range increase and diversity gain are referenced to 10- 2 average BER. Note that the increase in range is not strongly dependent on the modulation and detection technique considered, but will vary significantly with the power loss exponent and the BER. Specifically, the range increase will be greater than we show
in the next section if the power loss exponent is less than four or the required BER is less than 10- 2 . We considered both the low data rate case (no delay spread) and the delay spread case. For the delay spread case, the signaldelay for each scattered signal depends on the distance from the mobile to the scatterer plus the distance from the scatterer to each base-station antenna. For the spread-spectrum system with delay spread, we studied the use of a three-finger RAKE receiver for both the phased and adaptive arrays. To simulate the RAKE receiver, the computer program first convolved the delayed impulse of each scatterer with the spread-spectrum correlation function given by
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.
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where tel is the time delay corresponding to the distance from the center of the base station to the mobile. The responses from the 20 scatterers were then summed to obtain the signal at each antenna. These signals were weighted and combined by the phased array weights or the adaptive array weights (s;{'c, . i == 1.···.1\1). Note that the adaptive array weights vary as a function of delay. We then determined the three largest peaks in the output response that were separated by integer multiples of the code rate and combined these three signals to maximize the output SNR. That is, these three peaks were cophased and weighted by their signal amplitudes before combining. For the phased array, we considered three different models. In the first model, we considered a single beam pointed at the mobile, i.e., the phased array weights as given in (4). Thus, our model corresponds to phased array combining with a RAKE receiver after the combiner. followed by maximal ratio combining of the RAKE output. To model the 15-95 CDMA system with a phased array, we also considered a RAKE receiver on each antenna, followed by phased array combining of the RAKE outputs, with the beam direction optimized for each delay [rather than set to 90° as in (4)]. Thus, a separate beam was fanned for each of the RAKE fingers. Finally, we modified the second model to consider the beam direction optimized over M different, equally spaced angles, which models sectorized antennas. For the adaptive array, our model corresponds to a RAKE receiver on each antenna branch, with adaptive array combining of the antenna signals followed by adaptive array combining of the three highest output peaks, with the receiver timing optimized to maximize the output SNR. For the no delay spread case, in our simulations we used a 40000-ft range as the baseline case, with the scattering radius given by the required scattering angle. However, our results can be generalized to any range, as they depend only on the scattering angle and not the absolute values of the range and scattering radius. Therefore, in the next section, we present our results only in terms of the normalized range. Similarly, although we generated results for a one foot wavelength, our results can be generalized to any wavelength. Therefore, our results on antenna spacings are only in tenus of A. Also, for the delay spread case, our simulations used a 1.25-Mbps data rate (as in the 1S-95 CDMA system). The scattering radius was
529
set to 1200 ft (which is typical in mobile radio in suburban and urban areas) which results in a delay spread of three symbols. This radius was chosen because, as shown in the next section, this is the minimum delay spread for which the maximum diversity gain is achieved with the three-finger RAKE receiver. Thus, the scattering radius was chosen to maximize the RAKE diversity gain as well as the effect of a narrow beam width on the performance. Again, our results do not depend on the absolute values of the range and scattering radius and are therefore presented in terms of normalized range and scattering angle. Finally , note that by keeping the scattering radius constant as we increase the range (which would be typical in mobile radio), the scattering angle decreases. For example, a 10° scattering angle with the baseline case is only about 3° with a three-fold range increase. With fixed scattering radius, the predicted range increase discussed in the previous section must therefore be modified. It was noted before that, for a given scattering angle 0', the maximum gain is 360 /0' , and therefore the maximum range R, normalized to the omnidirectionalantenna range R o, is given by
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n
(7)
But since the scattering radius is kept constant, the scattering angle at range R is less than the baseline scattering angle no at Ro. specifically
(8) Therefore, from (7) and (8) , the maximum range increase is given by
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=
(360/0'0)1 / 3]. This increase is [with the corresponding M greater than the maximum range increase of (360/n)1 /4 for the fixed scattering angle case, e.g., the range increase is 4.9 for 0'0 = 3° versus 3.3 for 0' = 3° .
B. Results for Range Increase Fig . 6 shows the normalized maximum range versus the number of antenna elements for phased and adaptive arrays with >./2 antenna spacing, neglecting the delay spread. Results are shown for different fixed scattering radii, with the scattering angle for the baseline case of one antenna element given. We also show the theoretical range due to the antenna gain (M 1 / 4 ) without diversity, and due to antenna gain and M -fold diversity . Also, the predicted maximum range with phased arrays is shown. With the phased array, the range is shown to be limited to the predicted range limitation. However, the range improvement is degraded due to the scattering angle for M less than the theoretical value corresponding to the range limitation, and it requires many times more antennas to actually reach this limitation. For example, with a 20° scattering angle, the predicted range limitation is 2.6, corresponding to 46 antennas, but with 46 antennas the range is only 2.3 . Note that at a range
6
- _ . Adaptive Array . . . . . Phased Array Theory
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of 2.6, the scattering angle is reduced to about 80 for the 20° baseline curve. For the adaptive array , the range exceeds the no-diversity theoretical range for all scattering angles. due to antenna diversity . The diversity gain incre ases with the scattering angle and M, as expected. However, the diversity gain does not increase for scattering angles greater than about 20 0 • Thus, because the adaptive array has greater range with increased scattering angle, the difference between the adaptive and phased array increases dramatically with scattering angle. Next consider the effect of antenna spacing. With the phased array, our results show that the range does not increa se with wider spacing, and, in fact, the range decreases if the spacing is wide enough. With the adaptive array, the range increases with antenna spacing, up to that corresponding to the maximum diversity gain. Fig . 7 shows the increase in range with spacing for M = 2, 10, and 100 and baseline scattering angles of 3°, 10°, and 20°. Theoretical results for the range with maximum diversity gain are also shown. With baseline scattering angle s of 10° or more, the maximum range can be achieved with a spacing of about 10>'. Note that a baseline scattering angle of 10° corresponds to scattering angles of 6.20 , 3.4° , and 1.8° at the maximum range with J.\1 = 2, 10, and 100 , respectively. Consider the extreme example of a very large array. For-a baseline scattering angle of 30, with 100 elements a spacing of 10>' achieves a 5.IS-range increase versus the maximum 5.46 , even though the scattering angle at this range is only 0.58° (the array diameter would be 350 ft at 900 MHz and 160 ft at 2 GHz). Thus, with large arrays the antenna spacing can be much less than that required with two antennas to achieve nearly the full diversity gain . As a further example, a 100-element array increases the range about 2.8 times with a phased array and a scattering angle at the maximum range of 3° (about an 8.4°
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Fig. 9. Normalized maximum range versus the number of antenna elements for phased and adap tive arrays with .\/2 antenna spacing and a three-finger RA KE recei ver.
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Fig. 8. Diversity gain versus the maximum delay spread for a three- finger RAKE with a single antenna at the base station.
baseline scattering angle ) versus 5.5 times for an adaptive array with 10,,\ antenna spacing. Also, for this scattering angle , the range increase of a phased array with 100 elements can be achieved by an adapti ve array with only ten elements. For the delay spread case with the RAKE receiver , let us first consider the effect of the scattering radius on the diversity gain of the RAKE recei ver. Fig . 8 shows the diversity gain versus the maxim um delay spread for a three-finger RAKE with a single antenna at the base station . For our model, the maximum delay spread is given by twice the scattering
radius in symbol periods. That is, the minimum delay is given by the delay from the mobile to the base station. while the maximum dela y is given by a scatterer at the far edge of the scattering radiu s along the line between the mobile to the base station. The maximum delay is therefore the propagation time corresponding to twice the scattering radius. The diversity gain is seen in Fig. 8 to be within 0.1 dB of the maximum possible diversit y gain (three-fold diversity) for scattering radii corresponding to delay spreads of three symbols or greater. Therefore, in our simulation s, we set the scattering radius to three symbols. Note that with our model , the maximum delay spread does not decrease with the beam width of the array because the maximum delay variation is along the line between the mobile and the base station. Fig. 9 shows the normali zed maximum range versus the number of antenna elements for phased (with the IS-95 COMA system model ) and adapti ve arrays with ,,\ /2 antenna spacing and a three -finger RAKE receiver. As in Fig. 6, results are shown for differ ent fixed scattering radii, with the scattering angle for the baseline case of one antenna element given. However, in Fig. 9 the baseline case includes a three-finger RAKE with its 6.8-dB diversity gain. Thus, the actual range in the baseline case is 1.48 (= 106 . 8 /-<°) times greater than in Fig. 6. We also show the theoretical range increase due to antenna gain (lvI l /-<) and due to antenna gain and 3M-fold diversity (versus three-fold diver sity due to the RAKE receiver). With the phased array and a single beam pointed at the mobile , the range limitation is similar to that of the narrowband system (Fig. 6). However , with a separate beam for each RAKE finger, Fig. 9 shows that the range limitation is negligible for scattering angle s less than 20°, but there is
53 1
degradation in the range increase for scattering angles of 45 ° and 60° with more than about 40 antennas. This degradation is somewhat larger when fixed sectorized antennas, rather than continuously adjustable phased array antennas, are used, as Fig. 9 shows for the case of a 60° scattering angle. With the adaptive array, the range exceeds the theoretical range due to antenna gain and three-fold diversity. showing the additional diversity gain. Thus, there is a significant improvement with adaptive arrays for large scattering angles and large NI. Furthermore, in all cases the diversity gain of adaptive arrays increases with larger spacing, as shown in Fig. 9 for SA spacing with scattering angles of 3° to 0°. IY. CONCLUSIONS In this paper, we have compared the increase in range with multiple-antenna base stations using adaptive array combining to that of phased array combining. Our computer simulation considered a multi path model with a uniform distribution of scatterers within a given radius around the mobile, and determined the increase in range with arrays for 10- 2 average BER with coherent detection of PSK. -From our results we make the following conclusions. • Phased arrays were shown to have a range increase limitation given by the scattering angle. For scattering angles of a few tenths of a degree (typical in rural areas), this limitation is significant only for arrays with more than 100 elements, while with larger scattering angles (typical in suburban and urban areas), the range increase limitation can occur with far fewer elements. • For spread-spectrum systems, using a RAKE receiver with phased arrays, the maximum range increase degradation was much less than that of narrow-band systems. • In both narrow-band and spread-spectrum systems, adaptive arrays had no range limitation and could achieve diversity gain with A/2 antenna spacing with sufficiently many elements. Almost full diversity gain could be achieved with large arrays with antenna spacings of only a few wavelengths for scattering angles as low as 1°.
A. Effect of Antenna Spacing on Mutual Coupling With an NI-element array, the maximum gain is A1 without mutual coupling. Because of mutual coupling, however, this gain will vary with antenna spacing. Specifically, this gain is given by the directivity, i.e., the ratio of the peak to average gain for a signal arriving with a flat wavefront [20] maxli,'" IE(B,¢W
-
-1 41f
17I"1 0
0
27r
sin eI E (B, ¢) 12 dB d¢
E((),¢) == Ee(B)A(¢)
(A-2)
where E e (B) and A( ¢) are the variation in gain with elevation angle and azimuth angle, respectively. Thus, from (A-I) and (A-2), the directivity is given by
D ==
.l
l
IEe (Omax)A (cPlnax)12 l?7r sinBIEe(BW dB - jA(¢W d¢
1r
4~ 0
(A-3)
.0
where Omax and ¢max are the peak-gain elevation and azimuth angles, respectively. If we consider the typical base-station antenna with a very narrow elevation beamwidth, then the directivity can be expressed as
D
~G ~
IA(¢nlax)1
el
~
2
(A-4)
?
t11"IA(¢)1 2 d¢
21r .J0
where Gel is the gain due to the elevation beamwidth. Since we are interested in the effect of mutual coupling on the horizontal antenna spacing, we will set Gel == 1 for simplicity. Now, the voltage gain for a signal arriving at angle ¢ onto an M-element array (using maximal ratio combining to maximize gain) is given by
~ S,((P)Si'(¢maxl
A(¢) =
(A-5)
where s, (¢) is the complex signal gain (with phase relative to the other antennas) at the zth antenna element. We have assumed that the elevation beamwidth is very narrow, so that waves arriving at angles far enough from the equator to affect the relative phase between elements have negligible effect in the directivity calculation. Thus, the directivity is given by 1\/
Ll s i(¢ maxI
2
D ==
i==l
1
27r
Jo(27I" t;Si(¢)Si' (¢max) 1
AI
(A-6)
2 1
d¢
First consider a linear array of omnidirectional elements with narrow elevation' beamwidth, as with a vertical colinear array of dipoles (YCAD). In this case, the complex element responses are given by
ApPENDIX A
D -
i.e.,
(A-I)
where E ( e, 4» is the voltage gain at elevation angle e and azimuth angle 4>. For the base-station antennas, we will assume that the variation in gain with elevation angle is independent of azimuth,
for i
== 1, ... , NI
(A-7)
where d is the element spacing and 4> is the angle relative to broadside. Fig. 10 shows the directivity versus antenna spacing for an M -element array with the desired angle of arrival at broadside, cPmax == O. There are large fluctuations in directivity with antenna spacing (particularly at spacings which correspond to the onset of new grating lobes), showing substantial mutual coupling, with a spacing of A having about half the gain in decibels of a spacing of A/2. For a cylindrical array of equally spaced VCAD's with radius r and element 1 at 0°, the complex receive signal response at the 'ith element for a signal arriving at angle cP
532
M=100
20
Fig. 12. Cy lind rical array using cardio id-pattern antennas. with each e le ment pointed away from the center of the array.
20
iii'
15
::<:!-
Spacing (A)
z~ u
Fig. 10. Directi vity vers us antenna spacing for an .\I -ele rnent linear array with 0 mi'l..'\ 0° .
~
=
i:5 10
0.5
20
1
1.5
2
Spacing (A)
iii'
Fig. 13. Directivity versus antenna spac ing for an .\ 1-e lernent cy lindrical array with cardio id elements and C'nr n x = gO o.
::<:!-
~
.~
u
e
i:5
10
0.5
1
1.5
2
Spacing (A)
Fig . II. Directivity versus antenna spacing for an .\I -elernent cy lindri cal array with dipol e elements and Cl m a , = go o
is given by
s,(¢ ) = ej(2rrr / ,\)cos [d> -
(2 7C (i - l l /M l],
for i = I .·· · . /v£.
greater than >. /2 , the gain variation with spac ing is large for small M, but is less than 2 dB for M = 100 . Thus, with the cylindrical array, the mutual coupling becomes much less than that of the linear array as M increases. Thi s can be considered to be a result of adjacent elements being similar to endfire or broadside arrays, depending on their location around the circumference. Since grating lobe s arise at different spacings for endfire than for broadside array s. the mutual coupling fluctuations are somewhat reduced. To decrease the mutual coupling for small M , consider the use of cardioid pattern antennas (with narrow elevation beamwidth), rather than VCAD ' s, with each element pointed away from the center of the array (see Fig . 12) . With the card ioid antenna, the voltage gain for the zth element at angle dJ is given by [20]
(A-8)
v",(¢) = J2cos
(A-9)
Fig . 11 shows the directivity versus antenna spacing for an = 90 0 . Note that for spacings
M -elernent array with ¢ max
1)/M ] - 1)) ,
for i = 1. . " , iVI.
The spacing between adjacent elements is given by
d = 2r·sin ~ . M
(~ (COS [d> - 27r( i -
(A 10)
Fig. 13 shows the directivity versus antenna spacing for an 90° . Note that for spacing greater than >. /2, the gain variation with spacing is greatly reduced with small M . Our results show
M -element array with cardioid elements and ¢ max
533
=
that the directivity variation is approximately the same for other values of rPmax as well. Thus, with the cylindrical array of cardioid elements, the mutual coupling generates a gain variation of less than 2 dB for spacings greater than A/2 for all values of M. We will therefore ignore the mutual coupling in our simulations and assume a gain of 1Y1. ACKNOWLEDGMENT
It is a pleasure to acknowledge helpful suggestions by L. J. Greenstein. REFERENCES [1] S. C. Swales, M. A. Beach, D. 1. Edwards, and J. P. McGeehan, "The performance enhancement of multibeam adaptive base-station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Technol., vol. 39, pp. 56-67, Feb. 1990. [2] G. K. Chan, "Effects of sectorization on the spectrum efficiency of cellular radio systems," IEEE Trans. Veh. Technol., vol. 41, pp. 217-225, Aug. 1992. [3] 1. C. Liberti and T. S. Rappaport. "Reverse channel performance improvements in CDMA cellular communication systems employing adaptive antennas," in Proc. Globecom '93, Houston. TX, Nov. 29-Dec. 2, 1993, pp. 42-47. [4] S. P. Stapleton and G. S. Quon, "A cellular base station phased array antenna system," in Proc. Veh. Tee/mol. Conf., Secaucus, NJ, May 18-20, 1993, pp. 93-96. [51 B. Khalaj, A. Paulraj, and T. Kailath, "Antenna arrays for CDMA systems with multipath," in Proc. Milcom '93, Boston, MA. pp. 624-628. [6] A. F. Naguib and A. Paulraj, "Performance of CDMA cellular networks with base-station antenna arrays," in Proc. Int. Zurich Seminar on Digital Communications, Mar. 1994, pp. 87-100. [7] J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE 1. Select. Areas Commun., vol. SAC-2, pp. 528-539, July 1984. [8] _ _ , "Signal acquisition and tracking with adaptive arrays in the digital mobile radio system IS-54 with flat fading," IEEE Trans. Veh. Technol., vol. 42, pp. 377-384, Nov. 1993. [9] 1. H. Winters, 1. Salz, and R. D. Gitlin, "The impact of antenna diversity on the capacity of wireless communication systems," IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1740-1751, 1994. [10] T. Ohgane, H. Sasaoka, N. Matsuzawa, K. Tekeda, and T. Shimura, "A development of GMSKffDMA system with CMA adaptive array for land mobile communications," in Proc. Veh. Techno!. Conf, May 1991, pp. 172-177. [11] "Smart antennas," Northern Telecom product brochure. 1992. [12] W. C.- Y. Lee, "Effects on correlation between two mobile radio basestation antennas," IEEE Trans. Commun., vol. COM-21, pp. 1214-1224, Nov. 1973.
[13] W. C. Jakes, 1r. et al., Microwave Mobile Communications. New York: Wiley, 1974. [14] Y. Yamada, K. Kagoshima, and K. Tsunekawa, "Diversity antennas for base and mobile stations in land mobile communication systems," IEICE Trans., vol. E 74, pp. 3202-3209, Oct. 1991. [15] M. Hata, "Empirical formula for propagation loss in land mobile radio," IEEE Trans. Veh. Technol., vol. VT-29, pp. 3] 7-335, Aug. 1980. [16] 1. Salz and 1. H. Winters, "Effect of fading correlation on adaptive arrays' in digital wireless communications," IEEE Trans. Veh. Technol., vol. 43, pp. 1049-1057, Nov. 1994. [17] R. Price and P. E. Green, "A communication technique for multipath channels," Proc. IRE, vol. 46, pp. 555-570, Mar. 1958. [18] 1. R. Treichler and B. G. Agee, "A new approach to multipath correction of constant modulus signals," IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-31. pp. 459-472, Apr. 1983. [19] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. [20J W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: 1981, pp. 115, 116, and 141.
534
A Comparison of Two Systems for Downlink Communication with Base Station Antenna Arrays Per Zetterberg Abstract-In this paper, we will compare the performance of the downlink of two systems with antenna arrays at the base stations. One system uses a third of the available spectrum per cell 1/3, but reuses the channels three times within each cell. The other system reuses all the spectrum in all cells 111, but does not reuse channels within a cell. Thus, the maximum number of users in the two systems is the same. In order to account for the interference, the 1/3 system directs antenna pattern nulls against same cell cochannel users, while the 1/1 system directs nulls against cochannel users in other cells. The performances of the two systems are compared as a function of the azimuth angular width (seen from the base) of the multi paths generated at the mobile, using simulations and analytical derivations. The results herein indicate that the system with 1/1 reuse has a higher performance than the system with 1/3 reuse if fast intercell handover is used or if high dynamic range power control, nulling and synchronization is employed. Index Terms- Adaptive arrays, antenna arrays, array signal processing, land mobile radio cellular systems, radio spectrum management, space division multiplexing.
T
1. INTRODUCTION
HE USE of antenna arrays at base stations has been proposed as a means of increasing the capacity (spectrum efficiency) of mobile cellular networks. For high-mobility systems such as GSM, AMPS, and D-AMPS, the capacity enhancement achievable with an antenna array is likely to be given by the downlink enhancement. This is the case because the processing applicable to the downlink is limited to beam steering types of techniques, while in the uplink high-performance diversity combining techniques can be used. Downlink beamfonning has previously been studied in the papers [4]-[7], [10], (16], and [21]. The transmit techniques in [5J and [6] are based on instantaneous channel knowledge while the techniques in [4], [7], [16], and [21] are based on knowledge of the (complex) fading correlation between the antenna elements. Multicell systems are treated in [10], [16], and [21]. The three papers all show that substantial capacity gains can be achieved but that the downlink performance degrades with increasing angular dispersion (spreading of the signal energy in azimuth angle seen from the base). Reduced channel reuse distances (cluster size) are investigated in all Manuscript received December 15, 1995; revised March 30, 1998. This work was supported in part by the Swedish National Board for Industrial and Technical Development (NUTEK). Part of this work was done at the Center for PersonKommunikation (CPK), Aalborg University, Denmark. The author was with the Royal Institute of Technology (KTH), Stockholm, Sweden. He is now with Radio Design AB, S-164 28 Kista, Sweden. Publisher Item Identifier S 0018-9545(99)05739-4.
three papers but [21] also looks at allocating multiple mobiles on the same channel within the same cell. The results in [21] indicate that multiple mobiles in a cell are more efficient than reduced reuse distances. However, directed nulls against users in other cells is not considered in [21]. In this paper we compare a system which uses a third of the available spectrum in a 1200 sector cell (1/3) with a system that uses all the available spectrum (1/1) in all 1200 sectors. However, the 1/3 system reuses the channels three times within each cell and thus the maximum number of users in the two systems is identical. The 1/1 system is considered both in versions with and versions without intercell nulls (i.e., nulls directed toward cochannel users in other cells). The influence of uplink power control is also investigated. The uplink power control actually influences the downlink performance since it affects which interfering mobiles the base is able to identify. i.e., determine their presence and their direction of arrival. Simulations are performed assuming the air interface of GSM. Plots of power control ranges and downlink outage probabilities are given. The outage probabilities of the two systems are compared as a function of the azimuth angular width (seen from the base) of the multipaths propagation. This is important since the performance of beam steering arrays is very sensitive to the multipath propagation. The results herein indicate that the approach with 1/1 reuse has higher performance than the approach with 1/3 reuse if fast intercell handover is used or if high dynamic range power control, nulling, and synchronization is employed. This paper is original in that it compares different channel planning and beamforming strategies, and that it provides analytical results for systems employing nulling. While being fairly close to the simulation results the analytical expressions take only a few minutes to evaluate, whereas the simulations take several days to perform. Furthermore, the derivation of the analytical results gives insight into the systems. The paper is organized as follows: the cellular geometry and the propagation model are introduced in Section II. An overview of the signal processing performed at the base station is also given in Section II. In Section III, a transmission technique that exploits the propagation model is derived. The concepts of same sector frequency reuse and reduced cluster size (channel reuse distance) are introduced together with their associated algorithms in Section IV. Simulation and analytical results are presented in Section V. Details of the simulations and the analytical expressions are given in Appendixes A and C, respectivel y .
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 48, No.5, pp. 1356-1370, September 1999.
535
----r.
'
i
'/.
-'
0
;J
r
_
- r~::::~:~' ." " -'
·
/
__ _---.L(
• •
120 n
'\,
/
_ ----"'.
Fig. 2.
~
'- L
, . ~-.
--
120
0
The uniform linear array (ULAI and the polar coordinate system,
the complex-v ector valued "array manifold" functions a l ' L(H) and a DL(H) , in up- and downlink, respectively. In the case of a linear uplink array of uniformall y distributed identical antenna elements as depict ed in Fig. 2 below, al'L( H) is given by Fig. I.
Pan of the cellular sys tem.
a I'' L (H)
= P1'1'( H)
l'l . x [ 1. ex p] -.J" 2r.f ~ I' 'L slll (H )/ r·).. . . .
exp] -j2( 1TI.
II . PRELIMINARIES
The following three sections introduce the cellular geometry, the propagation model and the signal process ing at the base station, respectivel y. A. Cellular Systems
The coverage area of a mobile radio syste ms is divided into a network of cells , where each cell is covered by a base station . The cell s are assigned a subset of the available spectrum such that the same frequency cells are sufficiently separated spatially. In the theoretical analysis of such systems it is common to assume hexagonal cells with the base station in the center of the cell , as depicted in Fig. I. Contrary to the usual procedure, the cell labelin g in Fig. I does not refer to the spectra used in the cells but will be used as an aid in describing algorithms and results in the following sections. In this report , the hexagonal cells are also divided into three 1200 sector subcells or sectors. The subcells are covered by the linear arrays . The three subcells of a cell are labeled with the cell number as prefix and "a," "b." or "c" as suffixes . The "a," "b,' and "c" subcells are always oriented upward s, to the right and to the left, respe ctively. Two frequency reuse patterns denoted III and 1/3 are cons idered. In 1/1 reuse, all sectors use all the available bandwidth while in 1/3 reuse, "a," "b.' and "c" subcells use disjunctive subsets of the available spectrum. The sectors that are assumed to significantly disturb the downlink in subcell la are Ib , Ic, 2b, 2c, 3b, 7c, 4a, Sa, and 6a, and 4a, Sa, 6a when III and 1/3 reuse is employed, respectively.
B. Propagation Mod eling Consider a base station which employs the same or two different antenna arrays for up and downlink. The gain and phase of the antenna elements (the latter relative to antenna element number one) assuming a single emitter at azimuth direction B (as seen from the base), are assumed to be given by
-
1 )7rfn ~ 1 Lsill(H)/ (;) ] J
(I )
where pL'L(H), 2l 1.iL, l'": are the element pattern, antenn a spacing, and carrier frequency of the uplink. respectively. The param eter ITI is the number of antenna element s. which for simplicity is assumed to be the same in the up- and downlink array . The multidimen sional complex-valued base-band timedependent downlink impulse response between the antenna array at the base and the single antenna at the mobile is denoted ii Dl (t , T), where t is time and T is delay. The vector of signals transmitted on the multiple antennas of the base antenna array, xDl (t ), to a certain mobile . is given by '
(2) where w is a vector of complex weights and s(t) is the waveform modulated by the transmitted bits. From these assumptions it follow s that the signal received at the mobile, uDl(t ), is given by
where * denote s complex conjugate transpose. If the transmitted signal s( t) is generated from a linear modul ation, it can be written as
(4) 'I
where I(~l are symbols drawn from some complex alphabet, p8( t ) is the modulation pulse shape and Ti, is the symbol period, Combining (3) and (4), the following relation between the symbols transmitted from the base, I~l, and the signal I In the case of transmission to multiple mobile s on the same carrier at the sa me time, the signals to the differ ent mobile s are superpositioned, i.e .,
x DL (t )
536
=
L i w , ._ ,(t ).
sampled at times t = qTb + ~TDL at the mobile, is obtained:
-
1
E == -
(5)
T
n=-oo
where the "taps," h~L(t), are given by
h~L(t) =
1:0
iiDL(t, T)ps(nTb + L}.T
DL
- T)dT.
(6)
l
We define the local-mean symbol energy, at time to as
00
tO
+ (T / 2 )
t==to-(T/2)
E(t) dt
(14)
where T is a time-interval short enough for the ray parameters to remain fixed and long enough for the integration to converge. Combining (9) and (12)-(14) yields
Without loss of generality, the normalizations
(7) and
1:0
for all 6.T.
(9)
n=-oo
Equation (9) follows from (8) if ps( T) is band limited f S 1/2Tb . In practice this is not the case,2 however, we judge that the errors imposed by the assumption is minor. Assuming that the propagation can be described as the sum of JV rays, the impulse response flDL (t. T) can be written as
L ~V
g[ exp(j27TdPL t
+ 13pL)aD L ((-) t)8(T - Tt)
[=1
(10) where (h, 9l, dpL, 13PL, and T1, are the azimuth direction (as seen from the base), complex gain, Doppler frequency, phaseoffset and delay of the lth ray, respectively, and 8(T) is the Dirac function. Assuming that the azimuth angle of the lth ray is ¢l, as seen from the mobile (using the speed vector of the mobile as reference), the Doppler frequency dPL is given by
dPL = fDL COS(cPl)V C
h~L(t)
=L
9l exp(j27rdPL t
l==l
x ps (nT + 6.T D L
-
( 16)
(15) can be rewritten as
E == w*RDLw.
(17)
Let us now define the path gain G as the energy delivered at the desired mobile assuming transmission with a single antenna and unit power. Consider first transmission using antenna element 1 of the array. In this case it follows from (17) that the desired energy at the mobile is given by
[R D L ] L1
( 18)
where [M)r: e denotes element row == r column == e of the matrix M. If transmission is performed using antenna element r we obtain similarly ( 19)
We define G as the energy delivered at the kth mobile assuming transmission with a single antenna in the array-averaged over the elements of the array, i.e.,
G=
~ 1
f
r=l
[RDLL.r
== - Trace{R D L } m
(20) (21)
.
Similarly, (as in the downlink) the vector of uplink signals xUL(t), sampled at times, t == qTb + 6.T C L , at the base, is given by
L ex)
x
VL
(qT b + 6.Tl: L ) ==
(22)
where h~L(t) is given by
L 9l exp(j27rdfL t + j3f L) iV
(12)
h~L(t) =VpUL
Using (5) and (7) the following expression for the energy, per symbol, received at the mobile is obtained:
L
IglI2aDL(Bz)aDLo*(fJz)
n=-oc
+ !BpL)aDL(()z) Tt).
v
L 1=1
(11)
where f DL, u, and c are the downlink carrier frequency, the mobile speed, and the speed of light, respectively. The ray parameters N, ()l, 91, dpL, and Tl vary with time. However, they are assumed to be practically fixed during the time the mobile moves a couple of wavelengths. Combining (6) and (10) yields N
R D L ==
(8)
IpS(T)[2 dr = 1
are introduced. The following discrete time version of (8) is also assumed:
iiDL(t. T) ==
Defining the downlink channel covariance matrix R DL as
l==1
x aUL(fh)ps(nT
+ ~Tl:L
-
Tl)
(23)
00
E(t) =
Tblw·h~L(t)12.
(13)
n=-oo 2 Calculations we have made using the pulse shape of a linear approximation of the GSM modulation have shown that the left side of (9) varies between 0.0094 and 1.0015 using different ~ T.
and peL is the mobile transmit power. From the uplink signals x lJL (qTb + ~TUL), it is possible to estimate the uplink bit stream I~L by exploiting training sequences and channel coding [1], [20] (even if cochannel interference and noise is present). Given J qU L it is then straightforward to estimate
537
h~L(t). Unfortunately, it is impossible to estimate the instantaneous downlink impulse h~L (t) from the corresponding uplink entity h~L (t). The explanation to this can be found in the assumption that the up- and downlink carrier frequency differs by more than 10 MHz, which causes the instantaneous up and downlink impulse responses to become unequal and uncorrelated functions of time [22]. We therefore propose that the downlink beamfonning vectors are calculated using the downlink channel covariance matrix of the desired and cochannel users as input. We also propose that this matrix is obtained by first estimating the uplink channel covariance matrix defined by
R UL =
L
Next, it is assumed that the number of rays N is large the power of all rays given by 91 =
IglI2aUL(fJz)aUL, *(ez)
E{VDL(t)} ==0
(24)
n==-CX)
hUL( 1 ri t )h 'T/U L . *( t ) dt..
R
Trace{R
}
G==----m.
This implies that gain is the same in the up and downlink arrays. 1) Special Case: Flat Rayleigh with Gaussian-Distributed Power: The algorithms and systems introduced in the paper is based on the very general propagation model described above. However, the simulations and the analysis, are based on the assumption of frequency flat channels, Rayleigh-distributed signal envelopes, Gaussian distributed azimuth angular power distribution and uniform linear arrays in both up- and downlink. The frequency flat assumption appears mathematically as for l
== 1, . . . , L
(28)
using the framework above. Combining (12) and (28) yields
h~L(t)
==
vDL(t)ps(nT
+ ~TDL - TO)
(29)
where vDL(t) is given by
vDL(t) ==
L gl exp(j21rdPL t + (jpL)aDL((Jz).
==p
;.00 .-00
1
ICC. V 27fa
2
')
exp(-I/ j(2a-))
+ 1/) aDL. * (f) + 1/) d1/.
(35)
For the case of a linear array of uniformally distributed antenna
R DL
N
(30)
l==l
3This actually requires pUL to be known. Otherwise ROL becomes unknown to the degree of a scalar factor.
= GRDL(f).
a)
(36)
where the row == T, column == e element of the complex matrix-valued function R( f}, a) is defined as [R( fl.
a) ] r.
t!
== exp ( -
21 (J- 2 (e - r ) 2)
X eXI{j(e - T)
2:f~:L
Sill(IJ))
(37)
where ~DL and fDL are the antenna spacing and carrier is given by frequency of the downlink and
a
(27)
pUL'rn
(34)
(25) elements, R DL can be closely approximated as [22]
(26)
_ Trace{R UL}
DL
X aD L (f)
This relationship is obtained from (23) and (24). Similarly as in the downlink case, we define the path gain, as the power received in a single antenna element of the array, assuming unit power is employed in the mobile, averaged over the elements of the array. We also assume that the path gain is reciprocal, i.e., DL
(33)
The assumption of Gaussian-distributed Hi yields
and then employing one of the four methods listed in Appendix D to form an estimate of R DL.3 The uplink channel covariance matrix estimate can be estimated from estimates of the uplink channel by exploiting the following relationship: L:x:J
(32)
E{ vDL(t) (vDL(t)) T} = 0 E{ vDL(t) (vDL(t))*} = R DL
i=l
1 ~.tO+(T/2) R CL = T . f=to-(T/2)
(31)
and the angles fli Gaussian distributed with mean at the position of the mobile, fI, and standard deviation a. From these assumptions it follows that yDL (t) is a complex Gaussian zero-mean circular symmetric random vector with covariance matrix R DL. i.e.,
lV
pUL
1ft
a ==
7f2~DL fDL
c x 90°
cos( ())a.
(38)
The path gain G is assumed to be given by
G= (
~) 1p(e)12 L
(39)
'Y
where R is the cell radius, r is the base-mobile distance, 1 is the path-loss exponent, p( B) == cos( f}) is the antenna diagrams of the identical antenna elements (assumed identical in the up- and downlink), and L is a lognormally distributed random variable. This means that 10 log( L) is a normally distributed random variable with mean zero and standard deviation a L. The factor L represents the path-loss variations due to shadowing objects such as buildings and hills. In practice this loss varies with time. In the simulations and analysis of this paper, it is, however, assumed constant for the time-interval considered. In the uplink we obtain similarly
h~L(t) == )pULvUL(t)ps(nT + 6.T UL -
TO)
(40)
where yUL(t) is a zero-mean complex Gaussian random vector. The uplink channel covariance matrix is also given by (36)-(39), but obviously with "DL" replaced by "UL" and multiplied by the mobile transmit power pUL. However, due to the assumption of different up and downlink frequencies, vDL(t) and vDL(t) are uncorrelated.
538
C. Outline of the Signal Processing at the Base Stations
In the previous section, we have argued that the downlink beamforming should be based on the downlink covariance matrix R D L rather than the instantaneous downlink impul se responses. The reason being simply that the instantaneous downlink impulse respon se cannot be estimated by the base without feedback, i.e., using uplink signals only. The previ ous section has also given some indication of how the matrix R D L could be estimated. In this section, we will try to clari fy that issue further. Consider a system consisting of multiple base stations and mobiles . Let us denote the up and downlink channel covariance matrices associated with the propagation between the kth base station and the ith mobile R~,~ and Rr,~ , respectively. Let us further denote tap 1/ of the uplink impulse respon se between the ith mobile and kth base by h ~\ . t ). The path gain is denoted C k . ; . For the purpose of downlink beamsteering, the kth base needs the downlink channel covariance matrix of its desired user and of the cochannel user s. We therefore propose that the kth base station estimates the downlink channel covarian ce matri x of the users in its own and neighb oring cochannel cells by empl oying the "re ceiver module" sketched in Fig. 3. Thi s module estimates the downlink channel covariance matri x of the ith user given the training sequence and frequency hopping pattern of that user (in the case of a TDMA system employi ng frequency hoppin g). Thus, such a receiver module has to be applied for all users within the desired cell as well as those of the neighboring cell s. The input to the module is the sampled uplink signals, and the training sequence and frequency hopping pattern of the considered mobile . The module internally employs seven signal proce ssing units. The first of which is the mixe r. Thi s module down- converts and filters the received signals to produce a base-band signal. The frequencies and filter empl oyed are determined from the frequency hopping pattern and power-spectrum of the considered transmission. Given the base-band signal the "uplink-combiner" estimates the transmitted bit stream (possibly soft-outputs) by employing methods such as in [I] and [20]. The uplink combiner employs the training sequence of the considered user. The output of the uplink combiner is fed into a decoder which utilizes the channel coding to remove errors in the bit estimation process. The decoder also deli vers an indicator i k . i , which shows whether the decoded bits are reliable or not. Given the decoded bit stream, the "p ostdecoding channel estimation " unit forms an estimate of the tran smi ssion channel h~.\. i (t ), assuming that the actuall y transm itted symbols agree with those obtained from the deco der. With h~\ ! (t ) at hand R ~,~ is straightforwardly estimated usin g (2 5). Finall y, the R~,~ estimate is mapped onto an estimate of pF L Rr,~ . In Appendix D,four methods for doing this are proposed. With L known , an estimate of R r ~ can finally be obtained. Thu s, the kth base must somehow be informed of the power used by the ith mobile. In the following section , we introduce a transmission algorithm which uses the channel covariance matrices of the desired and interfering users as input. This method involves
Hopping Sequence-:-- - - - --{
,--
,(
PF
Fig. 3.
--- -- -tI
i k ,1
,
j UL q
.
pULIlDL I
k ,l
Block dia gram of the "receiver module ."
the solution of a generalized eigenvalue problem. In the case of frequency hopping, such a problem must be solved for each time- slot and each mobil e. As a result, the propo sed system will require an extreme amount of computational power. Thus, simplifications for the proposed algorithms have to be made before implementing the sys tem in practice. III. A TRA NSMISSIO N ALGORITHM
Con sider the situation where q base stations are transmitting simultaneously (on the same frequency) to Q mobiles." The three sec tors of a cell-site are regarded as three different base stations. Let the kth mobile be connected to the k base station. The base then transmit s using the weighting vector W k . The purpose of the section is to derive a good choice of W k, assuming that the downlink channel correlation matrix of the desired and some of the cochannel user s are known . The notations introduced in Section II-C are used . Seen from the viewpoint of the Q - 1 cochannel mobiles the signal transmitted from base k to mobile k is interference. Therefore, these mobiles are referred to as the interfered mobile s. Before givin g a criterion function for the choice of W k the ad hoc restriction is made that the "antenna arra y gain ," in the direction of the desired mobil e should be unity, i.e., _ Ek. k =
Ck . k
=
Trace {
Rr.l }
m
(4 1)
From (17 ) E k . k is given by E k. k
• DL = w kR k . kw/c .
(42)
Simulations have indicated that the constraint (4 I) is a good choice although no optimality is claimed. Assume for a moment that no other base station than base station k creates interference. Then the (mean) desired signal (or carrier) power 4This assumption seems to be inconsistent with the allocation of multiple mobile s on the same channel at the same base . However, by interpreting the results of the section pragmatically, the obtained algorithm can be used also in that case. see Section IV·A2.
539
to the (mean) interference signal power, SIR i , at the ith mobile, is given by SIR'i
== E i , i
Ek,i
w~RDLw .:I, 'I, 'l,1,
is made, i.e., the sum of the interference to carrier ratios at these mobiles is assumed to be proportional to II w k 11 2 . This approximation is reasonable if the positions of these mobiles are well separated. The constant in (49) will be obtained from loose reasoning and will vary between systems.
«;*RDL k,i w k Trace{R~t }
where the last equality follows since the ith base station also has a unity antenna array gain toward its desired mobile (all base stations employ the same transmission algorithm). Now, the criterion function for deriving W~: is selected as Wk
==
a~g min {. t
1,=1. :f.k
SIR;-l}"
(44)
Thus, in a sense, WI..: is chosen to minimize the sum of the inverse of the signal to interference ratios of the mobiles in the system. Combining (43) and (44) yields Wk
==
(45)
arg lllin{wZMwk} w
where the matrix, M, is given by
q
M =
'Tn
~
r:
{DL}
Trace R /,
l,=l,:j;k
t
DL
Rk,l,"
(46)
In practice, the interference at the ith mobile will of course not only come from base k, but from all bases. However, the criterion (45), which was derived at ignoring such interference, is still used. The solution to (46), subject to the constraint (41), is given by
W
where e maximizes
==
Trace { R~.\ } --~--e
me*RDL k,k e
DLe e*R k,k e*Me .
(47)
(48)
(RP\,
L.J
'i=p+1, #k
m DL { DL} R k i Trace R. . I
t , I,
rv rv
constant x I
(49)
In this section, we propose a number of systems which all use the transmission algorithm of the previous section. The systems differ in terms of channel allocation, power control, synchronization, and identification of interfered users. In terms of capacity enhancement strategy the systems can be divided into two approaches: same cell frequency reuse (SCFR) and reduced cluster size (RCS). The two groups of systems are described in the following sections.
A. Same Cell Frequency Reuse (SCFR) In this approach the capacity enhancement is achieved by allocating d mobiles within a cell, on the same channel. Thus, the base stations transmit to d mobiles simultaneously on the same channel. To make this feasible, dynamic channel allocation is applied to separate the cochannel users in the cell (with respect to the mobiles dominant direction of arrival). In principle, the deployment of SCFR in the downlink does not imply that the same capacity enhancement strategy has to be used in the uplink. However, if it is used, channel allocation and power control must be employed to combat near far effects in the uplink processing. Since some cellular systems dictate a one to one relation between uplink and downlink channels, e.g., GSM, we assume that the mobiles simultaneously accessing the same channel in the downlink also do so in the uplink. 1) Channel Allocation and Power Control: The mobiles in the cell are sorted with respect to their path gain to their desired base. Then, they are divided into power groups say r 1 ~ r 2, . . . such that all the mobiles in r 1 are stronger than all the mobiles in r 2 and so on. Power control is applied such that all mobiles on the same channel are received equally strongly (averaged over fast fading). The common power level is given by O.5G m a x +O.5G m in , where G In ax and G In in are the maximum and minimum path gain among the mobiles in a decibel scale. The power groups are allocated in timefrequency space such that their dissimilar power does not negatively affect the uplink processing. The dynamic channel allocation algorithm of [21] is applied on each of the power groups. This algorithm begins by sorting the mobiles in the power group with respect to their (}k, k angle, where (}k, k is the dominating azimuth direction in the propagation between the kth mobile and its desired base-station. In the special case treated in Section II-Bl, Ok, k is simply the mean of the Gaussian angular power distribution. Assuming that the sorted list of angular positions is "71, ... , tt«, x d» the mobiles corresponding to the angles T/i, "7i+n c ' ••• , "7i+(d-1)n c are allocated to the same channel. If frequency hopping is applied, the same channel users hop together from frequency to frequency.
CPr
It can be shown that e is the generalized eigenvector associated with the matrix pair M), corresponding to the largest eigenvalue, [13]. Methods to compute e may be found in [8], where it should be noted that R~\ is Hermitian and positive semidefinite and M is Hermitian' and positive definite (from the assumptions below). The parameters required to calculate W k are the downlink covariance matrix of the desired and cochannel mobiles and the desired signal power at the cochannel mobiles. In practice, the parameters required for the cochannel mobile will only be known for a subset of them, say mobile 1, ... , p. These mobiles will be referred to as the identified interfered mobiles. In order to account for the remaining mobiles the approximation Q ~
IV. TWO-CAPACITY ENHANCEMENT ApPROACHES
(43)
IrnwZRp,~Wk
-
540
2) User Tracking and Nulling: The users on the same channel within the cell are considered as the identified interfered users in the framework of Section III. Since the considered base station and the base station of the identified interfered user is the same base, it follows that R?~ == RP'~, in (46). The constant in (49) is chosen from the following crude reasoning: Assume that only mobiles in the first tier of cochannel cells are interfered. Assume that the path gains to these mobiles are given by Gk,i == (RID)r where D is the distance between the considered base and the first tier of cochannel base stations. Assume further that these mobiles have received desired power E; i = 1. Geometry yields that the number of significant unidentified interfered mobiles is approximately d x 1 where l == 6 and l == 3 using 1/1 and 1/3 reuse planning, respectively. With these assumptions we obtain q
'"
m
i=P~# Trace{RPr}
L q
(RjD)'Y
,=p+L # ~
"
Fig. 4.
User k Illustration of the considered situation.
p.UL
'mR D L
~
==
PoG~~ i •
(54)
~
k.l.
Trace{ Rr,~ }
is used, where e is a design parameter. If e == 1 then (50)
where the last approximation is accurate if the unidentified interfering users are well spread as seen from the kth base. The constant in (49) is thus chosen as constant == (RI D)r I d.
(51)
B. Reduced Cluster Size (ReS) In this approach the capacity enhancement is obtained by changing the frequency allocation plan, i.e., the cluster size. For instance, by going from a 1/3 to a 1/1 reuse pattern, a threefold capacity enhancement is obtained. The antenna array base stations are employed to compensate for the increased intercell interference. Several versions of this approach are investigated. The versions appear as different power control settings and with or without directed nulls. The development below assumes that uplink cochannel mobiles are downlink cochannel mobiles, i.e., mobiles that are interferers in the uplink are interfered in the downlink. For TDMA systems this implies that the timeslots of the base stations are synchronized. 1) Power Control: Consider the situation with two mobiles: the kth and the ith mobile, as illustrated in Fig. 4. The kth mobile is connected to the kth base station and the ith mobile to the ith base station. The path gain of the desired signal divided by the path gain of the interfering signal in the uplink at base k is given by Gk,kIGk,'i' If the kth and ith mobile use transmit power ?;:L and PiUL, respectively, the quotient of the effective uplink path gains at base k, QPGU L is given by
= Gk, kPi!L
Gk ,t.p.UL· t
(52)
The quotient between the downlink path gains at the i th interfered mobile is given by
Gi , i QPGPL == G ' k, i t
Basek
2
Herein, the power control law
Rr~
(RI D)~(l ell
QPG U L k
Base
QPG U L = k
Gi,i
Gk , ~
.
(55)
Thus, QPG~L == QPGPL. This implies that a mobile which is received strongly at base k, will be vulnerable to transmission from base k. This is a desirable property since it implies that the base will have a good chance of identifying (i.e., estimate the channel covariance matrix) the mobiles with poor downlink signal to interference ratio and subsequently suppress the interference transmitted toward these mobiles. Another advantage with e == 1 is that all mobiles within the cell are received equally strongly at the base and, the adjacent channel interference is therefore small. In practice, the use of e == 1 may be prohibited by the power control range allowed by the mobile. In that case a smaller e has to be employed. 2) Dynamic Channel Allocation: When e -j. 1 the power of the mobiles may vary significantly which causes adjacent channel interference problems. We propose that this problem is solved using dynamic channel allocation in the following manner: Sort the mobiles in the subcell with respect to their path gain to the base G k , k- Divide the mobiles into power groups say r 1, r 2, . .. such that all the mobiles in r 1 are stronger than all the mobiles in r 2 and so on. Finally, allocate the groups in time-frequency space such that the dissimilar power of the power groups does not negatively effect the uplink processing. If e = 1 is used, random channel allocation is simply employed. 3) User Tracking and Nulling: We will consider two versions with respect to nulling: with and without directed nulls. i) With directed nulls: The required entity of identified interfering users in (49) are Rr,~/Gi,i. When power control with e = 1 is used, the following equality holds:
(53)
541
(56)
Since PiULRr~ is obtained from the receiver module, see Section II-C, ~nd Po is a constant common in all cells, the required entity RP~/G"." is obtained without knowledge of
P iUL .
"
If e =I- 1 however, the kth base must be informed of the uplink power employed by the zth mobile. In order to choose the constant in (49) we assume that the base is able to identify interfering users with an effective path gain larger than F m in , and that the value of P 1n in is known by the base. This means that the users i which satisfy (57)
are not identified by the kth base. Equation (57) yields
G. C-t>G ru.,
fJ
1./
-
1
1.1.
T
I=p+l.-:j:.k
(J
<
L
k. i
(58)
RDL
n-1Gf'-1 k.l P IfllllrO Ti,1 -G T J.:. I
i=p+ 1. -:j:.1.. ~ T
R DL G
~
Pin in P()- I C'w- l I
(59)
(60)
T
where r is the number of significant unidentified cochannel users, and C is the median value of C t , ·i. The approximation (59) follows from the assumption that the number of unidentified cochannel users is large and that they are evenly distributed in azimuth angle as seen from the kth base. Based on geometry T is set to 11 in the 1/1 frequency reuse case and 5 with frequency 1/3 reuse. The median value G is assumed known. In the special case of propagation according to Section 11- B 1, it has been estimated that G = 0.1, under the assumptions used in Section V. ii) Without directed nulls: In this case, no interfering mobiles are identified in the framework of Section III, i.e., p = O. It follows that the constant in (49) can be chosen arbitrarily. With this approach synchronized bases are not a requirement. V. RESULTS
The downlink outage probability and uplink power control requirements of the proposed systems are investigated by means of simulations and analytical calculations in the two examples below. The propagation model of Section lIB 1 is employed. Other issues, e.g., uplink dynamic range requirements are treated in [22]. Example 1: A 1/3 same cell frequency reuse (SCFR) system with, d = 3, users per channel is compared with a 1/1 reduced cluster size (RCS) system. The comparison is made by simulations and analytical derivations. The RCS system is considered in four versions: with or without intercell nulls and using e == 1 or e 0.3 (Section IVB 1). The air interface is GSM with frequency hopping and discontinuous transmission [9]. Each simulation considers 48
=
users in each subcell using six 200-kHz carriers with eight time slots multiplexed on each. This is approximately seven times less bandwidth than the corresponding one antenna per sector system would require for the 48 users, [3]. The user activity factor is 0.5, i.e, a user transmits in 50% of its time slots. Linear arrays are of ten antenna elements m == 10, with half a wavelength spacing ti == O.5A are employed to cover the 120-degree sector cells. The propagation model of Section II-B I, applies. The propagation exponent used is 1 == :3.5 and the standard deviation of the lognormal fading is a L == 8. However, the lognormal fading between a mobile and two base stations is correlated with correlation coefficient 0.5, i.e., E{ 10 log(L i 1, JIO log(L i 2 . d} = a~I)/2. The angular spread a is independent of the distance from the base. The three values a = 0°, J O• and GO are considered. An analysis of measurements collected in an environment in an area of three-five stories high buildings and a high base-station installation (40 m) has indicated that this model yields reasonable performance predictions [22]. The handover is based on geometry and antenna pattern. This means that a mobile is connected to its geometrically closest base with a bias according to the antenna element patterns (a more precise description is given in item #1 of Appendix A). In practice this handover strategy corresponds to a case where the signal strength measurements are passed through a low-pass filter with a very large-time constant, before being used for handover decisions. The simulations are static in that the positions of the users and the lognormal fading is assumed fixed during the simulation. The SCFR system uses four power groups with twelve mobiles each (see Section IV-A 1). Group 1 uses the first and second time-slot, group 2 the third and fourth and so on. The RCS system with nulling use a P min which is half of the mean power of the six desired users in a TDMA time slot (see Section IV-B3). Two values of e are considered, e == 1 and e = 0.3 (see Section IV-Bl). When e == 1 random channel allocation is employed, whereas when e == 0.3 the channel allocation of Section IV-B2 is used with eight power groups consisting of six mobiles each. Group number one is allocated to the first time slot in the TDMA frame, group number two to the second and so on. More details of the simulation procedure is given in Appendix A. Simulations for the described systems are made 40 times according to the simulation method described in Appendix A. Histograms of the power transmitted from the mobiles using the power control methods described in Sections IV-AI and IV-B 1, are shown in Fig. 5. The upper, middle, and lower subplots consider the SCFR system, the RCS systems with e == 1 and the RCS system with e == 0.3, respectively. The powers are in all three cases normalized such that the mean power transmitted from a mobile is 0 dB. . In Fig. 6 the estimated outage probability, defined as the probability that the instantaneous signal to interference ratio is less than 3 dB more than 20% of the time (motivated in Appendix B) is plotted as a function of the multipath angular spread a, see Section 11-B 1. In Appendix C approximative analytical expressions of the outage probability are derived for the SCFR system, the RCS system with nulls (only the case
542
j10l :: 500 Ql
.0
E
:OJ
Z
0 -30
jl~1 :: 500 Ql
.0
E :OJ Z
Fig. 5.
0 -30
- 20
-10
0
10
Power transmitted from the mobile (dB)
Distribution of th e power control settings; upper: SCFR, middl e: RCS with e
=
I , lower: RCS with e
=
0.3.
0.16 0.14 0. 12 >,
~
:.0ell .D
0 •1
o
0:
S'o
0.08
ell
+" ~
0 0 .06
cl
c8
c3 X- - :: :: c2
*-
c4 __--.:~ __c8
0.04 c5
x-
0.02 c6 X" "
c7
0
0
234
Angula r spread
(J'
5
in degr ees
6
7
Fig. 6. Outage probability as a function of (J with geometric based handover. c l : RCS-WON , 1/1 analytical. c2: RCS-WON. 1/1. e = 1 simulation. c3: RCS-WON. 1/1, e 0 .3 simulation. c4: RCS-WIN. Ill . e 0 .3. simulation. c5: SCFR, 1/3, d 3. simulation. c6: RCS-WI N. Ill , e 1. simulation c7: RCS-WIN, Ill , e 1, analytical. c8: SCFR, 1/3, Ii 3. analytical. WIN with nulling, WON without nulling.
= =
=
=
=
=
e == 1), and the RCS without null s (e independent). Results using these expressions are also plotted in Fig . 6. Example 2: All the simulations and computations performed in Example I, are repeated but assumi ng signalstrength based handover. Thi s means that the mobiles are connected to the base station with the lowest path-loss (except for some hysteresis), see #1 of Appendix A. Figs . 7 and 8 are the counterparts of Figs. 5 and 6, respecti vely.
=
=
VI. CONCLUSIONS AND DISCUSSION The following sections list conclusions drawn from the ob servations made above, and discuss critical assumptions.
A. Power Control The results of Examples one and two show that the dynamic power control range in the mobiles must be larger than 50 dB in the e == 1 case, while ::::;30 dB is sufficient in the e == 0 .3
543
,=Jb= I 1~1 ~=~~::::B~, I -20
10
20
r~IL__-,-__p~ow=e, =-rt~"Eoon.
20
-30
-20
-20
-30
-10
0
-10
0
-10
10
0
10
30
I 30
(dB)
Power transmitted from the mobile (dB)
30
20
Fig. 7. Distribution of the power control settings: upper: SCFR. middle: RCS with (' = 1, lower: RCS with (' = D.3. 0 .06
r-----r----,---.---,-------r-r---.----.---, c2
c3 0.05
c4 c5 c6 c7
I I I
..,>, 0.04 ..0
2o
0'::
c2 /
0 .03
x c8
Q)
be
..,:tl
:: 0 0.02
0.01
.-'
c6
clx-
c3 c5 c4
x- - - _ ..
_
x-
,
- x
c7
c8
/
x- -
o
6
7
Fig. 8. Outage probability as a function of (7 with ~i gnal-.s tren gth based h~do.ver. cl : SCFR, 1/3, d = . 3, si m ulati~n. c2: RCS-W?~ I~- =_1 , S =:. 1, analytical. c3: RCS-WON, 1\ = 1, S = 1. e = 0.3 , simulation. c4: RCS-WIN, I~ = 1. S = 1. e = 0.3, simulation, c5. R,?S-W?N, I~ - 1.:-S -:- 1, e -:- 1, simulation. c6: SSFR, Il3,d = 3, analytical. c7: ReS-WIN, Ill, P = 1, analytical. c8: RCS-WIN, 1/1, e = 1, simulation, WIN - with nulling, WON = without nulling.
and SCFR case. As a reference, the GSM standard supports a power control range of 30 dB [9]. B. Dependence of Downlink Performance on Uplink Power Control The results of Examples I and 2 show that the uplink power control is critical for the downlink performance of systems with downlink intercell nulling . In particular, the results show
that the power control parameter e = 1 yields much better results than e = 0.3. Thus, the conjecture of Section IV-J~H appears correct, it stated that the base will be able to identify and null the mobiles with poor downlink quality if e = 1 is applied. Is this result general? If the identification threshold Pm in is made sufficiently small (i.e., the base can identify very weak mobiles), then e = 0.3 will perform equally well. This
544
means that the conclusion may not be true for any system. However, the result indicates the importance of an issue which is typically overlooked. It should also be noted that the beamfonning used in the paper takes the desired signal strength at the identified interfering mobiles into account in the criterion function (in order to achieve this, information has to be transmitted between the bases in the e = 0.3 case but not in the e == 1 case, Section IV -B3). If this is not the case, the effect may be worse since deep nulls will point toward users who already have a good signal to interference ratio. This problem does not arise in systems with only two users and two base stations, and analysis and experiments under such conditions can therefore be misleading. C. Downlink Outage Performance
The simulation results herein indicate that the RCS systems performs better than the SCFR system if signal-strength handover is applied, or if the uplink power control completely compensates for the path-loss, i.e., e == 1 (see Section IV -B 1) and nulling is applied. This would not have been so in the geometry handover case if more users and channels had been simulated [22]. The reason being that this would have separated the same frequency users in azimuth and thus made the SCFR system more robust against angular spreading. On the other hand, the simulation and analysis assume basically uniformally distributed users, which is favorable for the SCFR system. The simulation also assumes that all multipaths are confined to an area relatively close to the mobile. If this is not the case, a larger degradation is expected in the SCFR than in the RCS cases, since the SCFR system tries to separate mobiles in azimuth to avoid the influence of angular spreading. i\PPENDIX A SIMULATION PROCEDURE
The enumeration below describes the simulation procedure used in the paper. 1) The positions of the 11 x 3 x 48 users in cells 1-9. 12-13, are generated as follows: The position of user ~ is randomized with equal probability in the area 0
COS(30 ) ) 2/", ( COS((}i. i)
(~)
R:::; 2,
(61 'I
where ri,i and Oi,l are the distance and angle to the desired base station, respectively [the factor (cos(300)/cos( O',l) )2/1' models the cell radius as a function of angle 0 when the antenna patterns are given by p ( 0) == cos( 0)]. The lognormal fading to each neighboring base station is randomized and the corresponding path gain is calculated. The position of the user and the lognormal fading are regenerated (randomized) if a "mistake" is detected. In the geometryhandover case a mistake has occurred if the average path gain defined by (39), using L == 1, is larger for some other base than base i. In the signal strength handover case a failure has occurred if the strongest path gain, i.e., max, Gi,i is more than 3 dB stronger than the desired-
545
base path gain Gi, t. If there are n base stations which are stronger than O.5G·i , l (including the zth base), then a "mistake" is generated with probability (n - 1)In. Thus a random number is drawn to determine if a "failure" has occurred or not. This procedure is used to simulate the case with a fast handover and a hysteresis of 3 dB. 2) The channel allocation algorithms are invoked (Sections IV-A 1 and IV-B2). In the SCFR approach four power groups with 12 mobiles each are used. Group number one uses the first and second time-slot, group number two the third and fourth and so on. In the RCS approach, with e == 0.3 eight power groups (one for each time slot in the TDMA frame) are used. With e == 1 random channel allocation is employed. All simulations assume that the TDMA slots of the base stations are synchronized, although this is critical only for the reduced cluster size approach with directed nulls. However. the TDMA frames are desynchronized in the sense that each base has a random offset of one through eight bursts. 3) Weighting vectors (Section III) are calculated for all users in sectors I a-c, Zb-c, 3b, 4a, Sa. 6a. and 7c in the 1/1 reuse case and 1a, 4a, Sa, and 6a in the 1/3 reuse case. In the same-cell reuse approach only one weighting vector per user is necessary. This applies also to the reduced cluster size approach if nulling is not applied. With nulling however, multiple weighting vectors per user must be calculated. This is because frequency hopping is applied and the identified interfered users thus change between time slots. In order to calculate the weighting vectors it is therefore necessary to determ i ne which users are identified by the base. This requires, in turn, that the power control settings are calculated. Thus the power control at the mobiles are first calculated. Then it is determined which interfered users are stronser than Pmin' (and thus identified, see Section IV-B3). For subcell 1a only users inside subcells 1b, Ic, 2a-c, 3a-c, and 7a-c are candidates. Once the identified users have been determined the weighting vectors for all possible cases are calculated. 4) For each of the 48 users in subcell 1a it is investi sated whether they are experiencing acceptable speech q~lity or not. Based on the reasoning in Appendix B, we assume that this is obtained if the instantaneous signal to interference ratio exceeds 3 dB in at least 800/0 of the time slots. The fraction is calculated as follows: The mean desired power averaged over fadinzo G·I, I. for the considered user is calculated using (39). A random frequency hopping pattern is simulated by randomizing the cochannel user in cells 1-7 with nei bzhborin bo cells 10000 times. For each of the 10000 hops the cochannel users are drawn with equal probability among the mobiles allocated in the time slot. The mean interference (averaged over fading) at user 'i, is calculated for each hop using the formula
t, ==
L k
"7kGk,iw~Rvv(f)k,i,
ak,·dwk
(62)
where W k is the weighting vector of the kth user and G i, i, f)k, i, and o», i are the propagation parameters between the zth user and the kth users desired base (can be the same base in the SCFR case). The sectors selected in the sum of (62) are la-c, 2b-c, 3b, 4a, Sa, 6a in the 1/1 ReS case, and la, 4a, 5a, 6a in the 1/3 SCFR reuse case. Note that with SCFR, there is d cochannel users per sector (Section IV -A). To simulate discontinuous transmission the factor 'fJk is randomized independently for each hop (Pr{ 7]k = I} == 1 - Pr{ T}k == O} = 7]DTX == 0.5). The same cell cochannel users which are assumed to be active all the time constitute an exception. Note that the users which use the same frequency within the cell are the same in each time-slot (Section IV-A-I). When the mean desired and interfering signal has been calculated the probability for the instantaneous signal to interference ratio to exceed 3 dB is calculated using (63). This probability is averaged over the hops to produce the sought fraction. Finally, the number of users with acceptable speech quality are counted and the outage probability is estimated as the fraction of users in subcell 1a with unacceptable quality.
ApPENDIX C ANALYTICAL RESULTS
In this section, we derive analytical approximations of the outage probability for the SCFR system, the ReS system with intercell nulling (only the e == 1 case) and the ReS systems without nulling. The analysis uses the same assumptions as the simulations with a few exceptions. Among those are the antenna spacing, the number of users in the system and the spatial distribution of the users. The downlink antenna spacing is slightly increased to ~ == A/ J3, and the number of users is assumed large (infinite). The spatial probability density of the user positions (seen from the desired base) is assumed to be given by as shown in (65) at the bottom of the page, where the choice of TO is defined by the handover algorithm assumed. The reason for the choice of the distribution (65) is that it enables the derivation of an analytical solution for the outage probability while at the same time being very close to uniform at J == 3.5-4.0. In Appendix C-A-C below approximative expressions for the outage probability (probability of unacceptable speech quality) conditioned on the user position are obtained for the three cases. In order to obtain the unconditioned outage probability ~ the subcells are divided into "elements." ~1('il, 'i2 ) , defined by ~~('il ~ -l2) ==
ApPENDIX B INSTANTANEOUS OUTAGE PROBABILITY
Previous results have shown that 9 dB average signal to interference ratio is sufficient to provide reasonable speech quality in GSM (neglecting noise), [15]. We assume that the relevant property for the receiver is the probability that the instantaneous signal to interference is less than 3 dB. Assuming flat Rayleigh fading and one interferer this fraction can be computed using the formula (see [14]) 1
Pr{ SIRinstantaneous ::; SIRd = 1 + SIR/SIR
t
(63)
10 0 . 3 and SIR 10 0 . 9 . This yields Pr{SIRinstantaneous ::; SIR t } == 0.2. The formula (63) applies in the case of a single interferer only. However, with multiple interferers, the interference is usually dominated by the strongest interferer and we use (63) as an approximation in these cases. In Appendix C, analytical approximations of the outage probability are derived. In these derivations the following approximation of (63) is used with
SIR t
SIR o Pr{SIRinstantaneous ::; SIRt} = (1 + SIRo/SIRt)SIR
(64)
which is a "linearization" of (63) around SIR == SIRo. The natural choice of SIRo under the assumptions here is SIRo = 10°·9.
() .. )f( T 2.. , 2' 2,'1. -
const. x r.1.,2. COS(1-4f,) (6·2,1.' :)
{ 0,
cos(30 0 ) ) 2/ 1, O.05'i l ~ ( ~, (r 1.• { cos] f}i. 'L)
+ 0.05'£1
and 5'£2 - 60
t/ R)
:::; D.DS
< fI ::; S'i2 - 55}. (66)
This partitioning is illustrated in Fig. 9 using i 1 == 0, ... , 17, iz == o.... ~ 23, i.e., TO == 0.9. The outage probability is calculated for a central point in each element. Finally, the unconditioned outage probability is obtained as the sum of the central point outage probabilities, weighted by the fraction of users in the element. These fractions can be calculated analytically, [22]. The intention is that the elements should be small enough that the outage probability is approximately constant within an element. It is easily shown that the desired signal strength (disregarding lognormal fading) along the borders of the "annular elements" [where annular element i 1 is defined as Ui.., neil, 'i2)J is constant, when the element patterns are given -by p((}) == cos( fJ). If the user distributions of all subcells in the system are added, only small spots are left "empty" if TO = 0.9 is used. Thus TO == 0.9 will be used when "geometry based handover" is assumed. Previous results, [9], have shown that the gain of signal strength handover (described in item #1 of Appendix A) over geometry based handover is about 4 dB. We model this effect by choosing TO == 0.7, and thereby moving the mobiles (a distance corresponding to 4 dB), closer to the base. In Appendixes C-A, C-B, and C-C below, ReS with nulling, ReS without nulling, and SCFR are treated, respectively. The
if ~ (cos(300)/ cos((}i,i))2 f'(ri,i/R) ~ fa, elsewhere
546
I(}'i,il ~ 60°
(65)
0.8 0.6 0.4 0.2
o - 0.2 -0.4 -0.6 -0.8 -1 L.-_--'-_ _--L..._ _- ' - - - _ - - - '_ _::>.k=:--_ - ' -_ _L.-_--'-_ _-'--_---l 0.2 0.4 -1 -0.8 -0.6 -0.4 -0.2 0.6 0.8
o
Fig. 9.
The division of the subcell.
Res version with nulling is onl y considered in the case (' (e is defined in Section IV-8 I). A. Reduc ed Cluster Si;e (R CS) with Nulling and (:
=I
where D k . , and B (:r) are defined by
Dk
=1
We assume that all base s erroneously estimates their desired mobile to have zero angular spread. i.e., Il k . k = 0° for all /,; .5 Thi s yields
B (.I)
G i. , C ,' , R ,.,.(O . oi, cos(lh .,))
+ ( r - l )Pmi/lI
(71)
= Jiag (1.
ex p ] -):t) . . . . . exp ( - j (m - 1):1))
(72)
respectively . Using the equations above we obtain that the interfering power (averaged over fading) at the ith (from the kth base) is given by
= [1. exp ( -j J3Jr Sin( B)) . ex p ( - j J3( rn - I )Jr sin( I1) ) r
=
and
where arB ) is given by
arB)
,
G k . iwkR (B k ,. Il k . i )W k
=
Gk.,a *(O) B *(iid D;;l, R (O.
(68)
Il k . ,
cos(Bk , ;))Dk",ljB(O:k)a(O )
(a *(O) B *(a k )Dk',l,B (O:k)a( O))2
in (47) and (48) . With R k . k given by (67) it can be shown that the transmit vector at the kth base is given by
(73)
where
[(2Jr/J3) sin (Bk. d - (21l)J3) Sin Uh,j )] modulo 2Jr.
Assuming that only the kth and -ith user are identified by the kth base, the matrix M is given by M
(74)
=B (( 2JrI J3) sin( Bk, i) )Dk. jB* (( 2JrI J3) sin ( /h .;)) (70)
5 This is a pessimisti c assumpt ion if the angular spread of the desired user is large because in that case there exist possibilities to avoid transmitting toward the interfered mobiles by pointing the main beam toward the multipaths . However, the angular spreading considered herein is so small that this effect is negligible .
The impact of frequency hopping and discontinuous transmission is modeled by averaging (74) over the distribution of i'ik (which is shown to be uniform [0, 21r] in [22]) and assuming that the mobiles are active with probability TJDTX . Using the results of Appendix B, and assuming that all base stations have identified the -ith mobile but no other mobile, the probability
547
M, D k , i » and D(x, y) all be equal to the identity matrix i.e., M == Dk,i == D(x, y) == I.
that the instantaneous SIR is not exceeding SIR o is obtained as 1 - Pr{ Outage}
== Pr
{
SIRo SIR o
1+--
'TJDTX
c:
"
j
C. Same Cell Frequency Reuse (SCFR)
.
The derivation of the analytical approximation of the outage probability (conditioned on the user position) for the SCFR system is very similar to the derivation of [21, Theorem 11. The details of the derivation can be found in [22].
t
SIR t
(75)
~t
FOUR
::::: Pr { 10 log(G k , i) - 10 log(G i , , )
< 10 10 (
g 9
(-1
TJDTX
ApPENDIX
TO
R DL
D
TRANSLATIONS METHODS
1) If the up- and downlink manifolds are the same, i.e.,
(1 + SIRo/SIR t ) SIR t o
(81)
'Vi}
fJiCOS(Oi))),
R UL
(76)
it follows from (16) and (24) that the up- and downlink multipath covariance matrices are the same (except for the power scaling) i.e., RUL = pCLRDL, and the translation problem is thus eliminated. This requires two different antenna arrays for up- and downlink. The two arrays should have the same structure but scaled to their respective wavelength. This idea was first proposed in [16], and is referred to as "the matched array approach," in that paper. 2) If the same array is used in up- and downlink, i.e.,
where g(z , y) is defined as shown in (77)-(79), given at the bottom of the page, this approximation is possible as the interference usually is dominated by one base station. This is more true in systems with antenna arrays than otherwise. Assuming that the log normal fading (between a mobile and several base stations) is correlated with correlation coefficient, c, the outage probability at the 'lth user condition on its position is obtained as given in (80) at the bottom of the page. When (80) is used in Examples one and two, only neighboring cells are taken into account in the product. Furthermore, only the sector directed most closely toward the mobile of each cell is considered, since overly pessimistic results would otherwise be obtained. This is because (80) does not assume fully correlated lognormal fading between a mobile and the three sectors of a base station site. Also with ~ == A/2 there are fewer side lobes outside the ±60° region than with ~ = AI J3 [which (80) assumes]. The parameters SIR o and SIR t are set to SIR o == 9 dB and SIRt == 3 dB, respectively.
(82)
and (83)
and the relative duplex separation (fUL - fDL)/(fUL + is small then there may exist a compensation matrix Aconlpensate such that
f D L)
B. Reduced Cluster Size (ReS) Without Nulling see [22]. If (84) is valid R DL may be approximated as
By again assuming that all bases estimate the spreading of their desired mobile to be zero, i.e., o». k = 0° for all k,6 the results of previous section can be used by letting the matrices
pULRDL-"A -..
6 Simulations we have made have shown that the loss of neglecting the angular spreading in the ReS case without nulling is typically less than 1 dB, assuming linear arrays with eight-ten elements.
g(z, y) == max{f(x, y) z
~
compensate
RDLA*compensate'
3) If the spatial distribution of power is well approximated by a finite number of rays (which is less than the number
z}
(77)
f(x, y) =x (21f a*(O)B*(a)i>-l(x, Y)~v(O, y)f>-l(x, y)B(a)a(O) da }ii=O (a*(0)B*(a)D-1(x, y)B(a)a(O))2
(78)
D(x, y) == xRvv(O, y) + (r - l)Pmin I Pr{Outagelr"i,Oi,d=lx
~ roo
v 21r } x=O
II Q (m k'
exp ( i -
(85)
2-
(79) x2
20 dB (1 - c)
10 log
)
(g( 7JD~X (SIR ol + SIR;-l )t, O"k, i a dB v!1="C
k#i
548
cos( 8k ,i))) - mi, i-X)
dx
(80)
of antenna elements), i.e.,
lV
ftUL ~ ~ pULlhnI2aUL(fln)(aUL(enJ)*;
[10] T. Ohgane, "Spectral efficiency evaluation of adaptive base station for land mobile cellular systems," in Proc. IEEE Veh. Technol. Conf., 1994, pp. 1470-1474. [11] S. J. Orfanidis, Optimum Signal Processing, An Introduction. Singapore: McGraw-Hili, 1990. [12] B. Ottersten, M. Viberg, and T. Kailath, "Analysis of subspace fitting and ML techniques for parameter estimation from sensor array data," IEEE Trans. Signal Processing, vol. 40, no. 3, pp. 590-600. Mar. 1992. [13] B. Parlett, The Symmetric Eigenvalue Problem. Englewood Cliffs, NJ: Prentice-Hall, 1980. [ 14J R. Prasad and A. Kegel. "Improved assessment of interference limits in cellular radio performance." IEEE Trans. Veh. Technol., vol. 40, pp. 412-419, May 1991. r15] K. Raith and J. Uddenfeldt, "Capacity of digital cellular TDlVIA systems," IEEE Trans. Veil. Technol., vol. 40, pp. 323-332, May 1991. [16] G. Raleigh, S. N. Diggavi, V. K. Jones, and A. Paulraj, "A blind adaptive transmit antenna algorithm for wireless communication," in Proc. IEEE
N<m
n=l
(86) then the powers 1 hIT/. 12 and directions gri of these rays can be estimated from Rf~, using a conventional direction finding technique, e.g., [2], [11], [121, [17], and [18]. These estimates may then be used to calculate PtGLRDL using
JV
pULR D L ~ pUL ~ IlllnJ2aDL(Bn)(aDL(Hn))*.
(87)
Int. Conf Communications, 1995.
n=l
4) If a uniform linear array is used in the uplink, i.e., a UL (H) is given by (1), and the model described in Section II-B 1 applies, then the method of [19] may be employed to estimate the signal power, as well as H and a. With these estimates at hand, the transmit matrix pUL R DL may be explicitly calculated. REFERENCES [11 S. Andersson, U. Forsse n, and J. Karlsson, "Ericsson/Mannesrnann GSM field-trials with adaptive antennas," in Proc. IEEE Veh. Tee/mol. Conf., Phoenix, AZ. May 1997, pp. 1587-1591. f2J Y. Bresler and A. Mocovski, "Exact maximum likelihood parameter estimation of superimposed exponential signals in noise," IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, pp. 1081-1089, Oct. 1986. [31 C. Carneheirn. S. O. Jonsson, M. Ljungberg. Nt. Madfors, and J. Naslund, "FH-GSM frequency hopping GSM," in Proc. IEEE Veil. Techno!' Conf., Stockholm. Sweden. June 1994, pp. 1155-1159. [4] C. Farsakh and 1. A. Nossek, "Channel allocation and downlink beamforming in an SDMA mobile radio system." in IEEE Int. Symp. Personal. Indoor and Mobile Radio Communications, Sept. 1995, pp. 687-691. [51 D. Gerlach and A. Paulraj, "Adaptive transmitting antenna arrays with feedback," IEEE Signal Processing Lett., vol. I. pp. 150-152, Oct. 1994. [6] _ _ , "Adaptive transmitting antenna arrays with feedback." IEEE Trans. Veh. Techno!., 1995, submitted. [7] _ _ , "Base station transmitting antenna arrays for multipath environments," Signal Processing, vol. 54, no. 1, pp. 59-73, 1996. [81 G. H. Golub and C. F. Van Loan. Matrix Computations. Baltimore. MD: The Johns Hopkins University Press. 1983. [91 M. Mouly and M. B. Pautet, "The GSM system for mobile communications," Michel Mouly and Marie-Bernadette Pautet, 49 rue Louise Bruneau, F-91120 Palaiseau France, 1992. ISBN 2-9507190-0-7.
[171 R. Roy, A. Paulraj, and T. Kailath, "ESPRIT-A subspace rotation approach to estimation of parameters of cisoids in noise." IEEE Trans. on Acoust., Speech. Signal Processing, no. 34, p. 1340. 1986. [181 R. O. Schmidt. "Multiple emitter location and signal parameter estimation," in RADC Spectral Estimation Workshop, Griffiths AFB. NY, 1979, pp. 243-258; reprinted in IEEE Trans. Antennas Propagat., vol. AP- 34, pp. 281-290. Mar. 1986. [19) T. Trump and B. Ottersten. "Maximum likelihood estimation of nominal direction of arrival and angular spread using an array of sensors." Signal Processing, vol. 50, nos. 1/2, pp. 57-69, Apr. 1996. [20J J. H. Winters. "Optimum combining in digital mobile radio with cochannel interference," IEEE Trans. Veh. Technol.. vol. 33. pp. 144-155, Aug. 1984. [211 P. Zetterberg and B. Ottersten, "The spectrum efficiency of a basestation antenna array system for spatially selective transmission." IEEE Trans. veh. Technol.. vol. 44. pp. 651-660, Aug. 1995. [22] P. Zetterberg, "Mobile cellular communications with base station antenna arrays: Spectrum efficiency, algonthms and propagation models." Ph.D. thesis, Royal Institute of Technology, Stockholm. Sweden. June 1997.
549
Chapter 4
Implementation Issues
P
OSSIBLY the most challenging problem related to adaptive antennas is their practical implementation, from both a technical and a cost point of view. In realworld adaptive antenna systems there are a number of sources of random errors, ranging from antenna element misplacement and mutual coupling to amplitude and phase mismatches and quantization errors. This chapter includes work that deals with these issues from both a theoretical and a practical implementation angle. It starts with a tutorial paper from Dudgeon that describes mathematically and intuitively the fundamentals of digital array processing. Then an adaptive algorithm and its efficient pipelined architecture in the form of a triangular systolic array, particularly applicable to VLSI design, are described. A multiple input, multiple output orthogonalization algorithm, its systolic implementation, and its comparison with the wellknown Gram-Schmidt orthogonalization procedure are discussed in the paper by Yuen et a1. DuFort considers the design of optimum beamforming networks, and Er and Cantoni et al. present a unified approach to designing robust array processors. The article by Hansen discusses
551
design trade-offs and a procedure for selecting design parameters for Rotman lenses. Neural beamforming has been suggested as a means to increase the performance of an adaptive antenna (it has been shown that neural networks can control arrays in an accurate manner even with element and network errors) and reduce manufacturing and maintenance costs. The paper Mailloux and Southall presents a comparison between a neural network and a Buttler matrix performing the same direction finding task, and the paper by Southall et a1. discusses a direction finding system implemented with a neural beamforming network and presents some test results. Several papers in this chapter deal with mismatch problems with adaptive antennas. Among the issues discussed are nonlinearity effects in digital manifold phased arrays (Mathews), array imperfections and methods to cope with the reduction of the nulling capabilities (Jablon), mutual coupling compensation (Steyskal and Herd), forward-backward averaging methods for array manifold errors (Zatman), and the use of orthogonal codes for remote antenna calibration (Silverstein).
Fundamentals of Digital Array Processing DAN E. DUDGEON,
Abstract-With the advent of high-speed digital electronics, it has become feasible to use digital compu ters and special purpose digital processors to perform the computational tasks associated with signal reception using an antenna or directional array. The purpose of this paper is mainly tutorial, to describe mathematically and intuitively the fundam~ntal relationships necessary to understand digital array processing. It 18 hoped that those readers with a background in antenna theory or array pr~essing will.see some of the advantages digital processing can offer, ~hile those WIth a ~ackground in digital signal processing will recognize the array processing area as a potential application for multidimensional signal processing theory.
M
I. INTRODUCTION
UCH of the theoretical work being done today in the area of multidimensional ~ignal proc.essin g is motivated by the need to process signals earned by propagating wave phenomena. For radar to be successful, it was necessary to develop directional transmitting and receiving antennas so that azimuth as well as range and range rate information could be ascertained from the radar return. Similarly, this problem is also encountered in active sonar and ultrasonics applications. In applications where the source signal is not precisely controlled (such as exploratory seismology) or where the received signal is externally generated (such as passive sonar, bioelectrical measurements, or earthquake seismology), it is desired to elicit characteristics of the received signal (its signature) as well as its direction and speed of propagation. In recent times, it has become more and more feasible to perform the signal processing operations associated with array processing using digital compu ters or special purpose digital processing hard ware. Correspondingly, digital signal processing theory has grown to encompass these various applications. The following references are representative of recent articles of digital processing in the fields of radar [ 1] , seismology [2] , sonar [3] , ultrasound [4] , and bioelectrical measurement [5] . The point of this paper is to examine the fundamental array processing techniques, in particular the concept of bearnforming to determine the speed and direction of propagation of an incoming wave, from the point of view of a multidimensional Manuscript received July 26, 1976; revised November 12 1976. The author is with the Computer Systems Division, Bolt' Beranek and Newman, Inc., 50 Moulton Street, Cambridge, MA 02138.
MEMBER, IEEE
signal processing problem. We shall see the close relationship between conventional sampled-data systems and the sensor array as a receiver sampling the waveform in space. Accordingly, Section II reviews some essential points about sampleddata systems and digital signal processing techniques. In Section III, a linear array of sensors is used as a basis for discussing the weighted delay-and-sum beamformer with attention given to how to choose the appropriate weights. In Section IV, the relationship between the computation of beam spectra and the computation of a two-dimensional (2-D) discrete Fourier transform is examined. Section V looks at extending the results of Section III to higher dimensions. As an example of results from digital signal processing which can be applied to digital beamforming, the problem of designing the sensor weights for a multidimensional beamformer is discussed in Section VI. In the case of a Cartesian array of sensors, an ingenious mapping due to McClellan [6) can be used to design and implement beams with nearly spherically symmetric main lobes in a computationally efficient manner.
II. IMpORTANT CONCEPTS IN DIGITAL PROCESSING In this section, several important concepts from digital signal processing theory will be reviewed. These concepts will be presented in terms of a one-dimensional (1-0) signal for ease of understanding, but they are easily generalized to multidimensional signals. The reader is directed to [71 as a text on digital signal processing and to [8] as a review of 2-0 filtering concepts. The fundamental assumption of digital processing is that input signals are bandlimited to frequencies below one-half the sampling rate. If a continuous-time signal is sampled at a rate too slow (undersampling) for the frequency content of the signal, the Nyquist sampling theorem tells us that frequencies above one-half the sampling rate in the continuoustime signal will act like frequencies below one-half the sampling rate. This phenomenon is known as aliasing, and it is explained in detail in [7] as well as in a variety of texts and papers on sampled-data systems. Although we have been speaking of a l-D time signal, the same statements apply to signals which are a function of distance or other continuous independent variables.
Reprinted from Proceedings of the IEEE, Vol. 65, No.6, pp. 898-904, June 1977.
553
We can represent a I-D digital signal by s(n) where n is an integer. By doing so we are effectively normalizing the sampling rate to be unity. The Fourier transform of such a digital signal is defined by S(w)
=L s(n)e-j w n . n
se:
) =
~s(n) exp (_i
2
:
k
).
(2)
If the signal s(n) is zero for n outside the range from 0 to N - 1, then the sum over n in (2) extends only from 0 to
N - 1. In this case (2) defines a discrete Fourier transform (DFT) which is invertible; that is, the samples s(n) may be recovered from the values S[(21fk)/N] by s(n)
Nt S (21rk) exp (i 21rnk) N N N
=.!.
l
k=O
for n = 0, N - 1.
The DFT of a signal may be computed by an efficient algorithm (an FFT), the details of which are contained in texts [7], [9], [10], and papers [11], [12]. The advantage of the FFT algorithm is that the computation of S[(21Tk)/N] is proportional to N log2 N rather than N 2 as in a direct evaluation of the OFT (see (2)] . One type of digital filter which is important to the understanding of digital array processing is the finite impulseresponse (FIR) filter. Again we shall briefly review the 1-0 case which is covered in detail in [7], [9], [13] , and [14]. The name FIR refers to the fact that the impulse response of the filter is nonzero only over a finite domain of the independent variable. For example, if a filter has an impulse response h(n) such that h (n)
=0
for n
<0
and n
~
N
where nand N are integer-valued, then the filter is said to be an FIR filter. A special class of FIR filters are those which are said to be "linear phase." The impulse response of such a filter (assumed to be purely real) possesses even (or odd) symmetry about the midpoint of its nonzero region. An example of such an impulse response is
= 0 for n < 0 and n ~ N h (n) =h (N - n - 1).
hen)
(3)
Because of this symmetry, the phase response of such a filter is exactly linear and produces no phase distortion. A· further specialization can be made by requiring h(n) to be even about h(O). In that case, the frequency response of the filter is purely real. In addition to the perfect phase characteristics of linear phase FIR filters, they have the additional advantage of being easily and quickly designed by a computer program (15). This program approximates a given ideal frequency response optimally in a weighted mini-max sense; that is, the weighted maximum deviation from the ideal is minimized.
DIRECTION OF PROPAGATION
a
(1)
The reader will quickly recognize that the Fourier transform is continuous and periodic in the radian frequency variable w with period 21f. If only samples of S( w) are desired, (1) can be evaluated at the points w = (2trk)/N, where k is an integer variable taking on values from 0 to N - 1 and N is an integer constant k
~ / /
- - - - N SENSORS- - - -
Fig. 1. A plane wave impinges upon a linear array of N sensors at an angle Q.
III. BEAMFORMING The reasons for studying the formation of beams from an array of sensors are straightforward. In several of the applications mentioned in Section I, particularly passive sonar, the objective is to use the signals received by the sensors in a phased manner so as to preferencially detect signals coming from a particular .direction (i.e., signals coming in on a particular beam). In addition, by averaging over many sensors, the signal-to-noise ratio (SNR) is increased to aid in the measurement of other signal parameters [ 19) . An appropriate analogy is that beamforrning is related to multidimensional spectral analysis in the same way that bandpass filtering and I-D spectral analysis are related. Both beamforming and spectral analysis can be used to segregate received energy by frequency, direction and speed of propagation. (An excellent discussion of the latter approach is contained in [30].) Our discussion will concentrate on the beamforming approach, since it is the author's opinion that that formulation more accurately reflects the type of processing done in a realtime digital array processing system. In many physical situations, the signal one is interested in receiving and analyzing can be modeled as a propagating plane wave. In such a signal model, the value along a line (or plane) perpendicular to the direction of propagation is constant. If we assume that the plane wave is propagating with a speed c and in a direction at an angle a to the y-axis (Fig. 1), then the signal value r at a particular place (x, y) and time t may be written r(x, Y, t) = s ( t - (
x sin a + y cos a )) c
.
Note that this is really a function of one independent variable. Consequen tly, the function s(·) along with the direction and speed of propagation completely specifies the model signal. In order to focus on the structure of the beamforming computation, we shall assume that the signal s(·) is deterministic, not stochastic, and that the SNR is high enough that we may ignore the contribution of the noise. Later we shall indicate the way in which knowledge of the signal and noise statistics can be used in beamformer design. A simple way to try to measure s(') and the direction of propagation is by using a delay-and-sum beamformer (19). In Fig. 1, a plane wave impinges upon an array of N sensors uniformly separated by a distance D. If we let a denote the angle of incidence of the wave, then we would expect the signal received at the i + 1st sensor to be a delayed version of the signal received at the ith sensor (in the absence of noise and other waves). The amount of delay is D sin a/c. If we want to look for a signal coming from an angle a, we can form
554
the sum
g(a, r) =! Nt r, (t _iDa) l
N ;=0
C
where a = sin a and ri is the received signal from the ith sensor. Suppose, however, that an incoming wave has a different angle of incidence ao =1= Q and speed Co =1= c. Then
iDao) ( +~ .
Fig. 2. The array pattern for a delay-and-sum beamformer with N ::: 7 sensors.
ri(t) = s t
Substituting for ri(t) g(a, t)
=! Nil s(t - iD(~- ~)). N ;=0
C
Co
If we let s(t) be represented by its continuous Fourier transform Sew)
1
00
1 s(t)=-
2rr _
and let k
= waje, then g(a, t)
=-1
21T
L-ao
S(w)e1wtdw •
00
oo
Sew) W(k - k o ) e j •w
t
dw
(4)
where 1 N-l
W(v) = -
L
N ;=0
..
e-j 1v D
We shall call W(v) the array pattern. Note that (4) has the form of an output signal being equal to the inverse Fourier transform of the product of the Fourier transform of the input signal and the frequency response of a filter. Thus for particular values of a, D, c, and Co, the pattern W tells us how the frequencies in the input signal are weighted to form the output signal. For the case at hand, it is easily shown that
ao,
W(v) -
sin (NvDj2)
N sin (vDj2)
[.(N-
exp -J
I)VD] .
2
The magnitude of W(v) is plotted in Fig. 2 for N = 7. Notice that it is periodic in u with a period of v = (21T)/D. Since
lJ=w(~_ao) c
Co
(5)
the width of the central lobe of the array pattern decreases with increasing temporal frequency (w) and with increasing sensor spacing D. However, if w or D become too large, the array pattern will exhibit other large lobes in addition to the main lobe because of the periodicity of W(v). Historically, these extraneous lobes have been called "grating" lobes because of the analogy with optical diffraction gratings. We shall review the phenomena of grating lobes from the point of view of spatial sampling of a propagating wave. First, we shall assume that the wave-set) is of a single frequency Wo (monochromatic). This is no real restriction since a more general waveform can be decomposed into an integral of weighted sinusoids. If such a wave is traveling at a speed Co, then it will have a wavelength "-0 = 21TC0 I Wo. If the wave were incident at an angle a (Fig. 1), and we were to measure its value along the line connecting the sensors at one instant of time, we
would observe a sinusoidal variation as a function of position along the line of sensors. The spatial period of this variation can be shown to be Ao Isin a. If the spacing D happened to equal Ao/sin a, the sensor measurements would all be identical and one might mistakenly conclude that the wave arrived perpendicular to the array rather than at the correct angle Q. This is precisely the problem of aliasing described in the previous section except that here we are sampling a waveform that is a function of position (by choosing the sensor spacing D) rather than sampling a waveform that is a function of time (by choosing the sampling period D. Consequently, if it is expected that a signal will have a component with a wavelength as short as Amin' then the sensor spacing should be at most Amin/2 to avoid the effects of spatial aliasing (e.g., grating lobes). In designing a beamformer, the objective (in the most straightforward case) is to have W(v) be as close to an impulse as possible using only a finite number of sensors. Traditionally, measures of closeness are the width of the central lobe (the smaller the better) and the height of the side lobes (also the smaller the better). One way to decrease the central lobe width and the sidelobe height is to increase N, the number of sensors. Obviously, this expedient can be used only so much before economic constraints or a breakdown in the signal model come into play. Another way to alter the performance of the beamformer is to weight the sensor signals individually before summing them. If w(i) is the sensor weighting for the rth sensor, then the beamformer output can be represented by g(a,t):!Nt
l
N ;=0
W(j)S(t-iD(~-~))' C
Co
Following the previous derivation we can again write g(a, t) = -
1
21T
fao
.
S(w) W(k - ko) e1w
t
dw
-00
where now 1 N-l .. W(v) = w(i) e- j 1v D
L
N ;=0
is a generalization of the previous definition of W which includes the sensor weights w(i). The problem of determining the sensor weights so that. the array pattern has some desired characteristic is the same as designing a good data window for spectral estimation [16], or designing a prototype low-pass filter for use in a digital filter bank [29]. Either problem may be stated in terms of an FIR filter design problem and the FIR mini-max design technique [13] , as well as others, brought to bear on it.
555
Array patterns can also be designed to take advantage of knowledge about the expected distribution of noise which a array is likely to see. The sensor weights w(i) can be adjusted to maximize the SNR. This is analogous to' designing a Wiener filter given the spectral estimates of the signal spectrum and noise spectrum. Intuitively, the filter will have a frequency response which passes parts of the spectrum where the SNR is high and rejects the parts where the SNR is poor [1 7] . Furthermore, it is possible to adapt the sensor weights as the received signal varies, thus attempting to maintain a high SNR under nonstationary conditions.
signal, the sensor spectra, and finally to the multidimensional spectrum of the sensor signals. Recalling that T = 1, we see that the beam signal may be written as before g(
d nC
D' n
)
=
1 N-l
N j~
w(i) rj(N - idn) ·
The short-period spectrum of such a beam signal may be written as
m2;n
M+n-l (d C ) 21fk dnc ) G ( M'n,n = v(m-n)g ~,m
IV. DIGITAL BEAMFORMING AND 2-D DFT's Thus far we have treated time as a continuous independent variable and we have treated the spatial independent variable as being discrete, since measurements can only be obtained at the sensor positions. Now, however, we shall constrain the time variable to be discrete as well by insisting that sensor measurements be made at intervals of T seconds. In doing so, we must remain aware that there will be aliasing problems if received signals possess any frequency components above 1/2T Hz as seen by the sensor. We shall further assume that all of the sensors are sampled simultaneously. Thus we can denote the output of the N sensors ss r, (nT), i = 0, ..., N - 1. As before we can form beams by weighting, delaying, and summing the sensor measurements. g(a, nT)
1 N-l
=-
L
N ;=0
where v(·) is a spectral window as discussed in [16], [29]. Because of the limits of the summation, the FFT algorithm cannot be applied directly, but the formula is easily rewritten as
-jM
21fk dnC) [21fkn] G ( M'D,n =exp
dnTc
21fkm] -jM·
As indicated in [29], the FFT may be used to calculate the above sum. The exponential term external to the summation may even be incorporated into the FFT by making use of the circular shift property [7] . Using similar reasoning we may write the short-period sensor spectrum for the ith sensor as
s,
sin a = - - d n an integer. D Consequently, only a finite number of beams may be formed. It should be mentioned here that it is possible to interpolate other beams between these beams. This is equivalent to interpolating the sensor outputs ri(nT) to a higher sampling rate (18) so that T is reduced and the inter-sensor delay d n T can have a higher num ber of possible values. Making T small, however, increases the number of samples to be processed and correspondingly, the amount of computation per unit time. For any practical beamforming processor, a lower limit for T is dictated by computation speed and the amount of available data storage. For notational convenience, in the following derivations we shall assume T = 1. This may be viewed as taking our unit of time measurement to be one sample period. Consequently, frequencies will be measured in rad/sample period rather than rad/s. Quite often, one is more interested in the time evolution of the short-period spectrum of a beam signal than in the beam signal itself. This is equivalent to passing the beam signal through a bank of bandpass filters and examining their output. Early on it was recognized that the FFT could be used to make such computations efficiently [26] , (27). Recent work in the speech area has further demonstrated an efficient way of realizing a digital filter bank using the FFT [29]. We shall proceed to derive the formula for the time evolution of a beam spectrum showing its relationship to the beam
)
m2;o u(m)g'D,m +n -exp [
w(i) 'i(nT- idnT)
where a = sin a and d n T = (D sin ca!«. Because the time variable has been discretized, d n must be an integer. This puts constraints on the values of sin a which are allowed, namely
(dnC
M-l
L
[21fkn] M-l v(m)r;(m+n) 21fk ) ( - , n =exp i rt:: r
M
M
m=O
-exp [
21fkm] -iT·
Consequently, we may write the beam spectrum in terms of the sensor spectra as G (21fk dnc
M ) D'
n) •.N N~l w(i) R. (21fk n- id ) i~ 'M ' n
-exp
[_/1T:d
n
]-
(6)
The form of the above equation also suggests an FFT, but the form of the exponential term is a bit troublesome since kidn/M is not necessarily an integer multiple of lIN. To gain more insight into the structure of the computation of the beam spectrum, we shall turn to a formulation using the 2-D short-period spectrum of the sensor signals. The usefulness of thinking in terms of the multidimensional spectrum when approaching array processing problems has been previously recognized [28] t [30]. We define the 2-D shortperiod spectrum of the sensor signals as
556
21fk 21f1 ) R(- , -, n M N
L
1 N-l
=-
N ;=0
w(i)
L
n+M-l m=n
v(m - n) Ti(m)
N+ km)] M .
. exp [ - j21f( Ii
Based on the discussion of the last section, it is easy to imagine a 2-D or 3-D array of sensors located in space with the intention of detecting and recording propagating wave distur-bances. As before we shall assume that the signals we are interested in may be adequately modeled by a plane wave propagating . with . a speed c in a direction a =a x i x + a y i Y + azlz. . . whose amplitude vanes as a function of tune s(t) if we re.. cord the wave from a single stationary sensor [23]. (The vectors ix, i y , and i z are of unit length in the direction of the x, y, and z axes.) Thus the received signal for the rth sensor located at position Pi =xji x + yiiy + zii z will be
As before this can be written as
21Tk
2nl)
R (- , - , n =exp M N
[21Tkn] N-1 M-1
;=0 m=O
. ri(m
1
-w(i)v(m)
N
+ n) exp [-
j21T(~ + k:)].
In this form, R may be evaluated using a 2-D FFT in the same manner that G was calculated by a }-D FFT. [As an aside, note that the separable window function w(i)v(m)1N could be generalized to a 2-D window w(i, m).] Using the relationship for R i , the above equation may be written more concisely
R (21Tk, M
21TI ,n) =.!.N
i¥
"i: ;=0
w(i)R; (21Tk, M
n) exp [_
a . po)
';(Pi, t) = s ( t - ~ .
N
By comparing (6) and (7), the relationship between the shortperiod spectrum of the beam signal and the 2-D short-period spectrum of the sensor signals becomes evident. First, R must be evaluated for I = kdnNIM. This value for I may not be an integer, so we are faced with the problem of interpolating between FFT points as we were in (6). The problem can be circumvented by adjusting the FFT lengths so that M divides N evenly. Second, we must make the approximation
R . (21fk
' M'
g( a, t) =
M'D'
n)
':::::!..
R (21fk 2nkd n
M'M
,n
i
»
t
+ a0
.
Co
Pi)
Making the substitution
S(t)=~JS(w)eiWfdw 2n _OG and using the definition of the wave number vector [20 I ) [30] wale we see that
k =
1 JOG . Sew) W(k- ko)e 1w t dw 21T _co
(8)
g(a,t)=-
where the multidimensional array pattern is now given by W(v)
=L
w(i) exp (-jPi . v)
i
=L w(i)
)
i
relating the short-period beam spectrum and the 2-D shortperiod spectrum.
v.
i
=~W(i)s(t- Pi' ~- ::}
n) ': : :- R '. (2nM'k n- id n ) .
dnc
L w(i) r (p ;
By returning to the defining relation for R i , the reader will readily see that this approximation requires that the shortperiod spectrum of 'i(n) be the same over the M points from n to n + M - I as it is over the M points from n - id n to n - id.; + M - 1. (Note that there is no rotating phase factor in the approximation, since both short-period spectra are referenced to the same time origin at n = 0.) Intuitively, one would expect the approximation to be valid for well-behaved signals if id.; «M. The approximation will be exact if 'i(n) is periodic with a period of M samples. With these two points in mind, we can write
C(21Tk
a;
(Note that a is a unit length vector so that a; + a~ + = 1.) The quantity (a/c) is often called the "slowness" vector sin.ce it points in the direction of propagation with a magnitude of one over the speed [30]. If we want to look for signals coming from a particular direction ao at a speed co, we can add up the weighted and delayed sensor signals
j 2nli]. (7)
MULTIDIMENSIONAL BEAMFORMING
In the previous section, the fundamental techniques for processing signals received by a linear array of sensors was described. The advantages of spacing the sensor locations a uniform distance from one another were discussed, namely that a uniform spacing allows one to take full advantage of signal processing techniques (filter design algorithms, FFT's, etc.) developed for sampled-data systems where the sampling period is uniform. In this section, we shall outline the similarities between multidimensional array processing and the techniques of multidimensional signal processing. The primary emphasis will be to point out how existing techniques could be applied to the design of beamformers and how problems in multidimensional array processing can be approached by reformulating them in ways easily understood by 2-D signal processing researchers.
exp [- j(vxxi + vyYi + vzli)]
0
As pointed out by Kelley (20] , a wide-band signal processed by a beamforming operation will be altered since the argument of array pattern function depends on frequency. Equation (8) demonstrates this. It has the same interpretation as its counterpart (4), namely that the signal spectrum S is weighted by the array pattern W, which is a function of the difference in wave-number vectors k - k o , and implicitly the sensor positions Pi. The multidimensional beamforming operation is therefore performing the task of a multidimensional bandpass filter in frequency-wave number space, rejecting signals whose direction and speed of propagation are sufficiently different from the bandpass center. Contrast this to the processing method of Halpeny and Childers [301, where a spectrum analysis approach is used to the same end. Let us now assume that the sensors are located on a Cartesian grid. For simplicity, we shall further assume that the intersensor spacing is D in all dimensions. In general, the spacing can be different for each dimension. The sensors will be indexed on n x , ny, n z , and their positions will be
557
p(n x, ny, n z ) = (n x Di x + "» Diy + nzDiz).
The array pattern becomes W(v)
=2: 2: 2:
nx ny nz
w(n x, ny , n z)
exp [-jD(n x II x +n y v y +n z liz»)'
VI.
(9)
DESIGNING MULTIDIMENSIONAL DIGITAL ARRAY PATTERNS
In this section we shall borrow multidimensional FIR filter design techniques from the discipline of digital signal processing and apply them to the problem of designing an array pattern when the sensors are positioned at points on a Cartesian grid . As before, the array pattern should have a narrow central lobe and small sidelobes. Ideally, it would be an impulse. A variety of design techniques for 2-D FIR filters (which can be extended to higher dimensions) are reviewed in [8] , but we shall restrict ourselves to two techniques which should suffice in most cases. The easiest way to design a multidimensional array pattern is to consider separable solutions of the form
v,
Fig. 3. Contours fOT McClellan's transformation with [6 J, [24 I).
0 (after
Vz
- .'
then w(n x , ny, n;:) = wx(n x) wy (n y) w:(n;:) .
V
Co nseq uently the problem has been broken in a number of 1-0 design problems which may be solved as indicated earlier. A separable design technique is suited to the symmetry of the Cartesian grid (resulting in central lobes which are roughly rectangular) but not necessarily suited to the desired array pattern. In particular, for certain applications, it may be desirable to have an array pattern which exhibits circular (or spherical) symmetry. In the past this has been accomplished by using sensors arranged in circular (more accurately . polygonal) arrays [19]-[ 22]. To design the sensor weight s in general, an optimization must be applied to (9) to force W to approximate some ideal array pattern. However, an ingenious technique due to McClellan [61 was developed to design 2-D linear phase FIR filters from 1-0 linear phase FIR filters with nearly circular symmetry. McClellan's technique can be easily extended to higher dimensions as shown in the example below , and thus applied to the beamformer design problem when the sensors are equally spaced on a Cartesian grid. Let us assume that we have designed a 1-0 zero-phase array pattern (by zero-phase, recall that we mean that the sensor weights have symmetry) N-I
=0, .. ., -2-'
wU) = w(-i) for i
We shall further assume that N is odd so that the array pattern can be classified as a type 1 zero-phase FIR filter [ 14] . Then (N -1)(2
W(II)
= L
;=0
where a(O) =w(O) , and a(i) = 2w (i) for i = 1,2,· ·· , (N - 1)/ 2. Following McClellan's derivation, W can be written as W(II)
=
L
(N-l)(2
;=0
a' U) [cos
. II]'
(10)
by using the appropriate trigonometric identities. Now, for
Jr
2
I
4
I
I
I
I ".
..'
j ..'
.
.···· --8,v ' v e u Y
J:
,/~....,
- - - 8=
••~ •.,.
•
VI
IDEAL
".
Z
=uy , u z =0
8, v,. u y
'
v, e 0
".
T
4'
8
Fig. 4. Deviat ion from spherical symmetry of a 3·D McClellan transformation. The dotted line represents the ideal and the actual mapping along the axes. The dashed line represents the mapping along the path Vx = v y in the Vz = 0 plane. The solid line represents the mapping along the path V x = vy =v z .
the 3-D case, we make the substitution cos II = 1/4 cos II x + 1/4 cos lI y + 1/4 cos liz
+ 1/4 cos
II x
cos lI y + 1/4 cos II x cos
+ 1/4 cos II x cos lI y cos liz -
3
II;:
+ 1/4 cos lI y cos liz
4'
(11)
Equation (11) is such that if liz = 0, the transformation of variables is identical to McClellan's 2-D circular transformation. Furthermore, the expression is symmetric in II x , lI y , and VzSubstituting (11) into (10) will yield the 3-D array pattern W(lI x , lI y , liz)
aU) cos vi
-
=
(N -1)/2 (N -1)(2 (N -1)(2
L
nx=O
L
ny=O
L
n z=O
a" (n x, ny, n z)
. cos n x II x cos n y lI y cos n z
liz
(12)
where the a" coefficients are derivable from a' , the transforrnation (11), and trigonometric identities. Finally the sensor weights w(n x, ny, n z) can be derived from the a" coefficients by comparing (12) and (9) . A plot of constant values of II in the (lIx , lI y)-plane is shown in Fig. 3. Fig. 4 shows the fre-
558
quency variable u plotted against a parameter 8 for three paths in (v x ' vy ' vz ) - space. The dotted line represents the transformation tor e = vx , "» = uz = O. The dashed line represents the transformation for e = V x = v y, V z = O. Finally, the solid line represents the transformation for e = Vx = "v = lJz. Deviation from the dotted line is indicative of deviation from spherical symmetry along the three paths. Designing and implementing McClellan transformation filters have recently been studied in detail [24], [25]. The remarkable fact emerges that the method of deriving the 3-D weights w(n x , ny, n z ) from the }..D weights w(i) can be combined with the actual filtering operation so that the amount of computation needed to calculate the beam signals is significantly reduced (proportional to N rather than N 3 in this case). We see, therefore, that one advantage of locating sensors on a Cartesian grid is the availability of design and implementation techniques for array patterns which exhibit good circular symmetry and reduce computation. As before, in certain applications one may be more interested in short-period beam spectra rather than beam signals. If the sensor locations are on a Cartesian grid, then the FFT algorithm may be used to compute beam spectra similar to the way described in the previous section. The dimensionality of the FFT will be higher to reflect the num ber of degrees of freedom in direction-frequency space a signal may have. The use of multidimensional filter design techniques (in particular, McClellan's transformation) and multidimensional filter banks using FFT's represent two important examples of results from the field of digital signal processing which can be applied to digital beamforming and array processing. The reader should bear in mind that the preceeding discussion is more an illustrative than comprehensive presentation of digital signal processing techniques applied to array processing problems. Much work remains to be done. ACKNOWLEDGMENT
The author wishes to express his thanks to H. Briscoe and R. Estrada of BBN for several educational discussions of practical beamforrning systems. REFERENCES [ 1) P. E. Blankenship and E. M. Hofstetter, "Digital pulse com pression via fast convolution," IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP-23, pp. 189-201, Apr. 1975. (2] L. C. Wood and S. Treitel, "Seismic signal processing," Proc. IEEE, vol. 63, pp. 649-661, Apr. 1975. (3] F. J. Harris, "A maximum entropy filter," Naval Undersea Center, Rep. TP 441, Jan, 1975. [4] K. R. Erikson, F. 1. Fry, and 1. P. Jones, "Ultrasound in Medicine-A Review," IEEE Trans. Sonics Uttrason., vol. SU-21, pp. 144-170, July 1974. (5) L. J. Pinson and D. G. Childers, "Frequency-wavenumber spectrum analysis of EEG multielectrode array data," IEEE Trans. Biomed. Eng., vol. BME-21, pp. 192-206, May 1974.
(6) 1. H. McClellan, "The design of two-dimensional digital filters by transformations," in hoc. 7th Annu. Princeton Conf. Information Sciences and Systems, 1973, pp. 247-251. [7] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. Englewood Cliffs, NJ:' Prentice-Hall, 1975. [8] R. M. Mersereau and D. E. Dudgeon, "Two-dimensional digital filt ering," Proc. IEEE, vol. 63, pp. 610-623, Apr. 1975. [9] L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. [10 ) B. Gold and C. M. Rader, Digital Processing of Signals. New York: McGraw-Hill, 1969. [ 11 ] W. T. Cochran et al., "What is the fast Fourier transform?," IEEE Trans. Audio Electroacoust., vol. AU-IS, pp. 45-55, June 1967. (12) G. D. Bergland, "A guided tour of the fast Fourier Transform," IEEE Spectrum, vol. 6, July 1969. [13] J. H. McClellan and T. W. Parks, "A unified approach to the design of optimum FIR linear phase digital filters," IEEE Trans. Circuit Theory, vol. CT -20, pp. 697-701, Nov. 1973. [14 ) L. R. Rabiner, J. H. McClellan, and T. W. Parks, "FIR digital filter design techniques using weighted chebyshev approximation," hoc. IEEE, vol. 63, pp. 595-610, Apr. 1975. [151 1. H. McClellan, T. W. Parks, and L. R. Rabiner, "A computer program for designing optimum FIR linear phase digital filters," IEEE Trans. Audio Etectroacoust., vol. AU-21 , pp. 506-526, Dec. 1973. [16 ) G. M. Jenkins and D. G. Watts, Spectral Analysis and its Applications. San Francisco, CA: Holden-Day, 1968, Ch. 7. [17] J. P. Burg, "Three-dimensional filtering with an array of seismometers," Geophysics, vol. 29, no. 5, pp. 693-713, 1964. [18] R. W. Schafer and L. R. Rabiner, "A digital signal processing approach to interpolation," hoc. IEEE, vol. 61, pp. 692-702, June 1973. [19] J. Capon, R. J. Greenfield, and R. T. Lacoss, "Design of seismic arrays for efficient on-line beamforming," Lincoln Lab. Tech. Note 1967-26, June 27, 1967. (20) E. J. Kelly "Response of seismic arrays to wide-band signals," Lincoln Lab. Tech. Note 1967-30, June 29, 1967. (21) J. L. Allen, "The theory of array antennas," Lincoln Lab. Tech. Rep. no. 323, July 25, 1963. [22] D. K. Cheng, "Optimization techniques for antenna arrays," hoc. IEEE, vol. 59, pp. 1664-1674, Dec. 1971. [23] E. 1. Kelley, Jr., "The representation of seismic waves in frequency-wave number space," Lincoln Lab. Tech. Note 1964-15, Mar. 6, 1964. [24] R. M. Mersereau, W. F. G. Mecklenbrauker, and T. F. Quatieri, Jr., "McClellan Transformations for two-dimensional digital filtering: I. Design," IEEE Trans. Circuits and Systs., vol. CAS-23, pp. 405-414, July 1976. [25] W. F. G. Mecklenbrauker and R. M. Mersereau, "McClellan transformations for two-dimensional digital filtering: II. Implementation," IEEE Trans. Circuit and Systs., vol. CAS-23, pp. 414-422, July 1976. [26] J. R. Williams, "Fast beam forming algorithm," Acoust. Soc. Amer., vol. 44, no. 5, pp. 4154-55,1968. [27] P. Rudnick, "Digital beam forming in the frequency domain," J. Acoust. Soc. Amer., vol. 46, no. 5, (part I), pp. 1089-1090, 1969. (28) S. Haykin and J. Kesler, "Relation between the radiation pattern of an array and the two-dimensional discrete fourier transform," IEEE Trans. Antennas Propagat., vol. AP-23, no. 3, pp. 419-420, May 1975. (29) M. R. Portnoff, "Implementation of the digital phase vocoder using the fast Fourier transform," IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP-24, pp. 243-248, June 1976. [30] O. S. Halpeny and D. G. Childers, "Composite wavefront decomposition via multidimensional digital filtering of array data,t' IEEE Trans. Circuits Systs., vol. CAS-22, pp. 552-563, June 1975.
559
I
A Novel Algorithm and Architecture for Adaptive Digital Beamforming CHRISTOPHER R. WARD, PHILIP J. HARGRAVE,
A bstract-A novel algorithm and architecture are described whicb have specific application to high performance, digital, adaptive beamformiDg. It is shown how a simple, linearly constrained adaptive combiner forms the basis for a wide range of adaptive antenna subsystems. The function of such an adaptive combiner is formulated as a recursive least squares minimization operation and tbe corresponding weight vector is obtained by means of the Q - R decomposition algoritbm using Givens rotations. An efficient pipelined architecture to implement this algoritbm is also described. It takes the form of a triangular systolic/wavefront array and has many desirable features for very large scale integration (VLSI) system design.
T
I.
INTRODUCTION
HE OBJECTIVE of an adaptive antenna is to select a set of amplitude and phase weights with which to combine the outputs from the elements in an array so as to produce a farfield pattern that, in some sense, optimizes the reception of a desired signal. The substantial improvements in system antijam performance offered by this form of array processing has meant that it is now becoming an essential requirement for many military radar, communications and navigation systems. The key components of an adaptive antenna system are illustrated in Fig. l(b). The amplitude and phase weights are selected by a beampattern controller that continuous~y updates them in response to the element outputs. In some systems the output from the beamformer is also monitored to provide a feedback control. In all cases the resulting array beampattern is continuously adjusted to ensure cancellation of interference and jamming sources. The most commonly employed technique for deriving the adaptive weight vector uses a closed loop gradient descent algorithm where the weight updates are derived from estimates of the correlation between the signal in each channel and the summed output of the array. This process can be implemented in an analog fashion using correlation loops [1] or digitally in the form of the Widrow least mean square (LMS) algorithm [2]. The value of this approach should not be underestimated. Gradient descent algorithms are very cost-effective and extremely robust but unfortunately they are not suitable for all applications. The major problem with an adaptive beamformer based on a gradient descent process is one of poor convergence for a broad dynamic range signal environment. This constiManuscript received June 5, 1985; revised October 4, 1985. This work was supported by the Procurement Executive, U.K. Ministry of Defence.. . C. R. Ward and P. J. Hargrave are with Standard Telecommunication Laboratories Ltd., London Road, Harlow, Essex, U.K. eMl? 9NA. J. G. McWhirter is with Royal Signals and Radar Establishment, St. Andrews Road, Great Malvern, Worcestershire. U.K .. WR14 3PS. IEEE Log Number 8407019.
AND
JOHN G. McWHIRTER
tutes a fundamental limitation for many modern systems where features such as improved antenna platform dynamics (in the tactical aircraft environment, for example), sophisticated jamming threats and agile waveform structures (as produce; by frequency hopped, spread spectrum formats) produce a requirement for adaptive systems having rapid convergence and high cancellation performance. In recent years, there has been considerable interest in the application of direct solution or "open loop" techniques to adaptive antenna processing in order to accommodate these increasing demands. In the context of adaptive antenna processing, these algorithms have the advantage of requiring only minimal input data to accurately describe the external environment and provide an antenna pattern capable of suppressing a wide dynamic range of jamming signals. Open loop algorithms may be explained most concisely by expressing the adaptive process as a least squares minimization problem. In fact, the least squares algorithm may be considered to define the optimal path of adaptation. In this paper we describe a novel algorithm and architecture for high performance, digital, adaptive beamforming. The adaptive combiner function is formulated as a recursive least squares minimization process and the corresponding set of linear equations is solved using the Q - R decomposition algorithm. It is further shown how the Q - R algorithm can be implemented using an efficient pipelined architecture in the form of a triangular. systolic array.
II.
BASIC CONFIGURATIONS
The form of adaptive combiner which we consider in this paper is illustrated in Fig. 1(b). The inputs to the combiner take the form of a primary signal y(t) and set of N - 1 (complex) auxiliary signals x(t). The weight vector w is adjusted to minimize the power of the combined output signal which is given by e(t) = x T(t)w + y(t).
(1)
This type of adaptive linear combiner may be used in a wide range of adaptive antenna applications. It is well known, for example, how it may be applied to adaptive sidelobe cancellation. In this case the primary signal constitutes the output from a main (high gain) antenna while the auxiliary signals are obtained from an array of N - 1 auxiliary antennas. The adaptive combiner serves to modify the beampattern of the overall antenna system by directing deep nulls toward jamming waveforms received via the sidelobes of the main antenna.
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-34, No.3, pp. 338-346, March 1986.
560
?RIMARY
I NPU T . yll l
'UXiU ,, ,m INPU TS
: ~ :
~1 11
~
OUTPUT elll:{ (t)'!{+ ylll I
FEEDBACK
, SIGNAL _ J
(a)
Fig . I.
I
(b)
Key component s of an adaptive antenna processor . (a) Constraint preprocessor. (b) Adaptive combiner.
maintains a constant value J.I. in a given look direction specified by the vector c. It is worth pointing out that the " end-element clamped" configuration described above constitutes a particularly simple form of linearly constrained process in which the constraint vector is given by
It is also well known how this form of adaptive combiner may be used in conjunction with a suitable reference signal to control a more general antenna array in which all of the elements are essentially equivalent. The reference signal. which is assumed to be correlated with the desired signal. provides the (negative) primary input to the combiner while the signals received by the antenna array provide the N - I auxiliary inputs . In this case the weighted sum of the auxiliary inputs provides as close a match as possible to the reference signal and hence produces the desired output from the beamformer. The basic combiner illustrated in Fig . I(b) may also be used in the so-called " power inversion" mode which has particular application to communications . In this case the N antenna element s are assumed to be omn idirectional and of comparable gain . The received signal s are fed into the combiner. one of them going to the primary channel and thus having its weight coefficient constrained to unity . The other N - I signals enter the auxiliary channels with their adaptive weights initial ized to zero and so. prior to adaptation. the overall beampattern is determined solely by the (omnidirectional) response of the " primary element. " Thi s " end-element clamped" configuration provide s no inherent mechanism to inhibit the adaptive process from null ing the desired signal. However. the syste m is only allowed to adapt when the desired signal is known to be absent. When it is present. the weight vector is frozen thus allowing signal reception . Thi s is referre d to as the " power inversion" mode of operation becau se the differential interference power s received by the antenna elements are inverted by the combiner. A particularly important application of adaptive antenna arrays requires the power of an N clement combined signal
CT =
(4)
However. the incorpor ation of a general linear constraint is not so straightfo rward . A number of techniques have been proposed in the literature but in all cases the resulting implementation is extremely cumbersome. For example. Widrow [3] et 01. suggested the injection of an art ificial look direction signal into the antenna array receiver channels and introducing a corresponding reference signal into the adaptive process . This technique then requires an additional " slave processor" to apply the adapted weight vector . Frost [4] also showed how a general linear constraint could be incorporated into the adaptive process using projection operator techniques but the resulting algorithm is rather expensive in terms of computation . We will now show how the general linear constraint in (3) may be incorporated in a much simpler way. It may be assumed without loss of general ity that eN = 1 and so (3) may be expressed as (5)
where c and w denote the first N - I elements of the vectors C and w, respectively. Equations (2) and (3) can therefore be combined in the form
(6)
(2) to be minimized subject to a linear beam constraint of the form
(3) This constraint ensures that the gain of the antenna array
(0, 0------0 I).
where itt) denotes the vector of signals received by the first N - 1 channels of the N element array . Since the constraint has been absorbed explicitly by eliminating the coefficient WN and thereby removing the Nth degree of freedom, the power of the combined signal e(t) may now be minimized with respect to the unconstrained N - 1 element weight vector oW. The form of (6) is therefore identical to that of (1) and so the output
561
power minimization may be carried out using the type of adaptive combiner illustrated in Fig. l(a). The term + p,XN(t) corresponds to the primary signal y(t) while the transformed vector x(t) - XN(t)C corresponds to the vector of auxiliary signals x(t). This input data transformation may be implemented using a simple linear preprocessor array of the type depicted in Fig. l(a). In effect the Nth antenna signal is arbitrarily chosen as the primary combiner input. The corresponding antenna element is assumed to have omnidirectional coverage and the constraint preprocessor ensures that any signal which enters it from the required look direction is removed from the auxiliary channels before they enter the combiner. The adaptive nulling of this look direction signal is thus prevented. From the discussion in this section it should be clear that the type of adaptive combiner illustrated in Fig. l(b) has a wide range of applications in adaptive beamforming. In the remainder of this paper we concentrate on the development of a novel direct solution adaptive control technique which applies specifically to this basic configuration. III.
LEAST SQUARES MINIMIZATION
The function of the adaptive combiner in Fig. l(b) will now be formulated in terms of least squares minimization. We denote the combined array output at time t, by e(t i )
= x T(t,)W + y(t,)
For the sake of generality this unnormalized estimator includes a simple "forget factor" fJ which generates an exponential time window and localizes the averaging procedure. Introducing a more compact matrix notation the estimator defined in (8) may be expressed in the form
= I e(n)11
(9)
where e(/.) e(n)
= B(n)
e(t2)
(10)
e(t n )
and
B (n) = diag
{~n - I, ~ n- 2,
•.• ,
I}
(11)
with (32 = o. Now from (7) it follows that the vector of residuals may be written in the form e(n)
= X(n)w + y(n)
x T(t l)
X(n)=B(n)
(12)
x T(t2)
(13)
X T(t n )
and y(n) = B(n)
y(t l) y(t 2)
(14)
y(t n )
X(n) is simply the matrix of all data received by the weighted elements up to time In and y(n) is the corresponding vector of data in the primary or reference channel. The matrix B(n) takes account of the exponential time window and, for convenience, it has simply been absorbed into the definition of e(n), y(n) and X(n). Determining the weight vector w(n) which minimizes E1(n) is referred to as least squares estimation [5]. The conventional approach to this problem is to deri ve an analytic expression for the complex gradient of the quantity E~(n) and determine the weight vector w(n) for which it vanishes. Now from (9) and (12) we have for the complex gradient
V w(E2(n»
(7)
where x(t;) is the vector of (complex) auxiliary signals at time t; and y(t i ) is the corresponding sample of the (complex) primary signal. The residual signal power at time t n is estimated by the quantity E2(n) where
E(n)
where
= 2X H(n)(X(n)w + y(n)
(15)
and setting the right side of this equation equal to zero leads to the well-known Wiener-Hopf equation: M(n)w(n) + p(n) = 0
(16)
M(n) = XfI(n)X(n)
( 17)
where
is the (estimated) covariance matrix and p(n)
= X H(n)y(n)
(18)
is the estimated cross-correlation vector. The solution to (16) for nonsingular M(n) is clearly given by w(n) == - M -l(n)p(n)
(19)
and this provides an analytic expression for the optimum weight vector at time t ; In their classic paper, Reed, Mallet, and Brennan [6] suggested that the weight vector be obtained by solving (16) directly and showed that the problems of poor convergence associated with closed loop algorithms may be avoided in this way. This approach leads directly to the type of signal processing architecture which is illustrated schematically in Fig. 2. It comprises a number of distinct components-one to form and store the covariance matrix estimate, one to compute the solution of (16) and one to apply the resulting weight vector to the received signal data. These data must be stored in a suitable memory while the weight vector is being computed. The system also requires a number of high speed data communication buses and a sophisticated control unit to deliver the appropriate sequence of instructions to each
562
where R(n) is an (N - 1) by (N - 1) upper triangular matrix. Then, since Q(n) is unitary we have
PRIMARY CHANNEL
AUXILIARY CHANNELS
z
E(n) =
W....,
\I e(n)\I
=
II Q(n)e(n)\I
ARRAY
OUTPUT
=
II (R~n»)
wen)
+
(:~:~) II
(23)
where u(n) = P(n)y(n)
and SOLVE
t ~ "!. · -t't Fig . 2 .
yen) = S(n)y(n).
WE IGHT VECTOR
pen) and Sen) are simply the matrices of dimension (N - I) by nand (n - N + I) by n, respectively , which partition Q(n) in the form
Sample matrix inversion architecture .
component. This type of architecture is obviously complicated, extremely difficult to design and not very suitable for very large scale integration (VLSl) . Not only does the analytic solution given in (16) lead to a complicated circuit architecture. it is also very poor from the numerical point of view . The problem of solving a system of linear equations like tho se defined in (16) can be illconditioned and hence numerically unstable . Ill-conditioning occurs if the matrix has a very small determinant in which case the true solution can be subjected to large perturbations and still satisfy the equation quite accurately . The degree to which a system of linear equations is ill-conditioned is determined by the condition number of the coefficient matrix . The condition number of a matrix A is defined by
) . Q(n)= (p(n) Sen)
and so the condition number of the estimated covariance matrix M(n) is much greater than that of the corresponding data matrix X(n). Any numerical algorithm which avoids forming the covariance matrix expl icitly and operates directly on the data is likely to be much better conditioned. DECOMPOSITION
An alternative approach to the least squares estimation problem which is particularly good in the numerical sense is that of orthogonal triangularization [7]. This is typified by the method known as Q - R decomposition which we generalize here to the complex case. An n X (I unitary 1 matrix Q(n) is generated such that
A matrix A is defined in this paper as being unitary if AHA is termed onhogonal if ATA = I. 1
R(n)w(n) + u(n) = 0
(26)
E(n) = IIv(n)lI .
(27)
and hence
where AI and Ap are the largest and smallest singular values. respectively, of the matrix A . The larger Cn(Al . the more illconditioned is the system of equations . It follows from (17) that Cn(M(n» = Cn(X H(n)X(n» = Cn 2(X(n» (21)
Q(n)x(n)=(R~»)
(25)
It follows that the least squares weight vector wen) must satisfy the equation
(20)
IV. Q - R
(24)
(22)
= I. Matr ix A 563
Since the matrix R(n) is upper tr iangular. (26) is much easier to solve than the Wiener-Hopf equation described earlier. The weight vector w(n) may be derived quite simply by a process of back-substitution . Equation (26) is also much better conditioned since the condition number of R(n) is given by Cn( R(n» = Cn(Q(n) X(n» = Cn(X(n».
(28)
This property follows directly from the fact that Q(n) is unitary .
Givens Rotations The triangularization process may be carried out using either Householder transformations [7] or Givens rotations [8], [9], [10]. However the Givens rotation method is particularly suitable for the adaptive antenna application since it leads 1O a very effic ient algorithm whereby the triangularization process is recursively updated as each new row of data enters the problem . A complex Givens rotation is an elementary transformation of the form
(~s ~*) (g ..,' g: ~: ::::: '.'. ,) _ ( 0 , .. 0,
-
r;· .. r;
0 .. · 0, 0 ...
x; .
)
(29)
where the rotation coefficients, c and s, satisfy
-s . r;+c . x;=O s*s+c*c= I c*=c
(30)
and are synonymous with the cosine and sine of an angular rotation in the multidimensional complex space. These relationships uniquely specify the rotation coefficients as
c
(31)
.JXixi+riri
and
Direct Extraction of Residuals
Xi s=_· c.
(32)
r,
A sequence of such elimination operations may be used to triangularize the matrix X(n) in the following recursive manner. Assume that the matrix X(n - 1) has already been reduced to triangular form by the unitary transformation
Q(n-I)X(n-l)= ( R(n-l» 0 ).
(33)
Now define the unitary matrix
(Q(n-l) I~) . Q(n-l)=
o
(34)
11
Clearly
(I3X(n-l») Q(n-I)X(n)=Q(n-l) xT(tn)
Q(n)e(n)
(35)
and so the triangular process may be completed by the following sequence of operations. Rotate the N - 1 element vector x T(t n ) with the first row of (3R(n - 1) 'so that the leading element of x T(l n ) is eliminated producing a reduced vector x r'(l n ) . The first row of R(n - 1) will. of course. be modified in the process. Then rotate the (N - 2)-element reduced vector x T' (In) with the second row of 13R(n - 1) so that the leading element of x T, (In) is eliminated and so on until every element has been eliminated. The resulting triangular matrix R(n) then corresponds to a complete triangularization of the matrix X(n) as defined in (:2~). The corresponding unitary matrix Q(n) is simply given by the recursive expression
Q(n) = Q(n)Q(n - 1)
(36)
where Q(n) is a unitary matrix representing the sequence of
Givens rotation operations described above, i.e.,
.- (,BR(n-l») (R(n») 0 0 . =
XT(/ n )
-0-
(
1») == ( I3v(nu(n)_ 1) )
,Bu(n (jv(n - 1) y(tn )
a(n)
= (R~n)) w(n) +
u(n)
(
f3v(n - 1)
)
(39)
a(n)
and the weight vector must satisfy (26), it follows that the residual vector e( n) is given by
Q(n)e(n) = Q(n)Q(n - 1)B(n)
e(t l) e(t2)
= (f3v(nO- 1)). a(n)
(40) But Q(n) is unitary and so we have
e(ll)) e(t 2
= QH(n) A
(
e(t n )
) 0 (3v(n - 1) . a(n)
(41)
Considering only the nth element of the vectors in (41) it is then possible to deduce that the current residual e(t n ) is given by
e(tn ) = l'(n) . a(n) (37)
(42)
where N-I
It is not difficult to deduce in addition that
Q(n)
In many least squares problems, and particularly in the adaptive antenna application, the main objective is to compute the least squares residual since the corresponding weight vector is not of direct interest. Previous work by McWhirter [11] has described a modified version of the Q - R recursive least squares algorithm in which the least squares residual is produced directly at each stage of the recursive process without any need to derive the weight vector explicitly. The modified algorithm is much more robust since it avoids the solution of a system of linear equations which could be illconditioned. Furthermore, since the back-substitution circuit and the separate beamforming network are both eliminated, it offers a significant reduction in the complexity of the subsequent hardware implementation. The derivation of this technique may be summarized as follows. Since
=(f3R(;-I») x T(t n )
Q(n)
and this shows how the vector u(n) can be updated recursively using the same sequence of Givens rotations. The least squares weight vector w(n) may then be derived by solving (26). The solution is not defined, of course, if n < (N - 1) but the recursive triangularization procedure may, nonetheless, be initialized by setting R(O) = 0 and u(O) = O.
l'(n) =
= (u(n») v(n)
(38)
564
II c, ,=1
(43)
is the product of all cosine parameters generated during the sequence of Givens rotations used to elimina!e th~ ~ector xT(tn ) . Equation (43) follows from the fact that Q(n) IS SImply
the product of (N - 1) elementary rotations of the form
Q(n)
==
Q -1 (n ) Q N
N - 2(
n) . . .
Q1(n ).
( 44)
The ith elementary rotation is simply given by
o
o
Cl . . . . .
Sf (45)
: 1. 1 : rr
S; .•... C;
.where the only nonzero off-diagonal elements occur in the ith row and the ith column. The result may be obtained by considering the effect of a reversed sequence of conjugate elementary rotations on the nth element of the right-hand vector in (41). The parameter yen) may readily be computed during the recursive update of the matrix R(n) while the scalar quantity a(n) is available as a direct byproduct of the corresponding update for the vector u(n). The current residual e(t n ) may therefore be evaluated in a very cost-effective manner. In order to avoid complicating this discussion on adaptive beamforming, we have only considered the most direct form of the Givens rotation algorithm. However, it is important to point out that a very efficient "square root free" Givens algorithm has been derived by Gentleman. The square root free algorithm is equally applicable to the type of adaptive beamformer described in this paper and would almost certainly be used in any practical application. The essential details relating to its use may be found in [10] and [11].
Sensitivity to Arithmetic Precision An important aspect of any signal processing algorithm is its sensitivity to limited arithmetic precision. We have recently carried out a detailed computer simulation study to compare the effect of limited precision on the performance of two adaptive cancellation processors-one based on sample matrix inversion and the other on the recursive Q - R algorithm. The results indicate quite distinctly the improved performance offered by the data domain Q - R method under conditions of finite resolution arithmetic compared with the sample matrix inversion technique. Fig. 3(a) shows a simple schematic representation of the two computer simulations. In both cases the sequence of data samples was generated and applied to the constraint preprocessor. The preprocessor applied a look direction constraint toward the desired signal and was implemented at full computer precision. The transformed data were then truncated to the chosen arithmetic precision, this word length being retained throughout subsequent Q - R decomposition or sample matrix inversion computation. To ensure a fair comparison between the two basic approaches (i.e., covariance versus data domain), the effective sample matrix inversion solution was actually computed by performing a Q - R decomposition on the covariance matrix estimate in (16). In both cases the back-substitution was performed at full computer precision.
Fig. 3(b) shows a typical comparative result which corresponds to a 24-bit floating point word length (16-bit mantissa and eight-bit exponent). Here, we plot the expected signal-tonoise ratio at the output of an eight-element array as an increasing number of data samples are used to compute the adapted weight vector solutions. In this example, we have modelled the effect of three equal power jamming signals received individually at levels of 0 dB relative to a thermal noise floor of - 50 dB at the antenna array elements. The complex envelope of each jammer was described by an independent, narrow-band Gaussian process. The model also incorporated a desired signal received by the array at a level of 15 dB above the thermal noise floor but approximately 40 dB below the total received jamming. From Fig. 3(b) it can be seen that the initial rate of adaptation is extremely rapid for both sample matrix inversion and the data domain Q - R algorithm. In both cases a good level of jamming cancellation is obtained after about ten to 20 data samples. However, with sample matrix inversion there is clear evidence of an unstable weight vector as reflected by extreme fluctuations in the adaptive response curve. In contrast, the data domain Q - R method shows no sign of numerical instability and it is found that, over the timescale shown on these plots, the signal-to-noise ratio performance gets progressively better as the covariance information (in the form of the updated R matrix) gains more and more statistical accuracy with time. For this scenario it was found that the sample matrix inversion technique required a floating point word length of 32 bits (24-bit mantissa and eight-bit exponent) to achieve comparable performance with the data domain Q- R algorithm. It cannot be assumed, of course, that this word length would be sufficient for any arbitrary dynamic range environment. One should only conclude that the word length required by the sample matrix inversion approach will always be significantly greater than that for the data domain Q- R method. V.
SYSTOLIC ARRA Y IMPLEMENTATION
Kung and Gentleman [12] have shown how the Givens rotation algorithm described above may be implemented in a very efficient pipelined manner using a triangular systolic array. The implementation of a five-channel adaptive beamforming network using this architecture is shown in Fig. 4. It may be considered to comprise three distinct sections-the basic triangular array labeled ABC, the right hand column of cells labeled DE and the final processing cell labeled F. The entire array is controlled by a single clock and comprises three types of processing cell. Each cell receives its input data from the directions indicated on one clock cycle, performs the specified function and delivers the appropriate output values to neighboring cells as indicated on the next clock cycle. Apart from the introduction of an extra parameter into the boundary cell, the function of the boundary and internal cells is precisely that required to implement the Givens rotations described above. Each cell within the basic triangular array stores one element of the recursively evolving triangular matrix R(n) which is initialized to zero at the outset of the least squares
565
.
0
N=B
~
.
3 JAMMERS AT 000 DESIRED SIGNAL AT ·3500 THERMAL tJOISE FLOOR AT -5006
~
Covanance Domain
~
.
0
Full Precision
"iii
:o:!0
Vl, 0:: 2 0
Limited precision,-----",;;\;;;;-_-.,
l':: ,
Processing
COVARIANCE-------
,
DOMAI N
~ <:>
-to
10
20
30
40
50
60
NO. OF OATA SAMPLES
70
80
90
100
Full Precision (b)
(a)
Fig. 3.
Comparison of data and covariance domain algor ithms. (a) Simulation flow diagrams. (b) Typical signal-to-noise ratio response. I NP U T DATA I
I
II
WAVEFRONTS
t
I I
I
I
I
I I I I
I I
X22 X12
X21 Xl1
I I I I I
I I I
I
I I
I
I I I
I
X23 X 13
Y2 Yl
X24 X14 I I I I I I
I I I I I I I I
, I
I
I
I
INTERNAL CELL x,o(k)
C,o (k)~Cout(k)
-r;.?: S~,(k)
S,o(k)
xoul(k)
~(k) = - S,n(k). r(k-l)
+ C,n(k) x,o(k)
BOUNDARY CELL
I I I I
r(k) = / 1x.n(k)I' + Ir (k-1
I I
s"",(k ) =
c"",(k)
r(k)
C~(k) .
r(k-l)
+ S~(k) . x~(k)
E
)1'
C"",(k) = C~(k)
r (k-1)
F
= r(k) x,o(k)
S"",(k) = Sn(k)
BEAMfORMED
r(k)
Residual
I 8"",(k) = 8,n(k). coul(k)
Fig. 4.
=
Triangular systolic array for adaptive beamforrning.
calculation and then updated every clock cycle. As a result of this initialization the value of rn within each boundary cell is entirely real. Cells in the right-hand column store one element of the evolving vector u(n) which is also initialized to zero and updated every clock cycle. Each row of cells within the array performs a basic Givens rotation between one row of the
stored triangular matrix and a vector of data received from above so that the leading element of the received vector is eliminated as detailed in (29). The reduced data vector is then passed downwards through the array. This arrangement ensures that as each row x T(tn) of the matrix X moves down through the array it interacts with the previously stored
566
triangular matrix R(n - 1) and undergoes the sequence of rotations Q(n) described in the earlier analysis. All of its elements are thereby eliminated (one on each row of the array) and an updated triangular matrix R (n) is generated and stored in the process. As each element of the vector y moves down through the right hand column of cells it undergoes the same sequence of Givens rotations interacting with the previously stored vector u(n - 1) and generating an updated vector u(n) in the process. The resulting output, which emerges from the bottom cell in the right-hand column, is simply the value of the parameter a(n) in (42). The other value ')'(n) required for direct computation of the least squares residual e(t n) is generated recursively by the additional parameter ')' which appears in the definition of the boundary cell function. The value of l' (initialized to one) is simply multiplied by the "cosine" parameter in each boundary cell and passed on to the boundary cell in the next row two clock cycles later. The extra delay, which is a direct consequence of the temporal data skew, may be achieved by using an additional storage element which is indicated by a black dot in Fig. 4 and would be incorporated within the boundary processor. The required value -y(n) emerges from the final boundary cell and is simply multiplied by the corresponding output value a(n) to produce the desired residual. This operation takes place within the final processing cell F. A consequence of the highly pipelined nature of the systolic array and the need to impose a time-skew on the input data is the presence of an overall delay or latency in the system response. Each output residual e(t n ) corresponds to a data vector whose first element was input to the network 2(N - 1) clock periods previously. The systolic array described in this section clearly exhibits many desirable properties such as regularity and local interconnections which render it comparatively simple to implement. Furthermore. the control overhead is extremely low since the processing cells operate synchronously and the only control required is a simple globally distributed clock. However, the need to distribute a common clock signal to every processor without incurring any appreciable clock skew is one possible disadvantage of the systolic array approach particularly in large multiprocessor systems. It is possible, however, to implement the same basic design as a wavefront array processor of the type proposed by S. Y. Kung et al. [13]. In a wavefront array processor, the required computation is distributed in exactly the same way over an array of elementary processors as it would be on the corresponding systolic array. Unlike its systolic counterpart, however, the wavefront array does not operate synchronously. Instead, the operation of each processor is controlled locally and depends on the necessary input data being available and on its previous outputs having been accepted by the appropriate neighboring processors. As a result, it is not necessary to impose a temporal skew on the data input to a wavefront processor. Instead the associated processing wavefront develops naturally within the array. In order to operate in the wavefront array mode, every processing element must incorporate some additional circuitry to implement a bidirectional handshake on each of its input/output links and thus ensure that the necessary
communication protocol is observed. This represents an overhead which is not negligible but can easily be absorbed within the overall processing.
Obtaining the Weight Vector It is worth pointing out that, as well as being capable of operating in the direct, beamforming mode, the triangular array in Fig. 4 can also be used in conjunction with some additional circuitry to compute the weight solution explicitly. The scheme which was originally proposed by Kung and Gentleman [12] uses the triangular systolic array in conjunction with a linear systolic array which solves for the weight vector by back-substitution. This method could clearly be used with the circuit in Fig. 4 by providing suitable means for extracting the triangular matrix R(n) from the array. However, the weight vector, if required, can be obtained in a much simpler way as a further byproduct of the direct residual extraction technique. The method, which we refer to as "weight flushing" may be explained fairly simply as follows. As the nth data vector x(t n ) and the corresponding input y(t n ) pass through the triangular array in Fig. 4 they update the parameters of the system from their state at time n - 1 to the new state at time n. The vector x (t n) also undergoes a simple linear projection with the implicit updated weight vector w(n) to produce the corresponding output residual (46) Assume that the state of the system is subsequently , 'frozen" by preventing any further adaptation and define a simple N - 1 element projection vector of the form
:=
(0 ... 010 ... 0)
(47)
with unit ith element. If the vector q,j is now input to the array as though it were another vector of auxiliary samples and the corresponding primary input is set equal to zero it follows from (46) that the associated output "residual" must be given hv
cPTw(n) = w;(n).
(48)
It is therefore possible to "flush" the entire weight vector w(n) out of the array by inputting to the N - 1 auxiliary channels the sequence of vectors q,; (i = 1, 2, '.', N - 1)
i.e., by inputting a simple unit diagonal matrix. For the sake of brevity in this paper we have not explained in detail how the adaptive process may be "frozen" in practice. However, the technique is quite straightforward and may be implemented in a very direct manner. It is particularly simple when the square root free Givens rotation algorithm is being used.
567
VI.
CONCLUSION
This paper has described a novel algorithm and associated systolic/wavefront array architecture for high performance, digital, adaptive beamforming. The adaptive beamformer enjoys all the desirable architectural features of a systolic or wavefront array. As each row of data moves down through the
array it is fully absorbed into the statistical estimation process, the triangular matrix R(n) is updated accordingly and the corresponding residual is produced automatically. The circuit architecture is greatly enhanced by avoiding the need to derive an explicit solution for the least squares weight vector W (n). This leads to a considerable reduction in the amount of computation and circuitry required since it is no longer necessary to clock out each triangular matrix R(n), carry out the back-substitution or form the vector product x T(tn ) W(n) explicitly. The adaptive bearnformer described in Sections IV and V is also based on a very stable and well-conditioned numerical algorithm. Indeed the method of Q - R decomposition by Givens rotations is widely accepted as one of the very best techniques for solving linear least squares problems. However the final triangular linear system may, in general, be illconditioned and avoiding the back -substitution process also enhances the numerical properties of the adaptive combiner. In particular the systolic array implementation of the Q - R algorithm produces the correct (zero) residual even if n < (N - 1) and the matrix X is not of full rank. This sort of unconditional stability is most important in the design of real time signal processing systems. As part of the United Kingdom's research program into advanced algorithms and architectures for adaptive antenna array signal processing, Standard Telecommunication Laboratories and the Royal Signals and Radar Establishment are developing jointly an experimental wavefront array processor. This digital processor will be configured primarily as an adaptive antenna test-bed and will have the ability to process six input channels of data in real-time. Each node of the wavefront array processor will be based on an existing digital signal processor chip and hence will provide a useful degree of programmability whilst maintaining a node throughput rate which will allow a comprehensive range of real-time tests and trials. Eventually, the development of high performance processing nodes by VLSI design will permit the practical realization of such parallel processing architectures in extremely compact hardware form. In addition, the VLSI circuitry in conjunction with advanced technology will provide processing throughput rates far in excess of those obtainable by current nsp components and will therefore be matched to future wideband radar and communications applications.
[4]
[5] [6] [7]
[8] {9] [10] [11] [12]
[13]
ACKNOWLEDGMENT
The authors thank the Directors of Standard Telecommunication Laboratories Ltd. for permission to publish this paper. REFERENCES
[1] S. P. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 585-598, 1976. {2] B. Widrow and J. M. McCool, "'A comparison of adaptive algorithms based on the methods of steepest descent and random search," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 615-637, 1976. [3] B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode, "Adaptive "antenna systems," Proc. IEEE, vol. 55, no. 12, pp. 2143-2159, Dec. 1967.
568
O. L. Frost, "An algorithm for linearly constrained adaptive array processing," Proc. IEEE, vol. 60, pp. 661-675, 1971. G. L. Lawson and R. J. Hanson, Solving Least-Squares Problems. Englewood Cliffs, NJ: Prentice-Hall, 1974. I. S. Reed, J. D. Mallett, and L. E. Brennan, "Rapid convergence rate in adaptive arrays," IEEE Trans. Aerospace Electron. Syst., vol. AES-I0, pp. 853-863, 1974. G. H. Golub, "Numerical methods for solving linear least-squares problems," Num. Math., no. 7, pp. 206-216, 1965. W. Givens, "Computation of plane unitary rotations transforming a general matrix to triangular form," J. Soc. Ind. Appl. Math., no. 6, pp. 26-50, 1958. S. Hammarling, "A note on modifications to the Givens plane rotation," J. Inst. Math. Appl., vol. 1, pp. 215-218, 1974. W. M. Gentleman, "Least-squares computations by Givens transformations without square-roots," J. Inst. Math. Appl., vol. 12, pp. 329336, 1973. J. G. McWhirter, "Recursive least-squares minimization using a systolic array," Proc. SPIE, 1983, p. 431, Real-Time Signal Processing VI, 2983. H. T. Kung and W. M. Gentleman, "Matrix triangularization by systolic arrays," Proc. SPIE, 1981, p. 298, Real-Time Signal Processing IV. S. Y. Kung, K. S. Arun, R. J. Gal-ezer, and D. V. Bhaskar Rao, "Wavefront array processor: Language, architecture and applications," IEEE Trans. Comput., vol. C-31, no. 11, pp. 1054-1066, 1982.
Nonlinearities in Digital Manifold Phased Arrays BRUCE D. MATHEWS,
Abstract-In digital beamforming (DBF), the phase shifter is functionally replaced with a receiver and digital phase rotation. A Taylor series expansion of mixer nonlinearities is used to generate receiver intermodulation spectrums respective of the element position and the iso-Doppler wavefront directions of signal arrival across the array. The dominant intermodulation distortion at each element experiences linear phase errors across the array proportional to the harmonic number and the desired steering direction phase gradient. The array distortion signals are reduced relative to the desired signal by the array factor sidelobe isolation when desired collimation directions exceed a few beamwidths of scan off the array normal vector. The result of the nonlinear down conversion analysis is extended to inphase and quadrature imbalances and batch manufacturing tolerances for element receivers. I. INTRODUCTION
T
HE MOMENTUM AND expectations of advancing digital circuit technology [1]-[4] encourage attention to alternate phased array architectures. Digital beamforming (DBF) (see Fig. I) utilizes the conversion of the analog microwave signals to digital numbers for preserving the spatial phase information of a wavefront across the array [3]. Whether determined adaptively or a priori [4], the steering weighting and collimation processes are subsequently performed as digital complex number arithmetic. However, large numbers of receivers are necessary to realize such a mechaniManuscript received December 3,
1985~
revised April 18, 1986.
The author is with the Systems Development Division, Westinghouse Electric Corporation. P.O. Box 746, Baltimore, MD 21203. IEEE Log Number 8610030.
ME~BER, IEEE
zation. The nonlinearity subject of this paper follows from both the need to simplify these receivers for realizing affordable, producible systems and a phase error mechanism consequential to DBF which relaxes critical radar receiver design criteria. The array factor will depend upon design features of the receiver as a generalized phase shifter with nonlinear features from downconversion. Harmonic intermodulation of signals due to the nonlinearities of the final mixer is the principal source of distortion in the radar receiver [51. The superhetrodyne receiver uses several stages of down conversion and intermediate frequency (IF) processing to minimize undersirable signals and to efficiently transform the signal for digital formating. The final down conversion to baseband will take caution when the dominant signal from terrain backscatter, i.e., main beam clutter, has a nonzero Doppler spectrum component geometrically determined by the squint angle between the transmitted 1ine of sight and the velocity vector of the radar platform. Mixers are nonlinear devices. As viewed by a baseband spectrum analyzer, intermodulation products from nonlinearities in the final mixer relocate, as broadened replicas of the clutter spectrum, at harmonics of the offset error in positioning clutter to zero IF. In receiver development, a two tone test is performed. to observe the amplitudes of the intermodulation at the harmonICS of the difference frequency. This distortion determines the spurious free dynamic range [6], [7] and is particularlY, problematic for sensitive detection performance under large clutter conditions approach ing receiver saturation.
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-34, Vol. 11, pp. 1346-1355, November 1986.
569
I
I, I
I
,I I
L_
Nx1
--,
I
N x1
I
a priori
J
IL
···
B
U
~Stt Fig. 1.
Nx1
II.
F F E R
N x1
Beam Former
------~I&a
to IPP Buffer and Corner Turn Processor
With a digital manifold. any number of weighting/steering functions may be used to form the beam.
This paper first reviews in heuristic form the interchange of the phase steering and down conversion processes and introduces the mechanisms of distortion generation in the receiver. After an overview of the calculation, the clutter signal wavefront is modeled. The receiver down conversion and the digital collimation processes are described mathematically. The paper closes with summary extentions and graphic comparisons of the conventional and digital manifold phased array spectral results. These suggest DBF receivers are more tolerant of distortion and/or may employ alternate design approaches. DISCUSSION
Consider the receive operation of a linear array of elements in response to the signal wavefronts from terrain scattering of a transmit narrow beam from a nonstationary platform. In the conventional electronically steered phased array, an incident wavefront induces a linear phase error across the aperture. Commanded microwave phase shifters remove this linear phase error from a desired direction, and the element signals are accumulated through a precise analog phase and power combining network to yield a single output signal. A triple conversion superheterodyne, in-phase and quadrature sampling receiver using stable coherent local oscillator signals is employed to present an in-phase (I) and quadrature (Q) representation to a waveform digital matched filter processor
[8].
__
In digital beamforming, the order of the functions of the receiver and the array are interchanged, and the phase shifter replaced by a phase rotation arithmetic at each element. The
output from each element receiver includes distortion generated by the nonlinear processes of the receiver on the elemental signal. An I and Q digital sampling of the signal preserves the directional phase information of the incident signal component at the element. The element signals are collimated by furnishing a uniquely, judiciously chosen inphase and quadrature pair of digital numbers, i.e., the real and imaginary parts of a complex number to each element. These numbers have been called digital steering weights [3] and are applied to phase rotate the element signals so that the linear phase error across the array of elements for a specified direction of wavefront arrival is removed. In Fig. 1, these steering weights can be derived from and applied to the incident signals if a buffer can store the signals until the weights are computed. The collimation process is completed by a vector accumulation of the element signals. The distortion resulting from the nonlinear processes of the receiver may be analyzed, as are almost all nonlinearities, by power series expansions which lead to power products of the clutter signal and the local oscillator (LO). After low pass filtering, only certain combinations of clutter and oscillator products are important, in particular, those combinations which place the resulting center pulse repetition frequency (PRF) line clutter spectrum very near de. The desired portion of the mixer output corresponds to the first-order product of the clutter and the first-order product of the oscillator. Combinations involving higher order self-products of the clutter signal or LO may be termed self-mixing products. A feature of self-mixing is a harmonic relation in frequency and in phase, This may be illustrated by a simple nonlinearity and trigonometric identities. Consider a fourth-order nonlin-
570
earity
"-
I \
I \
(cos ex + cos {3) 4 = cos:' ex + 4 cos ' ex cos {3 +6 cos ? ex cos? {3+ 4 cos ex cos ' {3+ cos 4 {3 .
Gl
.,;
"' a ~
a:
u '" ~ e ...... '"
sin lJo
where
';.,
lJo
,1II
i
\," / \
I .-, -W~5
~
--t--t---r-~
,
\
I
I
if i
Yr
I
1\ • jI r >; .I I.
, .....\
-L/_ ..L C ' - _
\
\1 r
I
O~CTlON
",.....-THRESHOlO
I
•
,.
I
POS ITIONING HARMONIC
\
\
I .r ·,
if t
\
Fig. 2. For perfect positioning. the clutie r spectrum is well confined. Poor positioning leads to spreading by harmonic relocation. Higher level signals experience a more rapid proportionate increase in harmonic conten t.
The desir ed signal will be collimated by specifying rotation ph ase IjJ a nd inphase and qu adrature operations [3] (4) (5) where th e in-phase and quadrature voltage signals th e in-phase and quadrature steering weights W, cos IjJ WIJ sin 1jJ.
Vi. VIJ W" WIJ
(6)
(7)
Fr om (I ), the de sired downconverted component is Vi == cos (ex - f3)
(8)
Vq == sin (ex - f3)
(9)
hence
v: == co s (ex -
13 +
1jJ)
(10)
v ~ == sin (ex -
13 +
1jJ)
(11)
since
(3)
(12) Selecting
cPo arbitrary phase
x
II '1
,. Ii
,.
Q
When the signal is closely positioned to de (i. e . . small w, ). low pass filtering eliminates all terms in (2) except the final term on the RHS. and yields a downconverted version of the original signal located spectrally at 2 w, with a harmonicall y distorted phase. For radar systems on moving platforms. the intermodulation introduces broadened replicas of the incident clutter spectrum into the pass band relocated at harmonics of the frequenc y o ffset error. The harmonic intermod distortion co herently integrates . Fig . 2 dramatizes fast Fourie r transfo rm (FFT) folded PRF lines of a lar ge clutter signa l at the thre shold detect ion stage of a processor with a larg e dynam ic range for both A ID converte r and co he rent integratio n-a radar spectrum analyzer of perhaps 100 dB dy namic range. The received spect rum results from the illumi nation of terrain in relative mot ion. The 3 dB spectral width is proport ional to the antenn a be arnw idth and to the Doppler gradie nt at the pointin g geometry (see (2 1». The spectra l width typi call y is less than 1000 Hz and is much less than the PRF o f the sig nal waveform. Fig . 2 has been dr awn with ex agg erated position ing error to resolve the intermodul at ion produ cts . A goo d mixer may have con vers ion losses exceedi ng 60 dB below signal level. As the signal level increases . the relati ve levels o f the intermod products will increase. Contemporar y radars employ control loops to min imize desensitization of moving target detection and false alarms in presumed clutter fre e regions by precisel y positionin g clutter signa ls at zero IF . In the instance of the co nventional recei ver . the phase associated with the clutter sig nal is o f little significa nce . In the digital manifold pha sed array . this pha se co ntains the spatial information nece ssary for co llimatio n. T he sec ond harmon ic distorti on terms will have spa tial phase arguments ac ross the aperture of twice the linear pha se error of the fundamental . des ired term . This should be clear by letting . in the pre viou s example,
A
I!
'I II
~
+ 112 cos (2(ex + {3}) + 112 cos (2( ex -{3» l. (2)
cP == cPo + -
,.
~
6 cos? ex cos? {3 = 312 [ 1 +cos (2ex)+cos (2{3)
2 1l"X
Ii '\~1
..
z
Let ex = W it + et> be representative o f the return of a main beam iso-Doppler clutter patch. Suppose {3 = w~ [ = (W 2 + w,)t is the baseband positioning local oscillator. The middle term on the right hand side (RHS) of ( I) may be written
I ."\ II - - PERFECT POSITIONING. NEAR SATURATION · - .- POOR POSITIONING . lOW lEVEL I' I \I - - POOR POSITIONING . NEAR SATURATION
..
Q
(I)
the element relative location in the array wavelength the direction of arrival off-broadside,
and considering the final term of (2) as before .
- 27TX •
IjJ == - - sm A
eo
(13)
at all ele ments will place the desired clutter spectra from each element inphase. With distortion, digital beamforming removes only part of the element phase error so that the
571
CD Q
W
7) Retain dominant amplitude terms.
100
en
o
o t-
Z
10
~
10
IV.
a::
3
U
a
\AI
~
...S
.to
-
Clutter
PositionEnor,HZ
Conventional Receiver at Saturation
20
~
-20
-40
Fig. 3. Harmonic distortion is suppressed by the relative array factors for the harmonics.
distortion signal, (2), is not fully collimated across the array. Accentuating positioning errors will resolve the distortion components to the result of Fig. 3 for a small off-broadside scan angle. The above argument for the digital manifold phased array distortion is simplified. The use of local oscillators delayed TTl 2 rad between I and Q channels of the receiver introduces a nuance into the spatial interpretation due to the various selfmixing phases of the local oscillator in the steering arithmetic. Finally, the main beam Doppler clutter will defocus due to slight off-boresight angles of arrival which can be included in the result. The remainder of the paper deals with a rigorous derivation of distortion for the digital manifold phased array.
III.
The largest signal in the airborne radar receiver is usually clutter. i.e., backscatter from terrain. This signal is coherent within the time dwells of data collection and has well defined wavefronts of arrival. This section introduces a mathematical description of this signal which permits interpretation as an infinite number of wavefronts with a Doppler component related to the angle of arrival. This formulation permits a generation of distortion in the spectral domain and a representation of wavefront phase errors due to angle of arrival along the array. Clutter may be simulated as a finite superposition of sinusoids [9] . The amplitudes of these components are determined by scattering models, the radar equation, and the transmit illuminating antenna field pattern [8]. Such simulating sinusoid amplitude coefficients are calculated using area patches and imply a density generating function exploited with numerical integration [10]. The magnitude of such a simulating clutter signal is measured with a spectrum analyzer which effects a finite Doppler bandwidth and/or angle resolution on the clutter cell. For an illuminating transmit beam, the density function will acquire an amplitude taper about the direction of transmit collimation which is commonly approximated by a Gaussian function in angle. By expanding in a first-order Taylor series, the Gaussian argument may be either Doppler frequency or angle off boresight. Near the boresight direction 00 • for a Doppler frequency III and platform speed u.
OVERVIEW OF THE CALCULATION
The mathematics of harmonic distortion analysis produces a bookkeeping maze and a potential distraction. The following are the salient points in the calculation for the output spectrum of a nonlinear manifold phased array radar. 1) Model a single PRF line of the scattered signal spectral density function of a gated Doppler wavefront arrival direction and transmit illumination. Assume a Gaussian spectral amplitude function and justify a wavefront arrival phase argument linear with Doppler frequency. 2) Expand the time signals for down conversion in-phase and quadrature mixers using a Taylor series for the clutter and local oscillator signals. Form an infinite series over the mixer products in exponential notation. 3) Fourier transform into the spectral domain. Use convolution properties of the Gaussian terms for simplification of selfmixing terms. 4) Collect only those terms which peak within the low-pass filter band. 5) Apply the steering weights for a desired pointing angle using complex phase rotation arithmetic as prescribed in (4) and (5). 6) Identify phase error terms in the resulting spectra. Manifold accumulate the element signals into an array factor for each signal component using phased array linear phase error results.
SIGNAL MODELING
Id = 2v/X cos 00
(14)
O(!d)=arc cos (Xfd/2v)
(15)
sin 8(/) = sin 80 + (/ -!d) cos 80 d81
df
J=/d
.
(16)
Consequently. the wavefront phase delay may be approximated. for the main beam, by (17)
where main beam center Doppler the Doppler frequency the location vector for the nth element in the array the propagation vector in the direction of the Doppler frequency component and, for a linear array where
572
(18)
d = element spacing (19)
I3n u
trXn
tan (J
(20)
and f3n are the phase delay of main beam center and the phase delay slope for off boresight Doppler clutter for the nth element, respectively . The Fourier transform of the signal voltage input to the mixer is
CXn
;;;
L.O. " - - - PRF
--1_
0(
C
s
0(
w
...:J:::;
~: {e+ i[an+l3n(f-JI) +
1!2(f + 1t /~f)2}
: PHAS E
I
w
0
sn(f) =
LO
Z
--1
I
Ul
I
0(
I
Q.
I
Q.
I
:l;
I
0(
il
(21)
where
\ on\
\"""
Ao
the voltage amplitude density coefficient of element peak clutter as may be calculated from the radar equation Vo a scaling voltage magnitude Sf the spectral width of main beam clutter due to Doppler gradient tif = 2v/ D tan 80 for an electronically scanned transmit beam of aperture D = N d m a random component of phase, invariant in space and time, stable for short observations.
Fig. 4 . IF signals input to mixer. The Gaussian shape is due to the transmit illumination pencil beam . A well de tined spatial phase is due to the geometry of the Doppler effect in the main beam . DOWNCONVERT
MA NIF OLD
At the mixer input of the digital manifold element receiver, the signals are spectrally related as shown in Fig . 4 . The phase of the clutter signal depends upon the element location in the array through the subscript n . In the analysis of the mixer . the random component of phase is suppressed, although the reader may mentally note its effect. V.
N
rn
(TO MATCHED FILTERIFFTICFAR)
MATHEMATIC TREATISE OF NONLINEAR MANIFOLD PHASED ARRA y RECEIVER PROCESSES
In receiver operation, interest must ultimately focus upon the signal properties that affect the extraction of information . In radar , that information is often obtained from a digital signal processor which completes a matched gated Doppler filter for the signal and provides thresholded detection . The product of this analysis is the spectrum of an FFT process of known coherent gain . The analysis (see Fig . 5) begins with the generation of the intermodulation distortion through a final stage mixer ostensibly positioning the pulse/PRF line spectrum at baseband video. An ideal low-pass filter rejects out-ofband components . The collimation of the element signals begins with a complex arithmetic phase rotation on each element and is completed by a complex vector accumulation across the aperture analogous to classic phased array theory . Presumably the desired portion of the downconverted spectrum will be equivalent to the conventional phased array result, but the distortion components will have differing array factors . Mixers are nonlinear devices . The p-n junction semiconductor diode is the foremost example [5], [II] . As measured across a load, the junction current produces a voltage transfer
Fig. 5 . Digital manifold phased array clement processes include I and Q down conversion by a local oscillator and beamsteering phase rotation arithmetic .
where
Ao
the de bias voltage the local oscillator voltage magnitude the local oscillator frequency the clutter normalized time domain voltage signal at the jth element of the array due to a superposition of wavefronts , the clutter signal amplitude .
For the quadrature mixer, v,(t)=vs+vo cos (2·llfot+7rI2)+Aosj(t).
(24)
The subscript j, denoting the element location in the array, will be suppressed . The quadrature channel will be noted for how it differs from the inphase. Placing (23) into (22) and expanding the nonlinearity into.a Taylor series and using exponential notation for the eosin usOld [5]-[7]
(22) The voltage across the junction terminals VI will be due to a de bias, a local oscillator,. and the signal from the previous IF stages . For the in-phase mixer, V1r(t) = v~ + Vo cos (hfot) + Aosj(t)
(23)
573
(25)
For the quadrature channel, a is still replaced by a + tr/2, and a factor e1 (m - 2n )1r/2 is included in (34). When the argument (33) equals zero, the replica of clutter, i.e., the Gaussian amplitude envelope, reaches its peak value. For a pulse spectrum containing many PRF lines, the low-pass filter bandwidth is large compared to the PRF, and the clutter spectrum (see Fig. 6) is centered at
where
(m) 11
the Taylor coefficient evaluated at the bias point. (26)
m! =(m-n)!n!'
11=/0
l: x; QD
cos " (21C'lot ).
(28)
m=O
n - 2k+ m - n - 2/= 0
F«r the quadrature term, the bracketed term will include in the argument of the exponent, i(n - 2k)1C'/2, and a 1C'/2 in the argument of the cosine in (28). Equation (25) portrays the self-mixing and up/down conversion characteristic of mixers. A low pass filter will eliminate out of band components. Continuing the analysis in the spectral domain, the Fourier transform is V(f) =
[00 v(t)riz"f/ dt.
m
s"'(I)e- / 2 7i /
1
. (,n) [ ill ] m
dt=e+ I ;3/ ~
I
1=0
-(Xl
--
ili
(37)
The voltage spectrum out of the filter and input to the digital phase rotatation calculator is lJ3/(f)=c3/(f)+2Ao
'lm-l
n
I~I ~ ~o 00
k4 (2m) (n)k (2m I-n) 2m
m
n
where
m-I
.Jm
(36)
l=m-k.
(29)
e+ /(m - 21)(a - {3/ I ) _ ") _ _ _ _ _ _ e-I/21/-(m-2/)/I/~/~"'I"
= 2(/ + k)
will peak within the pass band. Evidently only even nonlinearities place signals in the pass band. Using (36) and rewriting 2m for m,
Attention will now be focused on the Fourier integral of the generalized power products of the signal and local oscillator. This integral simplifies for the Gaussian form of (21). By an induction of completing the exponential argument square, the reader may veri fy.
I:.
(35)
An ideal low-pass filter does not perturb the phase arguments, transfers in band components with no attenuation, and eliminates out of band components. It is very practical to ignore the tails of those spectra components of the distortion which peak outside the pass band. Since 11 and 10 are ostensibly equal, only those terms with indices
the binomial coefficient resulting from the expansion of the exponential argument in the Taylor series and the local oscillator. (27) C2[=
d/·
~ BW ~ PRF ~
x n ,k=1 - (n - 2k)(fo- 11) (30)
(39)
(40)
The spectrum into the low-pass filter becomes
=C1/(f) + 2A o l: QD
lJ1/(f)
",-1
l:
n
m-n
~ ~
m=J 11=0 k=O 1=0
. e + I «m -
n - 2/)a + {3xm ,n ,k ,l J
Where
2:
K
e- l.x7n,n,k,/12(m - n )Af2 J
(31)
For conventional radar receiver analysis, (38) is the spectrum presented to the digital signal processor. The array factor is incorporated into the definition of A o. Equations (38) and (39) indicate that the peaks of the distortion spectra are located at harmonics of the positioning error 10 - II and include so-called real and image spectra. A useful, semiempirical approximation considers dominant terms, (m = n k), (m = n, k = 0). Then QD
~ = ~ Aot:.j xm.n,k,/=I -
C2/(f ) =
(n - 2k)/o - (m - n - 2/)/1'
::;0 ~o K2: (m) n o[f-(m-2n)jo]· 00
m
~ F U3/ ~Ao ~ m con con m = I
(32)
t::. m - 1 _l;0_ 1 "
m
e-I/2(/±m(/I-/0)l~f~q2
(41)
where
(33) (34)
574
~o= ~ Aot:.f ~;
Usat
(42)
Q; (f) = ajco(f)[cos ,-/ LOCAL OSCILLATOR SINGLE PRF LINE OF CLUTIER
where Fig. 6. Spectral characteristics. The mixing products at IF near fo will place desired downconverted terms, as well as certain distortion, into the pass band. The low-pass filter bandwidth is larger than the spectral width of a single PRF line of clutter but smaller than the local oscillator at fo. This design rejects many undesired mixing by-products.
F
_[uovsatJ 4
m-
m
- 1- -d~; 2m (,)2 m. d VI
Gm,n,k=
u2
(43)
w~ =
Qj
sin
(2; (2;
Xj
sin
~2m-n
e- 1/ 2 [! + (n -
2k )(! t - ! o )/ tl! ..J2m - n J2
et>o)
e+ tn1/!=e+ / (N -
l )", / 2
(50)
(51)
sin (Nl/;/2) _
(52)
sin (l/;/2)
because, from (49), (50), and (20), (21), all the terms} depend upon
(53)
Xj=(j-l)d.
For uniform illumination, {I (f)
QJ
=
1 and
.'V
= co(f)
~ [cos OJ - sin OJ] ;=1
'Zm-l
co
n
2; 2; Gm,n,k
+ 1/2 ~
(45)
m= 1 n=Q k=O Xj
sin
et>o)
(46)
.
(1
[ [
+
+ in- 2k + J )e- /(N -
[<1 _i
Q' (f) = co(f)
n-
I lit n,k
sin (N'V -k.) .
:.
sin i' n,k
"k+ 1)e-,(N- I)"',i,k Sin. (N'iY ;k) SIn
N
2; [cos
'It:
k
(48)
575
J]
(54)
OJ + sin f)j]
OD
2m-l
n
+ 1/2 ~ ~ ~ c.c.: m= 1 n=O
k=O
Gm,n,ke+it/tn,k,j
k=Q
]
j=l
n
+ Qj ~ ~ ~ m= 1 n=O
k=O
n=O
the fundamental mixer outputs will be collimated across the array for the signals returning from the direction of transmit illumination. The prescription (45), (46), (4), (5) will be calculated for the general element. Recalling the notation differences between quadrature and in-phase channels, the post steering element spectra are
2m- 1
n=Q
~2m-n-1
N- 1 ~
(47)
OD
m= 1
Combine the brackets in (48), (49), i.e., the beam steering terms, into the exponential phase notation of the element signal. A straightforward summation over the elements, j, may be then undertaken since the exponential has a linear spatial phase error. Assume that all element amplitudes are identical except for the taper coefficients. The summation over elements uses the standard phased array result for uniform illumination
is the mixer conversion gain for the m th harmonic as measured for a two tone test with a receiver saturating sinusoid input Usal ' Equation (41) is useful because it describes the growth of the harmonics with approaching saturation through (42), and (44) provides an empirical quantity for the simplifying the multiple sums of (38). For digital manifold phased arrays, the phase arguments of (38) have yet to be accumulated into an array factor, and the amplitude A o is calculated using an element factor only. From each element, the ordered data pair. (VI, vQ) is to be phase rotated in the complex plane so that the signals from a desired wavefront of arrival will be in-phase and will sum into an array factor equivalent to the conventional phased array. The steering prescription given in (4), (5) is time independent and may be generalized to include an amplitude tapering coefficient Qj for controlling sidelobes. When the phase steering commands at the jth element are set to
cos
n
(2m) (n)k (2mm-k-n) n
2A o 2m 4m K
.
(44)
Qj
2m-l
l/;n,k.j = (n - 2k)cxj -l3j [f + (n - 2k)(fl - fo)]. I'
and
w~=
co
+aj ~ ~ ~ Gm~n,ke+i"'n,k.j
.:1t
L.PF.
1.0
OJ + sin OJ]
_
[
(1 - i n -
2k + l)e-i(N-l)'i'
sin (N~ n,+k) Itk. SIn
i':k
J]
(55)
~here
.;'k=
sin
:d
[(n-2k) sin (Jo - sin epo -
A
2u tan (Jo
[f + (n - 2k)(fl - fo)]]
sin (56)
sin
+ sin epo -
A
2u tan (Jo
[f + (n - 2k)(f, - fo)]].
(57)
The result of (54), (55) is just the electric field array factor of the aperture and should be regarded as very satisfying. If the weight coefficients are other than uniform, of course, another function is substituted for the sin (Nx)/sin (x) with an angle of arrival argument determined from the linear phase error. The phase error arguments (56), (57) determine the collimation properties of the spectrum components. When equal to zero, there are no phase errors across the array, and the array is said to be collimated. Otherwise, there is a residual linear phase error leading to the usual near and far sidelobe features of antenna patterns. The first two terms in (56), (57) indicate the alignment between the transmit illumination scattering and the phase rotation of the digital manifold. The final term models the linear phase error of mainbeam clutter and indicates that the relocated distortion spectrums harmonically defocus at frequencies lying in directions off boresight. A simplifying result will be discussed to clarify the impact of manifolding on intermodulation distortion. Consider as in (41) only the dominant amplitude components of the spectra as apparent in spectrum analyzer measurements with the collimation commanded for the direction of transmission. In general, Km is a decreasing function with increasing m, and, except near saturation, higher order terms are smaller from the multiple convolutions. Let
cPo = - 00
(58)
m=n=k
(59)
m=n, k=O
(60)
then,
(N7rd [(m + 1) sin 0 + AU- m(fl - fo)]J ) 0
A
(7rd
[(m + 1) sin (Jo + AU- m(f, - fo)]J) 2u tan 80
A
(N7rd
[(m+ 1) sin (Jo- A[f+m(f,-fo)]J)
A
. (7rd [(m + 1)
sIn
sin
-
A
(N:d
2u tan 80
L AF~m-1 m CD
0
m=l
sin sin
0
uo.Jm
(N1rd A
(1rd
H~
[(m-1) sin (Jo+AU-m(fl-fo)]]) A 2u tan 00
0+ A[f-+ m(/l - 10)]J) --0
2u tan (Jo
[(m _ l) sin (Jo _
AU + m(fl - fo)]J )
1\
sin
(7rd
2u tan 80
[(m-1) sin (Jo- A[f+m(fl-fo)]J) 2u tan 00
A
(61) where
= e: 1/2(/- m(/1 -/o)/a/..Jm1 2
(62)
H,;' = e- 1/ 2 (1 +111(/I-/O)/~f.J; ,2
(63)
H,~
and other terms are defined in (42), (43). In (61), for m = 1, i.e., the desired down conversion term, the second array factor term of the real spectrum is cancelled, since
(64) Likewise, for the image, the first term is cancelled. The remaining array factors have zero linear phase error except for the defocusing of off-boresight arrival. In general, for the distortion products, there will be array factors with a steering compensation (m - 1), and an anomalous, spoiled factor (In + 1). For odd harmonics, one of these beams is cancelled, while for even harmonics, both will be present. The directive gain is determined from the unnormalized array factor using angle arguments
(1 +;I-m)
[em _ l) sin (Jo + A[f- m(fl - fo)]J ) 2u tan 00
.
SIn
+ 1)
sin 00]
(65)
arc sin [(m - 1) sin 80]
(66)
arc sin [(m
l(f)~
2u tan 80
at the distortion relocation peaks. This nuance from the heuristic Section II arises from the phasing of the quadrature channel as perturbed by local oscillator self-mixing and transfered through the phase rotation steering prescription. VI.
RESULTS AND EXTENSIONS
In this section, results are delineated, and some further remarks are made regarding performance when some of the ideal assumptions of Section V are relaxed.
576
Consider a nose mounted array with a Gaussian array factor . Let the downconversion frequency error be 2v
11-10=- [cos Oo-cos Oil A
_ .- •CONVENTIONAL ARRAY. L OW LEVEL CONVEN TION AL ARRA Y. NEAR SAT - - - OBF. NEAR SAT. •
eo
(67) ui
'" is z
where
00 the direction angle to the main beam clutter 01
of transmit the direction eosin corresponding to positioning error.
~
(68)
\
j j
\
i
i
i
20
By choosing this error large, the distortion spectra may be resolved . For collimation in the direction of transmit , Fig . 7 compares the distortion spectra as calculated for the first five dominant terms of (61) with harmonic conversion power gains below des ired of a well designed mixer, e.g . ,
O 2 = -65 dB 0) = -75 dB 0 4 = -80 dB 0 5 = -85 dB .
40
:3 o
(69)
2V :;:;
I
i
\
I
/>.
if
I
I
\
I
II
16 67 KH z
/ '\ em /: __ \_.../-...~ _ ~.~'~ _ \
I
__
/ .......
\.
10 KILOHERTZ
\
1
1/
I
The higher harmonic distortion peaks for near saturation levels will lie about - 145 dB relative to the mainbeam clutter peak . For extreme downlook dynamic ranges and long coherent integration , the distortion is well below the noise limited threshold level. Fig. 7 shows the distortion level at the processor for DBF has been significantly reduced. The effects of increasing the scan angle near broadside are shown in Fig . 8. The nonlinear case at 10 mrad scan is essentially identical to the conventional array analysis . as expected . When scanned to larger angles , the clutter width and the positioning error increase , due to the geometry change. as expected . The feature of interest is the relati ve level of the distortion peaks . At each harmonic peak, the directive gain is decreased for the increased scan . It may also be observed that the even harmonic terms ar e broader than the odd harmonics . The even harmonic distortion has two beams, each of which will peak at slightly different frequencies with the net effect of broadening the lobe. The higher odd harmonics further illustrate this peaking of directive gain . Due to the selection of m + I or m - 1, however, the relative max imum in directive gain translates the lobe maximum slightly from a pure harmonic of the positioning error. This effect tends to narrow the odd harmonics lobe . The m - I selection for the fifth harmonic leads to a directive gain equal to the third harmonic m + 1 terms, and the spectra differ only due to the convolution differences . In the derivation of these results, a number of ideal assumptions were made . Of particular interest in specifying the receivers are effects of channel imbalances . Phase and amplitude errors which vary randomly from element receiver to element receiver will produce the same sort of effects on array factors as root mean square (rms) errors due to manifold tolerances or phase shifter quantization in conventional phased arrays . Premixer imbalances will be multiplied by the harmonic number, and the sidelobe level for the distortion products will rise . If the rms sidelobe level for the conven-
0 .0
u. \\ .I
I I
(70) (71) (72) (73)
\
.--- 41- - _1-
5 RAO
::
,
I \ i \
60
8 ex:
8,
. I
OJ
a
190:;:;
.\
I .
i
<,
\
Fig. 7. The harmonic distortion experiences a sidelobe direction gain. This result was generated using a velocity of 250 m/s. a wavelength of 0.03 m, an aperture of 0 .75 m. and a Gauss ian taper in azimuth producing a 0.058 rad bearnwidth.
CO Nv Eldl() NAl .. RRU . " ~ 10 "' 1'1
_ -
-
-
-
_
i..'9 F •
•
•U ... '"
0 8 F '0 3 )Q "'R ~e F 'o ~ ~ ,
'
t ·.,
" I I
I "
:11 ~ : I
\
\1..... : ' : I : ,
~
"
1,0 '" QAOI A N
I
\ \ '~./ I ~.:\"'" i 'iI''7\! I I
:, I
: 11 I
.,
1 . I ~ ,"
" I .
I ....
I .
:
, I , , 1' 1 I
I'
Fig . 8. Near broadside distortion spectra. With increas ing scan. the harmonic distort ion decreases as the linear phase error increases , and the array factor moves into the sidelobe region . These results were generated with a velocity of 250 rn/s using a wavelength of 0 .03 m, an aperture of 0.75 m. and a Gaussian weighted antenna patte rn in azimuth with a beamwidth of 58 mrad .
tional pattern is, from an rms error [12],
e = phase error standard deviation
N = number of elements,
(74)
then the relative gain for the mth harmonic distortion spectra peak for large scan angles is
577
(75)
and higher harmonics should completely defocus. Post mixer element mismatches are equivalent to the conventional phased array element error effects. Conventional receiver in-phase and quadrature channel mismatch tolerances are usually held tight to fully cancel the iJIlage portion of the downconverted spectrum [13]. An average amplitude or phase error in the I and Q channels will factor out of the array factor summation over elements «43)(54) and produce finite image cancellation. For a batch l11aaufacturing procedure, such a bias should arise only from the statistics of the sampled mean. The random I and Q imbalances for the total lot of manufactured receivers will lead to a probability that the mean of the smaller number of receivers collected into a radar system will be nonzero. Let Emu = the maximum tolerable imbalance, radians, allowable for a given image cancellation. The standard deviation of the receiver imbalances must be small to ensure a large confidence bound on the sampled mean statistic of the system imbalance. For a 0.99 probability that a system will pass the given image cancellation requirement,
.IN
receivers approaches the number of elements, image rejection is determined by the ensemble character of the receiver imbalances-a single receiver will not have a great effect. Tolerating imbalances with zero mean and finite standard deviation statistics becomes a permitted receiver manufacturing perspective. Receivers for DBF must become less conspicuous in size, weight, power and cost. In addressing receiver design, the array off-broadside cancellation of distortion relaxes specifications for many critical components. This relaxation is a necessity for entertaining increased scales of linear circuit integration and high volume production methods. ACKNOWLEDGMENT
The author is indebted to Dr. K. DeMartino, now with Dynamics Research, Wilmington, MA, for the clarity of his exemplary receiver analyses. REFERENCES
[1]
(76)
[2]
because the sampled mean is a standard normal random variable when normalized by the square root of the sample size (N is the number of receivers). When the in-phase and quadrature element channels are randomly mismatched, the effects appear in the array factor. A curious consequence of random imbalance is the appearance of finite anomalous beams for the odd harmonics. The array factor for the previously cancelled m + 1 term for the desired clutter would appear with a small amplitude coefficient more resemblant of an imperfect null. The other consequences are analogous to the understood element imbalance effects applying to quiescent and/or adaptive array factors [14].
[3]
- - € max nusmatch - < 2.572
(J
VII.
[4]
[5]
[6J [7J
(8)
[9J
CONCLUSION
In DBF radar, receiver generated distortion has accentuated linear phase error relative to the signal. The problem of this distortion to the radar is a desensitization of moving target detection near large clutter portions of the spectrum. For DBF, the array gain of any harmonic component of this distortion is fortuitously less as the spreading is greater. Nonlinearity specification may tolerate a relaxation approaching the relative sidelobe level of the array design, and the need for clutter positioning may be questioned. The final mixer stage involves the largest amplitude signals, highest percentage bandwidth, the largest consumption of power, and the greatest dissipation of heat. The result here argues that the specification of nonlinearities for this stage of the receiver may be moderated due to the array effects. The main consequence for "the receiver is a reduced local oscillator drive for the final mixer, perhaps by an order of magnitude. In DBF, this also means the relaxation of active manifolding for this LO. Components throughout the receive chain are additionally permitted lower intercept point and compression specifications. A further consequence of DBF for the receiver is the altered effect of I and Q channel imbalances. As the number of
[10] [11) [12] (13] [14]
578
S. M. Sze, "Semiconductor device development in the 1970's and 1980's-A perspective," Proc. IEEE, vol. 69, no. 9, pp. 1121-1131. Sept. 1981. Special Issue on Micron and Submicron Circuit Engineering, Proc. IEEE, vol. 71, no. 5. May 1983. P. Barton, "Digital beamforming for radar," Proc. Inst, Elec. Eng., vol, 127. pt. F, no. 4, pp. 266-277, 1980. H. Steyskal, "Synthesis of antenna patterns with prescribed nulls," IEEE Trans. Antennas Propagat., vol. AP-30. no. 2, pp. 273-279. Mar. 1982. W. R. Gretsch, . 'The spectrum of intermodulation generated in a semiconductor diode junction." Proc. IEEE, vol. 54. no. 11. pp. 1528-1535. Nov. 1966. J. R. Reid. "Spurious free dynamic range in wideband high sensitivity amplifiers.' Microwave J., pp. 26-32. Sept. 1965. J. W. Steiner. ,. An analysis of radio frequency interference due to mixer intermodulation products," IEEE Trans. Electromagn. Compat., Jan. 1964. pp. 62-68. M. I. Skolnik. Introduction to Radar Systems. New York: McGraw-Hill, 1962. p. 145. L. E. Brennan and J. D. Mallet. "Efficient simulation of external noise incident on arrays." IEEE Trans. Antennas Propagat., vol. AP-24, no. 9. pp. 740-746. Sept. 1976. . M. B. Ringel. "An advanced computer calculation of ground clutter in an airborne pulse Doppler radar." presented at NAECON '77, Dayton. OH. May 1977. S. M. Size. Physics of Semiconductor Devices. New York: Wiley. 1969. p. 105. R. J. Mailloux. "Phased array theory and technology." Proc. IEEE, vol. 70. no. 3, p. 261, Mar. 1982. H. Urkowitz. "Bandpass filtering with low pass filters," J. Franklin Inst., vol. 276. no. 1, pp. 1-13. July 1963. J. T. Mayhan and F. W. Floyd, "Factors affecting the performance of adaptive antenna systems," in Proc. 1980 Adaptive Antenna Symp., RADC, Rome, NY, pp. 154-179.
Adaptive Beamforming with the Generalized Sidelobe Canceller in the Presence of Array Imperfections NEIL K. JABLON,
Abstract-Antenna designers often employ linearly constrained adaptive beamforming as an antijamming measure. With minimal a priori knowledge of the signal environment, this technique nulls out jammers while simultaneously preserving the quality of the main lobe so that a friendly look-direction signal can be received with unity gain. Unfortunately, in the absence of special strategies, linearly constrained adaptive beamforming is hypersensitive to array imperfections when the input signal-to-noise ratio exceeds a certain threshold. This hypersensitivity manifests itself as a nulling of the friendly signal as if it were a jammer. Luckily, the signal nulling problem can be easily remedied by artificial receiver noise injection. A particularly simple and general structure for linearly constrained adaptive beamforming was proposed during the 1970's, and is known as the generalited sidelobe canceller. A detailed analysis of the generalized sidelobe canceller in the presence of array imperfections is discussed, and two new artificial receiver noise injection algorithms are proposed. Computer simulations are included to demonstrate that use of these new algorithms alleviates the signal nulling problem without seriously compromising jammer nulling. For the special case of the Capon maximum-likelihood beamformer, simple approximations are presented for: 1) the Wiener output signal-to-interference-plusnoise rat~o (SINR:), 2) tbe antenna element error variance that causes a 3 dB Joss of SINR: from its value for an ideal array, and 3) the optimal artificial receiver noise that maximizes SINR:. Manuscript received August 29, 1985; revised March 10, 1986. This work was supported by the Naval Air Systems Command under Contract NOOO 1985-C-0018, and by the Fannie and John Hertz Foundation Graduate Fellowship Program. This paper is based on a dissertation submitted by the author to the Department of Electrical Engineering, Stanford University, Stanford, CA, in partial fulfillment of the requirements for the Ph.D. degree. The author was with the Information Systems Laboratory, Electrical Engineering Department, Stanford University, Stanford, CA. He is now with the Data Communications Research Department, AT&T Information Systems, Middletown, NJ 07748. . IEEE Log Number 8609031.
MEMBER, IEEE
I. INTRODUCTION
A NTENNA DESIGNERS OFTEN employ linearly confistrained adaptive beamforming as an antijamming measure. With minimal a priori knowledge of the signal environment, this technique nulls out jammers while simultaneously preserving the quality of the main lobe so that a friendly look-direction signal can be received with unity gain. Unfortunately, in the absence of special strategies, linearly constrained adaptive beamforming is hypersensitive to array imperfections when the input signal-to-noise I ratio exceeds a certain threshold. This hypersensitivity manifests itself as beamforrner nulling of the friendly (look-direction) signal as if it were a jammer. This paper presents a detailed study of this hypersensitivity by considering a particularly simple and general structure for linearly constrained adaptive beamforming proposed during the 1970's, and known as the generalized sidelobe canceller (GSC). Luckily, the signal nulling problem can be easily remedied by artificial receiver noise injection. In this paper, two new artificial receiver noise injection algorithms are derived for the GSC. Computer simulations are presented to demonstrate that for a GSC with array imperfections, the use of these new algorithms alleviates the signal nulling problem without seriously compromising jammer nulling. The performance of the GSC is studied in the presence of I Throughout this paper, "noise" refers to additive receiver noise only, which does not include jamming.
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-34, No.8, pp. 996-1012, August 1986.
579
random element amplitude and phase errors, for an environment which consists of a look-direction signal (hereafter referred to as just the signal), one jammer, and additive white receiver noise. Both signal and jammer are assumed narrow band, which means that the reciprocals of their bandwidths are large compared to the transit times of their wavefronts across the array. Important additional assumptions made include omnidirectional antenna elements, random element errors that remain constant during the adaptation period, statistically independent and zero-mean wide-sense stationary signal, jammer, and receiver noise, and a linear, homogeneous, and isotropic propagation medium. The latter three properties mean that the medium characteristics are independent of signal magnitude, position, and direction of propagation, respectively. The hypersensitivity phenomenon is discussed in detail using Wiener filter theory to analyze. steady state behavior, and computer simulations to check the results. The Wiener analysis is largely based on the use of a quantity known as output signal-to-interference-plus-noise ratio (SINRo), which is a ratio of all wanted to unwanted power at the beamformer output. In the adaptive antenna literature, SINRo is widely accepted as a valid measure of output signal quality. Specifically, this paper contains the following contributions to the literature on the GSC in the presence of random element gain (amplitude and phase) errors. • The exact Wiener weight vector and steady state output signal-to-interference-plus-noise ratio. A simple approximate expression is presented for output signal-to-interference-plus-noise ratio of the Capon [1] maximum-likelihood beamformer. • A detailed explanation of look-direction signal nulling. Solid evidence is provided that when input signal-to-noise ratio exceeds a certain threshold, even a conventional delay-and-sum beamfonner outperforms the GSC for essentially all jammer angles of interest. • An equation for the antenna element gain error variance that results in a 3 dB decrease in output signal-tointerference-plus-noise ratio of the Capon beamformer from its value when the array is ideal. This equation should be helpful in setting reasonable antenna element error tolerances for linearly constrained adaptive beamformers. • Two on-line algorithms for GSC artificial receiver noise injection, derived from the novel viewpoint of imposing a large penalty on signal components that "leak" into the sidelobe cancelling signal [2]. A surprising result which falls out of this derivation is that the adaptive algorithm must be modified in such a way as to artificially add colored receiver noise. One algorithm extends the leaky least mean square (LMS) algorithm of Widrow and Steams [2], and the other is appropriate for applications where the state vector autocovariance matrix is estimated directly, such as the sample matrix inversion (SMI) method considered by Reed et 01. [3]. • Computer simulations to demonstrate the effectiveness of the above two algorithms.
• An equation for the "optimal" amount of artificially injected receiver noise, in the case of the Capon beamformer. • Suggestions for how the theory can be extended to the two important cases of multiple jammers and wide-band adaptive array processing. The analysis of linearly constrained adaptive beamforming in this paper has at least two major advantages over most previous approaches. The first is that although restrictions are introduced later on, the initial analysis assumes arbitrary size of element amplitude and phase errors. Thus, the correctness of the general approach is not dependent on "small" errors. The second is that by using the GSC, a linearly constrained adaptive beamformer that works with an unconstrained algorithm, the analysis is relatively easy to follow. Capon et al. [1] in 1967 were apparently the inventors of linearly constrained adaptive beamforming. They developed, analyzed, and provided experimental data to support the maximum-likelihood (ML) concept, defined as using a set of weights that minimizes beamformer output power subject to a simple unity gain constraint for signals coming from an assumed look direction. In practice, linearly constrained adaptive beamformers are most often implemented in ML form. In 1977, based largely on a beamformer first published by Applebaum and Chapman [4], Griffiths [5] proposed the GSC, which he and Jim [6] later showed was equivalent to a Frost [7] beam former under certain conditions. The GSC is able to implement look-direction unity gain (zero-order) constraints just like the Frost beamformer , but in addition is easily generalizable to deal with main lobe derivative constraints of any order. Griffiths and Jim [6] pointed out that, "new methods of adaptive beamforming are suggested by the generalized sidelobe cancelling structure," for example combined temporal/spatial constraints. Since the GSC uses an unconstrained rather than a constrained algorithm to adapt the weights, it may be possible to adapt much faster. The GSC also will be less sensitive to coefficient quantization effects, because the dynamic range of the signals in the adaptive portion of the beamformer is compressed. In general, the Frost beamformer and the GSC have different state vector autocovariance matrices and consequently different eigenstructures. Griffiths [5] stated that as long as the GSC signal blocking matrix has dimension one less than the number of antenna elements and its columns are linearly independent, then the Frost beamformer and the GSC will lead to the same steady state SINRo in a stationary environment, based on a comparison of Wiener solutions (i.e., infinitely slow adaptation). However, algorithm performance measures which are formulated in terms of eigenvalues, such as transient response time and misadjustment (due to weight jitter) [2] will be different for the Frost and GSC implementations. Griffiths and Jim [8], [9] wrote several reports dealing with the GSC. They pointed out that the GSC is a particularly suitable structure to use for studying the effect of random element amplitude and phase error effects on linearly constrained adaptive beamformers. By means of some simple
580
The GSC is shown schematically in Fig. 1, consisting of X" analysis and extensive simulation, they reached the conclusion that Gaussian amplitude and phase errors had the effect of elements which ideally would all be omnidirectional With reducing interelement correlation, leading to a lowering of identical amplitude and phase. A simple model for random main lobe gain and decreased ability to discriminate against antenna element imperfections is to let each element have a jammers. The negative effects of these errors were most random complex gain g;(i = 1, ... t K), assumed to remain constant during the period of adaptation. The use of a pronounced for high input signal-to-noise ratio (SNR;). Others who have made important contributions to analysis complex gain implicitly takes into account random element of the "imperfect array problem" include Zahm [10], Cox amplitude and phase errors. Denoting the zero-mean random [11], Takao et al. [12], Vural [13], [14], Mayhan [15], element amplitude error at element i by Sa, and the zero-mean Monzingo and Miller [16], Hudson [17], Compton [18]-[20], phase error by dpi, the complex gain can be written as Bar-Ness [21], Gupta and Ksienski [22], and Godara [23]. (1) Remedies to the imperfect array problem have been proposed by Applebaum and Chapman [4], Griffiths and Jim [6], Cox wherej £ ~, and dg; is the zero-mean complex gain error. [11], Takao et al. [12], Vural [13], [14], Hudson [17], Zahm The assumption of unity nominal gain for each antenna [24], White [25], Charitat [26], [27], Widrow and McCool element in no way hinders the generality of this approach. [28], [29], Er and Cantoni [30], [31], Ahmed and Evans [32J, The adaptive beamformer with and without array imperfecand Compton [33]. tions is illustrated in Fig. 2. Sp, includes the phase error due to Although Zahm' s [24] strategy was originally introduced as random element misplacement, which changes with signal a solution not to the mismatch problem, but to the problem of direction [35], so therefore D,.gj and gi change with signal preventing unconstrained adaptive beamformers from nulling direction. It is also conceivable that Sa, could be a function of out friendly signals, his technique is one of the most widely signal direction, for example if the antenna element pattern used ones for dealing with imperfect arrays. His idea was to was not truly omnidirectional. In the presence of errors, the artificially inject receiver noise in such a way that the weight adaptive beamfonner still adapts in such a way as to minimize vector was computed based on a higher receiver noise level mean square error (MSE), or equivalently output power, but than was actually present. However, the artificially injected the fact that it is unaware of the errors corrupting the data receiver noise does not actually appear at the beamfonner causes a degradation in performance. output. This prevents the beamformer from nulling out signals Later on it will be useful to work with the diagonal matrix G close to the look direction. If properly done, the effect on representing the complex element gains, and also the diagonal jammer nulling is minimal. matrix LlG representing the complex gain errors. Defining I as The outline of this paper is as follows: Section II derives the the identity matrix GSC Wiener weight vector. Section III uses the Wiener weight vector to derive the steady state SINRo, and mathematically G d diag {I +~gh demonstrates its hypersensitivity to array imperfections. SecG and LlG will be different for the signal and jammer, since tion IV derives the two new on-line algorithms for artificial receiver noise injection by borrowing the concept of penalty they are assumed to come from different directions. Therefunctions from optimization theory, and presents computer fore, subscripts will be used on G and ~G. A subscript swill simulation data to support the effectiveness of these algorithms correspond to the look-direction and j the jammer direction. For the narrow-band case, presteering the array to a known in making the GSC robust to array imperfections. In Section V look-direction is accomplished by use of a phase shifter at the it is explained how the results of this paper can be extended to output of each antenna element. In order to steer the array to the two important cases of a multiple jammer environment and wide-band adaptive array processing. Section VI contains the the look direction (Js, a presteering delay of - Ti,s is needed at element i. In the absence of presteering, a jammer arriving conclusions. from angle OJ would undergo a time delay at each element of II. WIENER WEIGHT VECTOR ti.). Thus, the look-direction signal after presteering can be In this section, the Wiener solution of the GSC in the treated as coming from the array broadside. The jammer presence of random array imperfections is derived, and undergoes a total time delay T; at element i of compared to the corresponding relation for an ideal array. The i= 1, ... , K. (3) analysis here is an extension of the author's work in [34], r, g Ti,) - Ti,s, where the GSC was analyzed in the absence of array Imperfections in the presteering electronics can also be imperfections using adaptive noise cancelling techniques included as part of the amplitude and phase error terms Sa, and (Widrow et ale [2]). Our analysis only considers steady state Wiener solution. In other words, adaptation is assumed to be Sp, After passing through the presteering delays, the signal infinitely slow, so that algorithm-dependent effects such as received at each element is corrupted by additive zero-mean misadjustment [2] and non-Wiener signal cancellation [2] do white noise, as shown in Fig. 3. This additive receiver noise is not come into play. assumed to be independent identically distributed (i.i.d.) A. Derivation of Exact Expression from element to element. In this subsection, we derive the GSC Wiener weight vector The beamformer consists of two branches. The upper is in the presence of array imperfections. termed the desired response branch, and its purpose is to form
581
CONVENTIONAL DELAY-AND-SUM BEAMFORMER
ADAPTIVE NOISE CANCELLER
I I
L
_
PREPROCESSOR "l.k
Reference I
B
"K,k
XK,k
Fig. I.
I I I
I
Block diagram of narrow-band generalized sidelobe canceller . Additive receiver noises following the steering delays are not shown.
I r-,
v
Adaptive
Output
Beamformer
r-,
K v
~
Fig. 3.
/
Algorithm to minimize MSE
(a)
Adaptive
Output
Beamformer
K
Algorithm to minim ize MSE (b)
Fig. 2. Adaptive beamfonner with and without array imperfections. (a) Without imperfections . (b) With direction-dependent imperfections.
To preprocessor
Model of ith receiver channel (i = I . ...• K).
the desired response dk> which is the primary input to the adaptive noise canceller. In the absence of array imperfections, the desired response branch is constrained to have a unit look-direction gain. In general , this branch is a conventional delay-and-sum beamformer, with K nonadaptive weights being fixed in such a way that the array beamwidth and average sidelobe level are both satisfactory [6]. This paper assumes uniform 11K weighting, but other weightings could be considered with minor modifications to the analysis. The lower branch of the beamformer is the sidelobe cancelling branch. Its purpose is to form the sidelobe cancelling signal Yle by providing K reference inputs to the adaptive noise canceller. Yic contains estimates of the jamming components in the desired response , so that after subtracting Yic from di, the beamformer output Zle is a "cleaner" representation of the signal. Note the use of complex conjugate weights Wi.IeO = I, "', 1<) in computing Yic. These weights can be updated by several different. methods, for example the complex LMS algorithm of Widrow et al. [2]. The .sidelobe cancelling branch is preceded by the signal 582
blocking matrix B, a preprocessor designed to block the signal so that the sidelobe cancelling branch cannot learn it. The preprocessor has K inputs and K outputs. In this paper it is assumed that K < K. The simplest example of a preprocessor is adjacent element differencing, yielding zero gain in the look-direction (in the absence of array imperfections). Griffiths and Jim [5], [6] showed that use of the latter blocking processor makes the converged GSC behave like a Frost beamforrner. For this preprocessor K = K - 1, and results in K - 1 degrees of freedom being available in the sidelobe cancelling branch to form nulls in jammer directions. The restrictions on Bare [6]
effect of the element gains on si: The same plane wave assumption can be used to represent ik in terms of the jammer i, at time sample k, the diagonal matrix Gj representing the effect of the element gains on Jk » and a diagonal matrix ~ (whose components should not be confused with the amplitude error terms da;) which accounts for the phase shift in components of ik due to presteering:
BIK=Ok rank (B) = K.
w is the common center frequency of the narrow-band signal and jammer, in rad/s. n, in terms of the individual receiver noises 0i,k is
s, = SkGs
A j g diag {e- i wT1,
(5)
f K is a vector of length K whose elements are equal to 1's, and OK is a zero vector of length K. The rank 'of a matrix is just the number of linearly independent rows or columns. In the sequel, matrices and vectors will be symbolized by bold uppercase and lowercase letters, respectively. Complex conjugates will be represented by overbars, transposes by a superscript T, Hermitian (complex conjugate) transposes by a superscript H, and steady state quantities which are based on using the Wiener weight vector by a superscript asterisk. £1[ . ] represents time expectation. Define the state vector u, and weight vector W k at time sample k as follows: T
wk,kl
T
ale represents the signal amplitude, ejwTsk the (noninformation bearing) carrier, and ejl/l le the phase. Ts is the sampling interval, in seconds. a 2, the signal power as measured at any element in the absence of array imperfections, then becomes
(19)
Papoulis [36] showed that the random phase
(7)
The unique Wiener weight vector w* (which minimizes MSE), is then [2] (10)
The complex snapshot vector at the kth time sample x., is defined as the vector of the received signal and jammer, following presteering and including the effects of both receiver noise and array imperfections: (11) Uk:
(12)
The snapshot vector is the sum of a component s, due to the signal, a component jk due to the jammer, and a component n, due to the receiver noise:
"'k
must be - Uta, b) represents a random variable uniformly distributed on the real interval [a, b]. Representing s, and i, in complex envelope notation, as.k and aj,k are the amplitudes of the signal and jammer, respectively. a; and a] are their powers. "'s.k and "'J.k are their phases. Finally, as,k and aj,k are statistically independent, as are "'sik and "'j,k' and the latter four quantities are all assumed to be varying slowly enough with respect to the sampling interval so that s, and l« can both be considered narrow band, since the narrow-band assumption for Xk implies that ale and l/;k are correlated over successive sampling intervals. Using (2), (4), and (12)-( 17), the state vector can be rewritten as - U(O, 21r) for Xk to be stationary, where
(9)
B then transforms x, into
(16)
(18)
(8)
*-R -I uu rude
e- i WTK } .
Sk and lie can both be represented in complex envelope notation. A sampled signal Xk in complex envelope notation is given by
Also define the autocovariance matrix R uu and crosscovariance vector rud:
W
".,
(15)
(17)
(6) .
(14)
jk=JkGjAjf
(4)
Uk,k]
r
(20)
It is worthwhile reemphasizing that in (20) the noise n, is affected by neither the element errors nor the presteering, because it was modeled as being added in after both the signal/ jammer reception and presteering delays. The appropriateness of this model will vary with the application at hand. From (8), (20), and the independence of signal, jammer, and receiver noise
(13)
Utilizing the plane wave assumption for the signal, It IS possible to write an expression for s, in terms of the signal Sic at time sample k and the diagonal matrix G s representing the
where the receiver noise power equal for all channels.
583
0'; £ E,[lni.kI2] was assumed
Equations (2) and (12)-(17) can be used again for the desired response d k (i.e., the adaptive noise canceller primary input) to obtain 1
1
-
dk=J( »: 1 =SkOls+lkOlj+J( nIl. T-
.
Making use of (23), (24) and (28)-(33), the exact expressions for Was and WOj are
(22)
as is the value of the "normalized" array (spatial) factor in the look direction, and Ctj is the analogous value in the jammer's direction: (23)
B. Discussion
(24)
The "normalized" array factor a in any general direction is
1 _ _ a ~ - 1 TGAI K
(25)
where the nonsubscripted steering matrix A is for any direction (J. A is formed by substituting (J for 8j in the expression for r., appropriate for the particular array geometry used, and then using (3) in (16). ex is both the "normalized antenna pattern of a conventional beamformer implemented with an imperfect array, and the ' 'normalized unadapted directivity pattern of the GSC with an imperfect array, assuming that the adaptive weights are all set to zero. Substituting (20) and (22) into (9)
- W Hk BdGs -1. 1k-Sk
t ,
(26)
Inverting (21)2 and multiplying by (26), the exact Wiener solution becomes
jammer. One would expect' this type of behavior, because the beamfonner only cares about minimizing output power (i.e., MSE), in any way it can. Therefore, without having extra information supplied to it concerning the nature of Sk (e.g., a pilot signal [2]), without advance knowledge of the imperfections, and without any other special remedies being taken, the beamformer has no recourse but to walk the plank of signal annihilation.
III. OUTPUT with Was and WOj given in terms of as, aj, SNRi , INRi, on OJ, Os), and 0)51 with the latter six quantities to be defined below. The input signa/-lo-noise and interference-to-noise ratios can be measured at any element to be (28)
(29)
The quantities as, OJ, OSj, and Ojs are also scalar, and will be called the S-, j-, sj-, and js-signal blocking matrix factors, respectively:
Os g (dG s f)HB T(BB T) -IB(dGs 1)
(30)
OJ ~ (GjAjr)HBT(BBT)-'B(GjAjI)
(31)
Os} ~ (aGsf)HBT(BBT)-'B(GjAjI)
(32)
T -I -Ojs = (G1Aj ... l ) H B T (DB) B(dGsl)=osj.
(33)
6
(36)
As long as Ik =I:: 0, the sidelobe cancelling branch uses it to try to estimate the signal. For "high" SNR i , that estimate will be fairly good, and the signal inadvertently gets treated like a
t t
_ 22rud-usasBaGsl +ujCljBGjA)l.
By studying the simpler expression for Uk that results when all array elements are ideal (i.e., ilg; = 0, i = 1, ... , K in (20», it is seen that the effect of array imperfections is to allow the signal to "leak" into the sidelobe cancelling branch despite the signal blocking matrix [6]. This leakage will be denoted by It, and with reference to Fig. I and (20) can be written as
2 Inversion of R ilil is accomplished by two nested applications of the matrix inversion lemma [37]. Considering the multiple jammer case, if there were N
jammers, an exact closed form expression for the Wiener weight vector would require N + 1 nested applications of the matrix inversion lemma, which would greatly increase complexity without adding much additional insight into the hypersensitivity phenomenon under study here.
SIGNAL-To-INTERFERENCE-PLUS-NoISE RATIO
The purposes of this section are fourfold. First, the exact formula for Wiener output SINR is derived. Second, we find the conditions when the signal will be nulled. Third, an approximate equation is presented for output SINR of the Capon beamformer. Fourth, the ideas in this section are clarified by carrying out a "Wiener simulation," which is done by randomly generating array imperfections, and then using (27) to compute the Wiener weight vector that minimizes MSE assuming that the beamforrner is unaware of the imperfections.
A. Derivation of Exact Formula The exact expression for Wiener output SINR is derived below. The output power Po of the GSC is
I E ,[ IZk 12]=21 (E,[ Idkl 2] Po £ 2 -Et[dk.Yk] -Et[dkYk] +E,[IYkI 2] ) .
(37)
With reference to Fig. 1, the sidelobe cancelling signal Yt is (38)
When each term in (37) is evaluated using (20), (22), and (38), it follows that Po can be expressed as the sum of three terms. The first is Pas, the output power due to the signal only,
584
the second term is P Oj , the output power due to the jammer only, and finally POn is the output power due to the receiver noise only. When w* is used in (38) to calculate the steady state sidelobe cancelling signal, and then is used in (37) to calculate the steady state values of Pas, POj , and POn (denoted by P ds ' P and P6n' respectively), one obtains
y:
enough for the jammer to be nulled out. For strong jammers, the second assumption will essentially always be true, since OJ == K away from the look direction [35]. Armed with the assumptions (44), (45), it is straightforward to show that [35]
y:,
w'
p*
.fl
Os -
I W [) W ;;"1 -1 (/21a 121-~-~ 2 s s as as
2
SINRri
== O.
(46)
This result provides solid evidence that the signal gets treated like a jammer. Furthermore, under the two assumptions made, the GSC performs even worse than the conventional beamformer, which at least will have nonzero SINR o given by (43).
(39)
(40)
c. Approximation by Output Signal-to-Noise Ratio This subsection will present a simple approximation for SINRti that is valid for the Capon beamformer. If the designer wishes to check the accuracy of the assumptions (44), (45), some B must be chosen so that (30), (31) can be evaluated. A class of signal blocking matrices known as central difference matrices is formed by using r cascaded columns of differencing, as shown in Fig. 4. Notationally, they are written as B ~- 1), where the subscript K indicates the number of antenna elements, and the superscript (r - 1) the use of a main lobe zero (r - 1)-st derivative constraint in the look direction (when the array is ideal). The quantity K (the dimension of the state and weight vectors) then becomes (K - r). The simplest case is a zero superscript, or r = 1, which is just adjacent element differencing, and configures the GSC to be maximum-likelihood, implementing a simple unity gain constraint in the look direction. The (K - 1) x K matrix B~) is
(41)
SINRti (i.e., the steady state SINRo) is then SINR* ~
o
p*
Os
P*.+P* OJ On
•
(42)
SINRo of a conventional beamformer, symbolized by SINRo,c, can be derived by setting W(b- and WOj to zero in (39)(41), and then substituting the result into (42). There is no need to explicitly indicate that SINRo,c is steady state, since no adaptive weights are involved in computing it:
SNRi las l2 SINRo,c = - - - -
INR'la'12+~ K I
(43)
J
Recall that the analysis so far involved no approximations
with respect to relative power levels of signal, jammer, and
-1
receiver noise. Additionally, no approximations were made as far as the magnitude of the array imperfections were concerned.
B(O) ~ K
0
B. Hypersensitivity to Array Imperfections This subsection will demonstrate signal nulling when SNR; is "high." Two assumptions will first be made. They are SNR·
~
INR·
~
I
I
1 Os 1 -
O.
0
(47) -1
From Fig. 4, it should be clear that the (K - r) x K matrix I) then becomes
B~-
B(r-l)
K
(44)
=
II' B ;=0
(0)
K-r+I'
(48)
Due to the importance of the Capon beamformer in practice,
(45)
J
where Os and OJ were given in (30), (31). The first assumption means that SNR; is high enough for the signal to be nulled out. Os measures the imperfectness of the array from the signal viewpoint, so that when Os is low (good array), only high SNR i will result in signal nulling, and when Os is high (bad array), even low SNR; will result in signal nulling.? The second assumption means that INRi is high
B~) will be chosen to illustrate the form (30)-(33) take. From
[35]:
which allows (30)-(32) to be written in purely scalar form:
) In an actual implementation, even though (44) may be satisfied, nulling of the signal may be prevented somewhat by the limited dynamic range of the weights. Throughout this paper, it is assumed that the weights have infinite dynamic range.
585
(50)
(51)
•••
2
Look-direction
r
signal
Jammer
(). J
K
...
Inputs
Fig. 4.
~
U
K
(0) _
sj -
~ £J
A -
~gi.s
~} ~ :.
K - r
• •• •
Outputs
e
Fig. 5.
• • •• •
Array geometry.
Letting Ea [ · ] represent expectation over an ensemble of antenna elements that are i.i.d., the gain error variance u 2 (which measures the variance of the fractional gain deviation from its nominal value) is defined as
i= 1, ... , K
jWT'
I
t= I
(54)
and the amplitude error variance and phase error variance in a similar manner:
where the superscript (0) has the same meaning as for B~). From these last three equations it is easy to see that as long as ~gi.s ~ 1 and tlgi,J ~ 1, which is realistic: (53)
As shown in Section III-A, SINRti is a complicated function of the number of array elements . the array geometry, the array imperfections, the look direction, the jammer angle of arrival, the input signal-to-noise ratio, the input interference-to-noise ratio, and the signal blocking matrix. Therefore, any assumptions which simplify the expression for SINRri while preserving its accuracy will help tremendously. Fortunately, for the Capon beamformer, there is such an assumption, and in addition to being intuitively pleasing, it can also be shown numerically to be quite reasonable. The assumption is that SINRti can be approximated by the Wiener output signal-to-noise ratio (SNRri). In a nutshell, this means that the Wiener output signal and receiver noise power are the same as if there were no jammer present, and the Wiener output jammer power is zero. For these approximations to hold, the jammer must fall outside the main lobe of the unadapted beampattem. In order to test the validity of the above approximations, Pris, P and Prin were plotted as a function of the gain error variance (cf. (54) below) for a ten-element equally spaced line array with d/). = 0.5, a broadside look direction, SNRi = 30 dB, and INRi = 50 dB. The array geometry is shown in Fig. 5. The jammer angle chosen was the worst one outside the 0 main lobe, at OJ = 17 • As with all other "Wiener simulations" in this paper, the computations were performed on a DEC VAX 11/780, using DOUBLE PRECISION program variables.
w'
~
10 9 8 7 6 5 4 3 2 1
Cascaded columns of differencing.
(1 + ~gi,j A ) e-
A 2
-
i= 1, ... , K
(55)
i= 1, ... , K.
(56)
The gain errors were randomly chosen as the sum of equal variance, independent, zero-mean Gaussian amplitude and phase errors: (57)
where both u a2 and ap2 must be "small" for (57) to hold. "Small" means (1 + Ila;)e j Ap ;
== l+da;+jllpi,
i = 1, .", K.
(58)
For this "Wiener simulation," only angle independent amplitude and phase errors were generated, so that dgi,s = dgi,j(i = 1, ... , K). In the plots referred to above, Pris, P and P6n were compared with and without the jammer present for in the range - 100 to - 20 dB, where
w'
a; (dB) £ 10
10glO
u;.
u;
(59)
The respective curves of Pris and P6n were so close to each other that they were virtually indistinguishable. P~ was at least 70 dB below both P
586
that SINRri can be approximated by SNRti, resulting in SNR· K SINR * == I o l+SNR~I K(K-l)a g2
TABLE I
THRESHOLD GAIN ERROR VARIANCE
(60)
which in the case of an ideal array is SNRiK, as it should be, and drops down 3 dB at the threshold gain error variance of (J2 =:: g, threshold -
1 -----SNR;K(K -1) .
(61)
The similarity between (60) and Hudson's equation [17, eq. (6.2.8)] is considered in [35]. The form of (61) is also consistent with Compton's [20] observations. This is elaborated on in [35]. Furthermore, (61) can be used to define the threshold input signal-to-noise ratio needed in order to have inadvertent signal nulling, for a given number of array elements and gain error variance. The result is a strong suggestion that the signal will be nulled under far milder conditions than given by (44) (it is shown in [35] that o~O) == (K - 1)<1;, which allows direct comparison between the conditions for signal nulling in (44) and (61». The simulation data in Table I (obtained by ensemble averaging 50 curves of SINRti versus (J; and then measuring (J~, threshold) shows that (61) is accurate to within 1 dB when the gain errors are the sum of equal variance, independent, zeromean Gaussian amplitude and phase errors that are angle independent, as discussed earlier. The array geometry is also the same as the one discussed earlier. The measurement error for the simulation data is ± 1 dB. For high SNR i , the threshold gain error variance is extremely small. In fact, since both Mayhan [15] and Schrank [38] have stressed that (J; < - 40 dB is usually considered difficult to achieve in practice, it is clear from (61) that for high SNR i , it will not be possible to control (J2 to a tight enough precision to avoid signal nulling. For example, if SNR i is only 20 dB, and a ten-element array is used, (61) says that (J2 needs to be about - 60 dB, which is an almost impossible specification. Therefore, when SNR i is high, some special
strategy must be taken to avoid signal nulling in linearly constrained adaptive beamforming, D. Wiener Simulation Example This subsection will contain a "Wiener simulation" example. In order to graphically illustrate the ideas in this section, a numerical example was done for a ten-element equally spaced line array with a broadside look direction and d/"A = 0.5, where d is the ideal interelement spacing, and A the radiation wavelength. The array geometry is the same as indicated previously in Fig. 5. Signal blocking was accomplished by adjacent element differencing (r = 1 in Fig. 4), which put the GSC into a maximum-likelihood configuration. The signal had a power level of <1; = 0 dBw, 4 the jammer (J~ = 20 dBw, and J the receiver noise (J2n = - 30 dBw . The imperfections were assumed to be due only to random element misplacement, obeying a two-dimensional zero-mean 4
Decibels relative to 1 W. The formula is: X(dBw)
=
K
1!..
10 10 10 10 10 10
0.5 0.5 0.5 0.5 0.5 0.5
~
8, (del)
8) (del)
SNR, (dB)
0 0 0 0 0
17 17 17 17 17 44
10 20 30 30 30 30
0
Ideally
placed
,.,--
I
/
/
I
\
\
Fig. 6.
"
(dB)
-
Simulation
-39.5 -59.5 -79.5 -79.5 -79.5 -79.S
-39 -59 -79 -79 -79 -79
SO 40 30 SO
" "-
---
\
0
}OII~ L1x
(JT '-....
threshold
Formula
SO SO
- --...
/'"
I Ay
a:.
Misplaced element:
•
element:
INR, (dB)
,,/ -."..,.....,
I
\
\
/
I
/
Geometry of random element misplacement.
Gaussian distribution whose radius had a standard deviation o, of 0.01 A (only 2 percent of the ideal interelement spacing, a specification that may not even be attained in practice for many applications), taking the element position in an ideal array as the centroid, shown in Fig. 6. These random misplacements cause a direction-dependent phase error at each antenna element. However, in [35] it is shown that for far-field signals, under the three assumptions of the random x misplacement ~ and random y misplacement ~y both being "small," statistically independent, and of equal variance, the phase error variance is independent of angle of arrival, 5 and is given by (62)
a;
In this particular simulation there were no amplitude errors, so is also given by (62), and is easily calculated to be - 24 dB, a rather large error for this system, in light of the roughly - 80 dB threshold gain error variance from (61). As a basis for future comparison, Fig. 7 plots SINRti for an array having ideally placed elements. This plot was generated by sweeping thejammer from 8j = - 90° to 8j = + 90° in 1° steps, and at each angle computing SINRti by (42), with SINRo,c (computed by (43» included as a reference in order to
10 log,o (X(W).
587
5 Although this result may seem strange, it [urns out to be a direct consequence of the fact that sin 2 8 + cos ~ 8 = 1.
50 . 00
Look-direction
•
,.> Adaptive
I
10 . 00
I
i
10 .00
I
. 00
-10 .00
-20 . 00
i!
I I
20 . 00
::
\
\"'-.J / \\ ;I\\ ) 1\, '-
I
I.
\ \
~
=
10 SNR i
= 30 dB \ .
d A
=
1 2
INRi
= 50 dB
Os
=
on
O'r
-1 00 . 00
A
=
\
;
\
"
\1)
-50 .00
. 00
i\
) \I Ij
l .j - ~
'\
a
I
II1\;1\\ ' \
i\
I
1,1
:i
I
II
!
i
1\
I\
I
/
i Ii
,i
::
K
- 30 . 00
~
·:·" ,,:.
30 .00
'--'
,I
/
\\ _- ,//
Conventional
50 . 00
100 . 00
Jammer angle of arrival, OJ (deg) Fig. 7.
Wiener output signal-to-interference-plus-noise ratio versus jammer angle for Capon beamformer implemented with ideal array . '
demonstrate the performance improvement due to adaptation , Clearly, in the absence of array imperfections, the GSC works very well. Fig. 8 uses the same parameters as Fig. 7, except that SlNR 1i was computed with the effect of the unknown random element misplacement included. Although the conventional beamformer is hardly affected by slight element misplacement, the GSC SINRri is seen to drop by over 50 dB at most jammer angles , which is a very serious loss. In fact, for the GSC, SINRri falls so low that even the nonadaptive conventional beamformer outperforms it for essentially all jammer angles of interest. These observations are consistent with the analysis presented earlier in this section . The array factors for both cases, computed by (25), are shown in Fig. 9. They are virtually the same for both the array with ideally placed elements , and the array with misplaced elements. The worst jammer angle outside the main lobe is 0 pointed out , which is OJ = 17 • A jammer coming from this angle is at the maximum of the peak sidelobe of the unadapted array. In Fig. 10, the far-field directivity pattern of the GSC for the above jammer angle is plotted, without the effects of element misplacement considered . In Fig. II , the far-field directivity pattern of the GSC for the above jammer angle is plotted, with the effects of element misplacement considered. From Figs. 10 and 11, it is clear that for the ideal array, only the jammer is nulled, with the gain in the look-direction satisfying the 0 dB constraint. However, for the array with slightly misplaced elements, not only is the jammer nulled, but due to the small pointing error resulting from element
misplacement, the signal is nulled as well, and in the process the main lobe has been destroyed. The high directivity shown at most angles for the array with misplaced elements does not violate any physical principles . It is merely indicative of the large weight values needed to null the signal. The beamformer chooses these large weight values because they minimize MSE, without regard for the effects on the directivity pattern. IV. ARTIFICIAL RECEIVER NOISE INJECTION
The goals of this section are to derive the new leaky algorithms for artificial receiver noise injection in the GSC by means of a novel approach , present an expression for the "optimum" level of this noise along with some "Wiener simulations," and discuss results of "data simulation " experiments which support the theory set forth in this paper. Contrasting the terms "Wiener simulation" and "data simulation," the latter means that the performance of the GSC was simulated not by computing the Wiener solution, but by using random input signals whose temporal and spatial characteristics were specified as input variables to a data simulation program. This type of simulation allows one to observe both transient and steady state GSC behavior.
A. Derivation of New Leaky Algorithms In this subsection , the new leaky algorithms for artificial receiver noise injection in the GSC will be derived. The Wiener weight vector computed in (27) solved the unconstrained optimization problem
588
minimize £( [i Zk 12] . w
(63)
10 .00
20 .00
\
\
......... CQ "'0
. 00
-....J
* 0 Pc:
~
-10.00
-10 .00
Look-direction •
\
Conventional
A ~
\
\
\
I
..\)UV\ jVU..
----- .-----.-- ----..-- --. \1/ -.-..--. .-- -. K = 10
SNR j
1 d = >.. 2 aOs
= 30 dB =
50 dB
= 00
~-· - -=~~p--~:~e
':"
:: '.
" "" "
" "
I
-60 .00 . 00
-50 .00
-100 .00
50 . 00
100 .00
Jammer angle of arrival, 8j (deg) Fig. 8.
Wiener output signal-to-interference-plus-noise ratio versus jammer angle for Capon bearnformer implemented with imperf ect array.
, y{ \
20.00
Look-direction
. 00
......... ~ "0
-....J
d
.-ca-
- 20 .0 0
/
~
0
u
ctl ......
-40 . 00
" "
..
Jammer at 8j
=
170
ctl
0. CI.l
>.
-60.00
ctl
K
~
< ~
- 80 .0 0
-d
10
Ideal array
1
Imperfect array
x =
2
=
00
6s
CI r
>..
= 0.01
-1 00 . 00 - 100. 00
- 50. 00
. 00
50 .00
Far-field angle, 6 (deg) Fig . 9.
Array spatial factors of ideal and imperfect arrays.
589
100 . 00
10 .00
Look-direction signal
•
..
Jammer
20 .00
-
.00
CQ ~
-
......"
-20 .00
\(
>.
0> °BU
-10 .00
~
is
-d A
-60.00
1
= -2
as = 0° aj -- 17°
-80 .00
- 100 . 00
SNR i
= 30 dB
INR I.
= 50 dB
-sc .oo
-100.00
. 00
Far-field angle,
10. 00
Look-direction signal
CO
"Cl
-
..
..
-~~V~\(
10 .00
......"
(deg)
100 .00
~
Single jammer far-field direct ivity pattern of Capon beamformer implemented with ideal array .
Fig. 10.
-
a
so.co
[I
.00
Jammer
/>.»>:
V
~
'--
-20 .00
>.
0>
.~
-10.00
a
-60.00
.... U
K
=
d A
1 = -2
10 SNR I.
= 30 dB
INRi
= 50 dB
as = 0°
-80.00
0" r
aj = 17°
-100 .00 - I ~ O . OO
A
= 0.01
-sc.oo
.00
Far-field angle, Fig. 11.
a
so.co
100 . 00
(deg)
Single jammer far-field directivity pattern of Capon beamfonner implemented with imperfect array .
590
where R gg is defined as the covariance matrix of element imperfections:
It was previously shown that the degradation of the GSC performance in the presence of array imperfections was due to leakage lie of the signal into the sidelobe cancetling branch, as given by (36). In order to eliminate this leakage, consider the constrained optimization problem minimize Et,a[1 Zk 1 w
2
(70)
Thus it is hoped that (69) represents a good approximation to the penalty function. Unfortunately, in an on-line implementation of stochastic steepest descent, the computation of P(Wk) as given by (69) could still prove to be quite a mess, due to the presence of the time expectation operator. Consequently, it is proposed that the instantaneous value of the weight vector at the time sample k be used to generate an approximation to the expectation on the last line of (69), in the manner
]
subject to Ik = 0
(64)
where Et,a[·] represents a double expectation, taken over both time and an ensemble of i.i.d. antenna elements. Equation (64) is ill-posed, and therefore cannot be solved directly. The reason is that knowledge of the signal leakage in advance is the same as knowledge of the unknown array imperfections, so the only way to be sure of satisfying the constraint would be to set all adaptive weights to zero. The result would be a conventional beamformer. However, it is well known in optimization theory that the solution of a constrained optimization problem can be approximated by solving a corresponding unconstrained optimization problem. From Luenberger [39], one learns that for the method of steepest descent, the procedure is to convert the constrained optimization problem minimize
!(Wk)
P(Wk) = a;wfBRggB TWk
where P(Wk) represents the instantaneous estimate of P(Wk). The approximation used to obtain (71) becomes succesthe biased Wiener weight sively better as W k converges to vector when the penalty function is used. If the element imperfections are assumed zero-mean i.i.d., then from (70)
w:'
R gg = a;I.
(73)
(65)
The complex LMS algorithm, which can be used to update the weights in the GSC, was shown by Widrow et al. [2J to be
to the unconstrained optimization problem minimize (j(w k) + CP(Wk» w
Wk..-'
(66)
= Wk + 2j.LE-k Uk = (I -
2J.LUkU~)Wk + 2J.Ldl\Uk
(74)
where J.L is called the adaptation constant, and fA; is the error signal at time sample k, which for the GSC is chosen to be the beamformer output z; = d, - Yk' If (74) is used for the GSC, the mean weight vector will converge to W* [2], and all the analysis done so far in this paper concerning hypersensitivity to array imperfections will apply. The complex LMS algorithm can be derived by considering the squared error function i; or performance surface estimate at time sample k of the form
where f(wk) , q(Wk), and P(Wk) all represent suitable functions for the problem. In system optimization theory, P(Wk) is known as a penalty junction, and c as a penalty constant, the latter generally chosen to be "large." For a stochastic problem, one usually chooses P(Wk) as E[lq(Wk)\2], with the expectation taken as appropriate. Then by making the penalty constant large in (66), the minimization of CP(Wk) dominates the minimization of (!(Wk) + CP(Wk». Therefore, the solution to (66) must have small q(Wk). Indeed
{k=~*+(Wk-W*)HUkU~(Wk-W*)
(67)
which leads back to the corresponding constrained minimization problem (65). Based on the above argument, the logical choice for the constrained optimization problem is the one in (64), namely (68)
By using (36) and the conjecture that for large penalty constant, the weight vector will be fairly independent of the array imperfections, the penalty function p( W k) can be written as [35] P(Wk) = Ec,a[\lk 1 2]
== a;Et[wfBRggBTWk]
(72)
Substitution of (72) into (71) results in
=0
subject to q(wk) = 0
(71)
(69)
(75)
with ~k denoting the MSE (i.e., performance surface) at time sample k, and ~* representing the minimum (attainable) mean-square error (MMSE). By taking Jl times the negative gradient of t, with respect to Wk, and adding it to Wk, Widrow et al. [2] developed the complex LMS algorithm. Due to the form of (75), the MSE is a K -dimensional concave upward parabolic bowl, which means that taking the negative gradient of tk on the average leads to descent in the bowl's steepest direction. Use of the estimated penalty function P(Wk) yields the modified performance surface estimate Pk, which is Pk={k+CP(Wk)=~*
(76)
with Pk denoting the MSE at time sample k when the modified performance surface is used.
591
Taking p. times the negative gradient of Pk with respect to and adding it to Wk in the manner of Chestek [40]:
weight vector and MSE by choosing p. in accordance with
Wk,
1 O
Wk+I=(I-r BDT)Wk+ 2P.Ek Uk
=
(1-21-{ukuf+f BBT) )Wk+ 2J.L dkUk
3 tr (R uu ) + o
(77)
From (77), one immediately obtains the equality (78)
The reader is warned that in general, the optimum c is a complicated function of the system parameters (cf. Section IVB).
The constant
r is generally close to zero and must satisfy r
> o. 6 This new algorithm will be called the GSe leaky LMS
algorithm, after Widrow and Stearns' [2] leaky LMS algorithm Wk+ 1 =
=
(1 -
t)Wk
+ 2p.ik U k
(I-2,.,.(ukuf +~ 1) )Wk+2,u1kUk
(79)
n
uu
r
)-1
+ 2,.,. BOT
rud
(80)
which is just like having equivalent receiver noise power a; (cf. (21» of (81)
where the injected receiver noise power
; =i.u:'
a2
T)
where tr( ·) denotes matrix trace [37], defined as the sum of the diagonal elements, and also equal to the sum of the eigenvalues. The bound in (83) is easily calculable in an on-line implementation, since tr(Ruu) is just the total input power to the sidelobe cancelling branch, and p., and B are all specified by the designer. The advantage of using an LMS-type algorithm in the form (77) over other adaptive schemes is that the computation per iteration required to update the weights is only on the order of K complex multiplications, since DB T is generally banddiagonal (e.g., B~)(B~»)T is tridiagonal). Sometimes one wishes to estimate R uu by other more computationally intensive means, such as in the sample matrix inversion (SMI) algorithm considered by Reed et al. [3]. Then artificial receiver noise can still be injected by means of the artificial noise covariance matrix estimation algorithm
r.
asc
with (1 termed the leakage parameter. When the leakage parameter is unity, leaky LMS simplifies to complex LMS, as given by (74). Widrow and Stearns' [2] analysis of leaky LMS can with minor modifications be used to show that the algorithm (77) has the effect of biasing the Wiener weight vector toward
w:= (R
i.tr (DB 2p.
(83)
af is simply (82)
The term (t/2p.)BB T in (80) is known as artificially injected receiver noise, because it is the same as the covariance of white noise after transformation by the signal blocking matrix. It results from solving the modified unconstrained optimization problem (66), and is not due to any noise sources introduced into the adaptation circuitry. Since the GSC leaky LMS algorithm is a special case of Chestek's [40] soft-constrained LMS algorithm, his results can be used to guarantee the convergence of both the mean 6 Strictly speaking, there is nothing that prevents the algorithm (77) from being used in the reverse mode, ~anin~ f < O. When u~ this way, th.e effect is to artificially subtract receiver noise, rather than add It. However, If one is not careful, some eigenvalues of (RUII +
2 T R- uu•i -_ R uu + (1;DB
(84)
where Ruu is the data-dependent estimate of R uu , and Ruu ,; is the data-dependent estimate of R uu , modified by the artificial noise injection.
B. Optimum Injected Noise Power This subsection will present an expression for the optimum artificially injected receiver noise power in the Capon beamformer, along with a "Wiener simulation" to verify that the artificial receiver noise injection strategy works. In order to use the algorithm (77), (17 must first be determined. If is chosen to be too small, the adaptive weights will hardly be affected and the signal will continue to be nulled. On the other hand, if af is chosen to be too large, the adaptive weights will all be driven toward zero and the jammer will not be nulled. In fact, as af tends toward infinity, the performance of the approaches that of a conventional beamformer. In [35] an expression is derived for the "optimum" artificially injected receiver noise power in the Capon beamformer, symbolized by opt. "Optimum" here mea~s ~at SINRti is maximized. The major assumptions in the derivation are that the Wiener output signal power is the same as if there were no jammer present, the Wiener output jammer power is the same as if there were no signal present, the Wiener output receiver noise power is the same as if there were no jammer present, the jammer is outside the main lobe of the unadapted beam pattern, and the leakage of the signal into the sidelobe cancelling branch is small compared to the equivalent receiver noise. Using these five major assumptions, and a few more minor ones
592
a7
asc
a7,
deviation equal to only 2 percent of the ideal interelemeor spacing. All parameters in this example are identical to those used in Section ill-D except for the strategy of artificial receiver noise iniection, Use of (85) suggests o'~It opl/U n2 = 32• 2 ~ dB, which was rounded up slightly to 0';/ o'~ = 35 dB in order to check the effect of an inadvertent 3 dB error when applYing the formula. Fig. 12 shows the resulting far-field directiVity pattern of this adaptive beamformer, using the latter amount of artificial receiver noise injection. Comparing to Fig. 11, it is seen that the signal nulling problem within the main lobe is now completely eliminated, while at the same time sacrificing less than 10 dB of jammer nulling. From the viewpoint Of
TABLE IT
OPTIMUM ARTIFICIALLy INJECTED RECEIVER NOISE POWER 2
O'"opc (dB)
K
10 10 10 10 10 10 10 10 10 10
d
r
0.5 0,5 0,5 0.5 0.5 0.5 O.S O.S
0.5 0.5
8, (deg)
0 0 0 0 0 0 0 0 0 0
OJ (deg)
17 17 17 17 17 17 17 17 30 44
SNR, (dB)
10 20 30 30 30 30 30 30 30 30
INR, (dB)
a~ (dB)
-40 -40 -40 -40 -40 -20 -30
SO SO
50 40 30 SO SO SO
-50
50
-40 -40
SO
a~
For-
Simu-
mula
latioD
18
23
28 26 23 33 31 26 29 30
18
23
28
25
23
34 31 25 29
Wiener filter theory, this result verifies that artificial receiver noise injection alleviates the signal nulling problem without seriously compromising jammer nulling.
30
Table II shows that this formula is accurate to within ± 1 dB when the imperfections are due to equal variance, independent, zero-mean Gaussian amplitude and phase errors that are angle independent, and a ten-element equally spaced line array with d/X = 0.5, a broadside look direction, and receiver noise power of O'~ = - 30 dBw are considered. The data in this table were obtained by ensemble averaging 50 curves of SINRti where versus 0';, and then measuring the two values of SINRti was down 3 dB from its maximum. 0'7. opt was then selected as being in between these two "3 dB points." This procedure assumes == which should be excellent for 0'; ~ 0';. The measurement error for the simulation data is ± 1 dB. The ± 1dB accuracy for the formula should be good enough for most applications, since a plot of SINRti versus u;/o'; has a fairly broad 3 dB width. For the cases in the table, this width was measured to be 36, 26, 16, 12, 10, 9, 13, 23, 18, and 20 dB, respectively. In addition, the loss in SINRti from its value when ideal elements were used was very small. More precisely, for the cases in the table this loss was measured to be only 0, 0, 0, 1, 1, 2, 1, 0, 0, and dB, respectively.
c. Data Simulation Examples "Data simulations" to support the analysis performed earlier can be found in [35]. These data simulations compared the conventional LMS algorithm with the GSC leaky LMS algorithm. The basic finding was that use of the GSC leaky LMS algorithm was effective in solving the signal nulling problem, and had the additional bonus of eliminating weight pathology due to finite precision effects. However, the number of time samples required for convergence was several times greater for the GSC leaky LMS algorithm.
u;
u; u;,
°
V.
EXTENSION OF RESULTS
The purpose of this section is to explain how the results of this paper can be extended to the two important cases of a multiple jammer signal environment and wide-band adaptive array processing.
A. Multiple Jammers
The only results in this paper which are totally independent of the number of jammers are the new leaky algorithms (77) and (84). All other results can be extended to the case of multiple jammers either by analytically inverting the more complicated expression for a multiple jammer autocovariance matrix, or by simply solving for the autocovariance matrix inverse numeriIn practice, although it is pleasing that an optimum exists, one does not have the a priori information needed to cally. The inverse of the autocovariance matrix is then compute it. There are two obvious ways to deal with this multiplied by the crosscovariance vector to obtain the Wiener situation. The first is to use an on-line search procedure to weight vector, after which it is straightforward to calculate any determine o'~ opt' Hudson [17] describes this type of procedure other quantities based on w*. If R uu is inverted analytically, certain approximations may become critical for obtaining as having, "overall speed comparable to matrix inversion," which means computation proportional to the cube of the tractable results, such as assuming that all jammers are large number of antenna elements. The second way to deal with the compared to the signal. Of course, another method of predicting performance in a lack of a priori knowledge is to either guess or substitute worst-case estimates for the unknown parameters in (85). multiple jammer environment is simply to write the data simulation software in such a way that the temporal and spatial Then it is hoped that the robustness of opt will permit this procedure to yield acceptable results. An example is provided characteristics of more than one jammer can be entered as input variables. This procedure was carried out by the author in [35]. In order to demonstrate how effective artificial receiver to confirm that even in a multiple jammer environment, noise injection can be, the procedure will be applied to the linearly constrained adaptive beamformers still null the signal imperfect array of Section Ill-D, which exhibited hypersensi- as long as array imperfections are present, no special remedies tivity to random element misplacement having a standard are used, and at least one extra degree of freedom is available.
u;
Therefore, one does not need to pick very accurately to obtain close to optimum performance, and secondly, a good choice of will essentially restore the output signalto-interference-plus-noise ratio to its value in the case of an ideal array.
u;
u;
at,
593
"0.00
++
Look-direction signal
Jammer
20.00
.00
,,-....
CQ
~
-20.00
..> .ij 0 ....
-40.00
'-"
/
~ .--e
0
-60.00
K
= 10
d A
=
SNR i = 30 dB
1 2
INRi = 50 dB
-80.00
u,
= 35 dB
= 0.01
-100.00 -100.00
.00
-50.00
Far-field angle, Fig. 12.
50.00
a
100.00
(deg) ...
Single jammer far-field directivity pattern of Capon beamfonner implemented with imperfect array, when approximately 3 dB more than the optimum amount of artificial receiver noise is injected.
direction signal as if it were a jammer. Modeling the array
B. Wide-Band Adaptive Array Processing The extension to the wide-band case is trivial, assuming that wide-band adaptive noise cancelling techniques are now used, which give the optimal Wiener weightings as a function of frequency. Although these optimal Wiener weightings are, "ideal, based on the assumption of an infinitely long, twosided (noncausal) adaptive transversal filter," Widrow et al. [2] showed that their performance could be closely approximated by using all-zero filters. Gooch and Shynk [41] recently demonstrated the potential for even better synthesis of the Wiener weightings by using pole-zero filters. When applying the results of this paper to the wideband case, one only needs to keep track of the change in wavelength Aas a function of frequency (since it affects both presteering and any possible random element misplacement), and 2 I CXj 1 , and 0-; must all be interpreted as functions of i», These conditions mean that the array imperfections must be viewed as frequency-dependent, and at some frequencies certain assumptions may no longer hold.
0';, 0';,
a;, 0';,
VI.
CONCLUSION
This paper tackled the problem of hypersensitivity of linearly constrained adaptive beam/arming to array imperfections for' 'high" input signal-to-noise ratio, by considering a particularly simple and general structure known as the generalized sidelobe canceller. The aforementioned hypersensitivity manifests itself as nulling of the friendly look-
imperfections as random element amplitude and phase errors constant during the period of adaptation, the hypersensitivity phenomenon was discussed in detail using Wiener filter theory to analyze steady state behavior, and computer simulations to check the results. Artificial receiver noise injection algorithms were derived for the generalized sidelobe canceller, and simulations were carried out to demonstrate their ability to provide the beamformer with robustness to array imperfections. For the special case of the Capon maximum-likelihood beamformer, simple approximations were presented for the Wiener output signal-to-interference-plus-noise ratio, the random element gain (amplitude and phase) error variance which leads to a 3 dB degradation in this Wiener output signal-tointerference-plus-noise ratio from its value when an ideal array is assumed, and the optimal amount of artificially injected receiver noise. Suggestions for how the theory could be extended to the two important cases of multiple jammers and wide-band adaptive array processing were discussed. Ideas for further investigation can be found in [35]. ACKNOWLEDGMENT
The author is grateful Widrow, for suggesting fect arrays" as a Ph.D. during the course of this 594
to his principal advisor, Dr. Bernard "adaptive beamfonning with imperthesis topic, and for his supervision research. The experience of working
with Dr. Arogyaswami Paulraj, who served as associate advisor, was equally rewarding.
[22] [23]
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I. J. Gupta and A. A. Ksienski, "Effect of mutual coupling on the performance of adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-31, no. 5, pp. 785-791, Sept. 1983. L. C. Godara, "The effect of phase-shifter errors on the performance of an antenna-array beamformer," IEEE J. Ocean. Eng., vol. DE-IO no. 3, pp. 278-284, July 1985. ' C. L. Zahm, "Application of adaptive arrays to suppress strong jammers in the presence of weak signals," IEEE Trans. Aerosp. Electron. Syst., vol. AES-9, no. 2, pp. 260-271, Mar. 1973. W. D. White, "Artificial noise in adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES-14, no. 2, pp. 380-384, ~1ar. 1978. J. G. Charitat, I r. , "The effects of error in the adaptive antenna reference," Proc. IEEE, vol, 70, no. 9, pp. 1128-1129, Sept. 1982. - - , " An algorithm for adaptive antennas and superresolution systems with faulty steering vectors," IEEE Trans. Antennas Propagat., vol. AP-34, no. 3, pp. Mar. 1986. B. Widrow and J. M. McCool, "A comparison of adaptive algorith~ based on the methods of steepest descent and random search," unpublished manuscript. J. M. McCool, "A constrained adaptive beamformer tolerant of array gain and phase errors," in Aspects of Signal Processing, pt. 2, G. Tacconi, Ed. Dordrecht, Holland: Reidel, 1977, pp. 517-522. M. H. Er and A. Cantoni, "Derivative constraints for broad-band element space antenna array processors," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, no. 6, pp. 1378-1393, Dec. 1983. M. H. Er and A. Cantoni, ., A new approach to the design of broadband element space antenna array processors," IEEE J. Ocean. Eng., vol. OE-IO, no. 3, pp. 231-240, Iuly 1985. K. M. Ahmed and R. I. Evans, HAn adaptive array processor with robustness and broad-band capabilities," IEEE Trans. Antennas Propagat., vol. AP-32, no. 9. pp. 944-950, Sept. 1984. R. T. Compton, Jr., "An adaptive array in a spread-spectrum communication system," Proc. IEEE, vol. 66, no. 3, pp. 289-298, Mar. 1978. N. K. Jablon, "Steady state analysis of the generalized sidelobe canceller by adaptive noise cancelling techniques," IEEE Trans. Antennas Propagat., vol. AP-34, no. 3. pp. 330-338. Mar. 1986. - - , "Adaptive beamforming with imperfect arrays," Ph.D. dissertation, Elec. Eng. Dept., Stanford Univ., Stanford, CA, Aug. 1985. A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1965, ch. 9, p. 303. T. Kailath, Linear Systems. Englewood Cliffs. NJ: Prentice-Hall, 1980, Appendix, pp. 655 and 658. H. E. Schrank, "Low sidelobe phased array antennas," IEEE Antennas Propagate Soc. Newsletter, vol. 25, no. 2, pp. 5-9, Apr. 1983.
D. G. Luenberger, Introduction to Linear and Nonlinear Program-
ming,2nded. Reading, MA: Addison-Wesley, 1984, ch. 7, pp. 221222. R. A. Chestek, "The addition of soft constraints to the LMS algorithm," Ph.D. dissertation, Elec. Eng. Dept., Stanford Univ., Stanford. CA, May 1979, chs. 5, 8, and 9, pp. 13, 19, and 30. R. P. Gooch and J. J. Shynk, "Wide-band adaptive array processing using pole-zero digital filters," IEEE Trans. Antennas Propagat., vol. AP-34, no. 3, pp. 355-368, Mar. 1986.
An Efficient Algorithm and Systolic Architecture for Multiple Channel Adaptive Filtering STANLEY M. YUEN, KENNETH ABEND,
SENIOR MEMBER, IEEE, AND
A bstracl-A multiple input-multiple output orthogonalization algorithm and its efficient systolic implementation are presented. The processing architecture is developed using a basic two input-two output decorrelation processing element (PE) as the primitive building block. Its features are discussed and compared to the recently published approach based on the well-known modified Gram-Schmidt (MGS) orthogonalizalion procedure.
' A
I. INTRODUCTION
DAPTIVE FILTERING has been a subject of intense research since the early work of the stochastic gradient or the least mean square (LMS) algorithm of Widrow [1], [2]. The virtue of the LMS algorithm lies in its computational simplicity. However. it suffers slow initial convergence and thus poor adaptivity in a rapidly time-varying environment. This drawback has served as motivation for deriving other methods of adaptive filtering which can provide faster cc nvergence and are not so sensitive to the signal statistics. One method of obtaining faster convergence is to adopt an exact least squares (LS) approach rather than the statistical approach. The family of recursive least squares (RLS) algorithms represents one group of techniques which are theoretically less sensitive to the statistical propenies of the data [31-[6]. There are basically two important differences between the gradient-based type of algorithms and the RLS (a.so known as Kalman) family. First of all. Kalman-type algorithms minimize an exact error criterion constructed from the actual input data . in contrast to the statistical error criterion for the gradient-based methods. Secondly, the error criterion is satisfied at every point in time for the Kalman family of algorithms. whereas the error criterion is achieved at convergence or steady state for the gradient -based techniques, as in the case of the LMS algorithm. Although the Kalman-type algorithms possess attractive convergence property, they have two major drawbacks. One is their large computational complexity and the second is their sensitivity to round-off noise. The latter may cause an algorithm to become unstable after a large number of iterations. To be more specific, the Kalman-type algorithms require O(N2) operations per time update for computing an Nth order filter, whereas only O(N) operations are needed for the LMS algorithm. To remedy the problem of round-off noise, a Manuscript received February 12. 1987: revised September 1~ 1987. S. M. Yuen is with the Electronic Systems Department, RCA Government Electronic Systems Division. Moorestown, NJ 08057. K. Abend and R. S. Berkowitz are with the Department of Electrical Engineering. University of Pennsylvania. Philadelphia, PA L9104. IEEE Log Number 8819825.
RAYMOND S. BERKOWITZ,
FELLOW, IEEE
family of algorithms based on orthogonal transformations can be used. They include the Givens, Householder, and modified G~am-Schmidt (MGS) transformations. These algorithms deal WIth data matrices with condition numbers equal to the square root of the condition number of the input signal covariance matrix. The condition number of a matrix of interest is defined as the ratio of its largest and smallest nonzero singular values and has an interpretation of being an error magnification factor. Consequently, these orthogonalization-based algorithms are less sensitive to round-off noise. In many advanced signal processing applications, the use of regularly structured processing is considered as the most feasible approach to obtain real time performance. The development of versatile processing nodes by sophisticated very large-scale integration (VLSI) design can lead to a new generation of adaptive processors which can achieve real time throughput rate as well as flexibility. As a result, researchers have investigated various implementation aspects of orthogo~alization-based algorithms in the context of parallel processmg. For example, time-recursive versions of the Givens transformation and the modified Gram-Schmidt algorithm have been developed and discussed in the context of systolic array implementation [7]-[10]. The MGS approach, in particular, h~s received a tremendous amount of attention in many radar SIgnal processing applications. Besides having a modular and regular processing architecture, the MGS algorithm possesses both time and order recursive properties [10]. Furthermore, it has been shown to yield good performance simultaneously in arithmetic efficiency, stability, and convergence times [11], [12]. The MGS procedure has been considered in the literature as an orthogonalization preprocessor for the LMS algorithm [13] , as a linear predictor for temporal input [14], as a sidelobe cancellor [15], and for clutter rejection in a nonstationary radar environment [16], [17]. More recently, the MGS orthogonalization algorithm and its corresponding triangular processing architecture have been generalized for efficient multiple channel adaptive filtering [18], [19]. The purpose of this paper is to introduce an alternative orthogonalization algorithm which results in a more efficient architecture for filtering applications in which there are as many output channels as there are input channels. One example is adaptive pulse Doppler processing in radar. For completeness, the basic theory and the architecture proposed by Gerlach [19] based on the MGS procedure are reviewed in Section II. We then systematically develop the new alternative algorithm and the corresponding efficient processing structure in Section III, starting with a.simple two
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 36, No.5, pp. 629-635, May 1988.
596
The vector X I - YW is called the residual vector and is orthogonal to the columns. of Y. Hence we can compute W from the normal equations
input-two output decorrelation processing element as the primitive building block. Finally, the differences between the two approaches are discussed and future work relevant to the new algorithm and processing architecture is addressed in Section IV. II.
The LS Problem
Y'YW=Y'X I
REVIEW OF BASIC THEORIES
9
(5)
Equation (5) is often called the LS estimator and is akin to the Wiener-Hopf equation derived using the criterion of least mean squares error [20]. Equation (4) can also be written explicitly as follows:
The LS problem is known by different names in different scientific disciplines. In the IEEE literature, the solution of the
K K K
L X!(k)Xl(k)- Wz L xi(k)X2(k)-··· -
WN ~ xi(k)XN(k) = 0
2: Xj(k)XI(k)- W L Xj(k)X2(k)-··· -
W N ~ Xj(k)XN(k)=O
k=O
k=O 1\
K
2
k=O
k:O A
k=O
k=O
K K K
2: X~(k)XI (k) -
W2
k=O
2: X:t(k)X2(k) -
[XI (k), x~ (k), ... ,
Since X. and X, - YW are not orthogonal, their inner product can be expressed as K
2: xi(k)x, (k) -
/':=(l
K
W2
2: xi(k)X2(k) /.:=0
A'
- . . . - ~V v ~ Xi ( k ) X\,( k ) =; }l
k=O,l'··, K.
where
(1)
that
J.1.
(4)
is a nonzero quantity. Next we combine (6) and (7) so is replaced with the matrix equation
We desire to determine a weight vector \V which minimizes the sum of squared errors defined as €(K)
==
L 1\
1
Rxxw ==
e2 ( k )
}l
o
(8)
o
1.:=0
==
(7)
k:..:O
xy(k)] ,
=[x\(k), yJ(k)]',
(6)
k~O
LS problem is associated with a number of equations and vector space concepts. The purpose of this section is to review brietly the essential equations and fundamental concepts, and to demonstrate that they are indeed interchangeable in the interpretation of the LS solution. Assume that we have an N channel system with the measurement vector at a given time instant k represented as X (k) =
... - W N ~ X~(k)XN(k) = o.
k=O
where w == [1, -W']' and R xx is the N x lv sample (2) covariance matrix of the input channels calculated based on K + 1 observations.
2: (xI(k)-y'(k)W)2 K
k=O
Wy'J.
The minimization of (2) is equivalent to the solution of the LS problem of minimizing the Euclidean length
According to the theory of signal-to-noise (SIN) optirnization in the field of adaptive arrays. the optimum weight vector W Op l is the value of w that satisfies
IIXI-YWII
Rxxw=J.lS*.
where W == [W2 , W 3 t
.",
(3)
where
XI =
Xl
(0)
X2(O)
X3(0)
XN(O)
XI
(1)
x2(1)
x3(1)
xN(l)
x2(K)
x3(K)
xN(K)
xI(K)
and y=
It is well known that the solution W satisfies the condition Y' (X, - YW) =0.
(4)
(9)
5 == [5., 52, ... , 5 N ) ' is often called the steering vector and J.L in this case can be an arbitrary constant. Equation (9) is known as the Applebaum maximum signal-to-noise criterion (21]. In a linear array antenna with equally spaced elements, the components of 5 are determined by the direction of the desired signal. Although Jl in (9) can be arbitrary ~ Jl in (8) is not, and is chosen so that the first element of w in (8) is a one. The key point in the derivation of (9) is the application of the CauchySchwartz inequality. The similarity between (8) and (9) is obvious. Although the approaches to deriving (8) and (9) are
597
and 12 ,1 is calculated so that (10) is satisfied. It is easy to see that
==== .
YIY1 12 1 , \Y2\2 DP
(12)
In an actual application, a finite number of samples would be taken for each input channel, thus (12) is estimated as s
I
v:1 Fig. 1.
Decorrelation processor [14].
~ Yl (k)Yi(k)
-x-----
_k=O
2,1 -
(13)
~ \Y2(k)1 2
k=O
LEVEL 1
LEVEL 2
··•
LEVEL N·2
LEVELN·l
Fig. 2.
~
DP
~
+OUTPUT
Modified Gram-Schmidt N-channel decorrelator [15].
somewhat different, it is clear that they provide the same solution of the LS problem. In fact. (8) can be considered as a special case of (9) with S = [1, 0, ... , 0]'. An alternative interpretation of (8) is that it is obtained by transforming the input signals of (9) such that the effective steering vector in the transformed signal space has the same simple form S = [I, 0, ·'·,0]'.
Multiple Channel Adaptive Filtering Using MGS The direct implementation of (8) corresponds to the inversion of the covariance matrix. However. it is well known that problems occur in the solution of the weights if R xx is illconditioned. A better approach is the use of the MGS orthogonalization technique which has been reported to have good numerical properties [22], [23]. Its processing architecture is shown in Fig. 2, using the simple decorrelation processor (DP) of Fig. 1 as the building block. To understand the operation of a single DP, we consider two channels of complex values input data: YI and Y2. The objective here is to form an output channel which is decorrelated with Y2. This is equivalent to setting
y;
Y; Yi=O
(10)
Where the overbar and asterisk denote the time average and the cOlnplex conjugate, respectively. We can also express (11)
In Fig. 2, X N is decorrelated with X., X 2 , " . , X N - I in the first level of the processing structure. In the second level, the output channel which results from decorrelating X N with X N - 1 is decorrelated with the other outputs of the first level of DP's, and the process continues as indicated in the figure. At the end, a final output is generated and it is totally decorrelated with the input X 2 , X 3 , ••• , X N • It is clear that the MGS decorrelation (orthogonalization) procedure is not unique with respect to the order in which X 2 , X 3 , • • " X N are decorrelated from XI. In multiple input-multiple output adaptive filtering, there can be as many output channels as input channels. Specifically, given there are N input channels, it might be desirable to generate N output channels so that each one of the N output channels is totally decorrelated with the rest of the N - 1 input channels. For example, this concept can be applied to radar Doppler processing in which a bank of filters is used to cover the entire Doppler band, and each Doppler sub band is processed such that it is totally decorrelated with the rest of the other subbands. Mathematically, this corresponds to solving the matrix equation
Rxxv=[
(14)
where V is the optimal weighting matrix and I is simply an N x N identity matrix. The SIN associated with each of the output channels is maximized by the corresponding column vector of if, and the nth output channel has a desired signal vector:
(0, 0, ... , 0, 1, 0, .'., 0)
I
i nth position. Based on the fact that there is no logic behind the ordering of the input channels in the MGS procedure, Gerlach was able to develop an efficient processing architecture for multiple channel adaptive filtering [15]. The design is illustrated in Fig. 3 for the case of eight input and eight output channels. The key point in the design is that arithmetic efficiency is achieved by taking advantage of computational redundancies and substructure sharing that can occur for different output channels. III.
DERIVATION OF THE NEW MULTIPLE CHANNEL ORTHOGONALIZATION ARCHITECTURE
Using the basic decorrelation processor of Fig. 1, it is possible to configure other orthogonal networks. To derive a
598
Fig. 3.
Complete realization of an eight-channel decorrelation network [15].
tive defined, we then proceed to construct an orthogonalization processing network . One structure which naturally takes advantage of the symmetry property of the PE is the tree-like network of Fig. 5. An eight input channel structure is illustrated in this example . The extension to an arbitrary number of input channels is obvious. Furthermore, the use of eight input channels also provides a one-to-one comparison between the newly derived tree-like architecture and the architecture based on MGS discussed in the previous section. The numbering system N 1(N 1 , N 3, • • • ) used at the output of each PE gives a clear picture of the orthogonalization procedure carried out by the tree-like processing architecture. The notation implies that the N1th output channel is totally decorrelated with the input: N 2 , N J , ••• etc . In the first row or y: y~ I J level of decomposition, XI is decorrelated with X 2 (and vice versa), X 2 is decorrelated with X J (and vice versa), and so on. E (Yj· V i) _"':'---Yj Next , the output channel which results from decorrelating Xl with X 2 is decorrelated with the proper output channel which E Y j \2) results from decorrelating X J with X 2• The decorrelation E (Y i • Yj) process continues as seen in Fig . 5 until two final output ----Vi channels are generated. It is important to emphasize that the E!lYi\2) two inputs to any given PE must be compatible, i.e. . the set of Fig. 4. Two input-two output building block. input channel indices enclosed by the two pairs of parenthe ses must be identical. Thus the first channel XI and the last more regular architecture for multiple channel adaptive channel X s are totally decorrelated with the input set (Xl , X J, filtering, we begin by considering possible modifications at the . . . , X s) and (X" X 2 , " ' , X 7 ) , respectively, in Fig. 5. primitive or building block level. An intuitive approach of The processing architecture of Fig. 5 generates two of the N achieving structural compactness is to employ the two input- desired output channels . In the case of multiple input-multiple two output processing element (PE) of Fig . 4 as the primitive output adaptive filtering , the remaining N - 2 channels can be building block. The only difference between the DP and the efficiently generated using the technique of " sliding-window" PE is the second orthogonal output associated with the PE substructure sharing as shown in Fig. 6. Four windows in our building block. This second output , however, requires a example of eight input channels correspond to the following smaller number of arithmetic operations than the only output four ordered input sequences : of the DP. This is especially true in the case of batch processing in which E{Yj*(t)Y;(t)} and E{Yi(t)Yj(t)} are obtained by time averaging. Once one of the two expectation operations is estimated by summing N time samples , the other is easily obtained by taking the conjugate . As N becomes large, the use of the PE as the basic building block would result in improved arithmetic efficiency . With the PE prirniy.
y. J
I
u
599
x,
PE
1 12~121
31~141
213'M131
V3
1l2 .~2.3 1
213
.t~(5)
31• . ~•. 5'
.• ,
' 15~5.61
",.,%?,.M,~.." "..~s." 112.3.• ~ 612.3'.51
~
5t6~6'
617~817 1
5 16'~16 .7I
".)d".'"
2 13 .4~.4 .5.6,31~1• .5.6.71
M,
~
112.3.•. 5.6~• .5.61 213 .~ 813.'.5 .6.71
~
~
112.3.• . 5.6~2.3 .•. S.6.71
/::~
8(1 ,2.3 .4 .5.6.11
112.3,4,5.6 .7.8'
CHANNEL I
Fig. 5.
CHANNEL 1
Tree-like orthogonaJization network.
x,
x,
\
\ \
\ \
. \
\ \
\
CHANNEL
8
CHANNEL 1
Fig. 6.
CH AN N EL
2
CHANNEL 3
CHANNEL
CHANNEL
•
5
CHANNEL
CHANNEL
6
7
" Sliding window" substructure sharing.
The first window generates the eighth and the first decorrelated output channels , the second window generates the second and the third decorrelated output channels, and so forth. In other words, the two output channels generated correspond to the first and the last elements of the ordered input sequence associated with a given window. The concept of substructure sharing using the "sliding window" approach is a useful tool in reducing the total number of PE's as well as the total number of arithmetic operations. Besides the "sliding window" technique, we can achieve further sharing of substructures by exploiting one other structural symmetry in Fig. 6. This symmetry is illustrated by the two dashed triangles
enclosing two identical substructures. As a result, one of the two substructures can be completely eliminated, and the final processing architecture requires JV2 - 3N12 (JV2 - (3N - 1)12) PE's if N is even (odd), where N is the number of input channels. The final design is given in Fig. 7. In the case of batch processing, we see that as the data are processed through a given row , the input data may be discarded and the output data become the new input data set for the next row of PE' s. Hence the two-dimensional structure of Fig. 6 can be collapsed to just a single row of PE's using simple feedback as shown in Fig. 8. Fig. 7 illustrates that as outputs leave at one side of the parallelogram structure they
600
CHANNEL CHANNEL CH A N N EL CH AN N E L 8 CH AN N EL 2 CHAN N E L 4 CH AN N EL 6 CH A N N E L 1 3 S 7
Fig. 7.
CHANN EL
8
Efficient orthogonalization architecture for multiple channel adaptive filtering .
CH ANNEL
1
CHANNEL
2
Fig . 8.
CHANNEL
CHANNEL
3
4
CHANNEL
5
CHA N NEL
6
CH AN N EL 7
Hardware compaction for batch processing.
enter at the other side. Thus we can imagine that Fig. 7 and Fig. 8 represent a cylindrical systolic architecture and a simple ring structure, respectively , in three dimensions . IV. SUMMARY AND FUTURE RESEARCH The main features of the newly developed multiple channel orthogonalization architecture are summarized as follows . 1) In contrast to the architecture based on MGS orthogonalization, it requires no broadcasting of data and any given processing node in the structure only communicates with its neighboring nodes in a pipelining fashion . Hence the design is "purely" systolic . 2) In terms of the total number of arithmetic operations, it is at least as efficient as the MGS approach. A detailed comparison will be given in a future paper. 3) The new architecture is developed in a very systematic and bottom-up fashion, starting with a simple two input-two output decorrelation processing element as the building block. 4) It is an extremely regular and compact processing structure. This is particularly true for batch processing, since in this case the original two-dimensional systolic array can be collapsed into a linear array with just N processing elements , where N is the number of input channels. 5) No unscrambling of the output channels is needed . The
MGS approach , on the other hand , requires a commutatio n algorithm so that the final output channels are properly aligned with the input channels . 6) The technique based on the MGS approach is most efficient for 2 m input channels , where m is a positive integer. The architecture presented in this paper , however, places no restriction on the number of input channels. In the field of adaptive filtering and estimation using VLS: parallel processing, many problems still exist. We are currently investigating the following research topics relevant to the new multiple channel adaptive filtering technique: 1) This paper focuses on the development of an efficient and compact processing architecture for multiple channel adaptive filtering based on the concept of orthogonalization. For simplicity of illustration in the development, batch processing is emphasized. The time-recursive version is yet to be investigated in detail. One advantage of the time-recursive version of the algorithm is that it only has a latency of N computing cycles for the first output to be generated , whereas N ·N, cycles are required in the case of batch processing. /If and N, correspond to the number of processing channels and the number of time samples used for decorrelation, respectively. 2) In real time applications, divisions take more time than
601
multiplications, since most of the special hardware components are optimized to perform multiplication and addition. It is desirable to modify the new algorithm and architecture so that the number of divisions is minimized. 3) It has been reported that the MGS algorithm possesses good numerical properties. It is important to examine the numerical properties of the new algorithm and compare them with those of the MGS algorithm. 4) The application of geometrical vector space concepts for deriving the rapidly converging recursive least squares adaptive filters is well known. Although the new algorithm derived in this paper is done from an architectural perspective, it is worthwhile to rederive the new algorithm using the geometrical approach.
[23]
REFERENCES
B. Widrow and M. E. Hoff. "Adaptive switching circuits," in 1960 WESCON Conv. Rec., pt. 4, pp. 96-140. [2] B. Widrow et al., "Stationary and nonstationary leamingcharacteristics of the LMS adaptive filter." Proc. IEEE, vol. 64, pp. 1156-1162, [!~
Aug. 1976.
[3] Lee et al., "Recursive least squares ladder estimation algorithms." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. [4]
[51
[6] [7] [8] [9]
[lC: [11] [12] [13] [14 1 [I5] [16]
[17]
[I8: [19] [20] [21] [22]
627-641. June 1981. J. M. Cioffi and T. Kailath .: 'Fast, first-order, least squares algorithms for adaptive filtering." IEEE Proc. ICASSP '83, Boston. MA. Apr. 1983, pp. 679-682. F. Long and J. G. Proakis, "A generalized multichannel least squares lauic algorithm with sequential processing stages." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, no. 2, pp. 381389. Apr. 1984. 1. D. Pack and E. H. Satorius, "Least squares adaptive lattice algorithms." NOSC Tech. Rep. TR423. Apr. 1979. J. Mcwhirter. ."Recursive least-squares minimization using a systolic array ... SPIE paper 431-15. 1983. H. T. Kung and W. M. Gentleman. "Matrix triangularization by systolic arrays," Proc. SPIE, vol. 298. 1981. S. Y. Kung. "VLSI array processors." IEEE Acoust., Speech, Signal Processing Mag., vol. 2. pp. 4-22. July 1985. F. Ling et al., .. A recursive modified Gram-Schmidt algorithm for least-squares estimation." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34. no ...L pp. 829-835. Aug. 1986. B. Friedlander. "Lattice filters for adaptive processing," Proc. IEEE, vol. 70. no. 8. pp. 829-867. Aug. 1982. I. S. Reed. J. D. Mallet. and L. E. Brennan.: 'Rapid convergence rate in adaptive arrays:' IEEE Trans., Acoust., Speech Signal Processing, vol. AES-IO. pp. 853-863. Nov. 1974. R. A. Monzingo and T. W. Miller. Introduction to Adaptive Arrays. New York: Wiley. 1980. Ch. 9. N. Ahmed and D. H. Youn. "On a realization and related algorithm for adaptive prediction;' IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-28, pp. 493-497, Oct. 1980. F. F. Kretschmer and B. L. Lewis, "A digital open-loop adaptive processor:' IEEE Trans. Acoust., Speech, Signal Processing, vol. AES-14. pp. 165-170. Jan. 1978. F. F. Kretschmer, B. L. Lewis, and F. L. C. Lin .: 'Adaptive MTI and doppler filter bank clutter processing." in Proc. IEEE 1984 Nat. Radar Conf., Atlanta. GA, Mar. 1984. A. Farina and F. A. Studer, "Application of Gram-Schmidt algorithm to optimum radar signal processing," Proc. lnst. Elec. Eng., vol. 131, pt. F, no. 2, Apr. 1984. K. Gerlach. "'Multiple channel adaptive filtering using a fast orghogonalization network: an application to efficient pulsed doppler radar processing, " NRL Rep. 8840. 1984. - - , "Fast orthogonalization networks." IEEE Trans. Antennas Propagat., vol. AP-34, no. 3. pp. 458-462, Mar. 1986. S. Haykin, "Nonlinear methods of spectral analysis," Topics in Appl. Phys., vol. 34, 1983. S. P. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-24, no. 5, Sept. 1976. C. L. Lawson and R. J. Hanson. Solving Least Squares Problems. Englewood Cliffs. NJ: Prentice-Hall, 1974.
602
A. Bjorck, "Solving linear least squares problems by Gram-Schmidt
orthogonalization." BIT, vol. 7, pp. 1-21, Jan. 1967.
Mutual Coupling Compensation in Small Array Antennas
aperture distribution is obtained in the presence of these parasitics, the mutual coupling can be compensated for. This compensation principle has been reported for a slot array [1] and dipole arrays [2]-[4]. The former is the only one that considers the case of scanning and presents some experimental data; all four rely on computed coupling coefficients. The present study differs in that it rephrases the approach for the receiver mode, appropriate for a digital beamforming antenna where the technique is most practical, and it describes an alternative method to determine the mutual coupling coefficients, that does not require analytically simple or reciprocal array elements. It also presents experimental data for a scanned waveguide array.
HANS STEYSKAL, MEMBER, IEEE, AND JEFFREY S. HERD, MEMBER, IEEE
Abstract-A technique to compensate for mutual coupling in a small amy is developed and experimentally verified. Mathematically, the compensation consists of a matrix multiplication performed on the received signal vector. This, in eifect, restores tbe signals as received b) tbe isolated elements in tbe absence of mutual coupling. The technique is most practical for digital beamforming antennas where tbe matrix operation can be readily implemented.
THEORY INTRODUCTION
We consider an array of single-mode elements, meaning that the element aperture currents (electric or magnetic) may change in amplitude but not in shape, as a function of radiation direction. In the receive mode, the signal at the output of the individual antenna element has several constituents: a dominant one due to the direct incident plane wave, and several lesser ones due to scattering of the incident wave at neighboring elements. As depicted in Fig. 1, we can write the received signal at element m as
The radiation pattern of an array of identical antenna elements is usually taken to be the product of an element factor and an array factor, based on the presumption that all elements have equal radiation patterns. Unfortunately, this may not be true for a practical array, where, due to mutual coupling, each element ., sees" a different environment. The nature of the error thus incurred can be displayed by expressing the individual array element pattern f n ( u) as the sum of one average array element pattern fa(u) and a pattern deviation o!n(u), which leads to the total array pattern
F(u) =
Vm(u) = cmmEm!i(u) +
LQnfn(u)eJnkdu
=f
n Q
(
u) L QneJnkdu + L Qnofn( u) eJnkdu. n
n
(1)
Here an = I a; I exp (jet>n) denotes the complex element weight, k the wavenumber, d the uniform element spacing and u the sine of the angle 8 from broadside, respectively. The first term on the right side of (1) represents the idealized pattern, and the second represents the error. One effect of this error pattern is to introduce a noise floor that precludes synthesis of high-quality patterns with very low sidelobes or deterministic pattern nulls. Other effects appear in signal processing arrays, such as adaptive or superresolution systems, which can be extremely sensitive to small errors due to the nonlinear processing involved. Since real-life signal processing arrays usually are comparatively small arrays, where element pattern differences are relatively large, this is a significant problem. It is clear from (1) that the element coefficients {an} always can be chosen such as to compensate for the pattern error at one particular angle. It is less obvious that the error normally can be corrected for all angles simultaneously. Furthermore, since this correction is scan independent, it also applies in the case of electronic scanning. It is the purpose of this communication to discuss such a technique and to present some experimental results. The key to the technique is an alternative formulation to (1), which recognizes that 1) any composite array pattern can be considered as a weighted sum of the isolated element patterns and 2) the effect of mutual coupling is simply to parasitically excite all elements, even though only one element is driven. Thus, by driving the array with modified element excitations, such that the desired array Manuscript received November 29, 1989; revised June 12, 1990. The authors are with the Electromagnetics Directorate, Rome Air Development Center, Hanscom AFB, MA 01731. IEEE Log Number 9038579.
L
n , m,*n
cmnEnfi(u).
(2)
The incident field Em at element m impresses an aperture current amplitude Em!i(U), where fi(u} is the. isolated element pattern, i.e., the pattern of the current mode assumed in the element aperture. This aperture current will produce an element output voltage cmmEmfi(u), where cmm denotes the coupling from the aperture to the output transmission line. The effect of the neighbor-
Vm Um at element m consists of a directly transmitted and several scattered components.
Fig. 1. The received signal
ing elements is described similarly, with Cm n denoting the COupling of aperture mode n to element output m. From a mathematical point of view, (2) simply expresses the linear relationship between the aperture excitations and the element output voltages. The physical meaning of the Cmn will be discussed below. We introduce the notation
(3) since this represents the desired, coupling-unperturbed signal received by the single element at the aperture. Thus for our unifonnly spaced array of identical elements Eoejnkdufi(u)
= u~(u)
(3a)
where Eo is the amplitude of the plane wave incident from direction
u.
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 38, No. 12, pp. 1971-1975, December 1995. U.S. Government work not protected by U.S. Copyright.
603
Substituting (3) in (2) leads to
iog antenna system. It then allows all subsequent beamforming operations to be performed with ideal element signals. such as are usually assumed in pattern synthesis.
(4)
DETEIUIlINA1l0N OF THE MU11JAL COUPUNG COEFFICIENTS
On the left side, the vector v represents the coupling perturbed signals { lin} at the element output ports, which via the coupling matrix C is related to the vector y d, representing the unperturbed desired signals {u:l. Thus compensation for the mutual coupling can be accomplished by simply multiplying the received signal v by the inverse coupling matrix C - I , yd
=
(5)
C-1y.
This concept is depicted in Fig. 2, where a network corresponding to C- I is attached to the array antenna. Note that the coupling. compensation is scan independent, i.e., the same matrix C - I applies universally for all directions of the incoming wave, as a consequence of our single-mode assumption. Multimode elements, as considered in [2]-[4], would require a scan dependent coupling compensation. When the received and compensated signals vd are weighted and summed in the conventional beamforming network, shown in Fig. 2, we obtain the array pattern F{u), defined as the ratio of the output voltage and the incident wave amplitude Eo,
L anv~ = f'{u) L anelnkd". o
1
F{u) = E
n
(6)
n
The array pattern (6) now has the desired form of a product of an element factor and an array factor. A comparison with (1) shows that, with the transformation performed, we have succeeded in dissolving the error pattern, the second term on the right side of (1). The matrix C- 1 may be difficult or impractical to realize by an analog network, but it can be readily realized in a digital beamform-
mutual coupling and feed line errors
n
11
coupling compensation desired signals
cf. (2).
and, recognizing that the cmn are the Fourier coefficients of these patterns, determine these coefficients numerically according to
(8) In order to do this, ji(U) must not have a null in the integration
interval. However, since the isolated element pattern normally is very wide, this is no serious limitation. Another restriction on (8) is that the element spacing be larger than A/2. Otherwise the integration interval extends beyond visible space, i.e., beyond the interval - 1 < u < 1 where gm(u) and li(u) are known. For the case of element spacings d < A/2 we can still perform a Spectral analysis of
gm(u) fi(U) -
LC n mn
11n d
scanning
beam shaping
A-
output beam Fig. 2. Illustration of coupling compensations and beam forming in an array antenna. Interelement coupling at the array face. represented by (c",n)' leads to received signals I} n at array element outputs" that are linear combinations of the desired, coupling-unpenurbed signals Multiplication by (c m n ) - I restores these signals. which are then weighted and summed to fonn the desired beam.
I):.
(9)
ejnkdu
to determine the coefficients Cm n' but the convenient orthogonality of the harmonic functions is lost and accuracy becomes a major issue. An advantage of this method is that it does not require reciprocal antenna elements. Thus, it is applicable to receive-only arrays. such as used for digital beamforming, where the element includes an entire microwave receiver. Furthermore, any channel imbalances, i.e., differences in insertion amplitude and phase between the element aperture and the element output terminal, manifest themselves in the self-terms em m and are also compensated for. In this sense the technique is similar to a conventional array calibration. In the second method, the matrix C is obtained from the related scattering matrix S ::: (smn) of the array. This relation is developed
desired signals vn d received on isolated elements
measured signals
There appear to be two di fferent methods to determine the coupling coefficients-one by Fourier decomposition of the measured array element patterns and another by coupling measurements between the array ports. The former requires driving the antenna only in one mode" either transmit or receive, and thus applies to nonreciprocal antenna systems. The latter requires driving each element in both modes and therefore is less practical, as discussed below. In the Fourier decomposition method we measure the complex voltage patterns g m( u) of the elements in their array environment.
8 -
-A'
- -
T T- - - T- - - - - - - 8' o~ b~
b~
Fig. 3. lliustration of the scattering matrices 5 and 5' of the array. Line sections between aperture plane AA' and terminal plane BB' are matched and reciprocal, with transmission coefficients tn.
604
here for the simplest case of a waveguide array fed by matched generators . For the general case the relation is complicated and not very useful. We consider a uniformly spaced array of waveguide elements, shown in Fig. 3, and determine the array element pattern of element m. This element is excited with a wave of amplitude am ' all other elements are passive. Assuming a reference plane A A ' for the antenna element terminals that coincides with the element apertures, the aperture voltages thus are ( 10) where the Kronecker delta wise. The radiated far field
0 mn
=
I for n
= m.
and
=0
other-
complicated than pattern measurements with Fourier decompos itions, in reality often is the less practical method. E XPERIMENTS
The coupling compensation technique outlined in the preceding section was applied to an eight-element linear array of X-band rectangular waveguides in a ground plane. Each element was in turn a column of 8 rectangular waveguides in a common H-plane . combined via a fixed 1:8 power divider. The array axis thus was parallel to the E-plane and in this plane the element spacing d = 1.25 cm = 0.5 17 A. The isolated element pattern corresponds to a normalized uniform apertur e distribution kl sin - u 2 kl
P (u )
( 17)
-u 2
where rn is the distance of element n to the observation point and the usual far-field approximations have been made. Comparing this expression with (7) and requiring that the transmit and receive patterns are identical, shows that
apart from a constant factor of no interest. Thus
C
=I+S
(12)
where I denotes the identity matrix. In real arrays , the scattering matrix cannot normally be measured directly at the element apertures , as assumed above, but only from a reference plane a certain distance behind the apertures. A more realistic case therefore is as shown in Fig . 3, where sections of transmission line are included between the apertures and the reference plane BB' , from which the modified scattering matrix S' is measured. These feed lines have different insertion phase and loss. However , for simplicity , we still assume them to be matched and reciprocal. so that they can be characterized by single transmission coefficients t m : Defining a diagonal matrix T , ( 13)
where I is the interior waveguide height. in our case I = 1.02 cm = 0.417 A. The complex voltage patterns g m( u) of the array elements were measured under matched load conditions , and recorded at 1/ 2-degree intervals with a digital rece iver. The coupling coefficients cm n were then numerically evaluated accord ing to (8) and the inverse matrix C- I was computed. In a second , similar measurement, the received voltages lI m ( u) were again recorded . Then , in an off-line simulation of a digital beamforming system, they were multiplied with C - I for coupling compensation, and amplitude and phase weighted for pattern shaping and scanning, as shown in Fig. 2. Examples of element patterns for a central and an edge element are shown in Fig. 4. We note that there is indeed a considerable difference in shape, which is attributable to mutual coupling effects , and also in overall power level , which mainly is due to a difference in feed line losses . In Figs. 5(a) and 5(b) we show synthesized 30-dB Chebyshev patterns as obtained without and with the mutual coupling compensation. Apparently the compensation technique gives about 10 dB improvement in sidelobe level , with the result that the actual pattern is quite close to the theoretical one. The remaining difference indic ates that the array excitation tolerance erro rs equ al ab out - 35 dB in ampllitude and 1° in ph ase. Figs. 6(a) and 6(b) show th e same 30-dB Chebyshev patt erns scanned to - 30°. Without compe nsation the side lobe level is still
it is easy to show that
(14)
S' = TST
o
and, from (4) and (12), that the received signals at plane BB' are /
( 15)
~
aI ~ II:
The modified coupling matrix C' at plane BB' thus is
C' = T + S'T -
1
-5
W
(16)
which shows that in this case we need to measure not only the scattering matrix S' but also the transmission coefficients {t m} . This mayor may not be possible, depending on the design of the actual array . Clearly for the general case , where the feed lines between the element apertures and output terminals are not matched, the required measurements become still more extensive . Thus , measurement of the network parameters, which intuitively would seem less
~Q. ·10
I
I I /
I
/
l/ ""\
""-
/ ./
./
,,- '"
-
-'
<,
,'\
.:>::::
./
r
'\
V
-
1' \
......
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11
~-
I
I
/
""
I i
i i
·15
-ao
·30
o
I 30
60
BO
ANGLE (DEG)
Fig. 4. Measured pattern magnitudes I gm(li) I for center (-) and edge (--) elements of the eight-element array.
605
·10
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~ ·20
a: w :!: 0. 30 a.
irl I
. 40
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·30
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-measured - - -theory
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-_ad
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f' l ~~
(
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·10
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1
ANGLE (DEG)
I
\ I
"I ,
\'
,'"
,
\ \ I \ 60
80
(b)
(b )
Fig. 5.
\I
\ \
1\[f,
ANGLE (DEG)
o
\
·30
1\ I
I
1
(a)
A~ ~\ ~, fl I
·50
/
I
iii'
I
,, , "
1\ I I
I
o
-30
-60
(a)
7
V II
h
Fig. 6.
3Q-dB Chebyshev pattern without and with coupling compensation. Scan angle O·.
limited to about -20 dB. When we apply the compensation, the same C - 1 matrix multipl y as for the broadside pattern, we again reduce the sidelobe level by about 10 dB and closely reproduce the desired patt ern.
3Q-dB Chebyshev pattern without and with coupling compensation. Scan angle - 30·.
and thus is valid for all desired pattern shapes and scan directions. Although it may be difficult to realize in analog form, it can be readily implemented in a digital beamforming antenna system.
CONCLUSION
We have developed and experimentally verified a technique to compensate for mutual coupling in an array. The technique should be helpful primarily for small arrays, where the array element patterns differ significantly due to edge effects. In large arrays, where each element sees essentially the same environment, these effects become negligible. Mathematically, the compensation consists of a matrix multiplication performed on the received signal vector. This, in effect, restores the signals as received by the isolated elements in the absence of coupling. An attractive feature is that this matrix is fixed
606
REFERENCES
T. Chiba, " On the realization of the synthesized pattern of scanning arrays , " presented at Int. Conf. on Radar, Paris, France, Dec. 1978. [2] B. Strait and K. Hirasawa, .. Array design for a specified pattern by matrix methods," IEEE Trans. Antennas Pro/Xlgat., vol. AP-17, Mar. 1969. [3] N. Inagaki and K. Nagai, " Exact design of an array of dipole antennas giving the prescribed radiation patterns, " IEEE Trans. Antennas Propagat., vol. AP-18, Jan . 1971. [4] Y-W. Kang and D. Pozar, " Correction of error in reduced side10be synthesis due to mutual coupling, " IEEE Trans. Antennas PrO/Xlgot., vol. AP-33, Sept. 1985. [I]
A Unified Approach to the Design of Robust Narrow-Band Antenna Array Processors MENG HWA ER, ME~fBER,
IEEE, AND
ANTONIO CANTONI,
Abstract-A unified approach to the design of robust narrow-band antenna array processors is presented. The approach is based on the idea of minimizing the weighted mean-square-deviation between the desired response and the response of the processor over variations in parameters. Three specific examples of robust design are considered: robustness against directional mismatch, robustness against array geometry errors, and robustness against channel phase errors. Initially a general quadratic constraint on the weights is developed. However, it is then shown that the quadratic constraint can be replaced by linear constraints or at most linear constraints plus norm constraint. These latter set of constraints are no more complex than those required for designs which do not incorporate robustness features explicitly. Numerical results show that the proposed approach appears to offer a unified treatment for directly designing narrow-band processors which are robust against various types or errors and mismatches between signal model and actual scenario.
1
SENIOR MEMBER, IEEE
>. i",,) ~
~I
"'1
y(t)
L
:
~ (c) JJ
L
O
Fig. 1.
I. INTRODUCTION
PTIMUM ANTENNA ARRAY processing with multiple linear constraints is now a well-known technique. In the simplest case, a single constraint is imposed, namely unity gain response in the look direction; the weight vector is then calculated by minimizing the array processor output power subject to this constraint. Under actual operating conditions, the assumptions of plane wave signals and an ideal propagation medium do not hold, and signal suppression can arise from causes such as beam steer angle errors, phase errors in the receivers, multipath propagation and wavefront distortion. Hence, the study and development of robust antenna array processors has long been an important topic of research. The studies of the effects of these errors on the performance of narrow-band array processors have been reported extensively in the literature [1]-[17]. To combat the problems due to beamsteer angle errors, the use of multiple linear constraints [3]-[6], [8], [22] and nonlinear norm-type constraints [8], [20], [23], [24] has been described in the literature. To combat the adverse effects due to sensor amplitude and phase errors, the use of norm constraints on the weight vector has been considered in [8],[18], [21]. The purpose of this paper is to present a unified approach to the design of robust narrow-band antenna array processors. The approach is based on the idea of minimizing the weighted mean-square-deviation between the desired response Manuscript received January 23,1987; revised June 21,1988. M. H. Er is with the School of Electrical and Electronic Engineering, Nanyang Technological Institute, Nanyang Avenue, Singapore, 2263. A. Cantoni is with the Department of Electrical and Electronic Engineering, University of Western Australia, Nedlands 6009, W.A. Australia. IEEE Log Number 8929472.
Structure of a narrow-band beamformer.
and the response of the beamfonner over variation in parameters. Three specific types of robust design are considered: robustness against directional mismatch" robustness against array geometry errors, and robustness against phase errors. Initially a general quadratic constraint on the weights is developed. However, it is then shown that the quadratic constraint can be replaced by linear constraints or at most linear constraints plus norm constraint. These latter set of constraints are no more complex than those required for designs not explicitly incorporating robustness features. The paper is organized as follows. In Section II, the complex notation for representing a signal incident on the array for narrow-band processing is introduced. In Section III, the formulation of optimum beamformers with robustness capabilities against various types of errors and mismatches is presented. Three examples are considered: robustness against directional mismatch, robustness against array geometry error, and robustness against phase errors. In Section IV, the optimization of the beamformer with robustness capabilities is considered. In Section V, an alternative beamforming structure based on a partitioned processor interpretation is formulated. In Section VI, some numerical results are presented and discussed. Section VII contains the conclusions.
II.
NOTATION
Fig. 1 shows a typical configuration of a narrow-band beamformer with L elements, L complex weights and complex output. The quadrature filter (QF), which is ideally a Hilbert transformer, can be approximated using a filter which provides 90 0 phase shift over the frequency band of the signal or simply a delay line with delay 1/4/0 if the signal is sufficiently narrow band.
Reprinted from IEEE Transactions on Antennas and Propagation. Vol. 38, No.1, pp. 17-23, January 1990.
607
where Q is the L x L dimensional Hermitian matrix defined by
Let W be the L-dimensional complex vector of adjustable weights defined by (1)
Q=
The response of the beamformer to a plane wave front of frequency fo with unity amplitude arriving from direction (6, c/» is given by
G(fo, 8, cP) = WHS(fo, 6, c!J)
p
(2)
(3)
where fo is the frequency of the plane wave and the {T;, i = 1, 2, ... ,L} are the propagation delays between the plane wavefront and the array elements. Let X(/) be the L-dimensional complex vector defined by
(4) where {X;(/), i = 1, 2, ... ,L} are the L complex signals at the outputs of the quadrature filters, then the weighted complex output of the beamformer at a time instant I is given by the complex scalar
(5)
For a stationary source field and a given weight vector W, the mean output power is given by
(6) where R = E[X(/)XH (I)] is the L x L dimensional array correlation matrix.
ill.
ROBUST PROBLEM foRMULATION
Assuming that the response of the narrow-band beamformer defined by (2) is a function of parameters {pi,i = 1, 2, ... .m}. The weighted mean-square-deviation between the desired response A and the response of the beamformer over variation in the parameter vector p ~ (P I , P2, ... ,P m ) can be defined as (7)
where Q(p) is a nonnegative weighting function for deterministic type of parameters or a probability density function (pdf) for those parameters which are modeled as random variable, {3 is a scalar given by (i =
c: pO-l/2~
=
e2
== WHX(I).
...
J
O(p)dp.
O(p)SSH dp
(10)
~J. r": ... In(p)A ·Sdp. Jpo -1/2tl
{3
(11)
Note that (9) is a quadratic function in the Wand it can be factorized as
S(fo, 8, q,) £ [expU21r!OTl),
y(t)
J
and P is the L-dimensional complex vector defined by
where H denotes complex conjugate transpose and S(fo, 0, c/» is the L-dimensional complex space vector defined by
expU2r!OT2),···,expU21rfoTL)]T
I11pO+I/26 ... {3 pO-1/2~
(8)
Substituting (2) into (7) and after some manipulation, one obtains
= (Wo -
W)H Q(Wo - W)
+a
(12)
where W o is the optimum vector which minimizes the meansquare-deviation defined by (9) and is given by
QW o =P.
(13)
a corresponds to the minimum mean-square-deviation defined by (14)
Robustness in the design can be achieved by introducing a quadratic constraint on the weight vector, namely
(Wo - W)HQ(W O - W) ~ e
(15)
where (16)
and 0 < ~ < 1 defines a normalized deviation over variation in parameters. Three specific types of robust designs are considered in the sections which follow.
A. Directional Mismatch The first example of the new approach is the design of a beamformer with beam-broadening capability. In many cases of interest, for example, in communications systems. the direction of the arrival of the desired signal is known only to within some angular tolerance. Also, in passive sonar detection application, when there are only a finite number of beams to span the total bearing angle, any signal that is not exactly matched to one of the beamsteer directions will be treated as an unwanted interference signal by the processor, and therefore will tend to be suppressed. Jt is desirable to have the ability to control the beamwidth of the beamformer in the look direction if required. The use of multiple linear constraints [3]-[6], [8], [22] and nonlinear norm-type constraints [8], [20], [23], [24] to achieve beam broadening has been. described in the literature. This section presents another approach for designing a, narrow.. band optimum beamfonner with robustness against directional mismatch. Beam broadening in cI> domain can be achieved by integrating the mean-square-deviation between the desired unity response (A = 1) and the response of the beam
608
• - o·
former over a spatial region of interest [cPl, cPu] as follows:
Source Direct:1on
e = ;4>1::' 11 - G(fo, 8,4>)12 dd: 2
== WH QW - (PHW + WHP) + 1
(17)
where (18)
p=
;4>I::S(fo,8,4»d4>
+---+-_ _ • • 90·
ac;.
(19)
where cPu == ¢o + 6.4>/2, cPl == 4>0 - 6.4>/2, and 6.<jJ defines the spatial region in the look direction over which the desired unity response is to be preserved. Fig. 2.
B. Element Spacing Deviation
The second example is the design of a beamformer with robustness against element spacing errors. An optimum beamformer can be very sensitive to errors in the assumed array element spacing. Recall that the steering vector S is a function of element spacing through the phase relation of the signal. Thus any deviation of the element spacing from its assumed design value could create an erroneous steering vector which is different from the one assumed in the constraint on the weight vector. Signal suppression may occur when the mean output power is minimized. Robustness in the design can be achieved by integrating the mean-square-deviation over some tolerance bound ~r about the nominal value rl as follows:
where
Azimuth plane of a double- ring circular array having ten sensor elements with five elements equally spaced on each ring.
5 Ii cos (cPO
-
(Xi) -
5 Ii COS (4)0 -
4 V cos (
-
(Xi) -
4 V COS (
5 -; cos (cPo -
(Xi) -
V COS (
vcos (4)0 -
ai) -
Ii cos (
4
4
5
I:S i , j :S 5
(X)),
- a j ),
6:S i , j :S 10
- a)),
1 S i :::; 5 6 :S j < 10 6 :S i
- ex)),
< 10
15:.JS5
(24)
and the ith element of P vector is given by i
== 1,2" ··,L (25)
where
(20)
5
where I" Q -_ -1- IfL +tu/2 . . . Ifl +~f/2 S( jO, 00 , (Llr)L f/. -~f/2 fl-~f/2
4>0
d i ==
)
where
P
1
== (Llr)L
Cij
Iff-
+M/2
f
L
-~f/2
...
f
r , + ~r/2 r,-~r/2
S([o, 00 , ¢o) ds, dr, ... dr.:
(22)
For example, for the double-ring ten elements circular array as shown in Fig. 2, the (i, j)th element of Q matrix when the mean-square-deviation is integrated over some tolerance bound Sr about the nominal value To is given by [Q l.. = exp [j 21T fOTOhij ]sinc (1T fO~Thij), i, j == 1, 2, ... , L
(23)
-
Cii),
1 S:. i
<5
{ ~ cos (cPo v
(Xi),
6 S:. J
:; 10
V cos (
== (i - 1)72°,
i==I,2,···,IO.
(26)
(27)
C. Channel Phase Errors The third example is the design of a beamformer with robustness against channel phase errors. The effect of channel phase errors on the performance of optimum beamformers is of great importance and has been extensively analyzed in the literature [6], [12]-[17]. Phase errors can arise in the array electronics or due to wavefront distortion in the propagation medium, and hence any signal that was originally in a plane wave may appear to the beamformer as an interference from some possibly nonreal direction. The beamformer will then attempt to null this signal even though it may be the desired signal one wishes to detect. Very little work has appeared in the
609
literature for combating phase errors. The use of norm constraint on the weight vector to combat against channel phase errors has been investigated in [8], [18], [21]. In this section a new approach based on the generalized response deviation constraint given by (7) is used to design an optimum beamformer with robustness against channel phase errors. Let S' be the steering vector in the presence of phase errors defined as
~
gee)
= {exp U(cP. + tl)], exp U(cP2 + r2)],
Fig. 3.
.. ·,exPU(cPL+fL)}T
(28)
IV.
where {cPi = 21r!OTj,i == 1,2,···,L} and {ti,i == 1, 2, ... ,L} are the phase errors associated with the L channels. It is assumed that {t;, i == 1, 2, ... , L} are independent random variables, uniformly identically distributed in [-&P/2,ocP/2] and are of zero mean and variance Robustness in the design can be formulated as follows:
e2
== -
13
minimize W H RW
where {3 is a scalar given by
==
subject to (W o - W)H Q(Wo - W) ~
J ... lxP/2
-04>/2
JOt/>/2 -04>/2
and n(tl,"', rL) is the joint pdf of {Si,i has the property that
(30)
== 1,2"" ,L} and
v
-
V)H R (W0
subject to V H QV
(37)
(38)
then the optimization problem defined by (36) and (37) be reformulated as minimize (W0
O(rl,"', sL)drl ... dSL
E.
' V :i: W o - W
-sen
(29)
(36)
w
2.,. j~/2 n(t.,···,rL)ll-~HWI2drl···drL Let
-lxP/2
(3
OPTIMUM BEAMFORMER WITH ROBUSTNESS CAPABILITY
The optimum weight vector is the solution to the following complex constrained optimization problem:
l1f.
1 J&/J/
Structure of the new partitioned narrow-band beamformer.
:s
-
V)
f.
can
(39) (40)
Using the standard primal-dual method [25], it can be shown that the optimal vector which solves the problem defined by (39) and (40) is given by (41)
Since {ri, i == 1, 2, ... ,L} are assumed to be mutually independent, it follows that
O(rl,
sz,' ", SL) ==
IT f2 ( t i )
where the optimum Lagrange multiplier ~ is the root of the following rational function of ~, namely
L
i
(32)
;=1
where Oi(ri) is the pdf of rio Substituting (28) and (32) into (29) and after some manipulation, one obtains
e 2 == w H QW - (pHW + WHp) + 1
The optimum weight vector is then obtained by substituting (41) into (38) and is given by (43)
V. AN
ALTERNATIVE STRUcrURE
The beamformer described previously can also be considered in terms of a constrained partitioned form shown in Fig. where 3. Wo is a coefficient vector of the upper filter. This vector ensures that a closest approximation, in the minimum-meanI, if i == j square sense, to the desired response is achieved in the beamsteer direction in spite of variation in parameters. W p is a 2 [Q]i j = { 5 ) exp U(tPi - tPj)] sine coefficient vector of the lowest filter, and is chosen to minimize the total mean output power. if i =1= j , i , j == 1, 2, . . . ,L (34) The output get) consists of the difference between a main beam Yo{l) and an auxiliary beam Yp{t). To prevent signal suppression, the adjustable weights of the lower filter must . . (o
r
(
(33)
t1
610
One pos sibility of ensuring that the lower filter has min imum response in the beamsteer direction is to introduce a quadratic constraint on th e weight factor as W~
QW p =s
>0. •
(44)
E.
--
,.. 18 . 9
This ensures that the output of th e lower filter contains little look direction signal component. Inst ead of quadratically constraining the weights of the partitioned beamformer, a set of linear constraints can also be used to approximate the effect of the quadratic constraint defined by (44) . Since ideally the desired response of the beamformer is primarily determined by the upper filter, to prevent signal suppression, the adjustable weight of the lowe r filter must be co nstrained to have zero response in the beamsteer direction and thus
§ ~
~
- - - - - _
.
O'
, \
\
\
••
a -1 1 11
\
\
\
, -,
-,
,
,,
- 2'a .1
..
.>0 •
...
I I ,'
lOX DD8:rTOoI
'5.'
".'
cD-t'
>S• •
(45)
Fig . 4 . OUlPUl SNR for the new quad ratically const rained optim um bea mforme r with j.
If Q has rank equal to n j, then it is clear that the necessary and sufficient co nditions for (45) to be satisfied are
constraint as well as the linear co nstrai nts defi ned by (47) . The new optimization problem is
E~Wp
= 0,
i
= 1,2, ' . . , n \
(46)
(52)
where {E" i = 1, 2, " ', nl} are the n \ orthonormalized eigenvectors corresponding to the n\ nonzero eigenvalues of Q. If Q has full rank or because in practice, the eigenvalue eva lua tion will not yield exactly zero eigenvalues, on e can impose no linear constraints of the form ErWp = 0, i = 1, 2, · · · , no
subject to
~ (~ A) /(~ A) X 100 percent
(47 )
subject to E ~Wp = 0,
i = 1,2, . . . , no.
IIWp W L L
;=no+ 1
A"
<5
i = no","I
»;
Se
(55)
by a suitable choice of no and <5 . VI.
(48)
(50)
The robustness of the processor is achieved through the eigenvectors {E" i = 1, 2, . .. , no} which are obtained from Q. It can be readily established that
W~{QWp =s
L L
W~ Q W p S
(49)
wp
(53)
The norm constraint defined by (54) ensures that
is greater than or equal to some threshold. The constrained optimization problem can then be expr essed as minimize E[g2(t)]
i = l , 2, ' '' ,no
=0,
(54 )
where no is the smallest integer chosen such that the percentage trace of the Q matrix defined by percent tr
Efwp
(51)
The choice of no according to (48) helps achieve a small uppe r bo und on W~Q Wp as required by (44) . However, complete co ntrol is not achieved because of the IIWpWin (51) . Thus one is led to an optimization problem which includes a no rm
COMPUTER STUDIES
T he purpose of this section is to present some numerical res ults to de mo nstrate the perfor mance achievable when the new approach is applied to the design of robust narrow -band arr ay processors . The array geometry used in the computer stud ies is the do uble-ring circular array shown in Fig. 2 . The beamsteer direction is assumed to be in the plane of the array and the 0° look direction is aligned with one arm of the array shown in Fig . 2. The parameter r is specified relative to the frequency of interest through the dimensionless spatial sampling factor Jl. which is defined by 6
r 'Ao
Jl.= -
(56)
where 'Au is the wavelength corresponding to fa . In the compute studies, p. was set at 0 .25 . Fig . 4 show s the array outp ut signal-to-noise ratio for the quad ratically constrained beamformer with I::i.cP = 0° and I::i.cP = 6° , respectively. The source scenari o was ass umed to consist of a 0° directional source of power 6 dB . The ambient noise field is spherically isotropic noise of 0 dB with un co r-
61 1
- - At-t .• -~-- -_.
30.'
_ . - _ . • At' .. 'I . •
_ - -:,,<:
' .
••
-,
-11iI. '
~ re.e
S.o
•1 .'
1'i.1
1NYtJI.4rDCsPAtDC URR
(I()
"'-:...-
- - - _.- _.-.- - - .- --- .. . .
0 .0
< .
-- .. _-
-19 .6
.
....
_"'.• L....~~........~~~"'-'-'---'-.....:::;~=---'- .............:L.--'-.......:.c..........J '.0
_ ._ ._ . • 6+" is"'
"'.
- - - - 61''' 2tII.BlC r,
·~--=~5 .-:-.:-,~, . - "':;' == - ~ --.... ~
It .•
- - - .. .. ... &t .. 10·
30.'
r,
. . 6r .. !;.1Ie r.
"'. ~
_ _ 6+- o'
f,
6C'·'. • r,
"'
s.•
Ie.'
SOD:R 'W.5E EJR:Pt
Fig. 5. Array gain in the 0° direction as a functionof interring spacing error for the new quadratically constrained optimum beamformer with E = 0.1 percent in the presence of a 6 dB 0° directional source, a 0 dB 180° directional source, a 0 dB spherically isotropic noise and - 30 dB white noise.
1'5.'
eor..)
"'.0
....
Fig. 6. Array gain in the ()" direction as a function of phase errors for the new quadratically constrained optimum beamformer with ~ = 0.1 percent in the presence of a 12 dB ()" directional source , a 0 dB 180· directional source, a 0 dB spherically isotropic noise and -30 dB white noise.
Qp is chosen so that related noise at level of - 30 dB. In the plots, an optimum beam is scanned through a number of bearing angles ranging 1~ _-)2_ (58) (~J ~j - 1. from 0°-25°. It can be seen that signal suppression problem due to directional mismatch is quite severe. The design of incorporating Llq, = 6° is to reduce the severity in the vicinity The source scenario was assumed to consist of a 0 dB 180° of the look direction . It is worth mentioning that with the new directional source and 0 dB spherically isotropic noise and approach, the width of the main beam can be specified, and a - 30 dB white noise. The power of the 0° directional source good compromise can be reached between a reasonable signal was assumed to be 12 dB. The array gain is averaged over acceptance angle and the ability of the beamformer to reject 50 different realizations of the random number generator. It is clear that the beamformer without 8q, incorporated is very directional interferences. Fig. 5 shows the array gain of the new beamformer as a sensitive to phase errors, especially at high input signal-tofunction of interring spacing errors. In the plots, the array noise ratio. On the other hand, the array gain is practically correlation matrix R is computed based on an array geome- constant up to 5° phase errors when 8fjJ = 15° is used in the try with difference interring spacings ranging from 0-25 per- design, even for a high input signal -to-noise ratio. cent smaller than ro = 0.25 hOI which is the value assumed VII. CONCLUSION in the constraint equation. The source scenario was assumed to consist of a 0 dB 180° directional narrow-band source, 0 This paper has presented a unified approach to the design dB spherically isotropic noise , and -30 dB white noise. The of robust narrow-band array processors. Three types of ropower of the 0° directional source was assumed to be 6 dB. bust designed have been considered; robustness against diIt can be seen that the processor without robustness incorpo- rectional mismatch, robustness against array geometry error, rated is very sensitive to interring spacing errors, especially and robustness against channel phase errors. Initially a gen at a high input signal-to-noise ratio . On the other hand, the eral quadratic constraint on the weights is developed. Sub beamformer with Llr incorporated in the design is able to sequently, the quadratic constraint is approximated by linear retain the array gain for small errors. constraints or at most linear constraints plus norm constraint. Fig . 6 shows the array gain of the new beamformer as a These latter set of constraints are no more complex than those function of sensor phase errors. In the plots, the array cor- required for designs which do not incorporate robustness fea relation matrix R is computed based on some random time tures explicitly . Numerical results show that the proposed apdelays at each sensor element as follows. proach appears to offer a unified treatment for directly designLet q,0 be the rms phase error, in degrees, atfo. A constant ing narrow-band processors which are robust against various phase error on all sensors will not degrade the system, so types of errors and mismatches between signal model and aca vector of sensor time errors OJ is assumed, where OJ > 0 tual scenario. corresponds to phase advance
L;;:
ACKNOWLEDGMENT
(57)
where ~j = ~j/Qp and ~j is a random variable between -1 and +1 with a constant probability distribution. The constant
612
The authors are grateful to the anonymous reviewers' comments, which helped to improve the presentation of the paper. REFERENCES
[1] C. L. Zaham, "Effects of errors in the direction of incidence on the
[2]
(3]
(4] [5] [6]
[7] [8]
[9] [10] [11] (12] [13]
[14] (15]
performance of an adaptive array," Proc. IEEE, vol. 60, no. 8, pp. 1008-1009, Aug. 1972. H. Cox, "Resolving power and sensitivity to mismatch of optimum array processors," J. Acoust. Soc. Am., vol. 54, no. 3, pp. 772-785, 1973. S. P. Applebaam and D. J. Chapman, "Adaptive arrays with main beam constraints," IEEE Trans. Antennas Propagat., vol. AP-24, no. 5, pp. 650-662, Sept. 1976. K. Takao, H. Fujita, and T. Niski, "An adaptive array under directional constraint," IEEE Trans. Antennas Propagat., vol. AP-24, no. 5, pp. 662-669, Sept. 1976. A. M. Vural, "A comparative performance study of adaptive array processors," in IEEE ICASSP '77 Rec., May 1977, pp. 695-700. - , "Effects of perturbations on the performance of optimum/adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, no. 1, pp. 76-87, Jan. 1979. R. T. Compton, Jr., "Pointing accuracy and dynamic range in a steered beam adaptive array," IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, no. 3, pp. 280-281, May 1980. J. E. Hudson, Adaptive Array Principles. New York: Peter Peregrinus, 1981. R. A. Mucci and R. G. Pridham, "Impact of beam steering errors on shifted sideband and phase shift beamfonning techniques," J. Acoust. Soc. Am., vol. 69, no. 5, pp. 1360-1368, May 1981. R. T. Compton, Jr., "The effect of random steering vector errors in the Applebaum adaptive array," IEEE Trans. Aerosp. Electron. Syst., vol. AES-18, no. 5, pp. 392-400, Jan. 1983. Y. Bar-Ness, "Steered beam and LMS interference canceler comparison," IEEE Trans. Aerosp. Electron. Syst., vol. AES-19. no. 1, pp. 30-39. Jan. 1983. R. N. McDonald, "Degraded performance of nonlinear array processors in the presence of data modeling errors," J. Arouse. Soc. Am., vol. 51, no. 5, pp. 1186-1193, Apr. 1977. D. J. Ramsdale and R. A. Howerton, "Effect of element failure and random errors in amplitude and phase on the sidelobe level attainable with a linear array," J. Aroust. Soc. Am., vol. 68, no. 3, pp. 901-906, Sept. 1980. A. H. Quazi, ., Array beam response in the presence of amplitude and phase fluctuations," J. Aooust. Soc. Am., vol. 72, no. 1, pp. 171-180, July 1982. D. R. Farrier. "Gain of an array of sensors subjected to processor
[16] [17]
[18] [19] [20] [21] [22] [23] [24]
[25] [26]
613
perturbations," Proc. Inst. Elec. Eng., vol. 130, pt. H, no. 4, pp. 251-254, June 1983. L. C. Godara, "The effect of phase-shifter errors an the performance of an antenna array beamformer," IEEE J. Ocean. Eng., vol. OE-IO, no. 3, pp. 278-284, July 1985. N. K. Jablon, "Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfection," IEEE Trans. Antennas. Propagat., vol. AP-34, no. 8, pp. 996-1012, Aug. 1986. E. N. Gilbert and S. P. Morgan, "Optimum design of directive antenna arrays subject to random variations," Bell Syst, Tech. J., vol. 34, pp. 637-663, May 1955. J. M. McCool, HA constrained adaptive beamformer tolerant of array gain and phase errors," in Aspects oj Signal Processing, pt. 2, G. Tacconi Ed. Dordrecht, Holland: Reidel, 1977, pp. 517-522. J. N. Maksym, "A robust formulation of an optimum cross-spectral beamformer for linear array," J. Acoust. Soc. Am., vol. 65, no. 4, pp. 971-975, Apr. 1979. H. Cox, R. M. Zesking, and T. Kooij, "Sensitivity constrained optimum endfire array gain," in Proc. IEEE ICASSP, 1985, paper 46.12. A. K. Steele, "Comparison of directional and derivative constraints for beamfonners subject to multiple linear constraints," Proc. Inst. Elec. Eng., vol. 130, pts. F, H, no. 1, pp. 41-45, Feb. 1983. J. W. R. Griffiths and J. C. Hudson, "An introduction to adaptive processing in a passive sonar system," in Aspects of Signal Processing, G. Tacconi Ed. Dordrecht, Holland: Reidel, 1977, pp. 299-308. K. M. Ahmed and R. J. Evans, "An adaptive array processor with robustness and broadband capabilities," IEEE Trans. Antennas Propagat., vol. AP-32, no. 9, pp. 944-950, Sept. 1984. D. G. Luenberger, Optimization by Vector Space Methods. New York: Wiley, 1969. M. H. Er, "Optimum antenna array processors with linear and quadratic constraints," Ph.D. dissertation, Dept. Elec. Comput. Eng., Univ. Newcastle, N.S.W. 2308, Australia, May 1985.
Design Trades for Rotman Lenses R. C. Hansen, Fellow, IEEE Abstract- The foundation of a satisfactory Rotman lens design is geometric. The ellects of the seven design parameters (focal angle, focal ratio, beam angle ratio, maximum beam angle, beam port curve ellipticity, array element spadDg, and focal length/X) on the shape, and on the geometric phase and amplitude errors of a Rotman lens are described. The adYantage of beam port shaping to reduce phase error, and of pointinl port horns at the opposite apex (instead of normal to the curve) to reduce oil-axis beam amplitude asymmetries, are shown numerically. A design procedure for selecting these parameters Is given, and a new calculation of lens gain Is presented.
M
I.
Yz
INTRODUCTION
ULTIPLE beam antennas have proved useful for various applications such as ECM, and the Rotman lens is often used . Design of these lenses must involve both geometric trades and mutual coupling effects between the lens ports. The latter is relatively difficult to control , but the former is crucial to the realization of an efficient and compact lens. Thus a careful geometric optics design should be accomplished first ; then adjustments must be made to reduce mutual coupling effects . This paper describes the geometric design trades . The Rotman lens has six basic design parameters: focal angle (x, focal ratio {3, beam angle to ray angle ratio 'Y, maximum beam angle Y,m, focal length f l , and array element spacing d. The last two are in wavelengths, and l' is a ratio of sines . A seventh design parameter allows the beam port arc to be elliptical instead of circular. Since the design equations are implicit and transcendental , with only one sequence of solution, the interplay of design parameters is difficult to discern . In this paper a series of lens plots is used to show the effects of each parameter. Geometric phase and amplitude errors over the element port arc vary primarily with (X and {3 , and with an implicit parameter which is the normalized element port arc height. Representative plots show how these errors depend upon the parameters. For lenses where the beam port arc and feed port arc are identical, resulting in a completely symmetric lens, the design equations are greatly simplified [8]. However, these lenses are seldom used because their design options are much more constrained. A new calculation of lens gain is presented , with the lens connected to an array of isotropic elements. Port spillover, and phase and amplitude errors are included in the gain calculation, but not impedance mismatches due to mutual coupling. Finally, a design procedure is outlined.
focus
Fig .!.
Ray geometry .
at the center. The focal angle (X is subtended by the upper and lower foci at the center of the element port curve . It is assumed here that the foci are symmetrically disposed about the axis , and that the lens is also symmetric. Then the parameter {3 is the ratio I of upper (and lower) focal length f 2 to f l :
(I) Clearly the lens width in wavelengths , f l lA, is another parameter. Now the angle of the beam radiated by the array is y" and if one of the off-axis foci is excited, then the ratio of lens ray angle (X to array beam angle y, is 1': sin y, 'Y = - - . sin (X
r.
An indirect parameter of utility is which relates the distance Y3 of any point on the array from the axis , to fl' This parameter controls the portion of the phase and amplitude error curves that the lens experiences. It is expressed: Y3'Y
r=· f
n. LENS PARAMETERS The lens equations equate path lengths from the foci to the array elements; see [7] or [6] for a derivation of these. Using the nomenclature of Fig. 1, it is convenient to normalize all dimensions by the principal focal length fl' This is also the lens width
(2)
(3)
l
Note that the line lengths w of Fig. 1 are an integral and essential part of the lens. The maximum beam angle, Y,m , is an important parameter, as is the array element spacing in waveI Note that this ratio tJ is the inverse of the ratio g used by Rotman, and by McGrath (41. As it is convenient to normalize all dimensions by Iv. the
Manuscript rece ived December 4, 1989; revised Septembe r 11, 1990. The author is at P.O. Box 570215, Tarzana, CA 91357 . IEEE Log Number 9041791.
ratio of
f 2 / f.
is more appropriate.
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 39, No .4, pp. 464-472, April 1991.
614
1. 0 . - - - - - - - - - - - - - - - - - , ALFA - 40
.8 ZETA
.6
.4
.2
o
L....-----'_~
. 80
Fig. 2.
lengths d IA. The
_
_
.L___J_
_'___ __ _ '
. 92
. d8
. 84
BETA
. 96
Upper limit on parameter zeta.
lmax that corresponds is given by lmax
(NE - l)-yd
(4)
= -----
2/ 1
where NE is the number of elements in the linear array , since Ymax = (NE - 0 /2. An upper limit on I occurs when the tangent to the element port curve is vertical ; this also gives w = O. This value of I is given by Fig. 3.
(5) Fig. 2 gives this limiting value versus {3, for several values of a . Since the useful range of I is roughly from 0.5 to 0.8, a range of {3 appropriate for a given a may be inferred . The geometric lens equation is a quadratic in the line length w that connects an element port to the corresponding array element:
(6) where the coefficients involve the parameters , a, {3 , and 'Y :
(1 - (3)2
0=1 2 12
c = -
r2 +
(7)
( I - {3c) 2 - (32
b=-2+-+ {3
12
2(1 - (3)
I - {3C
r 2S 2
I - {3C
12S 2 (l -
(3)
(I - {3C)2
14S4 4( I - fJC)2
(8) (9)
and C = cos o , S = sin«. Usually the number of beams and number of elements, the maximum beam angle and element spacing, are specified from
Effect of focal angle . (3 = 0 .9, "y = 1.1. psim = 50. /1 = 4 WV, d = 0.5 WV.
the system requirements . Thus , the task is to select the optimum a, fJ, 'Y, and /I/A . III.
EFFECT OF PARAMETERS ON LENS SHAPE AND PORT POSITIONS
The lens shape is important , both in conserving space and in reducing loss. For example , a wide lens tends to have path lengths that are more nearly equal, and allows the beam port curve and the element port curve to have different heights, and even different curvatures. As in Fig. I , the width is along the lens axis. Wide lenses have large spillover loss, and higher transmission line loss. A compact lens tends to minimize spillover losses; roughly equal port curve heights now become important, to avoid severe asymmetric amplitude tapers and large phase errors. Curvatures of the two port curves may be different; use of array element spacing greater than half-wavelength allows more beam ports than element ports to be used. For this case, the beam port curve may be more curved , and the element port curve flatter. The effects of the seven parameters will be shown through a series of charts . Six charts are shown in Figs. 3-8. Beam ports, which are ticked, are on the left. Element ports, also ticked, are on the right. Foci are indicated by asterisks . The focal length is normalized to unity, so that each tick mark on the axis (and on
615
l.0
VAL UES OF GAMA SHOWN
Fig. 4.
Effect of focal ratio. a = 40, r = 1.1, psim d = 0.5 WV.
= 50, I. = 4
WV,
the ordinate) is 0.05. From the ordinate scale the element positions may be inferred. Each lens curve is extended past the outermost port by half the width of that port. These examples have nine beam ports, and II element ports, and of course II elements in an equally spaced linear array. With all other variables fixed, increasing a opens the beam port curve, and closes the element port curve. Port positions are roughly unchanged. But the outer foci locations change markedly as expected. The three lens plots of Fig. 3 illustrate these effects. It can be seen that a value a can be selected that roughly equalizes the heights of the two curves. Of course, a must be selected in conjunction with the other variables, to minimize phase errors over the aperture. The outer foci should be comfortably inside the beam port curve. Increasing 13 has an effect similar to increasing a; the beam port 'curve opens, and the element port curve closes. Fig. 4 contains three lens plots to show this. Again, port positions are roughly unchanged. Also the focal locations change relatively little. Again, a value of 13 can be selected that roughly equalizes the curve heights. There are pairs of a and 13 that produce closely the same lens shape, and port positions. However, the foci vary with a, and the connecting lines (from element ports to elements) are different. Table I shows three a - 13 lens pairs that have common = 50, II/X = 4, and lens curves and ports, all for 'Y = 1.1, d = 'A/2. One may thus infer that the phase error over the
Fig. 5.
Effect of angle ratio. a = 40. {3 = 0 .9, psim d = 0.5 WV.
= 50, I I = 4 WV,
aperture for each beam will be different, depending upon a . This will be shown in the next section. For any set of the other four parameters, there are probably some a - (3 pairs that behave similarly. Increasing 'Y leaves both lens curves unchanged, but the beam ports are moved closer together, while the element ports are spread apart. A three-lens set in Fig. 5 shows this trend. Although the foci remain fixed, the ends of the curves change, so that the relative position of the foci changes. 'Y also can affect the relative heights of the two curves. Values of 'Y here are one or greater, as the cases used are all for large beam angles. When the beam cluster subtends a more modest angle, e.g., 30·, values of 'Y < I are appropriate as they allow a "fat" or curved lens. When is changed, only the beam port spacings change. Increasing '"m spreads the beam ports and extends the port curve, so that this parameter helps produce a lens with roughly equal heights of beam and element port curves. The three lens plots of Fig. 6 depict this behavior. Element spacing is critical as it controls the appearance of the grating lobes [2]. For a maximum beam angle of spacing that just admits a grating lobe is
"'m
616
"'m
"'m'
d /'A = 1/(2 + s~n "'m) ' In general, spacings are kept below this value.
(10)
. 50 .45 . 50 . 45 . 50 . 45
55 50 45
VALUES OF PSI-MAX SHOWN
Fig. 6.
Effect of maximum beam angle . Ct = 35. I, = 4 WV . d = 0.5 WV.
VALUES OF d/).. SHOWN
t3 = 0.92.
'Y
=
1.1.
When d is changed, only the element port spacings and the extent of the port curve change, analogous to Vt m changing beam port spacing. Fig. 7 uses two lens plots to show this. Increasing the lens focal length (width) in general increases the separation between the end ports as well. But changing f l / }., also changes all spacings, as the lens equations are normalized by fl' Thus as shown in the two lens plots of Fig. 8 changing fl /}., also changes the element port arc and element port spacings. The minimum value of f , is smaller for Rotman lenses than for other types of lenses [9]. Next, phase and amplitude errors will be examined . IV.
EFFECT OF PARAMETERS ON PHASE AND AMPLITUDE ERRORS
Aperture errors depend upon ex and (3, and upon eccentricity, but only indirectly on the other parameters. Thus the most insight results from plotting phase and amplitude errors versus the normalized parameter r; see (3). Since phase errors are zero at angles corresponding to the three foci, a satisfactory approach uses one beam position midway between the central and edge foci, and a second beam position beyond the edge focus. Amplitude errors occur at all beam ports, so more cases are needed to display amplitude error behavior .
Fig. 7.
Effect of array element spacing , Ct = 40. psim = 50. I, = 4 WV.
A . Phase Errors
t3 = 0.88 .
'Y = 1.1.
Figs. 9-12 show phase error versus r for lenses with ex of 30 and 40 deg., for the two beam positions. Note that to get phase error, the values from the figures are to be multiplied by f l / }.,. For the midfoci beams, the phase error is small, except for very large lenses. Phase errors for the wider angle beams are still modest, and will not be important except for large lenses, or designs with r> 0.75 . In general, the phase errors increase as ex is increased, for all beam positions. Although Rotman and Turner indicated an optimum value of (3, which was 2/(2 + ex 2 ) , examination of Figs. 9-12 show that an optimum (3 exists only for one range of and one ray angle. Best values of (3 are different for between foci rays and rays outside the foci. Since the latter usually have larger phase error, the designer could optimize, but the value would vary with both This old value also gives poor lens ray angle and with shapes when the number of beam and element ports are roughly equal. The results of using an elliptical beam port curve are shown in Fig. 13, where the phase errors are equalized at r = ±0.7 for a ray angle of 45 Note that the elliptical beam port curve is a simple way of realizing the optimum curve at Katagi et al. [3]. This gives a 13% reduction in phase error; the phase errors for the midfoci ray at 17.5 become slightly more asymmetric, but
617
r
rmax'
0
•
0
Fig. 8.
Effect of focal length . a = 40. {3 d = 0.5 WV.
TABLE I
a-{3 PAIRS
Number I
2 3
a = 30 a = 35 a = 40
{3 = .94 {3 = .92 {3 = .90
are still well below the 45° ray errors. Note that the ellipticity of -0.3 only changes the principal radius by 5%, so amplitudes are essentially unchanged. The beam port ellipse major axis is along the lens axis for this ellipticity.
B. Amplitude Errors
= 0.9 •
.., = 1.1. psim
= 50.
angles the near and far ends of the element port curve experience modest amplitude changes. Compared to the amplitude taper needed to produce 25 dB sidelobes, these amplitude errors are small. Actual lenses may have port widths larger than >../2. so the amplitude tapers can be expected to increase, especially for edge beams. The asymmetry of amplitude for the off-axis beams can be reduced by pointing each port hom at the opposite apex, instead of normal to the port curve [5]. For example, a nine-beam. ll-element lens with ex = 40, {:j = 0.9, 'Y = 1.1, !/1 m = 50, d = 0.5>" and I. = 4>" has amplitude taper for the outside beam as shown in Table II. Also shown is the taper for apex pointed horns. Use of apex pointing produces appreciable improvement. Gain is slightly improved.
Amplitude errors are calculated using beam port and element port hom patterns of sine ;r u, where horn widths are all set to a nominal >../2. Each port hom has its axis normal to the port curve. Fig. 14 shows amplitude error, normalized to 0 dB for the axial ray, for a lens with ex = 30, {:j = .94. Curves for ray angles of 0° , 15°, and 45° are given. Similarly, Fig. 15 is for a lens with ex = 40, {:j = 0.9, for ray angles of 0° , 20°, and 50° . These examples are two of the ex - {:j pairs of Table 1, and thus the amplitude errors are similar. As expected, for wide ray
618
V . CALCULATION OF LENS GAIN
Element and beam port spillover, phase and amplitude errors, port impedance mismatches, and transmission line loss all contribute to reducing lens gain. Note that, as in the case of a horn feeding a reflector antenna, there is no feed hom spreading loss, due to the equal path property through the foci. The small inequality of other paths is subsummed in the path phase and amplitude errors. Gain will be calculated here based on port
2. 0
1.
1. >
'"
<,
u,
<,
w '" 0
a:' a a: a: w w
til
-c a.
-0 .
-0 .
:I:
- 1.
-1. 5
-2. -O .B
- 0 .6
-0 . 4
Fig. 9.
-0 .2
0 .0
0 .2
ZETA
Phase error between foci. Rotman lens. ex
0.4
O. B
0 .6
= 30, ray angle = 15.
2 .0 <,
I I I I I
1.
1.
>
'"' ;:;:
<,
O.
<,
'"
ur 0
rr' 0 a: a: w ur
til
-c
-0 .
-0.
\
\
\
\
\
\
r
I I I I I
BETA =
\
'\
. 94
J
I
:I:
a.
-1.
-1.5
-,
/
J
I I I I I
/
/
/
/
<,
. 92
<,
/
---
- -
//
/'
--=-=""",~---,
..-/'.
-...::::::::-:- :-.. -...
'-'.
"\.
.
"
\
0 .2
Fig. 10. Phase error beyond foci. Rotman lens, ex
spillover and on aperture errors. Since the amplitude error calculation includes both beam port and element port hom patterns, spillover is included [10]. The phase and amplitude errors at the element ports are transferred to an array of isotropic elements. Then the problem reduces to that of calculating gain of a symmetric linear array of isotropic elements with complex coefficients. This is readily done [2]:
IL:A n I 2 L:L:AnAm*sinc(n _ m)27rd/>..'
"
\,
. 96
0 .0
=
-.
i
\"
ZETA
G
....." """".'-'. "
(11) 619
0 .4
""
I
I --- . /
/
\. 0 .6
O. B
= 30, ray angle = 45.
Actual gain is then that of (11) multiplied by the element gain, times the impedance mismatch factors. Variation of gain with parameters is very small. For example, for a typical small lens only 0.2 dB change occurs from the center beam to the edge beam. And the gain values are roughly independent of o , {3, 'Y etc. With a larger array the gain increases just as expected. Using the same nine-beam, l l-element example of the previous section, gain ranges from 10.2 to 10.4 dB; the latter value, for the center beam, is within 0.1 dB of the gain for a similar uniformly excited array. For this lens the range is ± 0.756.
r
2 .01...-
....,
I.
I.
>
x
<,
i:
O.
<,
l!l lU CJ
a:
-0 .
o
a: a:
lU
...
lU
Ul
-0 .
:I:
a.
-I.
- 1.
Fig. II.
Phase error between foci. Rotman lens. cr = 40. ray angle = 20.
2l
It L
I. ° t
~
<,
l!l
01
ur
CJ
a:'
-0 .
CJ
a: a: lU lU
Ul
-e
-0 .5
:I:
a.
- 1.0
-1.
- 0 .6
Fig. 12.
-0 .4
-0 .2
0 .0
ZETA
PROCEDURE
System requirements usually specify the frequency range, the number of beams and the angular coverage, and either the
0 .6
0 .4
Phase error between foci. Rotman lens. cr = 45. ray angle
Effects of feed hom spillover and internal lens reflections can be reduced by either employing dummy (terminated) feed horns adjacent to the edge horns, or through the use of absorber between the ends of the beam port arc and the element port arc [5]. Port impedance mismatches are outside the scope of this paper.
VI. A DESIGN
0 .2
O. B
= 22 .5.
beamwidth or adjacent beam crossover level. From these, a suitable combination of number of elements and d f A. may be inferred. The design process starts by the selection of a center frequency, at which all dimensions are computed. Then, a starting value of i. fA. is selected, to keep fmax well below 0.8 . The focal length will be somewhat less than the array length. Next, using the guidelines of Sections ill and IV. ex, (3, and 'Y are selected, to: locate the outer beam port a modest amount past the outer focus; produce beam port and element port arcs of comparable heights; and yield acceptable phase and amplitude errors at each port. Achieving this may require adjustment of
620
4 .0
\
>
E
\
\
\
3'
<,
\
u, <,
\
\
UJ
cr: a
-,
<,
<,
-0.
a: a:
s-:
UJ
/"
UJ
en
0
- .3
45
co 0
=
E -
s--:
~
z-: RAY
--
,.-
ANGLE' 17 . 5
I
a.
-2 0
-0 6
-0 2
Fig. 13.
2 .0
o
0 .0
0 .2
ZETA
Effect of ellipticity. Rotman lens,
Cl
I
0.4
0 .0
C 8
= 35, e = 0.92.
\. t~ \
~ \
Or
L u,
<,
rn a
a:' o a: a: -2 . C
\
RAY ANGLE - 0
\
"
........
........
........
<,
<,
~~- -- - - - - - - - - --- - - - - - - - - - - - ~~' 45
UJ
'", \
UJ
\
a
::0
\
-4 0
Fig. 14.
/ 1/'" or d / "', and of ex ,
Amplitude errors. Rotman lens.
{3 , and 'Y. Use of an elliptical beam port arc is usually not warranted except for large lenses, When a satisfactory design is realized at the center frequency, phase and amplitude errors at each port are calculated at representative frequencies, to assess performance over the frequency range. And of course the actual beam and element port hom widths are used. At this stage, calculation of a beam rosette (a set of beam patterns) at each frequency is appropr iate. Some compromise and iterative adjustment of parameters may be necessary to obtain good wide-band results, and to best accommodate mutual coupling effects.
Cl
= 30, e = 0.94.
Although the lens width / 1 is less than the array length, the lens height is always greater than the array length. Lens dimensions are reduced by the square root of effective dielectric constant for either stripline or microstrip implementation. See [1] for examples.
VII.
CONCLUSION
Guidelines have been given on how the seven Rotman lens parameters affect lens performance, and on how to select values for them, based on geometrical optics. Such a design must be tempered by mutual impedance considerations.
621
2 .0.-,-
-,
\
\
o.
\
u:
<,
\,
a:>
c
rZ 0 a: a: - 2 . w w
<,
~
y----
~~~
-".------- .............. ...
~
RAY ANGLE - 0
--
~~
.............. <, <,
------ ------- -------------.~.
~
50
0
:::>
I-
..J
o,
:>:
-c
- 4.
- 0 .6
-0 .4
- 0 .2
0 .0
ZETA
0 .2
0.4
Fig. 15. Amplitude errors. Rotman lens, cr = 40, {3 = 0.9. TABLE
n
BEAM ONE AMPLITUDE TAPER ;
fl =
4>-
Axes Normal to Arc
Axes through Apex
I 2
-8.49 -7.03
-2.26 -1.84
4
-4 .29 -3 .33
-1.46 -1.31 -1.37
-2.04 -1.70 -1.57 -1.68 -2 .08
-2 .73 -3.60 -4.71 -5 .95
Element Number
3
5 6 7 8 9
10 11
(dB)
-5.50
-2.58
(dB)
- 1.63 -2.08
REFERENCES
[I]
[2] [3] [4]
[5] [6] [7] [8] [9] [10]
D. H. Archer, "Lens-fed multiple beam arrays," Microwave J ., vol. 27, pp. 171-195 , Sept. 1984. R. C. Hansen, "Linear arrays," in Handbook of Antenna Design, vol. 2, A. W. Rudge et al., Eds. U.K.: Inst. Elec. Eng./Peregrinus , 1983, ch. 9. T. Katagi et al., "An improved design method of Rotman lens antennas," IEEE Trans. Antennas Propagat., vol. AP-32, pp. 524-527, May 1984. D. T. McGrath, "Contrained lenses," in Reflector and Lens Antennas, C. J. Sletten, Ed . Dedham, MA: Artech House, 1988, ch. 6. L. Musa and M. S. Smith, "Microstrip port design and sidewall absorption for printed Rotman lenses," Proc. Inst . Elec. Eng., vol. 136, pt. H, pp. 53-58 , Feb. 1989. D. M. Pozar, Antenna Design Using Personal Computers. Dedham, MA: Artech House, 1985, sec. 4.6. W. Rotman and R. F. Turner , "Wide-angle microwave lens for line source applications," IEEE Trans. Antennas Propagat.; vol. AP-ll, pp. 623-632, Nov. 1963. J. P. Shelton, "Focusing characteristics of symmetrically configured bootlace lenses," IEEE Trans. Antennas Propagat., vol. AP·26, pp. 513-518, July 1978. M. S. Smith, " Design considerations for Ruze and Rotman lenses," Radio Electron. Eng., vol. 52, pp. 181-197 , Apr. 1982. M. S. Smith and A. K. S. Fong, "Amplitude performance of Ruze and Rotman lenses," Radio Electron. Eng., vol. 53, pp. 329-336, Sept. 1983.
622
0 .6
O. B
Optimum Networks for Simultaneous Multiple Beam Antennas Edward C. DuFort, Fellow, IEEE The desired complex radiation patterns or aperture distributions are presumed to be specified but are arbitrary. The analysis which develops the maximum possible efficiency also suggests a synthesis procedure. The synthesis technique is completely developed for the case of the general linear MBA. Section II applies Stein's technique to the present situation. The correlation between beams is defined and the physical significance of the correlation matrix is developed. Eigenfunctions and eigenvalues of the matrix are discussed and the efficiency limit is derived. A general synthesis procedure is developed in Section III which is based on the eigenfunction structure of the correlation matrix. This BFN produces the maximum possible efficiency. Several practical examples are discussed in Section IV, and concluding remarks are contained in Section V.
Abstract- The design of passive microwave circuits for the formation of simultaneous multiple beams with arbitrary but specified shapes is considered. The maximum possible efficiency is derived from energy conservation and is determined from a Hermitian matrix whose elements are the correlation coefficients between all beam pairs. The eigenfunctions of the correlation matrix are the basis of a synthesis procedure for a practical network that will achieve the maximum efficiency. Several practical examples are given where unavoidable losses are typically 1 dB or more.
T
I. INTRODUCTION
HE problem of forming simultaneous multiple beams
from a common aperture is of considerable practical interest. For example, it may be necessary for a radar to illuminate a large elevation sector with several beams of different widths. A narrow high gain beam is used for the horizon. and successively wider beams are used at the higher angles. This provides total coverage from one multiple beam antenna (MBA) with a small number of beams. The gain reduction at higher angles can be tolerated when viewing altitude-limited targets because they are closer in range. Other examples arise in the design of multibeam feeds for dish antennas. Even though it is assumed that lossless transmission lines and ideal matched directional couplers of all values are available, it was shown a long time ago by Allen [11 that perfect efficiency often cannot be achieved. The b.arnforming network (BFN) must be dissipative in order to satisfy certain energy conservation relations. Stein [2] devised a method for calculating the maximum possible efficiency of the network knowing just the amplitude and phase of the desired radiation patterns or aperture distributions. His results are fundamental, and the loss cannot be circumvented by clever design of the linear passive BFN. A sensitivity to the problem can be obtained by referring to White's [3] work, which preceded Stein's. He studied some special cases of low sidelobe high crossover beams derived from a feed network attached to a Butler matrix. It was necessary to insert attenuation in the BFN to obtain the desired patterns. This author [4] studied the formation of a large number of identical low sidelobe beams pointing in the characteristic Butler directions. Networks were synthesized such that the Stein limit was achieved. In the present work, the problem is considered in general.
II.
EFFICIENCY LIMIT FOR GENERAL
MBA"s
The aperture of a linear array contains N elements which are connected through a linear passive BFN to M beam terminals. Excitation of beam terminal In with a unit wave produces a scattered wave Sn m at the nth aperture terminal as portrayed in Fig. 1. Perfect operation (for unit input) of the BFN would produce specified outputs B n m where ,V
L e.; I
1
2
n=l
(1)
= 1.
Since the BFN may be lossy, the scattered waves generally are of the form
(2)
where K ~m is the efficiency of the mth beam. In matrix notation I we have S
(3)
= BK
where the B matrix is specified but arbitrary and K is an unknown diagonal matrix. If the beam terminals are excited by an arbitrary input vector A, the output vector V at the aperture is
v == SA = BKA. The output power is obtained by multiplying
Manuscript received March 22, 1989; revised August 28, 1991. The author was with the Hughes Aircraft Company, Fullerton, CA. He is now at 2121 Domingo Road. Fullerton, CA 92635-3410. IEEE Log Number 9105265.
V
by the
I Upper case letters are matrices and overlined upper case letters are single column matrices (vectors). We use the dagger symbol to indicate conjugate transpose, so Ht = H means H is Hermitian. Lower case letters are scalars.
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 40, No.1, pp. 1-7, January 1992.
623
(4)
the efficiency of the best beam K n
2
= K~lwtew = KtlH or
H = WtCW = n',
BEAM FORMINGNETWORK(BFN)
AT~
Since ~e input power is power IS m
Pout
transpose
=
( l1a)
(Bt)lnBnm =
L
n=l
B:,Bnm = BJ
AlHA
o :5 K fl
. 8m. (6b)
(7b)
III.
where K;"m is the efficiency for the mth beam. The beam with the best desired efficiency is numbered one. The others are allowed relative degraded efficiencies m: In the radar example described earlier, the desired efficiency for the horizon beam is as large as possible (K II = 1, WI I = 1), but the high angle beam gain may degrade. The elements of the real diagonal matrix Ware of the form
o<
=
Wmm -s 1,
1
m
*" 1
(12)
S
1/ Al .
(13)
SYNTHESIS OF THE OPTIMUM NETWORK
The weighted correlation matrix H = W t C W has a sequence of eigenvalues AI' I = 1,···,M, and corresponding orthogonal eigenfunctions [5] ~t, which are solutions to (l lb), These eigenfunctions may be normalized
W,;
WI.
= A.nax;
AI
fl
(7a)
= KllW
:5
The maximum possible efficiency is the reciprocal of the largest eigenvalue of the H EIatrix. Now H I 1 = 1 in all cases; therefore the choice (A)m = 0ml in (12) shows that Al ~ 1. Thus, the efficiencies are always equal to or less than unity. Since H depends only on the correlation matrix of the specified beams and the specified weighting W~ no network design can exceed the efficiency limit (13) which is a fundamental energy conservation limit. It is shown in Appendix I ~hat the efficiency of the preferred beam number 1 always Improves by applying a weighting to the other beams. In the case W = I, Allen [1] showed that it is necessary that the beams be orthogonal, C = I, for K = 1. Stein [2] first derived the important results (13) for W = I. The problem of synthesizing a network which actually achieves the Stein limit for arbitrary specified beams is based on the properties of the H matrix.
where the equality holds when / = m. The desired distributions are orthogonal when C is the identity matrix; however, this condition often is not obtained, and this leads to the efficiency limit. Often the desired efficiencies are the same for each beam; however, to allow for bias, it is assumed that
K
(lIb)
therefore, the energy conservation relation (10) requires
The element elm is the scalar product between the lth and mth distributions, and is a measure of the correlation between these distributions. From the Schwartz inequality and the unit normalization (1) of B,
K mm == KllWmm
= A~
o < A.14 :5 AlA
(6a) n~l
A is
Since H is Hermitian and positive definite, it is well known that the A are real and positive [5]. These may be ordered with AM the smallest and AI and largest. Thus, for any A:
(5)
matrices. Consider the pair BtB whose elements are N
(10)
AA
H~
= VtV = AtKtBtBKA = AT · [Kt · (ntH) · K] . A
where we have relied on the well-known formula (AB)t
L
-
A=~
BtAt and indicated a particularly useful grouping of the
=
AT
for any and all inputs X Stationary values occur when an eigenfunction of the H matrix with eigenvalue ~.
Diagram of waves transmitted through BFN.
Pout
elm
the ratio of output to input
AHA
Pin
Vt:
N
2
(9)
-=KIl~SI
M
BEAMTERMINALS
Fig. 1.
fl :
Kt . BtB . K = KtCK
N
~:
(8a) (8b)
•
~I
= o/q.
(14)
~is set may be used to represent an arbitrary input vector A, which may be applied to the beam terminals of the BFN.
and are specified. The combination of matrices in square brackets in (5) is defined in terms of a new matrix H, and
624
A= at
M
Lat~,
(I5a)
= ~l· A.
( 15b)
1
composed of hybrid junctions where the network transmission coefficients m to I are are VI*",' The operation of the T matrix may be performed by attaching attenuators (AI/AI) 1/2 to the lth output terminal of the U'' network. Because the vectors E, are orthonormal, the operation of the t matrix also may be realized by a loss less network of hybrid junctions where the network transmission coefficients are (£,) n : A block diagram of the network is shown in Fig. 2. The initial synthesis problem for S reduces to synthesizing two lossless networks, the t network and the U t network. The procedure may be the same for both. The insides of the t and U t networks may be realized in a number of ways-series, parallel, or combinations of hybrid junction types. The Blass network is convenient for large arrays. These networks are described in the literature [6]- [8] from which the required synthesis can be deduced, but not without di fficulty. The synthesis of the {f network in Blass form, including limitations in coupling values, is contained in Appendix II. Note that in the radar case, the efficiency loss must be accepted in order to transmit three independent beams; however, on receive, identical low noise"amplifiers (LNA's) may be employed. If placed on either side of the U t network the noise input to the receivers will be the same as the noise produced by each LNA. The signals will be degraded by the factor Wm m IA t as before. With the LNA's placed between the attenuators and the tf/ network, the signals still are degraded, but the fate of the noise is more complex. Let each LNA produce unit noise power. Since the transmission coefficients through ir network are V/~n' the noise a~ at the mth receive terminal is
The response of the BFN to a unit input at beam terminal m is the specified function Snm = (BK)nm given by (3). The response to an input eigenfunction 'ItI is
2, = S~, = BK~, = K1IBW~/'
(16)
Note that the product of two different Z vectors excited by two different 'It is
ztz = ~tKtBtBK~ q
q
I
2-t
I
-
( 17a)
= K11'ltqHV,
where Q7) follows from the definition of the H matrix (9). Since V is an eigenfunction of H, (14) may be used to express (17a) as follows: -t . -
_
2
_
Zq Z, - K11A,tJ 1q
-
A,
-tJ 1q . Al
(17b)
Thus the output responses to eigenfunction inputs are orthogonal; however ~ there is loss, I 2, I 2 = AI/AI ::5 1 because o < AI ::5 AI' Define new orthogonal functions which are proportional to 2, and orthonormal. £,=
AI ) 1/2 _ _ B W ~, Z ,- 2 Al/( -A I
- t E I
( 18a)
I
.
E-
Q
=
~
U'q'
(18b)
Equation (16) may be written in terms of the orthonormal as follows:
E vectors
1= I .... ,M.
( 19a)
(21a)
In matri x notation these become: ( 19b)
(2Ib)
where U is a matrix whose columns are the orthonormal vectors ~" ~c) is an N x M matrix whose column vectors are the orthonormal vectors £" and T is a diagonal matrix with elements (A,/A)1/2. Note that UtU = I or U t = U-I; consequently, UU t = I as well, and both the rows and columns of U are orthonormal (U is unitary). Equation (19b) may be solved for S by postmultiplying each side by ir
suir = S = 6TU t .
Thus, received noise is always lowered in any channel by having the LNA's between the attenuators and the g network instead of at the receivers. It is not possible to say in general that the useful performance index, gain/noise or l/(Ala~), will recover from the loss produced by the attenuators; however it is easily calculated in any particular case using (21a).
(20a)
When beam terminal q is excited with unit amplitude, ( A') m = 0m q the first operation in (20a) produces the result i
(utA), =
M
L
m=l
(Ut)'mAm = (Ut),q = (~t)q = ~/~.
1= 1
n=I,···,N/2
I
v,~V,s = tJ q S '
q,s= I, ... ,M
(20c)
Bn2 = Dne- < n-
then output vectors of the U t operation due to two different beam terminal excitations are orthogonal. Thus the operation of the u' matrix can be performed by a lossless network 625
r r
(Neven)
e; = Dne
Since the rows of V are orthonormal or, M
EXAMPLES
The first example requires that two similar beams be formed with the same symmetrical amplitude distribution D n» but different linear phase progressions.
Dn=DN - n+ l ,
(20b)
L
IV.
~ D~
2
(22a)
2
(22b)
Neither beam is preferred; therefore, W = I, H = C, and the 2 x 2 correlation matrix has only one distinct off-diago-
the difference arm is
APERTURE
n
2
A,, ) 1/2 = tan a /2. ( A~
N
(26)
From (19b) the E vectors are
LOSSLESS ~ NEnNORK
-
(£1)
Bn l
n
B n2
(27a)
2 cos o /2
-
Bn l
(E,, ) = -
+
= ---
Bn 2
-
2 sin o /2
n
.
(27b)
Thus the iB network is a monopulse-type fee that may be realized in a number of ways such as: a dual series Lopez feed [9], a Blass feed, or a parallel feed. The form of the entire network is shown if Fig. 3 for the parallel feed 4' network. Those simple results may have been obvious from symmetry; however, it is reassuring that the general sy nthesis procedure produces the same networks. Another example is the case where the aperture distributions are cosine amplitude with the characteristic Butler matrix phase gradients: CI
LOSSLESS Ut NETWORK
M
m
BEAMTERMINALS
Fig. 2.
Diagram of the ideal network.
nal element C 12 .
el 2
=
L D,;[ eN
2 ) ( n - ( N + 1/2»4>0]
I
N /2
~
/lVL
s.; =
D;,
(23)
1) ]/ ~ D;,
N + - 2 - rPo
N /2
= cos a .
where Ii = (N + 1)/2, Iii = (M + 1)/2 and B is normalized to unity square magnitude. This case was treated in [4] for large Nand M. According to that analysis the efficiency for W = I should be the ratio of the average to the peak value of the amplitude distribution squared.
The homogeneous equations (II) for A and 'It may be written:
I K ,• I z =
1 N N ~
I e.; I ~ / I s.; I ~a,
l/[NI Bnml~l.a]
I - A ( cosO' The determinant must vanish; consequently A = 1 ± cos (J . Assume cos (J > 0 0'
A = 2 cos? I 2'
Az
]ej(n-ii)(m-m)(~"'''NI:
n) :
n = 1,··', N: m = 1,···. A1 (28)
I
D; cos [2( n -
~ cos[ (n -
a
= 2 sirr' -
2'
-t
(I/2' /2I)
(24a)
-t_
(I/2' /2 - I).
(24b)
if l =
v2 -
If cos (J < 0, the roles of AI and maximum possible efficiency is
Az
1/2.
(29)
This is confirmed by the present analysis as shown below. By writing B in terms of two orthogonal exponential functions
Bn m
e''" - fi)(m -
m+(1 /2»211"/ /v
= -------- +
Clm elm
With W = I, H
=
1,
= 1/2,
Clm
(25)
e)(n - n)(m - m - (1/2»271" /.Y
---~=----
It is easy to verify that B is properly normalized, and the correlation matrix elements are
are interchanged. The
- - - - < 1. I + cosa
=
= C,
= 0,
m
m
=I
=I
± 1
all other m.
(30a)
(30b) (30 e)
(11) are
(~)m+l + (~)m-l - 2(~)m(A - 1) = 0,
Since ex~tation of beam terminal 1 produces equal in phase outputs 'IrI' and excitation of terminal 2 produces equal out of phase outputs ~2' then the U t network is simply a magic T. The transmission coefficient of the attenuator attached to
626
m
= 1,2,···,M
(31a) (31b)
n=1
N/2
N/2+1
beamforming networks consist of two lossless networks joined by a bank of attenuators determined by the eigenvalues. The lossless network closest to the beam terminals is determined by the eigenvectors; whereas the other lossless network closest to the aperture is determined by the eigenvectors and the desired distributions. Both lossless networks alone produce orthogonal outputs; therefore they can be realized by interconnecting hybrid couplers. One standard form is the Blass network, which may be used for both lossless networks. Several examples show that the networks are relatively simple and results obtained earlier by other means are verified here.
N
ApPENDIX
I
EFFICIENCY CHANGE DUE TO THE DIAGONAL WEIGHT
W
MATRIX
=
When W == I, then H
C and
C'l' TYPICAL MAGIC TEE BEAM 1
SUM
Fig. 3. A
DIFFERENCE
BEAM 2
When W *- I, H
_it
-
+
Monopulse feed.
M
I
+
I
1
'
m,/= 1.···,M
satisfies the end conditions (31b) and is normalized square magnitude. It also satisfies (31a) provided A == AI
:=
.,
2 cos-
7rl
2M+ 2
to
(32)
2
1
==-
AI
2
-
A',
(~,t Wt)C( W~') == ---~~-- t
IW
(37)
it ' 12
2::1 Wnn(~')nI2
= h
lit'l:!
I
I:1(it')nI
2
= A -\W21~'12 nn n J
I: I i'~
1
2
or
-
2M+ 2
which is in agreement with (29) when M is large. The simplest network which achieves the efficiency of 1/2 is obtained by placing attenuation in the aperture of a Butler matrix [4]; therefore, the network derived by the present approach is omitted. This example does show that the present approach produces the same results as those derived by an independent method; however, the optimum network is not unique. V.
$
~ hll~'twI2
(33)
(34)
2C05 -
(36)
The first term on the right is the Rayleigh quotient applied to (35), and it is less than AI.
unity
Consequently, the efficiency of each of the beams is
I K II I
= Xl~'.
Regardless of W, the largest eigenvalue is greater than unity, and from the Rayleigh quotient applied to (36) A', satisfies the following inequalities:
DIFFERENCE
(-2 ) 1/2 sin (m / ). -- -7r M
(35)
WtCWand wtCW~1
solution of the form
( ')m -
=
= At'!'.
SUMMARY AND CONCLUSION
The maximum efficiency of a network that forms multiple beams is determined entirely by the beam correlation matrix. Innocuous appearing beam clusters can lead to surprisingly poor efficiencies. A class of optimum networks which achieve the Stein maximum efficiency limit is described based on the eigenvectors and eigenvalues of the correlation matrix. The
(38) The application of a weighting function always improves the efficiency of the first or preferred beam. ApPENDIX
II
RECURSIVE DESIGN OF BLASS MATRICES
The Blass matrix can be analyzed in terms of directional cross couplers and interconnecting transmission lines. The cross coupler has the scattering properties shown in Fig. 4. There may be an upper limit a < a m x < 1f /2 due to practical bandwidth or dimensional limitations of the coupler types chosen. This possibility is included in the calculations; however, efficiency will degrade beyond that due to beam correlation. The couplers usually are equally spaced, separation L, and are joined by transmission lines with wavenumber kg. A typical network has N antenna elements and M input ports (N > M) such as the network described in Section III. The network is shown in Fig. 5. The U network may be similar, but N = M. A typical section of network with 627
0
t-
j SINo(
Q
CD
----
~
COSO(
1
G)
s=
COUPLER an
Fig. 6.
0
0
COSa
-JSINa
0
0
- j SIN a
casu
a
0
casu -JSlN
Fig. 4.
-j SIN
a
cos e
0-
0
rn starting with rn = (Eq) n at the aperture. At each iteration we have z 1 = 0 since only the qth input is excited. Thus from (40) with Zl = 0 ri can be determined then Z2 from (39), and cycle back to (40) to calculate r~, etc. until r~ is calculated. Designate r; to be a new r n associated with next bank and repeat the cycle until the qth bank is reached.? The second application of these equations arises at this point where known are to be generated by the qth bank with r; = O. In this case (39) and (40) reduce to
c
Cross-guide directional coupler.
2
Typical section of Blass network.
'n
N
=
Zn+l
'ne-jt/>"
LOAD
I
z, cos
Qne-jkgL
(41)
= -jzn sin an.
(42)
If z. is chosen real, then from (41) and (42) the unknown phases are
(43)
-Zn- =e -j(n- I)k g L t z,
2
"
I
TYPICAL FIXED PHASE SHIFTER
(44) As in [10], the amplitudes are determined from inspection of Fig. 6 in conjunction with energy conservation or from the amplitudes in (41) and (42). 1
M
z, 1
2
=
I ZN+ I 1
2
+
N
Ln 1 '/1
2
(45)
.
Inserting z, from (42) produces the expression for the coupling parameter ex. n Fig. 5.
Blass network.
various traveling waves is shown in Fig. 6 for the transmit case. A known vector Ii with components (R)n = travels away from the coupler bank, after passing through phase shifters cJ>n. Incident waves zn and r~ are scattered by couplers each of which is characterized by an angular parameter an (a in Fig. 4). Referring to the typical coupler in Fig. and zn by 6, the outputs Zn+1 and 'n are related to inputs the following equations derived with the aid of Fig. 4:
csc
2a
n
= [
I .1 + ~ I r ]/1 r, 1~ esc? ««. ZN+
2
l 1
2
2
(46)
'n
r:
or
I ZN+I 12 =
(40)
max[ I r, 1
2
2 CSC
amx
(47)
n
The following choice minimizes the wasted power
(39) These equations are applied at each bank of couplers in one of two ways. First, assume the C network is to be designed and all the values of c/l n and an for coupler banks 1,2 q - 1 which generate vectors £),. ··, Eq _ 1 have been determined. The qth bank is designed by recursively calculating r~ from
N
L 1',1 2 •
IZN+112~ IrnI2csc2a.mx-
-
~ 1',1
1 2
Z N + 1 I 2: ] .
(48)
The coupling now can be determined from (46), and the input power is determined from (45) with n = 1. Since the Eq are
628
2 Although 4N+ 1 is not needed, it can be calculated and should be zero.. Thus curious result also follows from energy conservation.
nonnalized to unity, the efficiency is efficiency
= I Eq I 2 / I Z I I 2 = 11/ Z I I 2 .
[6] ( 49)
Thus, starting with q = 1 and r n = (E1)n, the entire rff network can be built recursively until q = M. The U network may be constructed similarly if desired. Note from (48) when Ct m x = 7r /2, I z.!V+ I 1 2 = 0 and the networks 1/ and U t will be lossless. Generally, the load losses will require adjustment of the attenuators in Fig. 2. Combine the efficiencies of the U t and {/ networks for the qth interconnecting channel to form an overall efficiency 11' q I 2. Then the attenuators in Fig. 2 are I Tq I 2 as follows:
I T q I 2 I Tq I 2 = 'A q Since
I Tq I 2 <
.
[7] [8] [9] [10]
(constant).
1,
I Tq I 2 = I ~ql 2 / ('f.. q / I i q I 2) max
( 50)
which reduces to 'A q / Al as before when I T q I 2 = 1. The overall efficiency derived in Section II must be reduced by the load losses in both the U t and (.r networks. REFEREI'CES
11] J. L. Allen ... A theoretical limitation on the formation of lossless beams in linear arrays." IRE Trans. Antennas Propagat .. vol. AP-9, p. 350. July 1961. [2] S. Stein. "On cross coupling in multiple-beam antennas." IRE Trans. Antennas Propagat .. vol. AP-IO. p. 548, Sept. 1962. [3] W. B. White. "Pattern limitations in multi-beam antennas." IRE Trans. Antennas Propagat .. vol. AP-I0. p. 430. July 1962. [4J E. C. DuFort .. 'Optimum low side lobe high crossover multiple beam antennas." IEEE Trans. Antennas Propagat .. vol. AP-33. p. 946. Sept. 1985. [5] G. Strang. Linear Algebra and its Applications. 2nd ed. New York: Academic. 1980. ch. 5 and 6.
629
R. C. Hansen, Ed. Microwave Scanning Antennas. New York: Academic. 1966, ch. 3, p. 247. R. C. Johnson and H. Jasik. Eds. Antenna Engineering Handbook. New York: McGraw-Hill. 1984. ch. 20, p. 56. Y. T. Lo and S. W. Lee. Eds. Antenna Handbook. New York: Van Nostrand Reinhold, 1988. ch. 19, p. 10. A. R. Lopez. "Monopulse networks for series feeding in antennas." IEEE Trans. Antennas Propagat., vol. AP-16, p. 436. July 1968. W. R. Jones and E. C. DuFort, "On the design of optimum dual-series feed networks," IEEE Trans. Microwave Theory Tech.. vol. MTT-19. p. 451, May 1971.
Direction Finding in Phased Arrays with a Neural Network Beamformer Hugh L. Southall, Senior Member, IEEE, Jeffrey A. Simmers, and Teresa H. O'Donnell
Abstract-Adaptive neural network processing of phased-array antenna received signals promises to decrease antenna manufacturing and maintenance costs while increasing mission uptime and performance between repair actions. \Ve introduce one such neural network wbich performs aspects of digital beamforming with Imperfectly manufactured, degraded, or failed antenna components. This paper presents measured results achieved with an adaptive radial b&4lis function (ARBF) artificial neural network architecture which learned the single source direction finding (DF) function of an ei&bt-element X-band array having multiple, unknown failures and degradations. We compare the single source DF performance of this ARBF neural network, whose internal weights are computed using a modified gradient descent algorithm, with another radial basis function network, Linnet, whose weights are calculated using 6Dear algebra. Both networks are compared to a traditional DF approach using monopulse.
S
I. INTRODUCTION
TANDARD antenna beamfonning algorithms, such as
monopulse, require calibrated antennas because they de-
pend on nearly identical antenna element performance. These algorithms do not perform well with uncalibrated antennas or unknown degradations. As phased-array antennas become
antenna measurement preprocessing, a radial basis function (RBF) neural network, and output postprocessing. We compare two variations of the RBF neural network, the first an adaptive network, ARBF [1], which uses gradient descent optimization training, and the second, a linear algebra based network, Linnet, which trains using a least mean squared (LMS) error solution. Comparisons between ARBF and Linnet and analysis of an error-weight surface show that the ARBF implementation converges to a near-optimal solution in only a few iterations.
Section III briefly describes a monopulse direction finding
(DF) algorithm whose performance we compare with the networks' performance. This algorithm, which is calibrated to partially compensate for nonideal element behavior and array misalignment, relies on near-identical array elements to form high-quality antenna beams for accurate results. Section IV presents and compares the experimental DF performance of the monopulse algorithm, Linnet, and ARBF at locating single sources in 10 data sets, taken under various signal-to-noise and interference conditions.
II.
larger and more highly integrated into physical structures, this uniformity requirement generates production and maintenance costs which are increasingly prohibitive for many military and civilian applications. The requirement for nearly identical elements results from a lack of adaptive beam forming algorithms capable of managing the complexities introduced by nonidentical elements with unknown behaviors. Traditional techniques synthesize aperture behavior as a mathematical combination of well-matched element and receiver channel responses. Neural network based beamfonning (neural beamfonning), in contrast, attempts to approximate overall aperture behavior from a finite number of observations of that behavior under varying circumstances. If we assume that the mapping between the received signals and the antenna's behavior is a continuous function, it is possible to model it with an artificial neural network trained at discrete samples along the function. After this learning process, the network can predict antenna behavior at points between the training points by generalizing. In Section II we present the architecture of a neural beamformer designed for single source direction finding and discuss Manuscript received February 1~, 1994; revised March 13, 1995. This work. was supported in part by the USAF Contract #F 19628-92-C-0 177. H. Southall and J. Simmers are with the USAF Rome Laboratory Electromagnetics and Reliability Directorate, Hanscom AFB, MA 01731 USA. T. O'Donnell is with AReON Corporation, Waltham. MA 02J54 USA. IEEE Log Number 9415634.
NEURAL BEAMFORMER DIRECTION FINDING
This section briefly describes the DF neural beamfonner components and structure, introduces and compares the adaptive ARBF and linear algebra based Linnet RBF networks, and explains the rapid convergence and near-optimal performance of the adaptive network.
A. Neural Beamformer Architecture The neural beamfonner architecture consists of antenna measurement input preprocessing, an artificial neural network, and output postprocessing. This section briefly summarizes the purpose and interaction of these functional clements. Network preprocessing exploits antenna expertise to simplify and enhance neural network inputs. It removes redundant or irrelevant information, eliminates artificial discontinuities in the input function space, and reduces problem inputs to a small set of relevant information, Although neural networks can learn to ignore irrelevant inputs, and discontinuities can be trained "across," if their locations are known and boundary points are available, these techniques usually create larger (and slower) networks than ones which utilize intelligent preprocessing. In the problem of single source DF. the amplitude of the received signal is not a strong indicator of the angle of arrival. The absolute phase of the received signal at each element also contains nonessential information, There is, however, a strong relationship between relative element phases and angle
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 43, No. 12, pp. 1369-1374, December 1995.
630
:9gjff' I
Anten",
InpolNodos
Gaussian
Processing Nodes Weights
Summation Nodes
Fig. I. The neural beamformer architecture consists of antenna measurement input preprocessing. an artificial neural network, and output postprocessing which gives an estimate. iI. of the angle of arrival, 8
of arrival. Therefore, we preprocess the measured phases at each element to determine the phase differences between consecutive array elements. These phase differences, however, contain artificial discontinuities caused by phase transitions (or branch cuts) in received phase measurements from -180 degrees to + 180 degrees. Discontinuities make it difficult for the network to learn the mapping from a small discrete set of training points, especially since the branch cuts are quasirandom (dependent on arbitrary receiver phase references.) To eliminate the branch cuts, we use the sine and cosine of the phase differences as final processed inputs. These functions have the added benefit of bounding the inputs between - 1 and 1 which is not essential for the RBF network but may be useful for other network architectures. It is important to note that we preprocess raw antenna measurements; there is no calibration or traditional antenna processing to correct for element mismatches. We chose RBF neural networks for this antenna application for a number of reasons. As mentioned earlier, the relationship between source angle and antenna measurements is generally a continuous function with small changes in angle yielding small changes in received measurements. After preprocessing, this is true for fully functional antennas and many forms of degraded ones. (Severely degraded antennas or those with intermittent errors such as phase shifter bit failures. however. may exhibit discontinuous behavior.) Therefore, we chose a neural network architecture which hac; proven successful at approximating continuous functions from a small set of samples and can be trained across discontinuities. Real-time processing requirements, independent of antenna size. also generated two guidelines for the network architecture. First, the network should have a constant processing delay regardless of the number of inputs. Second, the network should have a minimum number of layers (computations that must be performed sequentially). Thus as the number of inputs increase (larger antenna), the size of the network layers should grow at the same rate. with no additional layers required. The
RBF network architecture reportedly satisfies all of these constraints. The mathematical basis for these networks sterns from the fitting (approximation) and regression capabilities of RBF's [2], [3], combined with a feed-forward neural network architecture whose single "hidden " layer does not grow faster than the number of inputs [4], [5]. A three-layer REF network can also theoretically model any continuous function [2). The architecture of a three-layer RBF network. shown in Fig. 1, consists of an input layer, a hidden layer of Gaussian RBF's, and an output layer of summation nodes. The input nodes receive the preprocessed antenna data and broadcast the input vectors to each hidden layer node. Each input vector. x, is an element of the n-dimensional input space, ...t . For our experimental eight-element antenna, n == 14. Each input vector contains the seven cosines and seven sines of the phase differences between the elements . For these n-component input vectors, x== (Xl, %2, ' .. , X n ) , preptocessing ensures such that X/c E [-1,11, where k == 1,2 , · .. , n . The hidden layer maps X into a space ~ which consists of q-component Gaussian RBF vectors, rP. where q is the number of hidden layer nodes. The components of rP are m (I)
where rPi is the output of the ith hidden layer node for i == 1,2, · .. , q. rPi is calculated from the input vector, x, the Gaussian center. 'Tni , and the spread parameter, O'i . Initially, a, was a trainable parameter which varied for each Gaussian node. We achieved comparable results, however. by using the same value of the spread parameter for all hidden layer nodes. so now a, == (J . Each output node computes a weighted sum of the outputs generated by the hidden layer nodes q
YJ
== L
(2)
;= 1
where the w'J values are the output weights. Thus the output layer maps ~ into Y, the space of all possible angular directions to the source, where the output vector y= (Yl, Y2, ... , Yr) E Y for T output nodes. We postprocess the energy in the T output nodes to estimate the source angle of arrival. For single source DF we could train a single output node to emit a value proportional to the source angle. Since our goals include multiple source detection and DF, however, we chose multiple output nodes representing bins of energy in discrete angular regions of space. For an eight-element array, we use r == 13 output bins, centered at 10degree intervals from -60 degrees to +60 degrees, inclusive. We train the output nodes to emit values between zero and one, inclusive. which represent the presence and strength of a source within each angular bin. A bin output of "1" indicates a source exactly on the bin location and "0" represents no source. Values between zero and one on consecutive output nodes represent a source located between the bin angles represented by those output nodes.
631
Our postprocessing technique is used for both the single source DF networks presented here and for multiple source OF networks currently under development. We locate energy concentrations among the output nodes and consider each concentration to represent a source location. For each concentration, we determine the consecutive pair of nodes with the largest energy and interpolate between the bin angles to determine the angle of the source represented by that concentration. Although this technique performs well for single sources and multiple sources which are farther apart than two bin-widths, alternate output representations are necessary for multiple-source super-resolution. B. Network Training
In the previous section we presented the forward computational architecture of the RBF network. This computation will only produce correct results if the network has been adequately trained with pairs of inputs and their corresponding outputs. We investigated two training techniques for the RBF networks: a standard adaptive RBF training method and a linear algebra technique. Adaptive RBF training adds Gaussian RBF's to the hidden layer as needed. Initially empty, the hidden layer grows as training points are presented to the network. At each training point, either I) a new Gaussian function is added with its center on the training point and its initial network weights W1.] chosen to produce the correct network response for that point or 2) existing "close" Gaussian functions have their centers moved and weights adjusted to incorporate the new training data. Network weights are adjusted using a modified gradient descent algorithm known as "backpropagation." Additional details regarding the network architecture and training can be found in two articles by Lee [61. [7]. We refer to this implementation of the neural beamfonner as ARBF. Backpropagation of network error, however, does not ensure global optimization and introduces several vague network aspects. Often, the network designer must choose parameters for which the literature contains only general guidelines. Two important parameters include the step size and the stopping criteria. The step size dictates the convergence speed of the weights and whether they will converge or oscillate. One popular approach, which decreases the step size slowly, requires choice of initial step size, decreasing function, and stopping criteria. Even if these parameters are correctly chosen for the application, backpropagation may still stop at local minima. Unlike ARBF which adds basis functions as needed, the Linnet network places Gaussian RBF's at all training points and solves for the network weights using linear algebra [8], r9]. Linnet finds globally optimal weight values, in the LMS error sense. The weights are only optimal with respect to the training data used to construct the corresponding matrices, not necessarily for any other data. Consider a specific set of N input vectors from the input space X, labeled Xl , where l = 1,2"", N. For our data sets which measure a source at l-degree intervals from - 60 degrees to +60 degrees, N = 121. Each component of the
n-dimensional input vector is XLk, where k == 1, 2, ... ,n. Similarly, consider a specific set of N Gaussian RBF vectors
cIl1Vxq = [¢h) Q>li = e-
2:::-1 (:elk -m'l.k)2 '2(72
(3)
To train the network, we select t rows of ep to form a 13 x 13 (t = 13, q = 13 ) training Gaussian function matrix, if>t, which is used to solve for the weights. Output vector 111 has components Ylj for l == 1, ... , N and j = 1, ... , r, where r is the number of output nodes. The N rows of Y are the output vectors corresponding to the N input vectors. Each element of the output matrix, Y, is the weighted sum of Gaussian function values YNxr
= [Yli] =
[t¢liW'i]
Y=~W
~
Note that since we use the same number of output node "angle bins," r, as training angles, t, and Gaussian nodes, q, then r = t. == q == 13. For training, we know the desired outputs at the t training angles (-60 degrees, -50 degrees, "', +50 degrees, +60 degrees), corresponding to the rows of Y with I 1,11, ... , 111, 121 . We use these rows to form the desired output matrix Yd. Using 4lt and Yd. we can solve for the q x t weight matrix W from
=
(5)
where Yd is a txt identity matrix. Since we chose the number of training vectors, t, equal to the number of Gaussians, q, both Yd and CPt have the same dimension. Equation (5) implies that we have a determined linear system, i.e., q equations in q unknowns. If, however, the number of Gaussian nodes were reduced so that t > q , then the linear system is over-determined, and LMS can be used to solve for the weights. In either case, the LMS solution is the most general, since the pseudo-inverse of the LMS solution becomes a matrix inverse for q == t. All calculations in this paper assume q t. The steps for computing the optimal Ware:
632
=
1) Collect input data, i.e., the N known source angles.
X,k,
for N measurements at
I
TAB LE I
SUMMARY OF THE D ATA S ETS USED F OR DF ALGORITHMS
0.4
TuT07 TEsTG2 TuT87 TEsTsc nATAl nATA2 TEST80 TEST95 TEST60 TSST59
g0 35 1
03l 1 02 1 1 0' 1
1 w
0 25
i J O. 15
0.05J
0.1.
Data Sets
I SIN I Location I pen Bay -35 B < 1 B o en Ba)' _3lI B pen Bay -35 IB o en Bav -1 IB -cnamber >30 !B Chamber -TI B Open Bay -25dB Open Bay -5dB Open Bay -35dB Open Bay
Comment.
Absorber ill Elemellt 6 Metal Ground prane Screen Coffee Cana
Repeat of TEST97
1.5 0.5 Norma lized W(12.13)
Norm. izO
Fig. 2.
Name
Linnet performance versus weight changes .
2) Use a subset, q in length, of the N input vectors to Iorm a q x q matrix
(6) C. Rapid Convergence
One reason we use an RBF network is that they reportedly require orders-of-magnitude less iterations to converge (solving for weights) than networks based on the sigmoid neuron activation function [ II ]. Our results support this claim; ARBF achieved near-optimum performan ce after only three weight update iterations . The error surface shown in Fig. 2, generated by Linnet, explain s the reason for this rapid convergence. We arbitrarily picked two weights from the optimal Linnet weight matrix, tV. perturbed them from optimal, and calculated the resulting network performan ce to obtain the three-dime nsional error surtace . In Fig. 2. W (3. 3) and W ( I2, 13) represent normalized values for weights "connecting" the output of the third Gaussian with the third output vector component and the output of the twelfth Gaussian with the thirteenth output vector component, respectively. Normalized values fur each weight arc indicated along the axes. with optimal weights occurring at 1, 1. This plot is one of many we investigated; every one had the same steep sides and flat bottom. This held true eve n for data including severe ncar-field scattering which resulted in poor Dr; performance with large rID S angle errors (sec Da ta2 in Tables I and H). Thus if initial weights lie on a steep side or a fiat bottom, the modified gradient descent search should converge quickly to near-optimal performa nce in only a few iterations for the DF problem .
m.
M ONOPULSE DIRECTION FIND ING
Monopulse is a classic array antenna applicat ion for OF which uses high quality sum ami difference beams [12]. We selected aperture distributions developed by Taylor and Bayliss to synthesize sum and difference beams, respectively, with theoretical -30 dB near-in sidclobes, Of course, beam
quality depends upon element match and the received signalto-noise ratio, SIN [12]. We calibrated the array (element amplitude and phase mismatch compensation) by normalizing the ideal Taylor and Bayliss weights with the measured complex signal values obtained at each element for a broadside plane wave, Exact element mismatch compensation occurs at broadside; however , compen sation degrad es for source angles off-broadside. We measured received ampl itude and phase at each of the eight elements of the array for 121 source azimuth angles between -60 degrees and + 60 degrees at l- degree steps. The measurements are multipli ed by monopulse weights to form beams which supply inputs to the source angle estimation algorithm. Since we know the true source location, we can determine the rms angle error (calculated over all 121 source angles) from this angle estimate. We use rms angle error to compare performance for both the monopulse and neural network DF algorithm s.
IV. EXPERIMENTAL RE SULTS In this section we describ e our experiment and present monopulse and neural beamfonn er OF results using measured data from a variety of antenna and environment scenarios. We used a linear, eight-element array to gather data for monopulse and neural network testing. Element spacing was one-half wavelength at X-band (11.811 GHz), each "element" heing a combined column of eight horizontally polarized, open -ended waveguides (for increased eleva tion directivity ). Data was taken e very degree for azimuth angles from -60 degrees to +60 degrees by stepping a single source on a fixed radius in the far field of the array. At every angle, we measured and recorded element ampli tude and phase by connecting each element. in turn. to a phase/amplitude receiver through a coaxial RF switching matri x. We collected data for the ten scenarios summarized in Table 1. The row labels are data set names. Different sets represent changes in either received signal-to-noise ratios, S / IV , or the physical environment. The location "Chamber" is an anechoic chamber completely encased in microwave absorber, while "Open Bay" designa tes an open bay style room with the walls partially covered with absorber and an absorber "wall" built into the plane of the array. For Test87, the microwave absorber filled one complete column of open-ended waveguides (one "element"), The fine mesh metal screen used with TestSC was placed coplanar with
633
TABLE ill
Monopul.ePerformance
COMPARISON OF ARllF AND LINNET
Data Set TuTll7 Tu~2
1:ST87
'hsTSC
DATAl D4TA2 TEsT80 TuTil5 TEsTGO UST59
,
0.800 0.860 0.800 0.760
D.750
1.000 0.8&0 0.750 l.3oo 0.800
Belt, lOr LIHIfBT
RMs
AUF
lUiS Error RMS Error 0.41 2.01 l.4Cl 1.67 l.43 14.M u.47 1.37 U3 0.47
LOll
2.00 1.42 1.72 1.34 15.24 0.Cl7 1.31 2.37 0.88
ERROR (DEGREES)
Belt , for A1UIF
LI NNIn
LIlCNET
t1
0.540 0.835 0.760 0.705 0.750 0.810 0.700 0.735 1.000 0.665
RMS Error
UO 2.01 1.4G I.Cl7 1.43 15045 0.53 1.37 1.118 0.62
AUF
RMSEnor
0.63
l.ll7
l.41 UV 1.34 16. O. I. I. O. 6
-4~
, I -~:----:::-----==---:-----::':c----'-_---.J -40 -20 0 20 40 60
reasonable value of 2.5 degrees . The large error for Data2 is due to the severe near-field scattering environment which is not accounted for in the calibration.
Source Direction, Degrees
Fig. 3.
Monopulse OF performance for eight-element array.
B. Neural Beamformer Results
TABLE II U"CAUBRATED VERSUS CALIBRATED MONOPULSE PERFoRMA.'1CE FOR EIGHT-ELEMENT ARRAY
MODOp'" P.rorm&D<e KlIS AIlAIe Error DeD_ U_llr_ l,;aI1ll_ TEsT 7 l.37 0.73 E5~ 1.T.l \.J2 o,g7 1'UT87 1.13 TaTIc Ull 4030 OJllj DATAl 1.27 DATA2 U5 8.05 1.05 TaT80 l.ll0 TuTll5 l.37 0.71 lUll TEsTSO V.64 TuT5V 0.76 1.53
Duases
and surrounding the face of the array out to many wavelengths. A pair of two-pound metal coffee cans were stacked on top of one another and centered approximately four wavelengths in front of the array while collecting Data2 . Test59 was taken one month after Test97 under the same conditions.
A. Monopulse Results Fig. 3 shows an example of the performance of the monopulse algorithm. The data set used to generate this figure had an S/ N of approximately 35 dB and was collected in a low external reflection environment. The angle error is shown for all 121 angles . Table II shows the rms error metric for the different data sets. Each simulation uses a single data set, i.e., eight amplitudes and eight phases at 121 azimuth angles . Note that calibration improves performance for all data sets except Test87 and Data I . The calibration data were obtained from data set Test97 and used to calibrate the monopulse algorithm for all 10 data sets reported in Table II. Ideally , this removes element mismatch in the array . For Test87, however, one element was filled with microwave absorber, a condition not accounted for in the calibration data from Test97 . If we calibrate the Test87 data using the broadside wave data in Test87, the rms angle error becomes 0.59 degrees instead of the 1.13 degrees from Table n. A similar explanation also holds for Data 1. For Test60, the large rms angle error is due to large errors near ±60 degrees where the signal drops below the noise floor. Ignoring these outliers, the error drops to a more
Neural network training is similar to monopulse calibration since it uses known external signals to learn something about the array . While monopulse calibration attempts to compensate for antenna element mismatch, however, neural beamformer supervised learning, using pairs of function inputs and desired outputs, attempts to generalize the entire DF function . The ARBF and Linnet beamformers both approximate the function by locating Gaussian node centers and determining the weight matrix , W . Solving for W using the linear algebra approach theoretically yields superior performance (for the same training data) . We have discovered, empirically, that this is the case for training points which are evenly distributed across the input space . For poorly chosen training points (which can happen if the antenna or environment conditions degrade), however, ARBF can adapt by moving Gaussian centers to improve performance, while Linnet node s remain fixed. Therefore, under conditions which are less than "ideal" (i.e., Test87 and Data I), ARBF can outperform Linnet. Our empirical results verify this, if other network parameters (such as 0") are the same for both nets. Table III summarizes the performance of both networks for the 10 data sets used in Table II. Each network was trained with input and output vector samples from a single data set at lO-degree intervals and then tested across that entire data set at l-degree intervals . The columns under "Best 0" for Linnet" show the rms error for Linnet and ARBF with the networks' 0" set to the value which optimizes Linnet's performance. Similarly, the columns under "Best 0" for ARBP" report the performance of both networks at the 0" value which optimizes ARBF's performance. The (1 optimization is re-accomplished for each data set. The performance of Linnet and ARBF is very similar, especially for 0" values optimized for ARBF. The neural beamformer performance in Table III can be compared with the monopulse performance in Table II. In most cases, neural beamrormer pertorrnance is comparable [0 that of monopulse. For Data2, data with severe near-field scattering, the monopulse algorithm rms error is half that of the neural net algorithms . For TestSC , however, which also includes nearfield scattering since the large ground screen was not really flat, the neural net algorithms outperformed monopulse by almost
634
a factor of three. The neural net algorithms also significantly outperformed (by more than a factor of two) monopulse at relatively low SIN in a high reflection environment (Test80). V.
CONCLUSION
We presented two neural beamformers, ARBF and Linnet, which use modified gradient descent and linear algebra, respectively, to learn the single source DF function. We verified our hypothesis that ARBF finds a minimum in the error-weight surface after only a few training iterations. Examination of the error-weight surface explains this rapid convergence. We presented a comparison of the DF performance of the two neural beamformers and contrasted their performance with monopulse. Except for data with severe near-field scattering conditions, both neural beamformers performed very well. At very high SIN, the rms angle error for the neural beamformer was almost half that of monopulse for our experimental eightelement antenna. REFERENCES
[1] T. O'Donnell, J. Simmers, and D. J. Jacavanco, "Neural beamforming for phased array antennas," in Proc. 1992 Antenna Appl. Symp., Griffiss AFB, NY, Sep., 1992, USAF HQ Rome Laboratory. [2] M. J. D. Powell, "Radial basis functions for multivariable interpolation: A review," in Algorithms for Approximation, J. C. Mason and M. G. Cox, Eds. Oxford: Clarendon Press, 1987, pp. 143-167. [3] D. F. Specht, "A general regression neural network," IEEE Trans. Neural Networks, vol. 2, no. 6, pp. 568-576, Nov. 1991. [4] D. S. Broomhead and D. Lowe, "Multivariable function interpolation and adaptive networks," Complex Syst., vol. 2, pp. 321-355,1988. [5] T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE, vol. 78, no. 9, pp. 1481-1496, Sep. 1990. [6] S. Lee, "Supervised Learning with Gaussian Potentials," in Neural Networks for Signal Processing, B. Kosko, Ed. Englewood Cliffs. NJ: Prentice Hall, 1992, pp. 189-227. [7] S. Lee and R. M. Kil, "Bidirectional continuous associator based on gaussian potential function network," in Proc.1989 IEEE 11CNN Volume I, San Diego, CA, 1989, pp. 45-53. [8] A. C. Tsoi, "Multilayer perceptron trained using radial basis functions," Electron. Lett., vol. 25, no. 19, pp. 1296-1297, Sep. 1989. [9] J. Simmers, H. L. Southall, and T. O'Donnell, "Advances in neural beamforming," in Proc. 1993 Antenna Appl. Symp., Griffiss AFB, NY, Sep., 1993, USAF HQ Rome Laboratory. [10] B. D. Carlson and D. Willner, "Antenna pattern synthesis using weighted least squares," lEE Proc. - H, vol. 139, no. 1, pp. 11-16, Feb. 1992. [11] B. Kosko, Ed., Neural Networks for Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1992. [12] S. M. Sherman, Monopulse Principles and Techniques. Norwood, MA: Artech, 1984.
635
Application of Orthogonal Codes to the Calibration of Active Phased Array Antennas for Communication Satellites Seth D. Silverstein, Senior Member, IEEE
Abstract- This work describes two algorithms designed for remote calibration of an lYe-element active phased-array antenna. These algorithms involve transmission of N ~ N P. time multiplexed orthogonal encoded signals. The received signals are coherently detected, accumulated in vector forms, and decoded with the inverse of the orthogonal encoding matrix. The unitary transform encoding (UTE) algorithm is most suited for digital beamforming as it requires additional encoding hardware for an analog implementation. The control circuit encoding (CCE) algorithm is ideally suited for analog beamformers as it requires no additional encoding hardware. The CCE method encodes phased-array elemental signals using a Hadamard matrix to control the switching of intrinsic phase shifter delay circuits. The UTE and CCE algorithms can reduce the average measurement integration times for the complete set of calibration parameters by rv N, relative to the corresponding values for single-element calibration procedures. This is significant for satellite systems as calibration must be performed in a short enough time window that the process can be treated as being stationary. Proofs are given that the orthogonal codes satisfy the mathematical lower bounds for the asymptotic forms of calibration parameter estimation variances.
T
I. INTRODUCTION
HIS work introduces two algorithms that use time multiplexed orthogonal encoded signals to remotely calibrate transmitting and/or receiving active phased-array systems. Active phased-array systems belong to the class of smart antenna systems that possess the ability to perform programmable changes of the amplitude and phase of the elemental phased array signals in order to accommodate different beamforming scenarios. The first algorithm is referred to as the unitary transform encoding (UTE) method, whereas the second algorithm is called the control circuit encoding (CCE) method. The applications featured in this work relate to future generation geostationary (GEO) communication satellite systems that will deploy analog beamforming active phased-array transmitting antennas rather than present-day state-of-the-art reflector antennas. Extensions of the calibration algorithms to receiving and digital beamforming systems are straightforward. The UTE method is most effective for digital transmitter beamforming systems where the complex elemental phases required for the beamforming are introduced digitally at baseband with a high degree of accuracy. The digital signals are subsequently converted to analog signals and modulated up Manuscript received October 27, 1995; revised August 21, 1996. The author is with GE Corporate Research and Development, Schenectady, NY 12301 USA. Publisher Item Identifier S 1053-587X(97)00513-8.
to the desired RF carrier frequencies. The UTE method can be also used to calibrate an analog beamforming system. For analog applications, the UTE method requires some additional remote fail safe hardware above and beyond the beamforming phase shifters to provide accurate signal encoding. If the additional encoding hardware fails or is in error (including quantization errors), the calibration system can cease to function effectively. Alternatively, the CCE method is ideally suited and is the algorithm of choice for remotely calibrating an analog system because the encoding is exact (not subjected to quantization errors) and encoding hardware failures are not at issue, as the CCE method requires no additional encoding hardware. In an analog phased-array transmitting antenna with a p bit beamformer, there are p independent delay circuits that can be switched into the elemental electrical path to ideally provide 2P quantized phase levels corresponding to phase shifts of 21fm/2 P for m == 0, 1, ... , 2P - 1. We note that the process of representing delays as constant phase shifts over the relevant modulation bandwidth implicitly assumes a narrowband system. The gain and phase (complex gain) of the coherent elemental signals at the receiver is a function of the complex gain of the phase shifter delay circuits, the power amplifier, and the transmission path to the receiver. For a phased array with N e elements, a calibration system must be capable of accurately measuring the complex gains associated with all of the NeP delay circuits as well as both the relative gains and phases associated with the straight-through-path to the receiver with no delay circuits switched in. The straightthrough-path complex gains can also be referred to as the insertion complex gain as a natural extension of the commonly used term insertion phase. In order to extract the complex gain calibration data, one must perform coherent detection of the calibration signal at the remote receiver with one or more reference signals that are phased locked to the input calibration tone at the transmitter. For satellite systems, the coherent detection system architecture must be specifically designed to compensate for Doppler phase shifts due to satellite motion and independent phase noise effects due to nonsynchronized clocks on the satellite and the ground receiving station [1]. In order to obtain meaningful estimates of the parameters of interest, the calibration process must be performed in a time window that is short enough that the estimated parameters can be treated as being stationary (quasi-stationary). The relevant quasi-stationary time windows for GEO systems are
Reprinted from IEEE Transactions on Signal Processing, Vol. 45, No.1, pp. 206-218, January 1997.
636
determined by a number of temporally variable effects such as fluctuations of the array pointing angles (attitude), changes in the received signals due to variable atmospheric conditions, and changes in the received phase of the emitted elemental signals due to thermally induced effects such as phase offsets of the elemental semiconductor components and the physical warpage of the array panel. The thermal changes are caused primarily by diurnal variations of the solar irradiance on the phased array panel. The published literature on remote calibration procedures that can conceivably measure the relative ground truth of the full set of elemental complex gains, i.e., the Ne(p + 1) independent variables associated with the N e p phase shifters and the N e complete straight-through-paths for an N, element array, is somewhat sparse [2]. The suggested methods are predominantly variations on the theme of coherent detection of a single element (SE) under test while all the other elements are turned off. The SE methods are conceptually simple, but they unfortunately have some fundamental problems that make their usefulness questionable for the satellite mission calibration requirements. The first problem deals with the necessity of implementing a multipole microwave switch at the front end of the elemental electrical path for the purpose of directing the calibration signals to a single element at a time. This switch must be implemented in a manner that the relative complex gains of the straight-through paths for all the elements are not altered by the switching process. In principle, one could isolate a single element by turning off the power amplifiers in all the elements other than the one to be measured. This latter procedure is undesirable as it would interrupt communication service during the measurement time. The second problem arises from the fact that the SE procedures are relatively low effective estimation SNR (ESNR)I calibration procedures, which translates into relatively long measurement integration times. At practicable phased-array satellite link budget power levels, the integration times required to extract the calibration data for the SE procedures could be too long to satisfy the quasi-stationarity time window criteria described above. In principle, one could increase the SNR of the SE process by increasing the power of the calibration signals transmitted from each element. For efficiency purposes, systems are usually designed to operate at near maximum elemental power so that arbitrary increases in power levels are generally not feasible. Constraints on the maximum power are dictated, in part, by the I inear capacity of the elemental beamformer devices and the intermodulation levels of the power amplifiers. The UTE and CCE algorithms will significantly enhance the ESNR, thereby reducing the required average measurement integration times for the complete set of calibration parameters by rvNe relative to the corresponding values for single-element calibration procedures. We note that in systems subjected to high levels of multi path signals caused by secondary scattering, the straight-throughpath calibration estimates will suffer degradation. Although this is potentially an important issue for calibrating terrestrial1 The ESNR is defined as the ratio of the square of the mean to the variance of the parameter estimate.
based transmitter/receiver systems, it is not a problem for satellite calibration. II. MATHEMATICAL BASIS OF THE CALIBRATION ALGORITHMS We first develop the mathematical nomenclature. Let {s( n, i)} represent the individual coherent straight-throughpath signals that have been received at a single earth station receiver, demodulated, coherently detected, and sampled at times tie The algorithmic operations always involve coherent signals that are sampled at the same time point in their respective coherent bursts. This allows us to simplify the nomenclature by suppressing the sampling index i with the understanding that all operations refer to the same sampled time points. Accordingly, we use the notation
S == [s(I), s(2), ... , s(N)]T
( 1)
to represent the received, demodulated, sampled signal vector at the receiver when all the elements are simultaneously transmitting their straight-through-path signals. The calibration process is based on the following beamformer model. Calibration signal powers are assumed to be in the linear regime with respect to the beam former such that the effect of switching in the /Lth delay circuit of the nth array element with complex gain dJ..L (n) imposes a complex gain
(2) on the input elemental signal x( n). The effect of switching in multiple circuits generates the product of the complex gains
x(n)--td!"(n)
H dy(n) r
d!"(n)dy(n)x(n).
(3)
This model implicitly assumes narrowband signals. As an example, consider ideal loss less delay circuits with complex gains {dJl(n) == exp(j21r/2 Jl ) } for /-l == 1,2, ... , p. For a 5-bit beamformer, a phase shift of 21w/16 can be imposed on the nth signal by toggling the first, third, and fifth delay circuits, i.e., d1(n)d3(n)d;)(n)x(n) == exp(j21wj16);.r(n). The delay circuits corresponding to the different bits can be conveniently represented as a diagonal matrix dJ..L == diag [d/-L(l), ... , dJ..L(N)). The vector S/-L represents the coherent, demodulated, sampled received signal when the JLth delay circuit is toggled in on all the elements
SJ..L == dJlS == [dJ..L(l)s(l), dJ..L(2)s(2), ... , dp.(N)s(N)]T.
(4)
An obvious way to increase the overall recei ved signal power at constant receiver noise power is to simultaneously transmit signals from all N, elements with all elemental signals set at the maximum allowable power levels. If N ~ N, temporally multiplexing coherent encoded signals are transmitted, the individual elemental signals can be extracted by inverting the vector of received signals with the inverse of the encoding matrix associated with the IV codes. In principle, any invertible encoding matrix should work for UTE, and any invertible bipolar encoding matrix should work for CCE. The class of equal amplitude renorrnalized unitary encoding matrices such as 2-D discrete Fourier transforms
637
(Dl-Tj's and Hadamard matrices are optimal for the UTE algorithm, whereas the bipolar Hadamard matrices are optimal for the CCE algorithm. Optimal calibration encoder/decoders are defined herein as those that provide minimum variance estimates for the calibration parameters. A renormalized unitary matrix is defined as a matrix T, where the inverse is equal to a real-positive constant times the Hermitian transpose T- 1 ~f A-1T*. If the number of elements N, is less than the order of the encoding matrix N, then one effectively zero fills the encoded transmissions very much like a zero-filled DFf. Zero filling applies to both the UTE and CCE algorithms; it will not impair the desired information capture. The zero filling effectively acts as if the array has N rather than N, elements. The pseudo straight-through-path signals {s(n)} for n == N e + 1, ... , N will all be zero, and all signal components proportional to these pseudo-element signals will, likewise, be zero. Proofs that the 2-D DFf's and Hadamard matrices for the UTE algorithm and the Hadamard matrices for the CCE algorithm uniquely satisfy the mathematical lower bounds for the asymptotic form of the estimation variances are given in Section VI. These proofs also demonstrate that the covariance matrices associated with minimum variance encoders satisfy two important physical symmetry conditions. I) As the elemental parameters are statistically independent
prior to transmission, an optimal encoder-decoder should preserve this statistical independence. This symmetry condition is manifested by diagonal covariance matrices. 2) Identical systems must have identical statistics. In the context of the present problem, this property dictates that the variances, which are represented by the diagonal matrix elements of the covariance matrixes, should be the same when the deterministic elemental system parameters are the same. A schematic of the CCE beamformer is illustrated in Fig. 1. This particular architectural embodiment considers single reference and calibration signals at frequencies N 2 f o + f and N1fo + f, respectively. The reference channel is designed to bypass the beamformers so that the reference signal used in the coherent detection process is not altered by the CCE encoding.
N
R..(t)
= Ro +fn +
l
t
v(t')dt'.
(5)
+•
XR
REFERENCE CHANNEL
Nz
Fig. 1. Schematic of CCE transmission system architecture.
Here,
Ro
vector from the ground station to the center of the array panel at t == 0, stationary position vector of the nth horn relative to the array center, v( t') instantaneous satellite velocity relative to the ground station. The radiating horns are arranged in a plane; the inplane geometry is dictated primarily by the beampattern and grating lobe requirements of the communication signals. The single element signals {s( n, t)} received at the ground station can be represented by the far-field expansions of the spherical waveforms emitted from the radiating elements
exp
j27r [ct -IRo + fn + A
l
t
vet')
dt'lJ
it
s(n, t) ex - - - - - - - - - - - - IRo + 17" + v(t') dt'j exp far field limit --7
III. PHYSICAL BASIS OFTHE CALIBRATION ALGORITHMS The UTE and CCE algorithms share a common physical explanation related to the nature of their orthogonally coded interference (beam) patterns. The calibration procedures extract the relative complex gains of the phased array elements. These results represent the set of relevant parameters as the beam patterns depend only on the relative amplitudes and phases of the phased-array elements. Consider an RF transmission from a phased-array GEO satellite at wavelength A. Choose a coordinate system with its origin at a specific earth receiving station. At time t, the coordinates of the satellite elemental transmitting horns are given by
I'.
1--------------..-.< ro.'
{j27r [ct -
Ro
~
it
v,.(t') dt'] }
---------------
Ro
X
exp ( -J"21rT- n
'
RO)
AR
o
.
(6)
Here, v 1, ( t') is the projection of the instantaneous satellite velocity onto R- o . The phase terrns -27f[Ro + fot V 1,(t' ) dt/l/ A, representing both the distance from the array center to the receiver and the Doppler phase shifts due to the satellite motion, can be stripped from the demodulated I and Q signals at the receiver with coherent detection system architectures using suitably designed calibration and reference signals [1]. Typical parameters for OED satellites are Ka to C band, A "" 1-8 em, R; 34 X 106 m, and velocities with radial components V 1' up to ",,±1 mls. In the UTE and CCE algorithms, the encoded coherently detected signals associated with the {m} orthogonally encoded transmissions are a linear combination of the individual single
638
I"'V
element signals
S(rn,
Ro ) = K
LN
n=l
t(rn, n)a(n) exp
( -o
-j27rTn. R ) , ARo
form==1,2,···,N.
(7)
Here, K is a complex scaling factor that depends on the propagation path. In the far field, which is relevant to this case, K is the same for all elements. The relative values of the coherently detecte~ individual elemental signals {s(n) == K a(n) exp [- j27f7;n ·Rol (ARo)]}' s are obtained directly from the matrix inversion of the N simultaneous equations. In the UTE and CCE algorithms, the matrices of encoding coefficients with matrix elements (T)nln == t( m, n) are renormalized unitary matrices. In order to extract the!elative phases included in the s(n)' s, the projection angle of R; onto the uniform phase plane of the array must be known to a precision commensurate with the desired calibration phase angle accuracies. This requirement is generally not a problem as the orientation angle estimates obtained from conventional attitude measurements using earth, moon, and sun sensors can be refined, if necessary, to the desired calibration accuracy using the statistical properties of the elemental insertion phases observed in the calibration measurements [3]. Two frequently asked questions relate to the intensity and dynamic range of the coded patterns. The first question is Why don't you beamform prior to the encoding in order to concentrate the interference pattern and thereby avoid low energy interference nulls at the single earth station receiver? The answer is that the orthogonal coded phases will destroy any previously set beam pattern; if the existence of nulls at the receiver are part of the coded interference patterns, they have to be there to correctly invert the codes. The second question concerns the ability of the detection system to accommodate the dynamic range of the coded patterns. If the chosen Hadamard matrix includes a row with all row elements equal to unity ~ a spot beam interference pattern will occur along the boresight axis. This could result in dynamic range problems if the ground receiving station was placed along the boresight axis of the phased array. This potential problem can be avoided by a number of solutions. One, for example, would be to position the ground recei ving station outside the mainlobe of the antenna array associated with uniform elemental insertion phases across the array. For a typical antenna, the uniform phase mainlobe is r-:» 1° wide, whereas, for example, the continental US encompasses a longitudinal angular width of rv 7° .
implementation of the UTE method for either digital or analog beamforming systems. The UTE method codes the elemental signals associated with N successive transmissions with N orthogonal codes. These codes can be conveniently arranged in the order of their use, row by row, in an Nth-order renormalized unitary matrix. We note that although the signals can, in principle, be coherently encoded with any invertible matrix. the class of equal amplitude renormalized unitary matrices such as 2D DFT and Hadamard matrices are optimal as they provide minimum variance estimates of the individual elemental parameters in the moderate to high SNR regimes. Mathematical proofs of these claims are given in Section VI-B. For an analog beamforming system, the UTE calibration process requires N(p + 1) separate coherent transmissions, where N ~ N e , to generate the complete set of complex gains for the NeP delay circuits and the .J.Ne straight-through paths. Calibration of a digital beamformer involves the JVt' straight-through paths only. The UTE process increases the effective SNR's bv a factor of rvJV over an SE process at the same elemental power levels. The mth received coherent transmission of the straight through path (no delay circuits switched in) UTE encoded signals is represented by
yo(m,) ==
L t(rn, n)s(n).
Similarly, the mth received transmission of the UTE signals with the J1,th delay circuit toggled in on all the elements is given by J\T
YJ.l(m) ==
L
t(m, n)dJ.l(n)s(n).
As noted in the introduction, the UTE procedure is the algorithm of choice for calibrating the elemental insertion phases of a digital beamforming system. Here, the orthogonal codes used in the UTE algorithm are introduced digitally at baseband with a high degree of accuracy. For calibrating an analog system, the UTE algorithm requires additional encoding hardware. The discussion in this section is relevant to the
(9)
n=l
The t( m, n) coefficients correspond to the m, nth matrix element of the renormalized unitary matrix T. The successive coherent transmissions are accumulated in vector forms
Yo ==TS == [Yo(l)~ yo(2)~ ''', YO(N)]T. YJ.l ==TdJ.LS == [YJ.l(l), YJ.l(2), .. ', YJ.l(N)]T.
(10)
The estimates of the decoded signals are obtained by premultiplying Yo, Y J.l by the inverse T- 1 of the same encoding matrix that was used at the remote site. In the absence of noise,
S == T-1yo, SJ.l ==T-1yJ.l ==dJ.lS.
IV. THE UTE ALGORITHM
(8)
n=l
(11)
The components of S give the estimated relative straightthrough-path signals. We emphasize, again, that the relative complex gains are the important parameters for beamforming. The delay circuit complex gain estimates are extracted by taking the ratio of the recei ved decoded components
639
{dJ,(n)}
= {d~~~~s)(n)}.
(12)
As an example of the UTE method, we illustrate the procedural mathematics for calibration of p delay circuits and the straight through path for a four-element array. For simplicity in illustration, we use the natural form of a fourthorder Hadamard matrix [4], [5] as our renormalized unitary (orthogonal) transformation matrix
H
~f
Nat -
[
+1 +1 +1 +1] +1 -1 +1 -1 +1 +1 -1 -1 +1
-1
-1
V. THE CCE ALGORITHM The CCE algorithm encodes coherent signals from the elements of an analog beamformed phased-array system using controlled switching of the phase shifter delay circuits. In general, CCE switching is dictated by matrix elements of an N x N bipolar invertible matrix. The class of Hadamard matrices are optimal for this application as they are the only bipolar matrices that satisfy the minimum variance optimality criterion. Henceforth, discussions in this section will feature Hadamard matrices only, represented by H for the CCE process. The CCE process requires two sets of transmissions. The first set of transmissions uses H to control the switching. The second set of transmissions uses - H to control the switching. The difference of the two signal vectors associated with the two transmissions is proportional to the bipolar matrix H. This procedure effectively generates an exact orthogonal transform encoding of the signal vectors even though the control circuits may be imperfect. The CCE switching of the control circuits themselves provides the encoding; therefore, no additional hardware is required. The difference signals are decoded with the inverse of the same binary-bipolar orthogonal matrix H-l = N-l H T used in the control circuit encoding. In order to calibrate the full set of Ne(p + 1) independent elemental phase variables, the CCE procedure requires a total of 2N(p + 2) individual, or N(p + 2), transmission pairs and provides information comparable to a single SE measurement at an SNR effectively enhanced by a factor rv2N over a SE calibration measurement with the same maximum elemental signal power for each transmission.
(13)
.
+1
The first transmission represents the coding of the straightthrough path with no delay circuits toggled in.
[+1+1 The
HS
+1
-1
[S(I)] +1] -1 s(2)
+1
+1
s(3) s(4)
ATR~IVERyo, AT RECEI\'ER
~
[YO(I)] Yo(2)
Yo(3)' Yo(4)
+1 +1
-1
z~ro
subscript on Yo refers to the straight-through path.
+1
-1 -1
-1
+1
yo(l) == [+s(l) --+
(14)
+ s(2) + 8(3) + s(4)]
first coded trans burst;
yo(2) == [+s(l) - s(2) + s(3) - 8(4)] --+
second coded trans burst;
yo(3) == [+8(1) --+
+ s(2) -
third coded trans burst;
Yo (4) == [+ s(1) - s(2) --+
s(3) - 8(4)] 8 (3)
+ s(4)]
fourth coded trans burst.
A. Mathematics of Hadamard Control Matrices
(15)
Now, the sequence of coherent transmissions is repeated with a single delay circuit switched in on all the elements. For M == 1, ... ,p, the coherent signal vector at the receiver is y J.L == HdJ.LS. Written out in component form, we have
YJL(l) == [+dJL(l)s(l) + dJL(2)s(2) + dJL(3)s(3) + dJL(4)s(4)] --+
first burst;
YJL(2) == [+dJL(l)s(l) - dJL(2)s(2) + dJL(3)s(3) - dJ.L(4)s(4)] --+
second burst;
YJL(3) == [+d JL(l)s(l) + dJ.L(2)s(2) - dJL(3)s(3) - dJL(4)s(4)] --+
using
fourth burst.
+Hl\r +Hl\r]
H 2N == [ H + N
third burst;
YJl( 4) == [+dJL (l)s(l) - dJL(2)8(2) - dJl(3)s(3) + dJl(4)s(4)] ~
An Nth-order Hadamard matrix [4], [5] is an N x N bipolar orthogonal matrix with all elements [H]rnn == H(m, n) equal either to ±1. Hadamard matrices with N > 2 are known to exist only if the remainder of the order N after division by 4 is equal to zero. Hadamard matrices of a given order are not unique, as any permutation of the rows or columns produces a different Hadamard matrix of the same order. Hadamard matrices are orthogonal matrices with inverses H-l == N-IHT. The most popular forms of the Hadamard matrices are the symmetric, radix 2, natural forms that can be generated recursively from the trivial Hadamard matrix of order 1
(16)
The received signal vectors are decoded at receiver with the inverse of the encoding transforms as shown by (11) and (12). The performance analysis of the UTE algorithm is given in Sections VI-B and D.
-H 1V
•
(17)
The calibration method is based on the following mathematical procedure. Consider a diagonal matrix d of complex numbers d~fdiag [d(l), d(2), ... , deN)]. Construct matrices, DF, DR based on any Hadamard matrix with their (m, n )th matrix elements constructed according to the rules:
640
if H(m, n) == +1. if H(m,n) == -1'
DR(m, n) == {d+(1n )
if H(m, n) == +1 if H(m,n) == -1.
(18)
Matrixes of the differences of D F , DR are expressed In component and matrix form as
DF(rn, n) - DR(m, n) == H(m, n)(1 - den));
nF
-
DR == H(I - d).
Next, take the difference of the accumulated received signal vectors Y~~, Y~o, and decode the resulting vector by premultiplying by the inverse of the same Hadamard matrix that was used in the control switching at the remote site. In the absence of noise, the decoded vector ZjJ.o is obtained.
Z
(19)
J.L0
~H-l(yF _yR)
/iO
Here, I is the identity matrix. Multiplying the left-hand side of (DF - DR) by the inverse H- 1 gives a diagonal matrix
H-1(D F
DR) == H-1H(I - d) == I-d.
-
== (I - d/-L)S,
(20)
B. The CCE Process The first pair of transmitted encoded coherent signal vectors is based on CCE with the J.lth delay circuit with all other delay circuits switched out. These received transmissions are accumulated and expressed in vector forms
Y~~ == [Y~o(1), Y~o(2), "', Y~O(N)]T, == [y:O(l), Y:O(2), "', y:O(N)]T.
y:o
L
Y~v(m)
1\'
=
L
D~(m, n)dv(n)s(n);
L
D:(nl, n)dv(n)s(n).
n=l lV
y:v(m) ==
Here, again, the first J.l index reflects CCE with the p,th delay circuit, whereas the 1/ index reflects the fact that the z-th delay circuit is switched in on all the elements. The resulting decoded received signals are represented in vector form by (28)
The N complex gains {d v (11,)} are extracted from these decoded signals by taking the ratio of the decoded vector components
dv(n)
D:(m, n)s(n),
n=l
L D~(m, n)s(n). ]\'
(22)
n=l
The encoding coefficients D:Cnt, n), D~Cm, n) are dictated by the status of the delay circuits that are switched according to the Hadamard control rules previously discussed In conjunction with (18),
+1 F
D I' (m, n) =
{
dJl (n) dJ.L (n)
R
DI' (m, n)
= { +1
J.lth delay eire of nth elem switched out if H(m, n) == +1. Jlth delay eire of nth elem 1 switched in if H(m, n) == -1 (23) J.lth delay eire of nth elem switched in if H(m, n) == +1 Jlth delay eire of nth elem switched out if H(m, n)
==
-1. (24)
The differences of the encoding matrices are represented in component and matrix form as
D~(m,n) - D:(m,n) ==H(m,n)[l- dJl(n)];
D~ - D~ == H(I - d/i)'
(25)
(27)
n=l
f\/
y:OCm) ==
(26)
The second set of transmission pairs corresponds to CCE again with the J.lth delay circuit, whereas an additional delay circuit, say, the z-th, is permanently switched in on all of the elements. Here,
(21 )
The first subscript index J.l on Y:6 R designates the delay circuit that is toggled according to the CCE coded switching. The second zero subscript on these vectors indicates that these are the received signals with all delay circuits other than the j1,th switched out. Here, N bursts of coherent encoded signals corresponding to the N components of these vectors are transmitted and received for each calibration measurement. The mth received transmission pair of the first set of CCE signals is represented by
y:o(m) ==
/-LO
==H-l(D~ - D~)S
= zJlv(n).
(29)
zllo(n)
Repeating the above process using CCE with the p,th delay circuit with each of the other delay circuits singly switched in determines the subset of N (p - 1) complex gains d.;(n) for all ].J =f- tL The N values of {d ll (n)} have yet to be determined. These are obtained by repeating the procedure described above with the J.lth circuit permanently switched in on all of the elements, whereas any other circuit is used for the CCE. For example, if the ( f. p,th delay circuit is used in encoding, the {djJ. (n) } 's can be determined from dJl(n) == [z(Jl(n)]/[z(o(n)]. Now that all the {d,,( n)} are known for all ry == 1, 2, ... , p, n == 1,2, "', N, the straight-through-path signals {s(n)} are determined from either
s (n) = or
s(n)
=
Z JlO (n )
1 - dJl(n)
z(o(n)
1 - d«(n)
.
(30)
In the presence of noise, the best estimates of s(n) are obtained by using all available data and averaging the results from both terms in (30). The complete data set of N p delay circuit complex gains plus the N straight-through signals are obtained
641
with N(p + 2) CCE paired transmissions that are tabulated as follows:
Switching Action J,£th CCEj others switched out pth CCE; II I- p. switched in (th CCE; others switched out (th CCE Ilth switched in
Measured Result
(I-dl£}S (I - dl£)dvS (I - d()S (I-ddd~S
Required Transmission Pairs N
N(p-l)
We want to develop the expressions for the mean and covariance of the SE complex gain estimates. In Appendix A, we have shown that the expected values of terms involving the reciprocal of the noise-contaminated demodulated signal samples can be expanded in a Taylor series in the moments of the ATGN noise-to-signal energies
N N
It is important to note that elemental pathologies such as dead elements (usually due to power amplifier failure) and stuck delay element switches are both readily identified by the CCE process. VI. PERFORMANCE ANALYSES OF THE SE, UTE, AND CCE ALGORITHMS The calibration algorithms considered herein depend on coherent detection of demodulated signals. The noise at the receiver prior to demodulation can be modeled as zero-mean complex additive white Gaussian noise (AWGN) with a variance a 2 . The noise model after demodulation must be modified because the demodulation process both bandlimits and energy limits the noise. Moreover, indiscriminate use of AWGN in performance analyzes of coherent detection systems after demodulation leads to inherent analytical divergences. As discussed in detail in Appendix A, these difficulties can be resolved simply by choosing a noise model that is both physically more appropriate for the coherent detection process and is not hampered by analytical divergences. The coherent detection systems noise model used herein is referred to as additive truncated Gaussian noise (ATGN). ATGN is represented by independent amplitude and phase random variables with probability density functions (pdf's) characterized, respectively, by a truncated Rayleigh amplitude distribution and a uniform phase distribution. The minimum variance proofs for the UTE and CCE algorithm assume a power-bounded system with a maximum single element power E 1Uax ~f Is(k)/~lax.
A. Single Element Process Consider a single element measurement process where the complex gains {dl-L (k)} of the J.Lth delay circuit of the elements indexed by k are estimated from the ratio of the received signals with and without the delay circuit toggled into the elemental electrical paths. In the presence of demodulated ATGN receiver noise, the single element signal vectors with and without the J.Lth delay circuit toggled in are given by
= SI-L + iil-L' S =8 + no.
(33)
From the relations given in (32), with Ck ~f Is(k)1 2 , the mean and covariance of the dJl (k) estimates are given by
E{dJl(k)}SE ==dJl(k), SE
def
""
"
"
(RJLJl)kk = E{dJl(k)d~(k')} - E{dJ.L(k)}E{dJl(k')} 2 + Ck 1 E{ln(k)1 2}] == Dkk l
l{[ldJ.L(k)1
L
00
x
ckrnE { ln(k)/2rn} -
IdJl(k)/2}.
(34)
nl=O
From Appendix A, with aCk representing the cutoff for the ATGN noise energy,
c;l E{(ln(k)1
L
00
2
}
a2
== -
ck
ae- a c /o. / u 2 _ / 2' 1 - e- a c k U
ck71tE{(ln(k)1271t} = c~ (eC k /
m=O
a
x
q 2
_
[Ei(:~)
1)-1
- Ei([l - a] :~)]. (35)
Hence, the SE covariance has a closed-form analytic solution in terms of exponential integral functions. In Appendix A, we have also shown that asymptotically, with error terms of the order of a exp (-ack/a2), that the moments in the expansion can be evaluated using the simplified expressions for the moments of AWGN, providing that the infinite expansion is truncated at an appropriate integral order Me :s (ck/a2 - 1). In our simulations, we choose a to be 0.9. The asymptotic form of the SE variance with Ck set equal to E 1nax is
SJL
(31)
As discussed in Section II, the sampling indices have been suppressed. Here, nJl , no are the ATGN noise signal vectors, the components of which satisfy the statistical properties
E{[nJL(k)*nv(k')}t} == 6JlvbkkIE{lnl-L(k)12t}, E{[nl-L(k)nv(k')]t} == o.
(32)
The variances are the diagonal matrix elements of the covariance matrix.
B. UTE Algorithm Consider the general encoding case where the UTE encoding matrices T are invertible but not necessarily of the
642
Theorem 6.2: For a power bounded system, the column vectors of the minimum variance UTE encoding matrix t k ~f [t(l, k), t(2, k), ... , ti N: k)]T are: a) orthogonal and normalized to Ak == NE ln a x / ls(k )\2 and b) have equal magnitude components, {It(i, k)1 2 } == E ln ax / ls(k)12 . Proof' From (41), the asymptotic UTE variances for the dJL (k)' s estimates are represented as a nonnegati ve series in powers of the positive semi-definite diagonal matrix elements
renormalized unitary form. The noisy received signals vectors used in the analog implementation of the U1"t algorithm are represented by
Yo ==TS + no, YJl == TdJlS + nJl'
(37)
The signal estimates used to estimate the delay circuit parameters are obtained from decoding with the inverse of the encoding matrix
(T-1T-1*)kk
S==T-1yo
SJJ.. == T-1y JJ.
== diLS + T-1ii iL .
(38)
Using the notation, (T)krn == t(k, m) for the matrix elements, the kth components of these vectors are given by
It(m, k)s(k)1 2 == It(m, k)1 2 Is(k )12
:s E 1n ax ,
t-1(k, m)no(m),
rn=l
+L
\1m, k
t-1(k, m)niL(m).
L IV
(39)
nt=l
It(m, k)1 2
The estimates diL (k) are obtained from the ratio of the components in (39). Expanding the ratio in a Taylor series,
d (k) == BiL(k) - 11-
s(k)
- Is(k)12 == Ak.
(40)
AT
L n~=l
~
covariance with ei; ~f Is(k)1 2 are given by
E{diL(k)} ==diL(k),
E{[Bo(k)B~(k,)]nl}.
0-2(T)kk s(k)s*(k')
{Me
fl
It(m, k)1 2
rn=l
c1(k, m)t(m, k{
f
2
+
1
0- (T-1T- * )kk s (k )s * ( k' )
2 ) .
(46)
The minimum value of (T-1T- 1 * )kk corresponds to the lower bound of (46) f
(T - 1T - 1*) kk -_
]
1 1 ml[0-2(T- T - *)kk/ ]nt } s(k)s*(k')
- 15j,;k'O (~ C -<>~1. /(7
(45)
.. > 1 > ~ (T - 1T - 1 * ) kk - (T*T) kk - /\k \ .
nt==l
x
Itl
L
with (42) and (44), we obtain
CX)
+
AT
jt-1(k, m)1 2
== 1
+ [dJl(k)d~(k') + E{BiL(k)B;(k')}]
d (k)d* (k') [ tL tL
(44)
== I(T- 1T)kkI 2
(R~JE)kkl == E{ BiL(k)B: (k')}
1T- 1 *
== (T*T)kk < NEln ax
where the terms B,(k) ~f L:~~=l t-1(k, m)ii,(m)/s(k). The estimation mean, covariance, and asymptotic form of the
L
1,2, ···.1'1. (43)
Combining the Schwarz inequality
== [dJL(k) + Btl (k)][l - Bo(k) + B6(k) - ...]
x
==
Hence,
f\/
BJL(k) == diL(k)s(k)
(42)
As the variances are monotonic functions of the (T-1T- 1*)kk's, they will be minimized by the class of encoding matrices T that minimize the (T-1T-1*)kk'S. The condition that the power emitted from an encoded single element is less than En1ax. is represented by the inequality
IV
L
L It-1(k, 'm)12 ~ O.
nt=l
== S + T-1iio,
s(k) ==s(k) +
==
(41)
The methods for generating the asymptotic form given above are discussed in Appendixes A and B. The factor of N in the correction terms arises from Theorem 6.2 proven below, which proves that the asymptotic minimum variance form of (T-1T- 1 * )kk is 8kk, N- 1. These error terms are extremely small for the practicable system, e.g., for a IO-dB SNR, with N == 16 and a == 0.9, the correction terms are r-v7 x 10- 6 . We now prove the theorem that renormalized unitary encoding matrices are optimal for the UTE process.
1
(T*T)kk
1 .
Ak
(47)
This lower bound is satisfied when the Schwarz inequality (45) becomes an equality. These conditions are jointly satisfied iff t*(m, k) == Akt-1(k, m)~ t-1*(k, m) == Aklt(m~ k). The orthogonality and normalization of the column vectors of T follows from
(T*T)k1k ==
L t*(m, k')t(m, k)
l
643
rn==l
== Ak(T-1T)kfk == Akbk.tk·
(48)
This proves part a). The proof of part b) follows from combining the minimum variance normalization condition 2 Lrn It(m, k)1 == Ak with the power constraint on the maximum allowed values of It(m, k)\ from (43) Nlt(m, k)l~lax ~ Ak- These two relations dictate that the minimum variance encoding matrix elements must satisfy N
L
2
It(m, k)t ~ Nlt(m, k)l~lax·
The kth components of these vectors are given by
Zp.o(k) == [1 - dp.(k)]s(k)
+L
Zp.v(k) == [1 - dp.(k)]dv(k)s(k) + t- 1(km )[n3(m ) - n4(m)].
L
(49)
Tn=l
This equation can be satisfied only as an equality and, then, iff {It(i, k)1 2 } == AkiN. 0 Corollary 6.2: If the straight-through-path signals have equal power, then the minimum variance UTE encoder is a renormalized unitary matrix with all matrix elements having the same magnitude. Proof: If all the elemental powers are equal, the Ak' s are all equal to a constant A. From (48), T*T == AT-iT =} T- 1 == A-IT*, which is the definition of a renonnalized unitary matrix. At equal elemental power, from Theorem 6.2b), {It(i, k)1 2 } == AIN; hence, all matrix elements of the minimum variance form of T have equal magnitude. D Substituting the minimum variance form for T into (41) gives the asymptotic minimum variance form of the UTE covariance associate with a maximum received single element power E Illax
t-1(km)[nl(m) - ii 2 (m )],
rn
(53)
nl.
Let us define
Av(k) ~f [1 - dv(k)]s(k); Bvi(k) ~f Av~k)
L
(54)
C1(k, m)ni(m);
(55)
m
Cvij(k) ~f Bvt(k) - B vj (k)
for i
=I j.
(56)
Take the ratio of the components in (53) to get the estimate of dp.(n) and expand in an asymptotic Taylor series
dJ1.(k) = zVJ1.(k)
zvo(k) == [dp.(k) + C v34(k)] x [1 - C v12(k)
+ C;12(k)
- ...].
(57)
Proceeding in a manner similar to Section VI-B,
E{dJL(k)} = dp.(k) ,
(R~~~v)kkl ==E{Cv34(k)C~34(k')}
+ [dp.(k)d~(k') + E{Cv34(k)C:34(k')}]
L 00
x
E{[Cv12(k)C~12(k,)]nl.}.
(58)
nl.=l
Therefore, the minimum variance form of the UTE covariance matrix satisfies the physical symmetry conditions enumerated in Section II.
From Appendixes A and B, the asymptotic form of the CCE covariance is
C. CCE Algorithm
Consider a general CCE encoding case where the control matrices T are bipolar, invertible, but not necessarily of the Hadamard form. In the presence of ATGN receiver noise, the demodulated signals vectors in the CCE algorithm are represented by
Y~o =D:S +nl;
y:o ==
D~S + n2
D:
+ n3; Y~v == D:dvS + D.4. Y~v ==
dvS
(51)
The signal estimates used to estimate the delay circuit parameters are obtained from decoding with the inverse of the bipolar control matrix
ZJLO == T
"
-1
"
-1
F
R
(YJLO - Y JLO)
== (I - dJL)S + T- 1(nl - n2),
Zp.v ==T
F
Theorem 6.3: The minimum variance bipolar CCE matrix is a Hadamard matrix. Proof- The CCE variances, which are similar to the UTE variances, are represented as a nonnegative series in powers of the positive semi-definite diagonal matrix elements (T-1T-1*)kk. CCE variances are minimized by the class of encoding matrices that minimize the (T-1T- 1 * )kk 'so The required bipolar character of the CCE encoding matrix It(m, k)\ = 1 mandates that N
R
(Yp.v -¥p.v)
== (I - dp.)dvS + T- 1(n 3 - n4).
(52)
644
(T*T)kk ==
L
nl.=l
t*(m, k)t(m, k) == N.
(60)
From Schwarz's inequality (45)
ESNR - DELAY CIRCUIT ESTIMATE - THEORY & MC SIMULATION simula tion statistics based on N X 10K random trials
( T - 1 T - 1 *) . . > kk
1 - (T *T )kk 1 - N'
28
26
(6 1)
24
Again, the minimum variance form of T renders the Schwarz inequality an equality. Thi s equality is satisfied iff t * (m , k) = NC 1( k, m ). As T is a real -bip olar matrix, the minimum variance condition can be expressed as the matrix relati on
T-
1
=N -
1
=N - T T
(R/l /l,vv)kk' =t5,.:k'
+
2(T 2
NElll ax ll
-
d",(k)1 2
+
N
Ell axl~(T~ d", (k )12]
[ ld/l(kW
X{At, m =l
_ 0(
m! [
2 (T2
N Elllaxil
-
2
0.04
. x - theory - CCE + - theory - UTE
o-
Me Simulations
I
for the
7r
for the
"2 7r
2] m} (63)
64
ApPENDIX A
.
phase shift
NUMBER OF ELEMENTS
element power to the rec eiv er noise power E lll ax / (T2 of 10 dB . Both the theoretical and simulation results given in Fig . 2 co rres pond to the ESNR for the complex gain estim ate s of the 7r /2 phase shifter delay circuit. The MC sim ulation and theoretical results for the single element ESNR ' s of the 7r /2 phase shifte r are 6.121 and 6.120 dB , respectivel y for 320k random trials. The close correspondence between the the ory and the MC simulations serves to validate the theoret ical ana lysis . Th ese results also illu str ate the dramati c incre ase in the ESNR ' s that ean be obtained using the orthogon al codes , for d irect encod ing in the UTE pro cess, and for co ntro l co des in the CCE process.
phase shift
7r
32
Fig. 2. Theory versus MC simulation of the effective estimation SNR (ESNR) for the estimate of the complex gain of the 1l' /2 phase shift circuit.
d", (k)1
Qc-E~,./(2 ) }.
d",(k)1 = { :
e
16
Here again, the optimal form of the cov ariance matri x satisfies the physical symmetry co nditions enumerated in Secti on III. We note that the expansion parameter for the CCE vari ance has a fac tor of 11- d", (kW in the den ominator. As
11-
.
..
(62)
.
Thi s relati on is the definition of the class of bipolar Hadamard matrices. 0 For maximum elemental power levels , the asymptotic , optimal CCE covariance assumes the form {
.
SNR 10dB
.
T*
1
CC E
SE ESNR . theo ry 6 120 dB • simu lat ion 6 121 dB 320K trials
(64)
.
for the 16 phase shift
lower vari ance parameter estimates are obtained when the CCE encoding is performed with the phase shifter delay circuit that produces the largest magnitude phase shift.
D. Comparisons: Theory and Simulation We now compare the asymptotic ESNR ' s, which are represe nted by the ratios of the square of the mean s to the variance s of the parameter estimates, using the the oretical result s for the SE , UTE, and CCE varia nces given by (36) , (50), and (63), respecti vely, with Monte Ca rlo (MC) simulations of the corres pond ing ESNR ' s. For these examples, the erro rs induced by the trun cat ion of the asy mptotic expansions are so small that we truncated the expan sion s in (36), (50) , and (63) at their ninth -order expansion term s. Fig . 2 illustrates these comparative results. Hadamard matrices were used for the unitary encoding matrix in the UTE simulations. Hadamard matrices, as required, are also used in the CCE simulations. For these simulations, we have used a value of the single element SNR , which is defin ed as the ratio of received single
NOISE MODELS FOR COHERENT DETECTION
A typical approach to performance analyses of estimation algorithms is to develop analytical expressions for the variances of the parameter es timates under the assumption that the information bearing signals are co ntam inated by AWGN . Th e wid e use of AWGN has effec tively raised the model to the level of a de fa cto standa rd. However, one mu st pro ceed with caution, as the model is basically ad hoc , and generic applica tion can , at times, produce problematic results. The primary reasons for the popularity of the AWGN mod el are a) AWGN fairl y acc urately models thermal noise for ana lytica l expre ssion s invol ving low-ord er mom ent s o f the noise energy, and b ) the model is mathematicall y simple for both anal ytical analysis and Monte Carl o simulations . Unfo rtunatel y, the literal form of the AWGN model is inappropri ate for performance analysis of demodulated signal coherent det ecti on algorithms, which are the main topi c of thi s pap er. In the coherent detection proces s, the noi se-contaminated signals are passed through a series of filters in the process of demodulating the signals from RF down to a low en ough intermediate frequency for the signals to be sampled digitally . Demodulation both band limits and energy limits the signals. If one neglects the physical constraints on the noise imp osed by
645
the demodulation process and heuristically uses the AWGN model for the performance analysis of a coherent detection system, the mathematical results as shown below will exhibit analytical singularities. These difficulties can be resolved by picking a noise model that is more appropriate for the physical process. The physical energy constraint imposed by the demodulation filters can be mathematically captured by cutting off the noise energy at a large fraction of the signal energy. The signals are sampled at a Nyquist rate commensurate with the final demodulation filter bandwidth, and hence, the individual bandlimited noise samples will be statistically independent. Accordingly, the demodulated noise, for which the samples are represented by {ii( k) }, is modeled by independent amplitude and phase random variables with probability density functions (pdf's) characterized, respectively, by a truncated Rayleigh amplitude distribution and a uniform phase distribution
for 1'2 < oe: a otherwise
< 1; ()
. The moments of the ATGN noise energy WIth x clef == e / a-? are given by
E{lnI2A1 } AT G N == E{ln\
21\1.
1\1. ( ax )rn] C - ax ' " -m. ,- . [1 - 1 - c- ax L.J
}A\VGN
nt=l
These moments are mathematically more complex than the comparable AWGN moments E{lnI 2 A1 }A\VGN == M!a 2 1\1.. By changing variables in (A.2), 1oo( o , x) can be alternatively expressed in terms of exponential integral functions [6]
100(0, x) ==
cC-X
A
== A
(A.I)
loo(a, x) ==
L
Gaussian noise (ATGN). As discussed in Section VI-A, the variance of the singleelement complex gain estimate is proportional to
[Ei(x) - Ei([l - alx)].
(A.4)
nt=O
E-rnE{/nI2nt}A\VGN C
-ax
For moderate to high SNR, x asymptotic form
L
l\1.c
(A.2)
The closed contour in the above contour integral is counter clockwise on the unit circle. As the domain of t is 0 S t ~ a < 1 in the integral, the contour integral is equal to the residue at the pole at z == - Vi; the resulting integral is well behaved. On the other hand, for the AWON case, the domain of t is 0 ~ t < 00; the evaluation of the contour integral in this case produces a symmetric singularity in the neighborhood of t == 1 that causes a divergence in the subsequent t integral.
1, and foo(ax) assumes the
== fix(x) -
1.
(A.6)
The asymptotic form of the expansion is significantly simplified as it is an expansion in terms of the moments for the simplified AWGN model. One must truncate the expansion at an integral order Me for the asymptotic representation to be valid as the high order AWGN moments diverge. The maximum cutoff Me is chosen commensurate with the minimum value of £-nlE{\nI 2n1} A\ VG N , i.e., the lowest order that satisfies the inequality £-(1\1+1) E{lnI 2 ( 1\1 + 1 ) } A\VGN ~ M 2M E- E{lnI } AWGN. For Me == fix(x) - 1, using Stirling's formula for large M, In M! M In M - M, we see that the last term in the truncated expansion, e-1\1 E{lnI2A1 } A\ VG N ' is "-' exp (-x). For Q == 0.9, the fractional errors for different SNR's arising from the use of the asymptotic form for relatively low values of M are illustrated in Fig. 3. It is apparent from these results that the asymptotic form of the expansion is valid down into the moderate SNR regime, with SNR's down to "-'10 dB. In addition, for large effective SNR's, the errors are so small and fall off so precipitously that one can truncate the expansions at much lower orders than the maximum Me cutoff point described above. With these results in mind, we revisit the expression E{lsI 2Is+nl- 2 } and perform a Taylor series expansion in the ratio of the complex receiver noise amplitude random variable I"V
rn=O
»
k=l
c- n l. E { lnI2n l. }A\VGN - G(ae- a X ) ,
for Me
00
,nt ( )k m. ~ ax
n1=1
n1.=O
<1
00
~
- 1- c- ax ~ x nt ~~. (A.5)
ioo(a, x) "-'
e-17tE{liiI2nt}ATGN.
t
00
Is(k)1 2 ~f c. We call this noise model additive truncated
L
c-
-1
EC- X
Here, A is the pdf normalization constant A ~f a 2 [1 exp (-aE/a 2 ) ) with the deterministic signal energy
==
ja-1 - xt dt
From (A.2) and (A.3), we can write foo(a, x) in the form
E (0, 21r)
for a
(A.3)
646
the products of linear combinations of zero-mean circularly complex Gaussian random variables (CGRV's). This appendix features a mathematical theorem that greatly simplifies the evaluation of the perturbation terms in the expansion. This theorem should be generically useful for perturbation theoretic analyses of statistical and statistical signal processing problems. The proof makes use of the following well known lemmata. Lemma LB. 1: If A is linear combination of independent CGRV's, then A is also a CGRV. Proof LB.]: This is proven readily using characteristic functions; see, for example, [7]. Lemma LB.2: If Xi is a CORY, the moments of the square magnitude IXi 12 == X1,Xi satisfy the relation
ERROR USING TRUNCATED -AWGN- MOMENT EXPANSION O_------,r--------r-----,.------r------, SNRs lOdB
-40
en ...J
W
m -60 w
o
a
20dB
I I
~ -80
0:
a:
UJ
25dB
-100
-120
rn.n
- ----A-------J
_ 1 4 0 L . - - - - . . I - - - - - . - . l o - - -........
2 3 4
1
5
ORDER OF TRUNCATED EXPANSION
6
Proof LB.2: The proof follows directly from the wellknown fact that a CORY can be described in circular polar coordinates in terms of magnitude and phase random variables that are, respectively, Rayleigh and uniformly distributed. Therefore,
Fig. 3. Fractional error in decibels using truncated asymptotic expansions for different SNR's.
to the complex single element deterministic signal amplitude
E{l x iI2M 00
== ~ Z:: e -
11t
E {\.n. . \2rn} ATGN.
}
=
r: dB iroo
io
=e
(A.7)
nl=O
The expected value of the expansion gives the same result given in (A.2) for any SNR, whereas for moderate to high SNR, the asymptotic form given in (A.6) is applicable. As our interests are primarily in the moderate to high SNR regime where coherent detection is generally feasible, our performance analyses of the calibration algorithms will use the asymptotic form of the expansions where the terms in the truncated Taylor series expansion are evaluated using the AWGN moment relations. In the performance analyzes of the UTE and CCE algorithms, we have to evaluate terms of the form
M
1
+
C-z2~~~ dz ~~
yMc-y dy
== M!(E{lx'iI 2 } )1\1.
D (B.2)
Theorem TB.]: Let A and B be linear combinations of independent CGRV's. From Lemma LB. 1, A and B are also CORY's. The Mth power of the product AB* satisfies the relation E{(AB*)A1} == M!(E{AB*} )1\1.
(B.3)
Proof Tli.l : In terms of the independent CGRV's, {Xi}, .l4, and B have the form 1\/
A:= LaiX'l;
s(k)
s(k)
z2M+1
== M!~;A1
2
E
00
o
L t-1(k, m)n(m) lV
'i=l
(A.8)
B ==
n1.=l
L bix
z•
(B.4)
i=l
We follow the same procedure as discussed above to obtain the asymptotic form as a truncated expansion using the AWGN moment relations. In the asymptotic expansion for the covariances, we have to evaluate terms of the form E{[B(k)B(k')*]A1}A\VGN' where the B(k)'s are linear combinations of statistically independent AWON samples
From the independent character of the z,' s and Lemma LB.2
(B.5)
Using the multinomial expansion theorem
B(k) ~f I:~:=l t-1(k, Im)n(m)j s(k). These expressions can
be reduced to simple forms using the theorem proven in Appendix B. ApPENDIX
B
(B.6)
The asymptotic expansions of the covariance matrices for the UTE and CCE algorithms involve series expansions of
647
we express the moments of AB* in the form
By1(k) - B y2(k). From (B.IO), E{[Cv12Ck)C~12(k')]l\1}A\VGN == ,[2a2(T-1T-1*)kk' ] M. Av(k)A~(k')
At!
.
(B.12)
ACKNOWLEDGMENT
(B.7) From (B.5), (B.7) reduces to
The author would like to acknowledge discussions with his co-team members and consultants associated with the GElLockheed Martin Satellite Phased Array Program: J. M. Ashe, P. G. Bonanni, G. M. Kautz, F. W. Wheeler, W. E. Engeler, G. D. Mahan, R. L. Nevin, and G. Xu. REFERENCES
(B.8)
As
E{AB*} ==
L aibjE{xixj} L ai b:(,;, ij
::=
E{(AB*)M}
=M{~ aibie) ::=M!(E{AB*})A1.
(B.9)
M
0
(B.IO)
Using these results, the moments in the expansions of the covariance matrices for the UTE and CCE algorithm can be evaluated directly. For the UTE algorithm,
E{[BjtCk)B;Ck,)]A1}A\VGN =M! [E{BjtCk)B;(k')}A\VGN]A1 2(T-IT-l*) ] A1 -M' a kk' -. [ s(k)s*(k') . (B. II) For the CCE algorithm, the Cvij(k)'s are CGRV's equal to the difference of two independent CGRV's Cv12(k) =
648
[1] The design and analysis of the coherent detection systems are collaborative efforts of 1. M. Ashe, R. L. Nevin, G. Xu, and the author. These results will be published elsewhere. [2] 1. M. Howell, "Phased array alignment and calibration techniques," 10 Proc. Workshop on Testing Phased Arrays and Diagnostics, organized by the AMTA in conjunction with the IEEE Antennas Propagat Int Symp., San Jose, CA, June 30, 1989. [3] The orientation refinement algorithms are collaborative efforts of P. G. Bonanni, J. M. Ashe, and the author, based on an idea generated by the author; they will be published elsewhere. [4] H. J. Ryser, Combinatorial Mathematics. New York: Wiley, 1963 [5] W. K. Pratt, DigitaL Signal Processing. New York: Wiley, 1978. [6] Handbook ofMathematical Functions, M. Abromowitz and I. A. Stegun, Eds. U.S. Dept. of Commerce, Nat. Bur. of Std. Applied Mathematics Series, 55, 1965, p. 228. [7] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering. Reading, MA: Addison-Wesley, 1989.
The Analogy Between the Butler Matrix and the Neural-Network Direction-Finding Array R. J. Mailloux and Hugh L. Southall Electromagnetics and Reliability Directorate Rome Laboratory 31 Grenier St. Hanscom AFB, MA 0 1731-3010 USA Tel: (6 17) 377-3710 Fax: (617) 377-5040 E-mail: mailloux @maxwell .rl.plh.af.mil
3. The Butler matrix as RF beamformer
Keywords : Direction of arrival estimation ; neural network applications; antenna arrays; array antenna control
An N-element Butler-matrix multiple-beam system (3] forms N radiated beams, or receives signals from beams with maxima at
1. Abstract The primary goal of this paper is tutorial. Neural networks are beginning to have a role in array antenna control , and this paper compares the operation of a neural network and a Butler matrix performing the same direction-finding task.
Ante nna Ports (n)
2. Introduction
N
eural networks have been shown [I, 2] to be useful in control of phased arrays for detection and signal location. In particular, neural networks can control arrays with various types of element and network failures and can still perform accurate signal location, despite the errors. Since a trained neural network has output nodes that correspond to input waves from specific angular directions, and the Butler matrix has output beam ports that correspond to specific orthogonal input-wave directions, it seems that there should be some analogy between direction finding with a neural-network array and direction finding with a multiple-beam matrix.
Input Nodes (In)
Be am Ports (I) Weights (W kl
= 6(k.l))
Summ ation Nodes (Sk)
o A
Figure I a. A Butler matrix with post-processing as the direction-finding system.
Unfortunately, the language used by authors in describing neural-network control has to do more with function approximation and mathematics than with the physical parameters that concern antenna engineers . Consequently, the purpose of this paper is to use the antenna engineer's knowledge of the Butler matrix to lead to the kind of understanding necessary to contribute innovatively to the technolog y of array control with neural networks .
Antenna Port
To this end, the operation of the Butler-matrix network [3] of Figure 1a is described below, and compared with the neural network of Figure 1b. Figures 2a and 2b show the specifie implementation for a two-beam, two-element array. It is noted that the direction-finding neural network described in this paper is not technically a beam former, which is understood to be a network that passes information. The neural network is designed to output a digital estimate of the angle of arrival of a wave from an emitter. The classic Butler matrix is, however, a true beamformer: a passive network that passes the entire RF signal. However, like the neural network, the Butler matrix can be used to provide an estimate of emitter location, and so by considering this application we hope to shed light on the process involved in the neural-network directionfinding system .
(n)
Input Nodes
( In )
Center
( Mnl )
Gaussian (I) Proce ssing Nodes Weighls
( W kl )
Sum mation Nodes ( Sk ) "'::-:-L-:..:.l._..:..::..l--":'? ---":':..:..L- ' -'-L_
_
+--=5:..::.0~
Figure lb. A neural-network direction-finding system.
Reprinted from IEEE Antennas and Propagation Magazine, Vol. 39, No.6, pp. 27-32, December l'i97 .
649
angles Bt having direction cosines Ut, such that for a one-dimensional array with element spacing dx and wavelength it , (1)
for 1a half-integer, and N even; N-l N-l ---5:15:--.
2
(2)
2
These beam directions correspond to the peaks of orthogonal beams in space. The radiated beams have the shape of modified and, Sine ( sin( Nx )/[ N sin( x )] ) functions, orthogonality, the microwave network is lossless.
because
1
of
A response due to a wave from one of these "orthogonal beam directions" u/ appears as a signal at only one beam port, "1" (shown in Figure 1), corresponding to one of the orthogonal beams. For these incident angles, all other beam ports have zero output.
Figure 2b. A simplified two-elementJtwo-node direction-finding array. The phase difference is
~= 27rd.u , with
It input vector has two components, (sin
ters have two components, with
1
~,
ml
u=sinB. The
cos ~), and the cenand
= (sin~],cos~l)
ml=(sin~2,cos~2)' where ~l=mlul and uz=±O.5 are the Butler-matrix orthogonal beam locations.
1
The original Butler matrix is a passive analog power combiner. However, it is well known that one can perform the same functions digitally, and thus do digital beamfonning, in which case the equivalent digital processing is recognized as simply an FFT. In either case, signals An(t), incident from any direction u == sin B, are received by the nth array element. For an array of N elements, with uniform spacing d x '
Figure 2a. A simplified two-elementJtwo-beam Butler matrix for d = ),,/2. For the phase reference at the array midpoint,
/} = exp(- j 1r /2u) and I] = exp(+ j 7r /2u), with u =sin( B). The fixed phase shifts are calculated from Equation (4), with orthogonal beam positions u/ = 141/2 = ±O.5 . The ones (1) indicate unit weights.
(3)
These signals are combined in the matrix so that for the 1th beam, the normalized response is
650
node," a term that is explained more fully later. The system computes a set of center distances, nl , according to Equation (6) below, for each training angle, "l" These center distances are a measure of the difference between any new signal input, In' and the centers corresponding to that training point. The center distances are
e
(4)
where the phase center has been referenced to the array center, and we divide by N to normalize the response to 1(t ) at the beam peaks. This combines to form the resulting signal at the lth beam port.
. [Ntrd y x ( u-u[ )] . [mix ].
(6) where In is the nth component of the signal, and mnl is the center of the lth Gaussian node in the nth dimension. What this means, in the particular case of direction finding, is that each center mn / corresponds to a training angle, and so the distances enl are measures of the (angular) distance between each input-node phase or phase difference, "n," and each center, l. The center arcs are shown as straight lines between input nodes and Gaussian nodes in Figure lb.
SIn
G,(u) = I(t)
NSln
A
_ J
(u-u,)
This output includes the RF signal,
I(t) , weighted by
(5)
the angular
filtering of the antenna system. We will later derive the signal direction, U, from the response at adjacent beam ports.
The network identifies one Gaussian node with a center and its training angle, so that when the array later receives a signal from that angle, the Gaussian-node output is maximum, and decays monotonically for all signals from other angles. Selection of a second and subsequent training angles produce additional Gaussian nodes and centers only if that is necessary to obtain a sufficiently accurate output-angle estimate. The value of the lth Gaussian-node output is given by
4. A neural beamformer for signal location Figure 1b shows a neural-network direction-finding array that consists of a stage of preprocessing; a three-layer radial-basis-function (RBF) network that uses Gaussian radial basis functions; and a stage of post-processing. This architecture is chosen because it is simple and crafted to address the antenna-control task, and because it has been studied in detail by others [1, 2]. As depicted, the network collects preprocessed signals. It fans them out via the straight-line paths called "center arcs" to another set of nodes, called "Gaussian nodes." It then distributes the output of these nodes through a series of paths, defined by adaptive "weights," to the network-output nodes. The final stage of post-processing associates the output-node computed values with the desired angular parameters or source location.
Gl
:;:::
~N-l 2/ 7 2 e - c. 11--1 en' sa
,
(7)
with a being the spread parameter of the Gaussian node I. This expression shows the Gaussian-output maximum as unity when all distances enl are zero. The weight arcs, W, connect the outputs of all Gaussian nodes to the output nodes to obtain the best angle estimate. They are given by
The operation takes place as follows. Received signals from the array elements are preprocessed to eliminate extraneous data, and to convert the available signals into something that the neural network can efficiently use. For example, when the array receives a single incident signal, there is no expected amplitude variation between signals at each element (edge effects neglected), and so the preprocessor obtains the phase or phase differences between adjacent elements. The resulting data from this preprocessing are inserted into input nodes, In ,-of which there are usually two nodes used per signal-to account for in-phase and quadrature components.
(8)
where wkl is the weight imposed on the arc connecting Gaussian node I to output node k. The selection of these weights can be done in several ways. One method investigated uses gradient-descent-error backpropagation to iteratively determine optimum weights. A second method, called Linnet [2], places Gaussian-radial-basis functions. centered at every training sample, and uses linear algebra to solve for the weights that are the least-mean-square approximation to the correct angles. The selection of weights is an extremely important part of the adaptive process, because the Gaussian nodes overlap, yet the system is required to have only one significant output at each training point. Thus, the weights are chosen to make all the contributions from extraneous outputs sum to nearly zero (within the acceptable accuracy). This process is explained more fully later, with reference to Figure 3.
The RBF network then conducts a sequence of operations based upon the selection of a group of training angles, at which it will be required that the output nodes estimate the location of the training angles within some desired accuracy. This process is done adaptively, and is the essential feature of the neural network. Ultimately, the training process results in the set of output weights on the "weight arcs," described below. When the signal from the first trammg angle (or training point) is received, the network is required to identify one output or summation-node amplitude with that angle, with all other summation nodes taking on a zero value. This is done by creating a set of "centers," mnl' the members of which are initially equal to the received, preprocessed inputs at the first training angle. Each center is also identified as the location (or center) of a "Gaussian
Each output node then computes a summation of the weighted values generated by the M Gaussian nodes, and obtains the weighted sum at each kth summation, or output, node (9)
651
ment signals for any incident signal with direction cosine u. If the digital equivalent of the Butler matrix is used for direction finding, one could take as preprocessor inputs the phases of the input signals:
1
0.5
a
(10)
-0.5
and the centers here refer to the orthogonal-beam centers,
-1
-1
u
1
(11)
1
so, in general, one can again compute center distances with respect to the location of the orthogonal beams:
0.5
(12)
a
Using these center distances, the analog (or digital) Butler matrix sums each of the signals enJ so that at the lth output of the Butler matrix, there is a response with amplitude
-0.5 -1 l.--_ _- L_ _---I.
-1
UJ.
-0.5
a
....L...- _ _- - '
0.5
u
1
1
N- l
2
I
e
N-I n=- 2
1Cni
~
(13)
0.5
a
(14)
-0.5 -1 L--_ _.....L-_ _---L a -1 -0.5
This amplitude is analogous to the output of a neural-network Gaussian node. However, instead of the Gaussian basis function, it
....L...- _ _- - '
0.5
u
1
Figure 3. A comparison of representative Butler-matrix beamport responses and neural-network output- (or summation-) node responses, for two-, four-, and eight-beam systems (top, middle, and bottom, respectively). The Butler-matrix beamport response is the solid line, and the neural-network outputnode response is the dashed line. After the output nodes, the various outputs are post-processed in order to determine the angle of arrival of the incident signal. The post-processing consists of sampling adjacent output nodes and interpolating between them.
has the form sin(Nx )/[Nsin(x)]. The difference has interesting implications. Both basis functions have a maximum value of one, but the Gaussian basis function decays monotonically away from the center, and never quite gets to zero. Alternatively, the sin(Nx)j[Nsin(x)] has uniformly spaced zeros, and is a member of an orthogonal set that has the maximum of any member located at the nulls of any other member. This has an important impact on the choice of training points, and on the weights chosen to connect the central-node outputs and the output nodes.
If the input training signals correspond to the orthogonal beams, then for the Butler-matrix case of Figure 2z, there is no need to use multiple weights. This is because at each training point, only the one central node is activated, and the other nodes have their basis-function zeros aligned with the training angles. So, in this case, the Butler matrix is perfectly "trained" with orthogonalbeam training points if the central-node outputs, denoted by index " l," are connected to the network-output nodes, denoted by index "k," by a diagonal matrix of weights. With reference to Equation (9), the weights for the Butler-matrix case are
5. An instructive comparison The generality of the neural network is such that the input space containing the In inputs can include the recorded signals, or just their phases or phase differences. The inputs need not be taken from adjacent elements, although at some point there may be difficulties with 2JC ambiguities, unless extra care is exercised.
(15)
where c5"( k, i) is the Kroneker delta function:
Though far less general, and not adaptive, the Butler matrix performs some of the same functions as the neural network. The input nodes for the Butler-matrix case are the phases of the N ele-
(16)
652
SA =
M
L Wk/ = o, .
(17)
1= 1
Alternativ ely, for the neural-netw ork case, all the Gaussian radialbasis functions interact at all training angles, and so a full set of M weights connects each Gaussian node to every output node, in order to give the exact resulting output at each training angle. The matrix of weights is not simply diagonal, but is the full set determined adaptively by training the network.
6. Post-processing By noting the relative signal amplitudes, one can estimate the direction of any incident signal. This is the post-processing stage, and in the case of the Butler matrix it can, in fact, be done very simply and accuratel y. For orthogonal training points , one can show, as indicated in Figure 4, that by comparing the relative amplitudes of the two largest adjacent output-node values ( Sk and Sk-+ I), and by approximating the sin(Nx )j[N sin(x)] by the Sine function-which is extremely accurate for such closel y spaced beams-one can obtain a nearl y exact interpolation formula:
The Butler-matrix beam-port responses, G,(u), are phase shifted and combined RF-element signals, which give this direction-find ing beamformer its directional properties. The neuralnetwork output-node responses are not true RF beams , since the network has not passed the entire RF signal; however, the directional properties are still present. We can compare the beamport responses to the neural-network output-node responses, Sk(U), which, from Equations (7), (8), and (9), are weighted sums of the Gaussian processing-node outputs. The neural-network weights is the full set of weights determined by training the network at certain angles within the field of view of the array, but to illustrate the comparison , we can choose training points at the orthogonal -beam locations. This forces the output-mode responses to have the same maximum and zero locations as the Butler matrix. In Figure 3, we show a typical Butler-matrix beam-port response for two, four, and eight beam matrices, i.e., N = 2 , 4, and 8. For comparison, we show the output response of the corresponding summation (or output) node of a neural-network direction-finding array. For N greater than 4, the portions of both the Butler-matrix and neural-network responses between the zero crossings on either side of the beam peaks are almost identical. The adaptive weights, when required to conform to the orthogo nal training data, have thus produced an output response that is nearly identical to that of the Butler matrix.
(18)
It can be shown that this expression is valid for 1r/ N~ l.
The same interpolati on technique is used for the neural-network direction-finding array with adjacent output-node responses in Equation (18). 7. Conclusion The Butler matrix and the neural network have been compared to provide insights about the neural-network behavior for a direction-findin g array. The goal of the paper has been tutorial, since the two systems are only really comparable in the very limited case considered : an ideal array with equal element spacings, no failures, and using the orthogonal beam locations as training points . The comparison cannot explain issues that go beyond this special case, for the neural network is far more flexible than the Butler matrix, and doesn 't rely on the periodic arrangements of elements , or the proper operation of the feed lines. Even with a more generali zed array geometry, and in the presence of element and feed-line failures , the trained neural network builds the requisite number of Gauss ian nodes, with the appropriate spread and center locations , and optimizes the weights so as to best predict source location.
,,
0.8
\ Sk+1(U)
,, ,, ,, ,,
0.6 ",+04
e.t o 2: 02 e.t a j -- + --
-
-----;'-;-:--!---;;-:;---\--
-
-
,
(19)
However , within the constraints of this specialized case, the comparison illustrates the role of pre- and post-processing, the function of the Gaussian radial basis function, and the considerations in determ ining the weights applied to the Gaussian or modified Sine function node outputs . In addition, the comparison points out the basic similarity of the two procedures, and hopefull y reveals some insights about the operation of a neural network from the perspecti ve of antenna engineering.
,
-*---- -
-0.2
8. Acknowledgment
-a 4 L-_-'--_--'-_--'-_--'-_---''--_-'--_-'-_ - ' - - _ - - '_---.l -1 -0.8 -0 6 -04 -0 2 a 0.2 0.4 0.6 0.8 U =sin(8) I
The authors are grateful to Ms. Teresa O'Donnell of ARCON Corporation for the "trainin g" in the technology of neural networks , and the Air Force Office of Scientific Research for sponsoring this work.
Figure 4. An illustration of the post-processing required to find an unknown angle of arrival ( uo = sin 00 = -0.1) between training angles. For this four-elementlfour-beam example, the training angles for the two center beams shown are at ±0.25 in u space. The measured values of Sk(U) and Sk+I(U) are 0.87
A
9. References
_
1. T. O'Donnell, 1. Simmers, and D. Jacavanco , "Neural Beamforming for Phased Array Antennas ," in Proceedings of the 1992 Antenna Applications Symposium , University of Illinois.
and 0.38, respectively. Since uk+1- Uk = - = 0.5, we estimate Nd Uo = - 0.1 using Equation (17).
653
2. H. L. Southall, J. A. Simmers and T. H. O'Donnell, "Direction Finding in Phased Arrays with a Neural Beamformer," IEEE Transactions on Antennas and Propagation, AP-43, December 1995,pp.1369-1374. 3. J. Butler and R. Lowe, "Beam Forming Matrix Simplifies Design of Electronically Scanned Antennas," Electronic Design, 9, 12 April 1961, pp. 170-173.
654
Forward-Backward Averaging In the Presence of Array Manifold Errors Michael Zatman and Daniel Marshall
Abstract- In this paper, we investigate the use of forwardbackward (f I b) averaging for estimating the covariance matrix used for adaptive beamforming and space-time adaptive processing (STAP). We demonstrate that the estimation loss is reduced by the use of fib averaging and, for some STAP cases, fib averaging can even quadruple the available sample support. We also show that unknown array manifold errors have little effect on the effectiveness of f / b averaging. The gain from f / b averaging is demonstrated on data from the mountaintop database. Index Terms-Adaptive arrays.
F
I. INTRODUCTION
OR certain array geometries, there exists an invariance that allows each data vector to be used twice for covariance estimation in a process known as f [b averaging. Applications of fib averaging include an improved linear prediction estimator (1] and, in conjunction with spatial smoothing, a scheme to decorrelate coherent signals incident on an array for direction finding [2]. The application considered here is to improve adaptive beamformer performance by providing extra snapshots for covariance estimation [4]. There are samplestarved cases, e.g., when training over a small clutter discrete in an adaptive radar scenario, where fib averaging significantly improves performance. By demonstrating how fib averaging improves adaptive array performance, both in terms of the signal-to-interferenceplus-noise ratio (SINR) and the sidelobe level in the adapted array pattern, with the possibility of quadrupling the sample support in some scenarios, we extend previous results [4]. Furthermore' we prove that performance gains are still achievable with fib averaging in the presence of unknown array manifold errors. This paper is organized as follows. In Section II we briefly review the data model used for adaptive array processing, and introduce a model for array manifold errors. In Section III f [b averaging is reviewed and quadruple averaging appropriate for some space-time adaptive processing scenarios are introduced. In Section IV, the effect of array manifold errors on fib averaging is analyzed. Section V contains examples of f / b averaging used on data from the mountaintop database [5]
Manuscript received January 13, 1997: revised July 7,1997. This work was supported by DARPA under Air Force Contract FI9628-95-C-0002. M. Zatrnan is with the Lincoln Laboratory, Massachusetts Institue of Technology, Lexington, MA 02173 USA. D. Marshall is with Raytheon Systems Company, Tewksbury, MA 01876 USA. Publisher Item Identifier S 0018-926X(98)08892-9.
which demonstrate the utility of f / b averaging on real data. Conclusions are drawn in Section VI. II. DATA MODEL We assume the reader of this paper is familiar with the signal model used for array signal processing. This section is intended to briefly introduce the notation used later. Consider a uniform linear array of N sensors with an interelement spacing d though the analysis and all the results given here may be extended to any array, which produces a persymmetric covariance matrix, i.e., a matrix M such that
0
M* == JMJ,
where J =
[
~
o
~].
o o
(1)
The matrix J is known as the exchange matrix. The transfer function between bearing f) and the output of the array is represented by the Vandermonde steering vector
a(8) == [1,ej21rd>..-lsin(B)~
...
,eJ 21r( N -
l ) d>.. - l sin ( B) ] T
(2)
where A represents wavelength. Note that to within a constant phase term a( B) == J a( B) *. The data received at the array output is the sum of the K incident signals and the noise
tc
x(t) ==
L Qk(t)a(8
k)
+ n(t)
(3)
k=l
where ak (t) is the complex amplitude of the kth signal at snapshot (e.g. range sample) t and n( t) the noise vector at time t. We will define O-k (t) such that the signal to noise ratio (SNR) of the kth signal is lak(t)12 . The noise is assumed to be a zero-mean white complex Gaussian random process (though the Gaussian assumption is not a requirement for fib averaging). The steering vector a( 8) just described may be viewed as representing either a desired array response for which the array was designed, or a presumed array manifold, which due to modeling errors is different from the true manifold. In the latter case, the true manifold ah ( B) may be related to a( f)) by taking the Hadamard product of the latter with a vector of error coefficients h == [hI," . , hN]T, which preserves the output power of the steering vector, i.e. (4a)
and
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 46, No. 11, pp. 1700-1704, November 1998.
655
(4b)
o------.----.. . .
Except for the special case where h == Jh * for the true array manifold ah(O) 1= Jah(O)* to within a constant phase term, unlike the presumed array manifold a( 0). This inequality is assumed for the remainder of the paper. By inserting ah(B) into (3), data vectors for the actual array may be obtained. The elements of h are taken to be
h-i ==
C
+ .fj'i
L
1 ~ II R== L ~XlXl .
(6)
l=l
Adapted beamforming weights are computed using the inverse of the sample interference covariance matrix, i.e.,
(7)
w(fJ) == it-1a(H).
SINR loss, the ratio of output SINR to the SNR that could be achieved in the absence of interference by a matched filter, will be used as the main performance metric. For a target at angle fJ t (8)
where R is the exact interference-plus-noise covariance matrix. In this paper, the radar scenario in which the target r~nge gate may be excluded from the data used to estimate R is assumed so that the interference-only covariance matrix may be estimated and used to compute the adaptive weights. As an extension to the array processing problem, a space-time adaptive problem may also be formulated [6]. Given M temporal samples (e.g., coherent pulses) from an N -element array, the received space-time data at the tth snapshot is represented by the N by M matrix X(t). The temporal sampling is assumed to be uniform. Errors in this assumption are not dealt with here, though their effect may be analyzed in the same way as spatial errors. III. FORWARDS-BACKWARDS AVERAGING
F(x), the spatial power spectral density (PSD) of x (a single data vector), may be defined as
(9) for all values of 0 (the dependence of F(x) on f} is implicit here although it is not explicitly indicated in the notation). Given x*, the conjugate of x, and x", the reverse sequence of x, i.e., x r == Jx, then the following is true:
F(x*) == F(x·'·) == (F(x))r
and thus
iD
"
.......
.
-.;..- ....:., ...:........:.. .. -.:...:. --~~
-5
U) U)
o -10 a:
-I
(5)
where c is a real constant and .(It is an error term with a zero mean complex Gaussian distribution. Let the ratio of the variance of g'l to t: be €2. This model is used so that the expectation of some vector inner products can be constructed and used to give a "feel" for how performance varies as a function of (. The estimated interference-plus-noise covariance matrix of the data over L samples from the array is "
....... -.:..-- -.-:..-.:.-:,
-ORDINARY - -DOUBLE .... QUADRUPLE
~ -15
tn
-20 L -
50
........
~
_..I~
100 150 NUMBER OF SAMPLES
__'
200
SINR loss verses sample size for no, double, and quadruple ,./1> averaging computed using an eight-element eight-pulse STAP simulation.
Fig. I.
Since x and x r have the same spatial PSD, either or both may be used for estimating the signals' spatial parameters-e.g., in (6). The use of both is referred to as f / b averaging (double averaging). If the noise is uncorrelated between sensors, then from the covariance estimation perspective x and x r ' contain uncorrelated noise samples. To prove this, take the noise vector of (3). The noise samples are uncorrelated if E{n(t)*rn(t)H} == O(E{·} is the expectation operator). By simple algebra ?
E{n(t)*rn(t)H} == E{Jn(t)n(t)T}* == ()
(11)
for a zero mean complex Gaussian random uncorrelated process. In the space-time case, if X'" and x' are the reverse spatial and temporal sequences of X, respectively, (X8 == JX and X' == XJ) and F 2(X) is the two-dimensional PSD of X, then fib averaging may be used for space-time adaptive processing (STAP) because ( 12)
Furthermore, if there are only signals present with no temporal structure, (e.g., wide-band jamming in STAP) or with the same frequency structure either side of the center operating frequency (e.g., some wide-band adaptive beamformer structures employing tapped delay lines), then
F 2(X t ) == F 2(X*s) == F 2(X).
(13)
Hence, in some scenarios it is possible to obtain three extra samples when using f [b averaging to estimate a space-time covariance matrix (quadruple averaging). The extra samples, all of which possess the same spatial PSD as X, are X":", x-, and X*:>. The improved convergence properties for STAP through the use of fib averaging are demonstrated in Fig. 1 for an eight-element eight-pulse STAP simulation. Two jammers are present from bearings of 37° and 64°, each with .an interference-to-noise-ratio (INR) per element per pulse of 50 dB. Fully adaptive STAP [6] was used and average SINR loss curves are generated for a broadside target. With no f jb averaging the SINR loss as a function of samples is as predicted in [3]
F(xr ' ) == F(x). (10)
656
SINR loss
==
L+2-N L
+1
(14)
0-----------.. . . --.---. iD -a -10 .......
w
c
Proof:
-ORDINARY - -DOUBLE . .. ·QUADRUPLE
F (x~r) == I a(8)H x~r
::e ~
./
-40'----............w...;.---I-.-...---"""'-........................,;""....... 50 o -50
BEARING
Fig. 2. Adapted responses obtained using 200 samples.
For STAP, N is replaced with -.N" 1\11. The use of double or quadruple f [b averaging improves the SINR loss performance to levels equivalent to doubling or quadrupling the ratio of L to ». The normalized adapted response p( 8) of a spatial array is computed as
and p(f}) == max C( p f} )}
(18)
The second effect of array manifold errors is that a covariance matrix estimated from the true data vectors with f [b averaging may have a higher interference rank than one estimated without f / b averaging. This will limit the improvement available due to fib averaging. The rank of the interference increases because the signals in Xh and x~r (the forward and backward samples) are not spatially identical and it is easily shown that xj, (t) [from (3) and (4)] and x~r (t) are uncorrelated. Applying fib averaging to (3) results in
-30
p(H)
== \(a( f))*r)H xj, 12
== la(B)Hx hI2 == F(Xh)'
~ -20
Z e".
2 1
Xh(t)*r ==
t:
L CYk(t)* Jah(8J.:)* + In(t)*.
(19)
k=1
Note that in (19) the modulating signal rY.k has been conjugated. The correlation between ali: and n~ is (20)
( 15)
since the mean values of the in-phase and quadrature components of Ct'.k are the same. Since each signal in Xh is uncorrelated with its equivalent in x~,r and will be slightly different spatially, a covariance matrix produced using f / b averaging will have two signal subspace eigenvalues/eigenvectors for each signal present I. However, the second eigenvalue/eigenvector will only affect adaptive beamformer performance if it is above the noise floor. Given two spatially similar but otherwise uncorrelated signals, it is possible to determine the eigenvalues of their covariance matrix. The following equation for the smaller of the two eigenvalues was derived by Hudson [8]:
The sidelobe levels in the spatial response are a function of the number of samples [7]. Plots of the spatial adapted pattern for the example above are shown in Fig. 2 for a beam formed at broadside and zero Doppler with a 30-dB Chebychev taper in each dimension. The sidelobe levels of the adapted responses improve due to the addition of f / b averaging. IV. ARRAY MANIFOLD ERRORS The effects of the differences between the true and presumed array manifolds on fib averaging should be addressed in order to quantify the practical usefulness of fib averaging. Although the analysis in this section is for spatial adaptive beamfonning, the results also apply to STAP. Two effects of the differences between the manifolds may be found. The first effect appears when the spatial PSD is computed with the true array manifold. The true spatial PSD' s of Xh (a single true data vector) and x~,r are different, i.e.,
/\ 2
~ N p 1 P2(1 - \4) \2 )
~
PI
(21)
+ P2
where PI and P2 are the powers of the two signals, and '~) 2 is the "spatial cross-correlation" of the two signals. For the case of f [b averaging using the true array manifold with a single signal present I
1
(16) (22)
where
(17) over all (). If fib averaging is used with the true array mainfold, then, as a result of this inequality, the adaptive beamformer will adapt to a spatial PSD that does not correspond to the actual environment and performance will be degraded. However, in practice, the true array manifold is unknown, and presumed array manifold is used instead. If the spatial PSD's of Xh and x~r are the same for the presumed array manifold, then this first effect will not appear when the presumed manifold is used. Theorem: If a presumed array manifold where a(0) Ja(8)* is used, then F(Xh) == F(x~r), where F(Xh)
la(B)H x hI2.
Given the model of h from Section II, it can be shown that the expectation of the right-hand side (RHS) of (22) for small values of ~ can be approximated as 1
(23)
Assuming that ~ «: 1 the term ~4 in (23) may be ignored and the RHS of the equation may be approximated as 1 - 2~2. The term 2~2 may be viewed as the power in the error terms of the presumed array manifold. Due to the random nature of the errors modeled, on average the fraction N -1 of the error I This is why f / b averaging has a decorrelating effect in the presence of coherent/correlated multipath [2].
657
.. - - - - - - - - . 0 , - - - -.......- - -
0 , . . - - - - - - - -.....- - - - - - -
_
-2
in :Eo.
-4
~
-6
en
~
V--ORDINARY -50 dB - - ORDINARY -10 dB ...... FIB -50 dB . _. -F/B-10dB
a:: -8
Z
en
m ~-10 -5
en
I
/
FIB
<,
,
I' -
~ -20 \ I
a:
~ -25
en
-10
SINR loss versus sample size with array errors.
I4) I- ~ ?
1-
.)
Fig. 4.
may be made. Substituting this into (21) and noting that == ]J2 == 0.51 n 2 for the fib averaging case being considered here, the following equation is derived for the magnitude of the second eigenvalue, which appears due to fib averaging 1
A2
, ,) 112(Y. 2 N 1
~ 2~-
==
1.) 2 e-1nl-lV. 'J
(25)
If A2 is above the noise floor, then the performance of the adaptive beamformer will be compromised due to fib averaging. From (25), it can be seen that when 2 2 ~ <, a 1')7\.T
(26)
A2 is below the noise floor, and there will be no loss in adaptive beamformer performance. Instead of quantifying the loss for the single signal case when A2 is above the noise floor, we will look at the more general case when fib averaging causes the rank of the interference covariance matrix to increase from v to v + T, (r ~ v). Here, u is the interference rank prior to fib averaging and r the increase in rank due to fib averaging in the presence of array manifold errors. Apart from being orthogonal to the rest of the signal subspace, each new signal subspace eigenvector will be random in nature (due to the random nature of the array manifold errors modeled). In this scenario, the following expression for the SINR loss due to the increased interference rank is derived in Appendix A:
SINR loss
~
N -v-r N. -1)
o
(27)
e.g., for N == 20 and v == r == 2 the SINR loss is 16/18, which is about -0.5 dB. This loss will decrease as the number of array elements increases. From the combination of (14) (the gain due to improved sample support) and (27) (the loss due to array manifold errors) it may be determined when fib averaging will improve performance, Simulations for a 20-element array with array manifold errors of ~ = 10- 5 and ~ == 10- 1 were run. Two jammers from 37° and 64° are present, each with a 50-dB INR. Fig. 3 is a plot of mean SINR loss (averaged over the unjammed
BEARING
50
Azimuth cut of SINR loss for file IDPCA65.
angles) versus the number of samples. For ~ == 10- 5 fib averaging always improves performance.. as would be expected from consideration of (26), (the additional eigenvalues due to fib averaging are below the noise floor). However, with ~ == 10- 1 fib averaging only improves performance below about 90 samples. Equation (26) shows that for ~ == 10- 1 the rank of the interference will double from two to four with fib averaging. Hence, from consideration of (14) and (27) fib averaging should improve performance below about 90 samples. Simulation and theory agree.
(24)
2~-
-30
-50
power is spatially correlated with the signal. Providing N is large this may be also be ignored. Hence, the approximation
PI
-
~ -15
-12 '-------"---~_ _......__ _......J o 50 100 150 200 NUMBER OF SAMPLES Fig. 3.
- - ORDINARY
v.
REAL DATA EXAMPLES
Results from the application of .fIb averaging to data from the mountaintop radar data base are also presented here. A 14-element nominally uniform linear array is used. Although all of the receive channels were equalized, the array itself is not perfectly calibrated. The radar transmitter array is capable of simulating platform motion (see [5] for more details on the mountaintop system). PRI-staggered STAP [6] with a threepulse sub-CPI was used to process the data. The first data set (file IDPCA65 collected on 3/9/94) contains ground clutter returns, which form a ridge that is an unambiguous function of bearing and Doppler. 100 training samples were used to compute the adapted weights, with f [b averaging doubling the sample size. Fig. 4 shows an azimuth cut of the estimated" SINR loss at 64-km range and O-Hz Doppler. Assuming no array manifold errors, the expected improvement due to averaging given by (14) is about 1.25 dB. Excluding the clutter null at broadside, the average improvement is 2.9 dB. This measured figure suggests that the range samples are not completely independent. The second data set (file STAP2017 collected 011 3/28/94) contains two wide-band jamming signals at 5° and - 30° from broadside. Fifty training samples were used to compute the adapted weights with fib type averaging quadrupling this figure. Fig. 5 shows an azimuth cut of the estimated SINR loss at 164-km and O-Hz Doppler. Assuming no array manifold errors, the expected improvement from (14) is about 6 dB. The average improvement for the plot due to f Ib averaging (excluding areas near the jammers) is 5.7 dB. 2 Since for exprimental data the exact covariance matrix is unknown, an estimated covariance matrix was used in the denominator of (8) to compute the SINR loss.
658
Thus, the expected value of the SINR loss in this scenario IS about 1 - 1 IN, which for large N becomes insignificant. Now consider the case of T random eigenvectors and v nonrandom eigenvectors, (as may exist after fib averaging in the presence of random array manifold errors). Since all of the eigenvectors are orthogonal to one another the v random eigenvectors lie randomly in the N -v-dimensional vector subspace orthogonal to the v nonrandom eigenvectors. Then for each of the T random eigenvectors the expected value of the quotient on the RHS of (30) is
10 ..--......- -.-.---..--,.-_..~-........---..-.,
in
0
~
/
en en
o -10 -oJ
J\
,
/
a:
,
\ I , I
~ -20 en
--ORDINARY
-
\I
I -30 '"-......- ...............-......_...... -60
Fig. 5.
-40
- FIB
......_ ......... 40 60
--~_
-20 0 20 BEARING
e. E{ a(la(B)H B) a (H)
i I2}
H
Azimuth cut of SINR loss for tile STAP2017.
Extending this to the case of an SINR loss of
VI. CONCLUSION We have shown that for STAP fib averaging can double or even quadruple the sample support. Furthermore, we have proven that there are many scenarios where f / b averaging improves adaptive beamformer performance despite the presence of unknown-array manifold errors. These cases are when a medium to large array with only poor or adequate sample support is used. The improvement due to fib averaging was demonstrated by simulation and on experimental data from the mountaintop database. ApPENDIX A
SINR Loss Due to Random Eigenvectors: Noting that and a(H)Ha(H) == N, (8) for SINR loss becomes
RR- 1 ~ I
SINn. loss ;:::;
a(H)Hft-1a(H) a( &)H a( &) "
(28)
Providing the eigenvalues of all of the interference subspace eigenvectors are above the noise floor, then (29)
where E == [e1,"', e(v+r)] is the matrix of interference subspace eigenvectors. This approximation will overestimate the influence of eigenvectors with eigenvalues near to the noise floor. However, simulation shows that this approximation is adequate for eigenvalues at least 10 dB above the noise floor. By plugging (29) into (28), the following equation which assesses the effect of each interference subspace eigenvector on the SINR loss is produced:
SINRloss;:::;l-
L
i=r+v
la( e)H e.i 12 a(B)Ha(B) "
(30)
1,=1
It is instructive to first consider the case of an interference subspace of rank 1 where the interference eigenvector is "random." By consideration of two vectors lying on an Ndimensional sphere, it is easily shown that the expected value of the quotient on the RHS of (30) is
E{ a(la(B)
He 2 -i 1
())H a( B)
}
==
~. N
(31)
659
SINn. loss
~
t:
==
(32)
_1_ N - v.
random eigenvectors results in
r 1 - --
lV - v
==
N
-V-'1'
N -
'(J
(33)
REFERENCES [1] S. W. Lang and 1. H. McClennan. "Frequency estimation with maximum entropy spectral estimators." IEEE Trans. /scoust.. Speech. Signal Processing, vol. ASSP-28. pp. 850-861. 1980.
[21 S. U. Pillai and B. H. Kwon. "Forward/backward spatial smoothing for coherent signal idcnnfication." IEEE Trans. Acoust., Speech. Signal Processing, vol. 37, pp. 8-15. 1989. f31 I. S. Reed, 1. D. Mallet. and L. E. Brennan. "Rapid convergence in adaptive arrays," IEEE Trans. Aerosp. Electron. Svst.. vol. AES-9. pp. 853-863, 1973. [4] R. Niztberg, "Application of ML estimation of persyrnmetric covariance matrices to adaptive processing," IEEE Trans. Aerosp. Electron. Svst., vol. AES- I 6. pp. 124-127, 1980. [5] G. W. Titi and D. Marshall. "The ARPA/NAVY mountaintop program-Adaptive signal processing for airborne early warning radar:' in Proc. ICASSP '96, Atlanta. GA, May 1996. pp. 1165-1 168. [6J J. Ward. "Space-time adapnve processmg for airborne radar," MIT Lincoln Lab. Tech. Rep .. vol. 1015. 1994. [7] B. D. Carlson. "Covariance matrix estimation and diagonal loading in adaptive arrays." IEEE Trans. Aerosp. Electron. Svst.. vol. 24. pp. 397-401, 1988. [8] J. E. Hudson, Adaptive Array Principles. London, U.K.: Peter Peregrinus, 1981.
Chapter 5 Experiments
A
series of experimental systems from academic institutions and the mobile communications industry have demonstrated the viability of adaptive antennas to provide the claimed advantages. The results of some of these trials are presented in this chapter. The first paper presents an experimental system for multiple source direction finding, while the second deals with an adaptive antenna that employs the CMA algorithm with a OMSK system. The next three papers were selected from a very significant (from a technical point of view) session on "Adaptive antennas: measurements and results" of the 47th Vehicular Technology Conference in Phoenix, Arizona, USA, in 1997. Therein are presented results for the performance of a four-element antenna test bed from AT&T and Lucent Technologies using the DMI algorithm for the IS-136 system, Ericsson's OSM adaptive antenna test bed and the trials with Mannesman Mobilifunk, and the direction finding adaptive antenna test bed from the Aalborg University for GSM/UMTS (partly funded by the European project TSUNAMI II).
The next paper presents results for a multibeam smart antenna system by Metawave, and the last two papers present results from possibly the most influential research project in the field of adaptive antennas for mobile communications, the TSUNAMI I project (Technology in Smart Antennas for Universal Advanced Mobile Infrastructure). Field trial results demonstrate both transmit and receive digital beamforming and the spatial division multiple access (SDMA) concept in terms of tracking and BER performance. Also, practical issues that play a key role in the performance of a direction finding adaptive antenna system, such as calibration errors and the linearity of the transmit amplifiers, are considered. Following work like that presented in this chapter, the research community now faces the challenge of taking this technology and the associated standards closer to efficient full-scale network deployment.
661
Multiple Source DF Signal Processing: An Experimental System RALPH O. SCHMIDT,
MEMBER, IEEE, AND
INTRODUCTION
M
ANY PHYSICAL PROBLEMS in sonar. radar. seismic, etc. signal processing utilize the records of multiple sensors to locate one or more sources of coherent energy-generally assumed to be point source, omnidirectional emitters. Typically, the sensor outputs contain the source waveforms as modified in amplitude and phase by the medium between the sources and the array elements and the transfer function of the array elements themselves. Essentially all of the geometric information is encoded in the set of pairwise cross correlations associated with the sensor outputs and arising from the propagation of the directional wavefronts across the array. Typical direction finding (OF) systems utilize techniques which are derived assuming that only a single source is being received. Some of these DF systems are nonetheless employed in multiple source environments by extending the implernentaR. O. Schmidt is with Saxpy Computer Corporation (formerly GuilTech Co.), 255 San Geronimo Way, Sunnyvale, CA 94086. R. E, Franks is with ESL/TRW. Inc., 495 Java Drive, Sunnyvale. CA
94088.
IEEE Log Number 8406823.
MEMBER, IEEE
tion rather than the theoretical basis. A DF system designed to handle a single source at a time may indeed be expected to work in a multiple source environment if one of the following conditions is true.
Abstract-The multiple signal characterization (MUSIC) algorithm is an implementation of the signal subspace approach to compute parameter estimates of multiple point-source signals from the observed voltages received on an array of M antennas. In it, the solution to the multiple source direction finding (DF) problem is provided by the intersection of the signal subspace (obtained from the received data) and the array manifold (obtained via array calibration or prior knowledge of array directional characteristics). The MUSIC algorithm was implemented to experimentally verify the performance of the signal subspace approach to DF under very general scenarios and conditions which are regarded as difficult to impossible in traditional systems. The results of those experiments are described herein. The experimental system consisted of an eight-element antenna array 13 wavelengths in diameter, an eight-channel receiver and digitizer, and a minicomputer with disk storage to process the digitized data. With ideal instrumentation, the MUSIC algorithm provides performance that, as the amount of data collected increases without limit, is asymptotically ideal. However, with finite precision and finite data collection, the performance of even an ideal system can be a sensitive function of source and scenario parameters. Tests demonstrated the resolution of three sources all within one beamwidth (5°), even when the closer two were spaced less than 0.2 beamwidths, Sources that were polarized differently could be resolved at closer spacings. Experimental DF accuracy was limited by the uncalibrated scattering of source energy from the test range support tower and from the ground. The measured direction of arrival of one source changed by less than 0.01 beamwidths as the other two sources were switched on and off in all combinations. In general, results indicated that all parameters of a source can be measured and the signal waveform can be recovered as well in the presence of other sources less than a beamwidth away.
Manuscript received April 12, 1985; revised August 29, 1985.
RAYMOND E. FRANKS,
1) The signals are well separated in frequency; i.e., by more than the reciprocal of the observation time. 2) The signals are well separated in time; i.e., by more than the reciprocal of the bandwidth of the data. 3) The signals are well separated in direction of arrival (DOA); i.e., by more (in radians) than the reciprocal of the diameter (in wavelengths) of the array collecting the data. However, if the system is receiving signals from several sources that overlap in frequency, overlap in time, and have DOA's within an array beamwidth (i.e., overlap in DOA), more capable processing techniques, such as those based on the signal subspace approach, are needed. This paper is intended as a performance report on an experimental system based on the signal subspace approach [1], [2], [3] to multiple signal OF and source parameter estimation. (The work was done as part of the ESL and IR&D program.) There are other approaches to processing array data but which were not implemented in this system. Most are not available in the literature with the capability to treat the completely general problem, thus they would not actually be compatible with the implementation here. Most can be categorized as based on one of the following. 1) Traditional beamsteering, i.e., either physically or electrically turning an array with a known main beam response and regarding a maximum in the perceived power as a DOA estimate. 2) Nullsteering, i.e., either physically or electrically turning an array with a known null response and regarding a minimum in the perceived power 1 as a DOA estimate. 3) Computational signal processing, i.e., the sensor voltages are subjected to computations which separately detect the presence of multiple signal sources, estimate the parameters (e.g., DOA, strength, polarization, correlation) among the multiple sources, and design sets of weights to be applied to the sensors which will suppress all the sources but one in order to READ 2 (reconstruct the waveform of) the selected source. I Plotting the reciprocal of a steered-null response provides a very "sharp" maximum when compared to a steered-beam response. Thus, the approach has led to what are known as "high-resolution" methods. Of course, no actual beams are formed which are any sharper than traditional beams. 2 The all capital letters version of the word is used to place the function of reconstructing the signal waveform of a selected source (i.e., to READ the source) on the same level as the function of finding the direction to the source (i.e., OF).
Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-34, No.3, pp. 281--290, March 1986.
663
The signal subspace approach can be said to be a high resolution, computational approach. The principal high resolution approaches include those of Capon [4], Burg [5], and Pisarenko [6]. Many variations on these approaches have appeared. 3 THE DATA MODEL
The received data vector is modeled vectors and a noise vector. A signal confined to the signal subspace; i.e., DOA vectors. A received data vector
as the sum of the signal vector is by definition the span of the source x can be written
XI
X2
0(8 1)0(02 ) " . 0(8 K )
II /2 +'
iK
WI
W2 WM
or x = A f + w. Then N received data vectors can be arranged as the columns of an (M x N) data matrix X with X = AF + W. This is an additive noise model and therefore applies almost directly to radar, sonar, etc., but not necessarily to autoregressive data such as speech or stock market prices. It should be noted that the underlying data model for MUSIC is equivalent to assuming that received signals of interest are from point sources .... If the source vector f and the noise vector ware independent, the expected data covariance matrix S is
They are stored according to azimuth (Az) and elevation (El), The DF problem consists largely in solving for the DOA vectors of the signals in the observed data vectors. The' computation consists of evaluating the DOA spectrum on the grid of stored array manifold points. Assuming that the grid is not fine enough to satisfy final DOA accuracy requirements, the final step is to interpolate among the candidates nearest the peak. Traditional means of interpolating between samples of a function to find the peak by, for example, fitting polynomials, is quite undesirable due to the sharpness of the peaks and the intrinsically incorrect model. Further difficulties arise when signal polarization is unknown since the DOA vector fully depends on source polarization. 5 Since the grid of array manifold points is the grid of points where the DOA spectrum gets evaluated, the precise DOA will be found by interpolation. But the DOA spectrum depends on the source polarization which is unknown. What is needed is a means of computing the DOA spectrum at the grid points without 0 priori knowledge of source polarizations and the means to interpolate between them without that knowledge. The expression for the DOA spectrum given below meets the first requirement. The second is met by a multilinear interpolation algorithm described in [7] and implemented for use in this experiment. Multilinear array manifold interpolation has proven to be accurate, efficient and completely consistent with the requirements of the polarization aspect of the DF algorithm [2], [3], [8].
S=APA*+ a2So
THE
where P = (l/N)FF* and a2So = (liN) WW*. (The noise covariance matrix So is usually normalized such that tr (So) = M. Then (11M) tr (0'2So) = (12 so that (12 is the average sensor noise power.) THE ARRAY MANIFOLD
The signal subspace method [I], [2], [3] characterizes the directional response of the array in terms of an "array manifold' which is made available either analytically or by "calibration." The array manifold is the response of the system to a single signal source as a function of position or DOA. In the completely general case where the array elements are quite arbitrarily positioned, with arbitrary response patterns and polarizations, the array manifold completely characterizes and fully represents the array response. The array manifold is stored in the computer as a collection of vectors of (complex) voltages, each vector corresponding to a particular DOA. 3 Consult almost any of the recent issues IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION or TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING for references to high-resolution
direction-finding. Especially noted are the works of Bienvenu and Kopp, Cadzow, Gabriel, Gething, Haykin, Johnson, Kumareson, McDonough, Owsley, Reddi, Tufts, Wax, Widrow, and others. 4 Alternative models include that of the autoregressive model which appears implicitly in the maximum entropy method. In this model, noise-free signal data is generated by driving a system having an all pole transfer function with an impulse. To generate noisy data, one must drive the system with noise. The model is not an additive "noise model. DF methods based on this model are necessarily (slightly) mismatched to the additive noise data of this experimental system.
MUSIC
ALGORITHM
The multiple signal characterization (MUSIC) algorithm [1], [2] is an implementation of the signal subspace approach to multiple source location and signal parameter estimation. The signal subspace itself is a vector subspace to which the received data vectors, if noise were absent. would be confined. With K signals present. the signal subspace is a Kflat (i.e., a K dimensional subspace) with "zero thickness." The data matrix X and the data covariance matrix. S are of rank K. With a small amount of noise present, the K-flat becomes "thick with noise. " The matrices become full rank. With noise present, the signal subspace must be estimated from the received data. The intersection of the signal subspace and the array manifold then provides the DOA vectors which can be converted into the desired DOA's. The algorithm consists almost entirely of geometric computations, such as vector dot products, matrix transformations, projections onto subspaces , lengths of vectors, eigenvectors, etc .. To solve the multiple signal DF problem, the MUSIC technique makes use of the matrix S of cross correlations between the individual sensor (antenna) signals. By means of an eigenstructure analysis of the S matrix, it is possible to isolate the complex vector subspace in which all the single signal DOA vectors must lie. Thus, the correct DOA's for all
664
S A DOA spectrum, as characterized above, is a search throughout DOA space for a best fit. If the signal subspace DOA vectors depend on signal polarization, will it be necessary to expand the dimensionality of the search space to include polarization parameters? Fortunately, no! Refer to [2] for details that show that polarization is a parameter similar to amplitude in that it may be estimated after DOA.
the signals may be found simply by determining which vectors in the array manifold lie in (i.e ., intersect) this subspace. In addition to OF, it is often desirable to reconstruct a selected signal source waveform in spite of the presence of other (interfering) signals . That is, one may select from the available OOA's the one to be used to compute the "signal recovery filter" vector of M weights to use in summing the M antenna voltages in a way which best reproduces the selected signal while suppressing the effects of noise and the other (interfering) signals. The MUSIC algorithm is a technique making use of the fact that the eigenvectors of S define invariant subspaces such as the signal subspace . That is, the eigenvectors can be used to compute a spectrum with DOA as the independent variable. The peaks of the spectrum correspond to signal directions of arrival. The signal subspace approach leads to a "high-resolution " method in the sense used by Capon [41. To explain this . think of conventional DF methods in terms of turning an antenna or array with a highly directional known response . The received energy leve l will peak at the DOA desired . Now think of a dipole or loop antenna with a broad peak response in one direction but a sharp null in another known direction . Turning this antenna until the received power level nulls (i .e.. until the reciprocal of the level peaks) leads to . ' high resolution" as compared to the conventional approach . The algorithm . as implemented for this experiment. operatcs on the .\1 x M complex matrix (S matrix) of correlations between the M antenna array elements to produce the DOAs of up to (,\1 - 1) signals . 6 There arc alternative implementations of the MUSIC algorithm which . for example. operate on the data matrix itself. but which will not be discussed here. Fig . 1 is a block diagram of the MUSIC DF /READ process. indicating that the data matrix X is the matrix of antenna voltages and that the S matrix is the data covariance matrix which can be directly obtained from the data matrix as
STA RT (N)
- 10" < II < 10" - 30" < . < 30*
Fig . I .
MUSIC OF /READ algorithm .
nullspace has been found. If this were a case of sensors in a scalar field instead of a vector field. the DOA spectrum would be computed using
P(8, rjJ) ...... - - - - - - a*(8, ¢)E"Eta(8, ¢)
where the matrix Es has for columns the eigenvectors of S spanning the orthogonal complement of the signal subspaceoften referred to as the noise subspace though. since noise is also in the signal subspace. it may be more accurate to call it the signal nullspace . In the present case. the sensors (antennas) are immersed in a vector field . There are two array manifolds, al(8, rjJ) and a2(8, rjJ) corresponding to two different test polarizations used to calibrate the array . (For details , consult [2], [3], or [81.) For this experiment. the array manifold was stored every 0 .25· spanning the range - 10· to + 10· in elevation and - 30· to + 30· in azimuth and was calibrated with vertical and horizontal linear test transmitters. The DOA spectrum was computed using
S ...... (l IN)X X* .
The eigenvalues of the S matrix represent the power 7 in the data. If the noise were zero. there would be as many nonzero eigenvalues as there were sources. A test to decide the number of sources present consists of deciding how many eigenvalues represent noise alone versus noise and signal. This is the detection part of the problem . The detection test can be assessed experimentally by collecting statistics on ., false alarms" and "missed detections." Under simplify ing assumptions about noise and signal statistics, array geometries and sensor response patterns. the test may even be assessed analytically . The DOA spectrum can be computed once the signal
P(8, rjJ)
+-----------------------
, The M sensors may be im me rsed in a scalar field (e .g . • (he pres sure in air Or water or earth associated with an acoustic wave) or in a vector field (e .g . . the electric or magnetic vector associated with a radio wave ). Scalar field sensors lead 10 a vector array manifold where in up to (M - I) sources can be proccssed. Elcctrornaunctic field se nsors lead 10 a bivcctor array man ifold (see [3]) which allows up to (.'vf - 2) sources . in gene ral. or ( M - 1) if POlarization diversity is not util ized . An array of identically polarized antennas is nondiverse and may as well be o pe rating in a scalar field. 7 The eigenvalues are the powers in the orthogonal directions specified by the corresponding eigenvectors . The sum of the eigenvalues is the total power in the data because the eigenvalues are onhogonal contributors but the indiVidual eigenvalues do not specify individual signal powers .
l\min( [ :;~:: :~ ] ENEtv [
a,(8, rjJ)a2(8, rjJ)
J)
which is a reciprocal of the minimum eigenvalue of the 2 x 2 matrix in parentheses . The algorithm "detects" the number of signals present with their individual DOA 's and it will also produce estimates of the received power for each signal, the polarization of each signal, and the time correlations between the signals (to indicate multipath; i.e ., multiple DOA reception from one source).
665
The source covariance matrix P was computed using
P<-(A *So IA) -IA *So I(S - AminSO)So IA(A <s; IA) - I. In this experiment, both azimuth and elevation are produced since both coordinates are represented in the array manifold. (Indeed , complete source location would be available if range were also represented.) Source polarization is also "observable" since there is polarization diversity among the receiving antennas. If. on the other hand. all antennas had been dipoles arranged in parallel, there would have been no means for sensing an electric field component other than that in the direction of the dipole . Thus. the array manifold collected and stored for this experiment had to include the fact that, for each DOA, there is a two-flat (twodimensional linear subspace) containing all possible DOA vectors corresponding to all possible source polarizations. See [2], [3], or [8] for a discussion . The array manifold is actually two array manifolds; one for each of two differently polarized incident waves. At each DOA, the two array manifold vectors define the two-flat corresponding to the DOA.
Fig . 2.
Antenna dement .
AN EIGHT-CHANNEL EXPERIMENTAL SYSTEM
The MUSIC algorithm has been under development on IR&D at ESL for several years . DF systems have been built and tested that operated with four vertical dipole elements in an array with less than half-wavelength spacing. Operation against up to three signals has been demonstrated. including the development of special software checks to resolve ambiguities which can appear when a system with M antennas is used to resolve a full complement of (M - I) signals. Polarization resolution has been demonstrated using an array of four slant 45· polarized elements against both vertically and horizontally polarized signals. A version of the algorithm that will resolve 100 percent correlated multipath has been demonstrated with a four-element array. The most recent experimental system employed an array of eight antennas and an eight-channel receiving system . This system has been used to demonstrate DF and READ (i.e .. generation of a separate array output for each signal reconstructed) on up to three signals. The remainder of this report describes the results of tests on the eight-channel experimental system operating in moderate to high signal-to-noise ratio (SNR) environments.
A. The Array and Antenna Elements The antenna array consists of circularly polarized cavitybacked spiral antennas fed by a PC board Roberts balun, shown in Fig. 2. Eight antennas were used, with four left hand and four right hand circularly polarized. The array diameter is 86 in which is about 13 wavelengths at the operating frequency of 1.8 GHz. This was the maximum size that could be conveniently built and tested . The elements were somewhat arbitrarily placed on the array frame as shown in Fig. 3, with five antennas on the circumference and three within the circle.
set during experimentation to drive the digitizers to full scale or other desired level. The digitizers convened all eight channels of analog to eight-bit binary digital simultaneously . Sampling rate was 16 kHz. giving a usable IF bandwidth of 6 kHz and a maximum data collection time of 300 ms.
B. The Receivers and Digitizers
C. The MUSIC Algorithm Software
Eight parallel channels of mixers, amplifiers, and filters were constructed to act as the receivers. The signal levels were
The block diagram of the MUSIC algorithm software used with the eight-channel hardware is shown in Fig. I. The
Fig. J.
666
Eight-clement receiving array .
computations were carried out in an HP2100 minicomputer with direct memory access (DMA). Approximately 5 Mbyte of disk storage was available for program storage as well as for storage of the array manifold and the actual data. The basic computational sequence depicted in Fig. 1 is: 1) collect the (real) data and convert to complex analytic by
constructing the quadrature channel. of the data and compute the eigenstructure-decide the number of sources; 3) compute the DOA spectrum at the stored array manifold grid points-find the peaks; 4) interpolate between the grid point peaks for precise
2) form the S matrix
DOA's~
5) calculate the source parameters: strength, polarization,
correlation: 6) compute READ channel optimal data weights for desired signal(s)-output data channelis): 7) provide interactive graphics to portray data and results.
1) A Three-Signal Experiment: Three sources were positioned within the field of view of the array. For convenience the three radio sources were numbered 1, 2, and 3. Source 1 was the same source used to collect the array manifold. The sources, all placed within a beamwidth of each other, turned on/off in all seven combinations while received data were processed for the purpose of estimating the source parameters. Source 1 transmitted a carrier at 1.8 GHz. less 2 KHz. It was approximately 100 percent AM modulated with a 400 Hz. sinusoid. Source 2 transmitted a carrier at 1.8 GHz. plus 2 KHz. without modulation. Source 3 was also unmodulated with a carrier frequency of 1.8 GHz. Fig. 4 is a small portion of the DOA spectrum resulting from such processing. (A much larger portion is given in Fig. 5.) The number of sources is "decided" by the algorithm in this case to be three, and the three peaks are then found to the nearest array manifold grid point. Array manifold vectors are available from storage for a grid of Az/EI' s spaced every 0.25 (i.e., about a twentieth of a beamwidth). Interpolation between grid points is then performed using the multilinear interpolation algorithm [7] to get the final DOA' s, which are listed in Fig. 4 along with the other source parameters estimated (power, polarization, and waveform correlation coefficients) . The spread on the measured DOA estimates for each of the three signal sources over the seven on/off combinations possible is given in Table I. It was suspected that multipath on sources 2 and 3 was causing greater DOA spread for these signals than for source 1. but it was not possible to eliminate the multipath on the test range. To verify that multipath was the source of error. a computer simulation of the three-signal case was made, adding in multipath at a level of - 26 dB for sources 2 and 3. and adding in noise at a level of - 35 dB. The spread on DOA estimates from the simulation for each of the three signals for the seven combinations is given in Table II. The simulated signals were all vertically polarized. while signal 3 was circularly polarized for the experiment. Even with this difference, the spreads for the simulation were similar to that for the experimental measurements, confirming the suspicion that multipath associated with the experimental set-up was the major source of the DOA variation. The array of eight antennas used in the experiments is a "sparse" array. The spacings between adjacent elements are substantially greater than a half-wavelength. This array diameter of 13 wavelengths could support 26 antennas across or about 600 elements to fill out the circle at half-wavelength spacings. Sparse arrays are subject to ambiguities, i.e., false indications or peaks in the DOA spectrum. Fig. 5 portrays the situation for this experiment. An "extra" peak appears toward the back of the 20° by 20° patch as shown but only when source 3 is "on." Since the number of signals is known from the eigenvalues to be three, and the ambiguous peak is about 10 dB below the others, there is no problem selecting the correct peaks and finding the correct DOA's. (Note also that source 3 is a circularly polarized source and the array elements are half right hand and half left hand polarized. Thus, to this source, the array acts like a mere four-element array.) 0
The MUSIC algorithm as implemented for this experiment did not include the steps necessary to treat 100 percent correlated multipath. See [2). [3] for a discussion of how a subspace interpretation of the data reveals that multipath is present and how it can be algorithmically treated. Some ·'uncalibrated" multipath was received along with the direct path signals during the experiment. which was a small source of error.
D. Instrumentation Error Sources Limitations imposed by the test hardware and antenna range conditions include: 1) eight-bit quantization of data . ..w dB equivalent SNR: = - 112 dBm. maximum signal = 70 dbm, 42 dB SNR: 3) variable SNR down to about 30 dB on each signal to prevent digitizer overload with three signals present: 4) test range multipath 20 to 30 dB below direct path: 5) mechanical array directional repeatability = 0.02 2) thermal noise
0
.
E. Results of Multiple Signal Tests Tests were conducted with two. three. and four signals incident on the eight element array. The inherent beamwidth of the 7 ft array at 1.8 GHz is 5°, and source DOA separations will be stated as fractions of the beamwidth. The array manifold was collected using a 6 ft reflector antenna as the source, designated as source 1. Source 1 had a bearnwidth of about 7 0, so it did not produce multipath from the surface of the parking lot between the sources and the array. It did generate some multipath from scattering off the array tower support structure, but this was a part of the total incident field during calibration. Thus, multipath did not cause any significant error on source I. Source antennas 2 and 3 were low-gain antennas that did produce multipath from the parking lot and also from the tower. Field probing in front of the array showed 1 to 2 dB field variation, corresponding to multipath levels of - 20 to - 26 dB relative to the direct path.
667
3 PEAKS FOU ND AT THE FOLLOWING GRID POINTS AZ EL
-5 .25 - 0.75 -3 .00
.1 '2 '3
- 5.00 -4.25 -5.00
Max Spec tr el Peak
= 28 .3 dB
2. 25 INTERPO LA TED DO AS AND SIGNAL PARA ME TE R ESTI MATES AZI MUTH ELE VATIO N POW ER (DB) POL . TILT
- 5.207 -0.718 - 3.163
II
'2 '3
- 4.888
-4.366 -5.055
o
/
- 10.470 - 4.90 1 -8.152
43 .611 - n.276 - 88 .68 9
ELLIPT 0.048
0 .865 0.02 1
t
CORRE L ATION COEFFI CI ENTS (NORM ED TO UNIT PWRS )
1.00 0 .12 0.07
0 .07 0 .14 1.00
0.12 1.00 0.14
Fig . 4 .
Three-signal field experiment.
A mb igui t y? No! It is abo ut 10 dB be lo w source pe ek s.
.1
.2
.3
Source DOA Peaks
Fig . 5.
Three-signal DOA spectrum; 20· x 20· Az/EI patch .
TABLE I MEASURED SPREAD ON DOA ESTIMATES Source I
Azimuth Elevation
0 .036·
0.015·
Source 2
Source 3
0.124·
0.067· 0.036·
0.070·
TABLE II SPREAD ON DOA EsTIMATES FOR SIMULATED COMBINATIONS OF SIGNALS
Azimuth Elevation
Source 1
Source 2
Source 3
0.035·
0.095·
0.129·
0.001·
0.021·
0.019"
The signal recovery function is shown in Fig. 6. The bottom trace is the result of processing with only the desired signal (source 1) "on," i.e., without interfering signals present. This result is stored for use as a reference against which the comparisons can be made to evaluate various signal recovery algorithms. The bottom trace need not line up, therefore, in time with the other traces. In the top trace, a single antenna is selected and the signal received is plotted . The combination of signals from all three sources are received; the individual signals cannot be distinguished. The second trace is the output of a conventional beamformer steered in the direction of the desired signal. Since the 5° beamwidth of the array , while narrow, is stiII broad enough to include all three signals, the result is again expected to be a combination of the three signals. The desired signal (the 400 Hz amplitude modulated signal shown as the bottom trace) is now just visible . In the third and fourth traces, the eight antenna waveforms are weighted and summed using a vector of weights designed to maximize SII + N (i.e., the ratio of desired signal-powerto-power-in-noise-and-interfering signals). The weight vector h is obtained by solving Sh = 00 for h where ao is the DOA vector of the desired signal. The conventional max S/(1 + N) beam former is the result of applying the vector of weights h = S - lao to the data and is often referred to as the WienerHopf solution. But the DOA vector 00 is estimated rather than known exactly . Ideally. ao lies entirely in the signal subspace-it has no signal nullspace component. As a practical matter. the estimated ao vector will generally include a component in the signal nullspace . Such a component. however slight. can be the source of an unexpected and inordinately large loss in S/(1 + N). That is, performance can be "hypersensitive" to any nonzero signal nullspace component in ao if the weight vector h is the result of solving Sh = ao. To avoid hypersensitivity, the weight vector h can be found another way. The Wiener-Hopf solution does not apply to the pointsource signals plus noise model that generates our data. The matrix inverse we seek is the inverse within the signal subspace . That is, we wish to solve an equation similar to Sh = ao where S is replaced by a matrix whose rank and range space reflect the number and DOA's of the point sources thus far estimated and ao is replaced with a legitimate array manifold vector which also lies in the range space of the matrix. Such a solution for h would not be hypersensitive in the case of point sources. However, the above solution has not been described in directly implementable terms. Our experimental implementation of this solution is motivated by practicality. It is easily implemented. We have chosen to solve Sh = ao after "correcting" 00 by projecting it onto the signal subspace. Hypersensitivity does not arise in this "subspace-corrected" beamformer. The fourth trace of Fig. 6 is the result of solving the Wiener-Hopf equation for the weights using the best DOA vector from the array manifold. The third trace is the result of
668
80.0
240 .0
160.0
DESIRED SIGNAL DOA USEO AS ESTIMATED FROM REC 'D DATA :
Fig. 6.
=
- 5.21,
EL
=
-4.89
Comparison of signal recovery algorithm s; three signals. READ source # 1.
correcting the best vector by projecting it onto the signal subspace implied by the data covariance matrix S. Thus, the signal subspace approach employed in MUSIC has, as a part of its implementation, a straightforward way to avoid the hypersensitive term which leads to the problem. This is demonstrated in the third trace where the result is essenti ally the interference-free signal derived using the MUSIC version of the filter. 2) A Two-Signal Experiment: Two sources were initially set almost a beam width apart with both sourc e antennas vertically polarized, and then moved together to see how closely the two sources could be placed before the DOA estimated degenerated. The results of several measurement s from 0.80 to 0.16 beamw idth spacing are summarized in Tabl e III, and the DOA spectra are shown plotted in Fig. 7. Table III gives the DOA offset from the true mechanical angle for each condition of sources "on." The far spacing produced good results, but as the spacing was reduced to one -third beam width with both signals identically polarized , performance becam e marginal. Th e DOA error on source 2 rose to 0.72 (0.14 beamwidth) and the spectrum for this case (Fig. 7(b)) shows that the two peaks were so bro ad as to have almost merged. At this same spacing, changin g the polarization of source 1 improved resolution , as shown in Figs. 7(c) and 7(d) . In particular , orthogonal polarization sett ings gave marked improvement. Spacings down to one-fifth (Fig. 7(e)) and one-s ixth beamwidth (Fig. 7(f)) gave good results for orthogonally polarized signals. Moving the two sources closer togeth er than one-sixth BW was not possible for two reasons. First, the sources were as close as they could be without one antenna being in front of and shadowing the other. Second , even though it is possible in principle to locate two source s with different polarizations but with the same DOA, the algorithm would need to be expanded 0
AZ
to search in polarization rather than DOA when needed. As implemented , it searches only DOA space for spectral peaks . The results for the one sixth beamwidth spacing are shown in Fig . 8. One source modulation was AM while the other was FM . Figs . 8(a) and 8(b) portray the attempts to READ source I and source 2. respectively. Note that the conventional beamformer is reading the sources just as well as the subspacecorrected beamformer . Although the two sources are quite close in DOA , they are nearly orthogonal in polarization. The DOA vectors are therefore nearly orthogonal . Steering a conventional beamformer toward either source (i.e., using the DOA vector of one source as antenna weights . then summing) will maximize the corresponding signal and simultaneously the other. C ONCL USION
The experim ental results reported here show that DOA and all parameters of a signal source can be accurately measured and the signal waveform can be recovered as well in the presence of other sources within a beamwidth . Performance characteristics demonstrated include the following . I) Multiple closely spaced sources can be detected, located, and read with practical system hardware. Some results obtained in our experiments with three signals located within a beamwidth were:
669
• 0 .05° (0.0 1 BW) DOA estimat ion errors , • 0 .5 dB power estimation errors , • polarization estimation errors of 2 tilt and 0 .05 ellipticity, • at least 10 dB signal-to-interference-plus-noise S/(I + N) improvement in signal recovery without performance loss due to algorithmic hypersensitivy. 0
1.00 INTERPOLATED DOAS AND SIGNAL PARAMETER ESTIMATES AZIMUTH ELEVATION POWER (DB ) POL. TILT -5.066 0 .112 - 9 .016 - 8 9.351 -9.002 - 0. 175 - 8.812 -84 .887
II .~
t
- .25 ELLIPT. 0 .007 - 0.009
t1 '2
.2
t1
INTERPOLATED DOAS AND SIGNAL PARAMETER ESTIMATES AZI MUTH ELEVATION POWER (DB) POL . TILT - 5.107 0.103 - 6 .091 - 89 .616 -5.9 53 -0.092 - 3 .7 53 - 86 .738
t
t1
(a)
t
.2
(b)
- .25
- .25
t1 '2
ELLIPT. 0.001 - 0.035
INTERPOLATED DOAS AND SIGNAL PARAMETER ESTIMATES AZIMUTH ELEVA TION POWER (DB) POL . TILT - 5.075 0 .136 -8.290 50.469 - 6.4 36 - 0. 335 -7.498 88 .072
ELLIPT . - 0.002 0.0 39
t1 ' 2
INTERPOLATED DOAS AND SIGNAL PARAMETER ESTIMATES AZIMUTH ELEVATION POWER (DB) POL. TILT - 5.09 7 0.090 - 8.944 - 1. 009 - 6.603 - 0 .337 -4.2 58 - 86.0 51
ELLIPT. 0.014 0.053
t1
(c)
1.00 - .25
t1
'2
INTERPOLATED DOAS AND SIGNAL PARAMETER ESTIMATES AZIMUTH ELEVATION POWER (DB) POL . TILT - 5.06 1 0.055 - 10 .885 - 1.074 - 5.841 - 0. 170 - 5.876 -85.347
- .25 ELLIPT. -0.035 0.036
t1 '2
INTERPOLATED DOAS AND SIGNAL PARAMETER ESTIMATES AZIMUTH ELEVATION POWER (DB) POL . TILT - 5 .0 45 0.070 -8.544 -1.215 - 6.005 - 0. 151 -4.38 1 - 83 .702
ELLI PT. - 0.041 0 .013
'1
' 1
(e)
(f)
Fig. 7. Two-signal experiment. (a) Far spacing (0.8 BW), both vertical. linear polarizations . (b) Medium spacing (1/3 BW), both vertical. linear polarizations. (c) Medium spacing (1/3 BW), both linear polarization. 40° tilt difference. (d) Medium spacing (1/3 BW), orthogonal polarizations . (e) Close spacing (1/5 BW). orthogonal polarizations. (f) Extra close spacing (1/6 BW), orthogonal polarizations.
670
TABLE III
-
TWO-SIGNAL EXPERIMENTS TWO
SIGNAL
EXPERIMENTS
" FAR" SPACING
E!J
..
O.aBW
''MID" SPACING BOTHVERT. LIN. POL
CD.. CS)
-e
HORIZ. LIN. POL. 1!1 VERT. LIN. POL.
e
-
PEXTRACLOSE" SPACING
-
HORIZ.lIN·I'O\8 VERT. LIN. POL.
-
0.03"
-
O.or 0.7r
-
2
-
O.or
1.2
o.or
0.22"
0.03"
-
1,2
O.OS-
1
0.03"
-
1.2
O.OS-
1
0.03"
2
-
1,2
1/1 BW
0.03"
1
2
1/SBW
0.01·
O.OS-
2
HORIZ. LIN. POL. VERT. LIN. POL.[IJ
-
o.or
1,2
1
113BW
'CLOSE" SPACING
SOURCE.2 OOAOFFSET
-
0.021·
2
113BW
''MID" SPACING
o.or
1
113BW
-
1 1,2
- CD
VERT. LIN. POL.1!1
SOURCE., OOAOFFSET
2
[IJ
"MID" SPACING 60· TILT, LIN. POL.
SOURCES "ON"
-
o.or
120 .0
-
DESIRED SIGNAL DOA USED AS ESTIMATED FROM REC'D DATA:
)
0
AZ
~
- 5.06,
240 .0 EL
=
300 .0
0.05
(a)
OW 0.011"
O.or 0.10"
0.10"
o.or
2) Polarization can be exploited for detection. location, and READ performance improvement. Polarization can easily mean the difference between success and failure to resolve in DOA and signal recovery . In this particular experiment, two signals one-sixth of a beamwidth apart with the same polarization were not resolved , but changing one source to the orthogonal polarization provided immediate resolution. 3) Arbitrary arrays can be calibrated with polarization diversity. A rather sparse calibration grid spacing (i.e., onetwentieth BW or 0 .25 can be used to obtain accuracies on the order of 0.01 in OOA, via an interpolation algorithm using a multilinear technique [7]. 4) The READ feature implemented is new and not hypersensitive, and makes full use of polarization information. The experimental results consistently showed superior signal recovery over traditional means . 5) The multipath-free algorithm can be used to treat scenarios with multipath as large as 20 dB below the direct path signal. Larger multipath can also be treated if decorrelated upon arrival at the array . Highly correlated multipath can be handled with the extension to the MUSIC algorithm [2], [3]. In conclusion, we note that MUSIC and the signal subspace approach introduces new, distinctly geometrical thinking about general OF/signal systems, antenna design, and considerations such as equipment imperfections and other sources of 0
180 .0
60 .0
120.0
DESIRED SIGNAL DOA USED AS ESTIMATED FROM REC 'D DATA :
180.0
AZ
=
- 5.84,
240.0
EL
300 .0
-0.17
(b) Fig. 8. (a) Comparison of signal recovery algorithms; two signal experiment. extra close spacing (1/6 BW), orthogonal polarization. READ source # I . (b) Comparison of signal recovery algorithms; two signal experiment , extra close spacing (1/6 BW). orthogonal polarization, READ source #2,
671
system errors. In handling quite difficult scenarios, such as multiple polarizationally diverse sources located closely (i.e ., within an array beamwidth) , the MUSIC algorithm directly addresses the problem and thereby offers good performance, generality, and flexibility. Also, due to its algorithmically well-structured form, it lends itself well to an exploitation of state-of-the-art computing architectures utilizing VHSIC, VLSI, and systolic arrays of processing elements. ACKNOWLEDGMENT
It is a pleasure to acknowledge the expert assistance of Tom Harris and Rich Yamamoto of ESL, Inc. Tom programmed the algorithms, developed and operated the system software,
and collected the data. Rich designed and constructed the array and implemented the receiver and data collection system. REFERENCES
[I]
[2] [3] [4] [5]
[6]
[7] [8]
R. O. Schmidt, "Multiple emitter location and signal parameter estimation," in Proc. RADC Spectrum Estimation Workshop Oct. 1979, pp. 243-258 (also IEEE Trans. Antennas Propagat., this issue, pp. xxx-xxx). - - , "A signal subspace approach to emitter location and spectral estimation," Ph.D. dissertation, Dept. Electrical Eng., Stanford Univ., Stanford, CA, Nov. 1981. - - , "New mathematical tools in direction finding and spectral analysis," in Proc. SPIE 27th Ann. Symp., Aug. 23, 1983, San Diego, CA. J. Capon, "High-resolution frequency-wavenumber spectrum analysis," Proc. IEEE, vol. 57, no. 8, Aug. 1969. J. P. Burg, "Maximum entropy spectral analysis," Ph.D. dissertation, Dept. Geophys., Stanford Univ., Stanford, CA, 1975. V. F. Pisarenko, "The retrieval of harmonics from covariance functions," Geophys. J. Royal Astron. Soc., no. 33, pp. 347-366, 1973. R. O. Schmidt, "Multilinear array manifold interpolation," to be submitted for publication. E. R. Ferrara and T. M. Parks, "Direction finding with an array of antennas having diverse polarizations," IEEE Trans. Antennas Propagat., vol. AP-31, pp. 231-236, Mar. 1983.
672
An Implementation of a CMA Adaptive Array for High Speed GMSK Transmission in Mobile Communications
Takeo Ohgane, Member, IEEE, Takanori Shimura, Naoto Matsuzawa, and Hideichi Sasaoka, Member, IEEE
Abstract-The hardware implementation of an adaptive array as a technique for compensating multipatb fading in mobile communications is described. The number of the antenna elements is four. The target communication system is modulated by 256 kbps Gaussian-filtered minimum shift keying and has a timedivision multiplexing architecture with 24 time slots. Based on the digital beam forming concept, all of the signals and the array weights are digital-signal processed. The constant modulus algorithm (CMA) is employed for weight optimizing. In an additive white Gaussian noise channel, this system bas 5.6 dB gain in an energy-per-bit-to-noise-density ratio (EbINo), at a bit error rate (BER) of 1.0 x 10- 3, compared with a single-antenna system. The result of the basic field test shows that the gain at a BER of 1.0 x 10- 3 reaches 22.3 dB in a nonselective, slow Rayleigh fading channel given a 5 Hz maximum Doppler shift. These results certify our successful implementation. I. INTRODUCTION
R
E CENTL Y , the number of subscribers for mobile radio communications has been increasing rapidly, and new services are being demanded; for example, facsimile and high-speed data transmission. It has been suggested that wideband time-division-multiple-access (TDMA) could extend the capacity and facilitate multispeed data transmission services. However, in digital mobile communications, use of higher speed data transmission widens the required frequency bandwidth, which leads to much greater susceptibility of system performance to multipath fading. Thus, to realize such a wideband system, multipath fading compensation techniques must be implemented. An adaptive equalizer is often employed for reducing the effect of multipath fading by means of compensating the channel frequency response [1]. However, the range of the time delay can be compensated is limited by the total time length of the taps. Adaptive arrays have been studied as a means for canceling jammers that are uncorrelated with the desired signal by directing nulls towards them [2], [3]. Several papers have described the canceling performance for interferers [4]-[7] and for delayed signals [8], [9] in digital communications. In addition, the bit error rate (BER) performance in a multipath fading channel is reported for high-speed Gaussian-filtered minimum shift keying (GMSK) transmission [10]. The imporManuscript received May 27,1992; revised September 15, 1992. T. Ohgane and H. Sasaoka are with Communications Research Laboratory, Ministry of Posts and Telecommunications, Koganei, Tokyo 184, Japan. T. Shimura and N. Matsuzawa are with Central Research Laboratory, Hitachi Ltd., Kokubunji, Tokyo 185, Japan. IEEE Log Number 9207174.
tant characteristic of adaptive arrays is that the desired-toundesired ratio (DUR) increases as the correlation between the desired and undesired signal falls [9]. That is, adaptive arrays can improve the BER performance for long delayed signals without architecture changes, in contrast to adaptive equalizers [10]. This becomes advantageous with higher speed digital transmission. However, previous papers have dealt only with theoretical analysis and/or computer simulations. Considering practical use in the near future, a hardware implementation trial is required. This paper deals with the implementation results of a four-element adaptive array for a 256 kbps GMSK transmission system [11]. The system employs the constant modulus algorithm (CMA) [12] for weight adaptation. CMA does not require a reference signal and automatically chooses the desired one from multiple incident waves. These characteristics provide easy implementation and stable performance. The digital beam forming (DBF) concept is applied to control the array beam pattern. The block processing technique realizes real time execution. Section II briefly discusses CMA in comparison with ordinary algorithm. The specifications and hardware architecture of the system are described in Sections III, and IV. In Section V, BER performance in an additive white Gaussian noise (A WGN) channel is shown. In addition, the BER performance in a nonselective, slow Rayleigh fading channel with 5 Hz maximum Doppler shift, measured by field test in our laboratory site, is shown as a check of system implementation. A typical array pattern for the field test is also shown.
II.
CMA ADAPTIVE ARRAY
CMA has been proposed by Treichler [12] as a suitable algorithm for compensating fading or jamming of constant envelope signals. Fig. 1 shows the basic architecture of the C:rviA adaptive array. Define an n-dimensional received signal vector X(k) and weight vector W(k) at a sampling time k'I, (k = 1,2, ... ), where Ts is a sampling period, as
X(k) = [xl(k)
x2(k)
W(k) = [x{(k)
w2(k)
Xn(k)]T w,lk)]T
Here, T denotes transpose. Both X and Ware represented by complex vectors. Then, the output signal of the array is obtained by
y(k) == XT(k)W(k).
Reprinted from IEEE Transactions on Vehicular TechnoLogy, Vol. 42, No.3, pp. 282-288, August 1993.
673
(1) (2)
(3)
TABLE I SPECIFICATIONS OF THE SYSTEM
Item Array
Output
Specification
Radio Channel Center Frequency TX Power Modulation Modulation Bit Rate
1431.5 MHz 37 dBm GMSK (BbT = 0.25) 256 kbps
Channel Access Fig. 1.
Multiplexing Channel Bit Rate Number of slots Slot Length
CMA adaptive array.
Under the assumption that the transmitted signal has constant envelope, the array output y should have constant envelope. However, the multipath fading can cause amplitude fluctuations in the received signals. The objective of CMA is to restore the array output y to a constant envelope signal on average. This is accomplished by adjusting the weight vector W to minimize the cost function J, which provides a measure of the amplitude fluctuation. J is given by (4)
where E[ ] is ensemble average operation. Due to a processing time constraint, we chose the simple gradient search algorithm to minimize J. The parame~ers p and q affect the sensitivity and the convergence behavior of CMA [8], [10]. Usually, lower p and q provide higher noise tolerance and stability under slower convergence. Here, (p, q) are set to (1, 2) based on the computer simulat~on results [10]. By replacing the ensemble average operation with an instantaneous value in (4), we obtain the update equation of W as W(k
+ 1) = W(k) - lLe(k)X*(k)
(5)
e(k)
(6)
= y(k) -
y(k)/ly(k)1
where JL is the step constant and * indicates the complex conjugate. Equation (5) is similar to the LMS algorithm. However, they are very different, in that CMA requires no reference signals for estimating the error function e(k). Equations (4)-(6) show no requirements for a reference signal and an insensitivity to phase fluctuations. Conventio~al algorithms using the mean square error (MSE) cost function [2], [3] can compensate both amplitude and phase of t~e signal using a reference signal that is a replica of the transmitted signal. A training signal sequence is often employed as a reference. However, the received signal in the multipath environment consists of several incident rays with different time delays. Therefore, in the receiver the reference training sequence must be generated with timing .coincident t~ one of the multipaths. Generally, we can consider the maximum ray to be the desired signal, as estimated from the complex impulse response of the channel using a Cox type sounder [13]. However, it is very difficult to estimate the position of the maximum ray from the instantaneous impulse response, since the response at each time delay is expressed by a vector sum of rays having the same time delay. Also, missynchroni-
TDM 8 kbps 24 512 bits
Adaptive Array Number of Elements Update Algorithm Processor Sampling Frequency
4
CMA T1320C25
1 MHz
zation can easily lead to degradation of the array performance [9]. In contrast, CM uses no reference signal, but automatically selects one or several of the multipaths as the desired signal. Thus, CMA does not need to know the arrival timing of incident rays when the array weights are updated. Moreover, we need not synchronously sample the received signal with the clock timing, since the constancy of the transmitted signal envelope is independent of the clock timing. Therefore, CMA can significantly reduce implementation complexity. However, due to the phase insensitivity and the unsynchronous sampling timing, we must recover the adequate detection timing and carrier phase after the adaptive processing. These problems can be solved easily by a Costas loop [14], [15] and clock recovery using over-sampled data [16]. III.
RADIO CHANNEL SPECIFICATIONS
Table I shows brief specifications of the system. The transmitted signal is modulated by GMSK, which is a well-known constant envelope modulation [14]. The carrier frequency is 1.4315 GHz. The modulation data rate is 256 kbps, so the bit duration T is about 4 JLS. The normalized 3 dB bandwidth BbT of the premodulation Gaussian low-pass filter (LPF) is 0.25. The system accommodates 24 users per carrier by timedivision multiplexing (TDM), and each channel data rate is 8 kbps. Fig. 2 shows the data format of a time slot. The frame-synchronization word, the data-synchronization word, and the preamble consist of different pseudo noise (PN) sequences. The guard space is a fixed non-data sequence. Thus, the whole time slot of 512 bits is continuously GMSK-modulated. IV.
ARCHITEcruRE OF DIGITAL SIGNAL PROCESSORS
Fig. 3 is a block diagram of the receiver. A CMA adaptive array of four antenna-elements is implemented by using the
674
__- - - - 48 msec
I
I-----J 8 kbps
384 bns
D.~ SlQnng
Pulu
2'
\
...
AdapllY'
Proc:. sc"",,
OMSK
Oemoc1J~cln
~'-,. o
1S
75
106
256 kbps
Fig. 4. Timing chan of the block processing.
.....~
(DPRAM). The interrupt signal also works as a queue that directs each processing unit to start the execution . In the adaptive control unit, the 2048 data samples in DPRAM are read out one by one , and the array output is evaluated. Each array output is passed to the GMSK demodulation unit during every weight updating iteration. By this pipeline architecture, every unit can obtain a 48 ms processing time for one time slot, and it reaches 234 operation cycles for each sampled data . That is enough to execute the adaptive processing routine for CMA based on the simple gradient search algorithm.
490 511
512 bus FS PA OS IW GS
Frame Sync Word Preamble Data Sync Word Intormahon Word Guard Space
Fig. 2. mMA , 101
r··········_····_····_····_···············_;
!
f---i~;----"'-"" r
!
: : i
B. Down and A ID Converting Unit
[ O'T'7'----JO
i
;
Fig. 3.
Block diagram of 4-element adaptive array.
~~F co.ncept. Also, the array output is demodulated by digital signal processing" These operations are performed by the digital signal processor (DSP), TMS320C25, which executes a 16-bit, fixed-point operation every 100ns. As seen i? Fig. 2, each processor has only 2 ms to deal with the target time-slot data . This permits less than 40 operations per bit, which IS not enough to execute the whole adaptation or demodulation procedure. In the system, we utilized block processing to solve this problem . At first we present a brief view of the block processing; then we describe the function and configuration of each block of the receiver.
A. Block Processing
As state above, a maximum of 40 operations can be per formed on each bit. To complete the digital signal processing for the adaptive control in real time would require an enormous number of DSP's and complicated parallel-processing software . Block processing is the easiest way to avoid this hurdle. Fig. 4 is a timing chart describing the block processing. The duration of the target time slot is only 2 ms. However, we can utilize the waiting time of 46 ms until the next target arrival if we expand the processing time of the target data . This is achieved by storing all the data of the target in RAM . Here , there are 512 bits in one time slot, and the sampling frequency is set to four times the bit rate for over sampling. Thus, after receiving the interrupt signal from the frame synchronizing unit, 2048 complex-samples (i.e., both I and Q channel) of one time slot are stored into dual-port RAM
675
Each received radio frequency (RF) signal is down-converted to intermediate frequency (IF) band and band-pass filtered to eliminate noise in out-band. The optimum value of the normalized 3 dB bandwidth of the predetection bandpass filter (BPF) BT is 0.63 for coherent detection [14]. However, a BPF narrower than BT = 1.0 degrades the performance of the CMA adaptive array, since it causes amplitude fluctuations. Therefore, the 3 dB bandwidth B is set to 320 kHz, so BT = 1.25. The received power variation of each branch is roughly regulated by an automatic gain controller (AGC). The gain is determined by the maximum power of the four branches, and that gain is used on all branches (common AGC) . If the AGC were controlled independently in each branch, each gain would change independently according to the Doppler shift. However, considering the weight control , it is desirable for the variation speed of the gain to be slower than adaptation. The gain of the common AGC method is expected to be almost constant when the received power variation of all the branches is uncorrelated, and this improves the stability of the CMA adaptation. The IF signals are quadrature down-converted to I and Q channels by a local oscillator which is independent of the tran smitter (quasi-coherent detection). The carrier phase difference is not important here, since it will be recovered later by the GMSK demodulator. Both I and Q channel are sampled every 1 /LS, which corresponds to about quarter-bit duration, and AID converted with lO-bit accuracy. This oversampling method is needed to find the exact detection timing, since clock timing is not recovered when the data are sampled, as discussed briefly in Section II. To store all of the samples for each channel, two 4-KB DPRAM's are implemented in each branch. C. Frame-Synchronizing Unit
Frame synchronizing is very important in a TDM system. Even if we employ block processing, the frame-synchronizing
""~
s..mphng Pulse
Fig. 5. Frame synchronizing unit.
unit must search all of the received signal for a unique target word in real time. This requires a high-speed DSP suitable for performing complex correlation. We have employed the special purpose DSP, IMS-Aloo, which contains an array of 32 high-speed accumulators and has a dataflow architecture. Fig. 5 shows the configuration of the frame-synchronizing unit. The branch providing maximum power is chosen, reducing the possibility of mis-synchronizing. Envelope limiting is used to maintain a constant threshold level while searching for the correlation peak. The optimal threshold level is determined from the indoor measurements in an AWGN channel. The frame-synchronization word is a 15-bit PN sequence, so we must complete the complex correlation of 60 samples within a sampling period of IIJ.S. Correlation is implemented by four cascaded IMS-Aloos, which can process up to 63 samples in only 400 ns if an 8-bit coefficient is chosen. The complex correlation output, the squared amplitude of which is above the threshold level, is regarded as a candidate for the correct correlation peak of the frame-synchronization word. Then, the correlation window is centered on this peak . Thus, spurious peaks outside of the window, caused by similar sequences in the information, can be eliminated. When the real peak cannot be captured, the correlation window is reset and the searching process is restarted. The interrupt signal, as the queue for each processing unit, is generated after adding some delay to the frame synchronizing signal, in order to adjust the queue timing to the top of the slot.
Samp ling Pulse
Queue
BroadCaStBus •••.••.•:.
Fig. 6. CMA controlling unit.
D. CMA Controlling Unit
Due to adoption of block processing, the CMA adaptation procedure requires just 2 DSP's. However, in the system we used a 4-DSP configuration to ease the implementation. Fig. 6 shows the configuration of the CMA controlling unit. One DSP works as a master processor that calculates array output and error function; the others work as slaves. These communicate asynchronously using first-in-first-out memories (FIFO 's). Brief flowcharts for the master and the slave are shown in Fig. 7. All processors start the procedure on queue from the frame-synchronizing unit. First, the weight of each branch is initialized. In mobile communications, there is no information about the arrival angle of the received signal. So, one branch is initialized to one, and the others to are initialized to zero to get an omni-directional pattern. The branch initialized to one is that having the maximum power in the first sample, as this method is reported to improve the BER performance [10]. Each iteration begins with weighting of the input sample. Here , the weights are expressed in double precision to de-
676
Fig. 7. Flowchart in CMA controlling unit.
crease the round-off error. The slaves send the weighted inputs to the master as indicated by dashed line 1 in Fig. 7. The master evaluates the array output, and it is written into the DPRAMs for the GMSK modulator. The error estimate shown in (6) is obtained using a look-up table and broadcasted to the slaves as indicated by dashed line 2. Weight updating occurs at the end of each iteration. The preamble shown in Fig. 2, consisting of four PN sequences of 15 bits, is used for CMA training time. The weights must converge to the optimum value within the 6O-bit duration of the preamble. This is achieved by choosing an appropriate step constant JL. However, the global optimum convergence is not guaranteed. Note that the receiver does not use a priori knowledge of the preamble sequence, since CMA requires no reference signal.
o o
WiD amIy w i array
- - simulated CMA Ccmrol
wiD array
10"
Bursl/
Sertal
"""y Output 0 ... QJ
Recovered Sipl al 0 &. Q) Recovered Clock
a: W
Fig. 8. GMS K demodulator unit.
'0·.--- ........--~- ........- --.
10 .3
o measured <,
10 .2
--slmutale
-,
"\ \
'0" 1O·L--_~
o
5
\
'. \
Fig. 10.
\
\
\ \ _ _.-.....J_",,:,::-_-:
20
Fig. 9. The BER performance without the adaptive a rray in a n A WCiN cha nnel measured in IF band .
E. GMS K Demodula tor Unit In this un it, shown in Fig. 8, the array output is demodulated . Th e ca rr ier phase and the clock are recovered by a Costas loop [14), [15]. This process is just the same as an ordinary an alog Costas loop used in IF band. Phase shifting and estimatio n of the ph ase erro r require sin, cos , tan " , and division ope ratio ns. (T hese are referred to the look-up tabl e.) Du e to the processing time constraint, detection is carried out by anothe r DSP . Th en , th e ph ase-recovered I and Q da ta and th e clock are deliv er ed to the detection processor. The appro priate det ection timing and corresponding sample value are interpolat ed from the over-sampled data. The demodulated data is buffered by FIFO and output at the spe ed of 8 kbps by a Burst/Serial transfer unit having one DSP. V. BER
15
20
Th e HER per fo rmance in an AWG N chann el.
Fig. 10 sho ws the BER performance in an AWGN channel, wher e the RF output of the tr ansmitter is connected directly to th e receiver using pow er dividers. E b/ No of 0 dB approximat ely corresponds to an RF input power of -115 dBm. G en erally, the adaptive arr ay tries to maximize the SNR of th e array output when there are no interferers such as delayed paths. T ha t is, in an A WGN channel, the signal component of each branch is synthesized coherently, while the noise is synthesized incoherently. Therefore, an adaptive array o f fo ur antenna e lements can provide a 6 dB gain , compared wit h a one-e lement syste m. The measurement result shows that E hl N o, at a BER of 1.0 X 10-\ is 5.9 dB with (w I ) the ada ptive array and 11.5 dB without (w /o) it. The gain in E bl No is about 5.6 dB , which is almost equal to the th eoretical value . Th is shows that the syste m works acceptably. B. A Slow Rayleigh Fading Channel
PERFORMANCE
In this section, the fund amental BER performance of this syste m is des crib ed. Th e step constant JL is set to 0.0" in these measurem en ts on th e basis of earlier simulation result s [10]. To provide a comparison with the CMA array system , we also measured th e BER per form ance using only one ante nna eleme nt.
Fig. 11 shows th e BER performance in a non-selective, slow Rayleigh fading channel measured inside our laboratory site . T he receiver is installed in a measurement van . On the roof of the van, about 2.5 m high, four A/4-monopole antennas are posit ioned in a square with antenna spacing of 0.444.\. Th e vehicl e speed is about 3.6 krn/h , so the maximum Doppler shift is about 5 Hz. Such slow speed provides the lower bo und of BER performance in the Rayleigh fading channel. Th e me asur ements wer e rep eated for various transm itter power s o n a straight course behind a four-storied building wh ich block ed th e line-of-sight path. The BER is averaged every 10 m. The dash ed line sho ws the lower bound est imated by Murota 's fo rmula [14], for a single antenna system , as I
A. An A WGN Channel To check the GMSK dem odulator, the BER pe rform an ce of one b ra nch dir ectl y co nnec ted to the tr ansmitter in th e IF band is me asur ed , as shown in Fig. 9. The dashed cu rve indica tes the simulated BER performance for th e case of BT = 1.25. The degradation from this is distributed between 1.5 and 2.3 dB .
677
P.( f) == 4af
(7)
whe re I' is th e average E bl No and P, is the average BER. a is esti ma te d as 0.55 for B bT = 0.25 and BT = 1.25 from the result in [14]. The degradation for a single antenna measurement is about 3.3 dB , which demonstrates the good performanc e of th e digital Costas loop .
10
0 r----.,-----~-----.
o •
w/o array
mea sured course.
w i array - - Murola's eqn.
.
10.2 II:
W CD
10 -3
10m 10-$''-----'--- ' - -10 o
EblNo
-'--
20
-
o
Fig, 1J. The BER performance in a flat fading channel.
From comparison of the two curves in Fig. 11, it can be seen that the use of the adaptive array considerably reduces the degradation of the BER performance due to Rayleigh fading. In a Rayleigh fading channel, only fully coherent paths exist. Therefore, minimizing the cost function J in (4) is almost the same as maximizing the SNR of the array output, corresponding to combining the signals with weights in proportion to the SNR of each branch. The gain at a BER of 1.0 X 10-3 and 1.0 x 10-2 is about 22.3 dB and 14.9 dB, respectively, which is comparable with that of maximal ratio combining diversity. The results suggest that the CMA adaptive array is equivalent to the usual space diversity in a Rayleigh fading channel. The outlier point at around 28 dB is probably due to the lack of the high SNR data. In a Rayleigh fading channel , switching diversity can obtain just 3 dB worse gain than maximal ratio combining diversity with far less complexity [17], [18]. (The gain of the adaptive array is similar to that of maximal ratio combining diversity, as the result of this measurement shows.) However , the advantage of the adaptive array becomes apparent when we assume frequency selective fading and! or cochannel interference due to its capability to null out such interference. This area should be the subject of a future experimented study. When analyzing the array performance, the directional pattern provides a useful visual tool. Fig. 12 shows (a) one particular location on the measurement course and (b) the directional pattern measured there. There is no line of sight; therefore, it is expected that the diffraction path at the edge of the building would become the largest path arriving at the receiver. The main beam is directed to 220degrees, which is almost coincident with this expected diffraction path. This result shows that the CMA adaptive array considers this to be the desired path. VI.
'measured point
---' 30
Fig. 12. Directivity pattern at the measured point. (a) Location; (b) directivity pattern.
measurement is done using a 256 kbps GMSK transmission system having TDM architecture with 24 time slots. In an AWGN channel, the array has a 5.6 dB gain in E b ! No at a BER of 1.0 X 10- 3 compared with a single-antenna system. The result of the field test shows that the gain at a BER of 1.0 X 10-3 reaches 22.3 dB in a slow Rayleigh fading channel with 5 Hz maximum Doppler shift. This performance indicates that the system was implemented successfully. In addition , it confirms that, from the array -pattern analysis, the array works to direct the main beam to the major path. In order to evaluate the array performance in a selective fading channel, further measurements are required. ACKNOWLEDGMENT
CONCLUSION
We have implemented a four-element adaptive array based on the DBF concept to reduce the multipath fading effect in high-speed digital transmission . The weight adaptation by CMA and demodulation is executed by several DSP's. The
o
678
The authors wish to thank M. Koya of the Central Research Laboratory, Hitachi Ltd., and M. Yokoyama and M. Mizuno of the Communications Research Laboratory for useful comments. In addition, the authors should like to thank S. Sampei, Y. Kamio, and S. Sekizawa, all of the Communications Research Laboratory, for helpful suggestions and support during the experiment.
REFERENCES
[1] S. U. H. Qureshi, "Adaptive equalizer," Proc. IEEE, vol. 73, pp. 13491387, Sept. 1985. [2] B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode, "Adaptive antenna systems," Proc. IEEE, vol. 55, pp. 2143-2159, Dec. 1967. (3] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. [4) R. T. Compton, Jr., "An adaptive array in a spread spectrum communication system," Proc. IEEE, vol. 66, pp. 289-298, Mar. 1978. [5] J. H. Winters, "Spread spectrum in a four-phase communications system employing adaptive array," IEEE Trans. Commun., vol. COM-30, pp. 929-936, May 1982. [6) J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE Trans. Veh. Technol., vol. VT-33, Aug. 1984. [7) Ganz and R. T. Compton, Jr., "Protection of a narrow-band BPSK communication system with an adaptive array," IEEE Trans. Commun., vol. COM-55, pp. 1005-1011, Oct. 1987. [8] R. Gooch and 1. Lundell, "The CM array: An adaptive beamformer for constant modulus signals," in Proc. /CASSP'86, vol. 4, pp. 2523-2526, Apr. 1986. [9] Y. Ogawa, M. Ohmiya, and K.ltoh, "An LMS adaptive array for multipath fading reduction," IEEE Trans. Aerosp. Electron. Syst., vol. AES-23, pp. 17-23, Jan. 1987. [10] T. Ohgane, "Characteristics of CMA adaptive array for selective fading compensation in digital land mobile radio communications," Electron. and Commun., in Japan, Scirpta Technica, Inc., Part 1, vol. 74, no. 9, pp. 43-53, Sept. 1991. [11] T. Ohgane, H. Sasaoka, N. Matsuzawa, K. Takeda, and T. Shimura, "A development of GMSK/TDMA system with CMA adaptive array for land mobile communications," in Proc. 199/ Con! on Veh. Technol., pp. 172-177,1991. [12] J. R. Treichler and B. G. Agee, "A new approach to multipath correction of constant modulus signals," IEEE Trans. Acoust., Speech, Signal Process, vol. ASSP-31, pp. 459-472, Apr. 1983. [13} D. C. Cox, "Delay Doppler characteristics of multipath propagation at 910 MHz in a suburban mobile radio environment," IEEE Trans. Antennas Propagat., vol. AP-20, pp. 625-635, Sept. 1972. [14] K. Murota and K. Hirade, "GMSK modulation for digital mobile radio telephony," IEEE Trans. Commun., vol. COM-29, pp. 1044-1050, July 1981. [15] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1983. [16] Y. Higashida, M. Hagiwara, and M. Nakagawa, "Timing extraction and carrier estimation for PSK signal block demodulation system," Trans. IEICE Japan, vol. J71-B, no. 4, pp. 540-546, Apr. 1988, (in Japanese). [17] C. Sundberg, "Block error probability for noncoherent FSK with diversity for very slow Rayleigh fading in Gaussian noise," IEEE Trans. Commun., vol. COM-29, Jan. 1981. [18] F. Adachi and J. D. Parsons, "Unified analysis of postdetection diversity for binary digital FM mobile rad.o," fEEl:, T;a. IS. \--·eh. Technol., vol. 37, Nov. 1988.
679
A FOUR-ELEMENT ADAPTIVE ANTENNA ARRAY FOR IS-136 pes BASE STATIONS Robert L. Cupo··,Glenn D. Golden··, Carol C. Martin·, Karl L. Shermans", Nelson R. Sollenberger", Jack H. Winters·, and Peter W. Wolniansky·· * AT&T Labs • Research •• Lucent Technologies • Bell Labs Holmdel, New Jersey 07733 USA Abstract: We describe the algorithms, implementation, and laboratory performance of a real-time four element adaptive anteDDa array testbed for the uplink of a 1.9 GHz 18-136 PCS base station. Using our enhanced Direct Matrix InvenioD algorithm, experimental results show nearly a 6 dB higher laiD at a 10- 2 bit error rate (BER) with four venus the typical two receive antelUUlS and operation at close to a 10- 2 BER even with an interferer of equal strength to the desired signal at 60 mph fading rates. These results demonstrate the feasibility of using adaptive arrays to increase both the range and capacity of TDMA ceUular systems.
II. WEIGHT GENERAnON ALGORITHMS Figure 1 shows a block diagram of an M antenna element adaptive array. The complex baseband signal received by the ith element in the kth symbol interval Xi (k) is multiplied by a controllable complex weight w; (k). The weighted signals are then summed to form the array output y(k). Thus, the output signal is given by y(k) = w T (k) x(k)
(1)
where the weight vector w is given by W
I. INTRODUCTION
=
[W\W2oooWM]T .
(2)
the superscript T denotes transpose, and the received signal vector x, which consists of the desired signal, thermal noise, and interference, is given by
Antennaarrays using optimum combining reduce multipath
fading of the desired signal and suppress interfering signals, thereby increasing both the range and capacity of wireless
x
systems.
= [XIX2oo.XM]T
(3)
Previous work [1-3] has shown the potential improvement in gain and interference suppression using adaptive arrays in the TDMA mobile radio system 18-54/136. Computer simulation results [ 1] for a two-antenna array showed that, using the Direct Matrix Inversion (DMI) weight generation algorithm, the array could achieve significant increases in range and capacity with less than 1 dB loss from theoretically achievable gain due to weight tracking degradation at vehicle speeds up to 60 mph at 900 MHz. However, this paper did not consider other implementation issues such as errors in the data-derived reference signal, and recent interest has been focussed towards more antennas and PeS (1.9 GHz) frequencies, which causes increased tracking degradation (see, e.g .. [2]).
Fig. 1 Block diagram of an M element adaptive array.
In this paper, we study the performance of a four-element adaptive array intended for use on the uplink of an 15-136 pes base station at 1.9 GHz. We present computer simulation results for the tracking performance of the OMI algorithm as presentedin [1] and propose enhancements to that algorithm to improve tracking performance, We then briefly describe the implementation of a real-time adaptive array testbed, and show experimental results obtained with this testbed using an RF multipath fading emulator.
The weights that minimize the mean squared error in the output signal, where the error is the difference between the output signal y(k) and a reference signal d(k), which ideally is the transmitted data (which also maximizes the signal-tointerference-plus-noise ratio (SINR) and minimizes the bit error rate (BER) with Gaussian noise), are given by [4],
(4)
In (4),
Reprinted from IEEE 46th Vehicular Technology Conference, pp. 1577-1581, May 1996.
680
R.a
= E [X·XrJ
=
u;u~
2
+ 0' 1 +
j~ Uj·uJ.
which is nearly half (in dB) the 6 dB theoretical gain one would achieve by going from 2 to 4 antennas. Furthermore, the loss is even greater with interference, as shown in the computer simulation results of Figure 3 for an equal-power interferer (i.e., signal-to-interference ratio, SII = 0 dB) with 0, 92, and 184 Hz fading. (These curves are not smooth because of the BER measurement error with computer simulation using only 178 time slots.) With DMI, the required SIN for a 10- 2 BER is increased by 2.1 and 11 dB at 0 and 184 Hz, respectively.
(5)
(assuming the desired signal, noise, and interfering signals are uncorrelated), where Ud and Uj are the desired and jth interfering signal propagation vectors, respectively, 0 2 is the noise power, I is the identity matrix, L is the number of interferers, and rxJ(k)
= E[x(k)d·(k)] = UJ
.
(6)
Let us now consider the implementation of optimum combining using DM! as in [1]. Using a K-symbol rectangular averaging window, the weights are given by -I
"
w(k + 1) = R xx (k) r xd(k) A
•
TOMA FRAME 40 ms
•
(7)
where
. . - TIME SLOT8.687 ",. (182symbcil)
3 3
(8)
8
14
81
8
8
-...
81
G R 0. . --_.. . ,.... _....
and rxJ(k)
=
J...
r.
x(j)d·(J)
Kj=k-K+l
Fig. 2 The TDMA frame and time slot structure for 15-136.
(9)
To reduce this degradation we propose the following techniques. First, since OMI is based on optimum combining, this algorithm uses equal weighting for interference and noise suppression in the weight calculation. However, the estimate of the interference can be poorer than the estimate of the noise variance (which is typically known or can be accurately estimated). Thus, equal weighting of the noise and interference may not give the weights which result in lowest BER. Indeed, when interference is not present, our computer simulations and real-time experiments using a fading emulator show significant improvement when maximal ratio combining (MRC), which ignores interference, is used.
Figure 2 shows the frame and time slot structure used in IS-,136. For uplink (mobile to base) transmission, each time slot includes a 14 symbol synchronization sequence, and 6 symbol CDVCC sequence [5]. The synchronization sequence is intended for use in acquiring initial tap weight settings for an adaptive equalizer, while the CDVCC sequence identifies the mobile's associated base station. Both of these fields are known at the base station, and thus it is convenient to utilize them for adaptive array weight acquisition as well. Specifically, we use the synchronization sequence as the reference signal for initial weight acquisition, and then use the coherently-sliced detected data as the reference signal thereafter, i.e., d(k) == quan(w T (k) x(k»)
An enhancement to OMI which accommodates an adjustable relationship between the effects of noise and interference is diagonal loading (DMUDL) (also known as "diagonal augmentation", and "eigenvalue shifting" [6]). The weights are given by
(10)
where quan() denotes 7tl4 QPSK coherent detection. Note that we consider coherent detection for the reference signal, but
differential detection for the base station. performance as described below.
w (k + 1)
This impacts
= [( 1 -
~ ) Rxx ( k) + ~ (J 2 I] - 1 rxd ( k )
, ( 1 1)
where the constant ~ ~ 1 is the diagonal loading factor.
The performance obtained using OMI (7) is worse than that with the ideal weights (4) because of three factors: data detection errors, channel variation over the finite window, and noisy estimates for R u and r xJ with the finite window. Note
As verified by our results, when the interference-to-noise ratio (lIN) is very large, (i=O, i.e., (7), gives the lowest BER, while when lIN is very small, ~= 1, i.e., MRC, gives the lowest BER. However, most importantly, our results show that for l/N' s between these values, there exist intermediate values of ~'s which give lower BER than either (7) or MRC, with the largest improvement over (7) and MRC when the interference and noise powers are comparable. Our results also show that the optimum ~ (for lowest BER) is nearly independent of SIN, although it does depend on the fading rate. Therefore, let us define ~ opt as the ~ which results in the lowest BER in the presence of worst case fading, i.e., 184 Hz. OUf results indicate that an appropriate choice for ~o as a function of lIN (in dB) is
that the effect of channel variation is reduced by decreasing K.
while the effect of noisy estimates is reduced by increasing K. Fortunately, the best performance in IS-136 with 4 antennas is achieved with K== 14, the length of the synchronization sequence. However, because of these effects, OMI has significantly degraded performance compared to ideal with 4 antennas in 15-136, particularly at high fading rates. Specifically, with noise only, the required signal-to-noise ratio (SIN) for a 10- 2 BER is increased by 1.2 and 2.7 dB at 0 and 184 Hz (corresponding to 60 mph at 1.9 GHz), respectively,
681
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Next, consider the problem of errors in the data-derived reference signal, where any symbol detection error made increases the subsequent weight estimation error. Since increased weight estimation error increases the BER. error propagation can occur, resulting in a complete loss of tracking. However, in IS-136, the 6 symbol COVCC, located near the middle of each time slot, can be used to reduce the effect of error propagation. Recall from above that the CDVCC is known at the base station, so in principle it can be used as the reference signal for weight generation during the corresponding symbols of the time slot. However, it cannot be used directly because, as is the case with all data in the IS-136 time slot, the COVCC symbols are encoded as differential phase shifts, rather than as coherent constellation points as required for the reference signal. We therefore use the detected phase of the data symbol prior to the CDVCC as the initial phase reference for CDVCC. This technique reduces the required SIN for a 10- 2 BER about I dB for the case of an interferer with S/1 =0 dB .
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Finally, consider the problem of channel variation over the estimation window. The adaptive array output signal (I) at the kth symbol is computed based on a uniform sliding window extending K symbols backward in time . The kth array output is then used to estimate the kth detected symbol d(k), which is then used to compute the k + lth weight vector w(k + I), according to (7), and so on. Thus, the weights generated by (7) at symbol k are based on a received data vector x(k) which is time-centered about symbol k - KI2 (K assumed even) . The kth weight vector is thus really more representative of the ideal weight vector in the middle of the window than at the end of the window. We cannot use this fact to improve the weight computation itself, because the weights depend on the d(k), which are not available until the most recently computed array output is at hand. However, we can use it to produce a better signal for the purposes of data detection by the base station : Instead of driving the base station with the array output signal y(k), we instead use a time -shifted version z(k) that is computed based on the received data for which the weights more closely correspond, that is,
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Figure 4 shows the performance with an equal-power interferer with the combination of the algorithms discussed above. This combination includes time-shifted weights, use of COVCC, and diagonal loading, with and without the subspace method, which we refer to as enhanced subspace and enhanced OMI, respectively. The tracking degradation at a 10- 2 BER is reduced to about I dB at slow fading rates and 3 and 7 dB with the enhanced subspace and enhanced OMI methods , respectively, at 60 mph fading rates.
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682
timing adjustment. The implemented algorithm was equivalent to OMI with diagonal loading. using a userselectable ~ and weight time shifting. Note that the use of CDVCC. the subspace method, and a variable ~ based on estimated UN as described above, were not implemented. Furthermore, an exponential window (see [ 1D. rather than a rectangular window, was used.
III. REAL-TIME ADAPTIVE ARRAY SYSlEM The above techniques were implemented in our real-time four-antenna Adaptive Antenna Array (AAA) system, which is capable of both cellular-band IS-54 and PCS-band IS-136 operation. The basic system components are four UQ receivers, a baseband processor, and a transmitter for remodulating the baseband array output signal back up to RF for use by the base station. The system is controlled and monitored from a networked workstation, using a customized graphical user interface (GUI) which permits real-time parameter changes and provides extensive performancemonitoring outputs.
• Data logging: Signal-moise-einterference power on each subchannel is logged for later analysis, along with indications of special events such as AID overloads. • Operational parameter monitoring: Mean squared error. error vector magnitude display, selected timing epoch. weight trajectories. signal powers and more are available (remotely) at the workstation GUI in quasi -real-time.
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IV. EXPERIMENTAL RESULTS The real-time experimental results in this paper were derived using our AAA real-time system with an RF multipath fading channel emulator under laboratory conditions in order to provide repeatable results. The multipath emulator can produce four independent channels with three paths each to support cochannel interference testing with one or two interferers. with Doppler rates up to 740 Hz and delay spreads up to 100 JJ.s. and with Rayleigh, Rician, or log-normal fading.
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The Baseband Processor performs all of the AAA signal processing functions described in this paper, and numerous ancillary functions in addition, as summarized below : • AID sampling ksamples/sec
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Figures 6 and 7 compare the experimental results using the laboratory RF multipath emulator with the computer simulation results . These results are for DMI with weight time shifting and diagonal loading with a ~ fixed for each figure: ~ = I (MRC) in Figure 6. and 13 = 0.43 (optimum 13 for lIN = 10 dB) in Figure 7. (Note that the experimental algorithm differs from the simulation algorithm in the use of an exponential rather than a rectangular window to reduce computational complexity. Our simulation results indicate that this should have a negligible effect on the results, however.) There is good agreement between the experimental and computer simulation results. except for the case of an equal-power interferer at 184 Hz fading. In this case. the experimental system has a higher BER, probably due to the significant additional lSI power induced by an equal-power interferer. This lSI is due primarily
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• Channel filtering: Square root raised-cosine, a = 0.35. • DC compensation: Mitigation of DC component independently on each subchannel, with user-selectable time constant. • Symbol timing estimation: Determination of best timing epoch, within TIS symbol of eye center. • Adaptive array processing: User-selectable choices among basic algorithms and parameters: OMIJDL. MRC, slicing method (coherent or differential detection), adaptation stepsizes, diagonal augmentation, and frame 683
COVCC, diagonal loading with tl based on estimated IIN. and/or the subspace method, at the cost of increased computational complexity.
to analog filtering effects which are not modeled in the simulations. • - - Expanmant -
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V. CONCLUSIONS In this memorandum we have described the implementation of a prototype four-element adaptive antenna array for IS-136 base stations. Experimental results using our enhanced adaptive array algorithm for range extension showed that we could achieve more than 5 dB of the theoretical 6 dB higher gain at a 10- 2 BER in a Rayleigh fading environment than a two-element array using postdetection diversity combining. This corresponds to a 40% increase in range in a typical mobile radio environment. At slow speeds in the presence of interference, the four-element array realized more than 3 dB of the theoretical 4 dB gain advantage relative to a two-element array in the absence of interference, even with a cochannel interferer having the same average recei ved signal power as the desired signal. At 60 mph fading rates, the results show that an interferer can reach nearly the level of the desired signal while maintaining a 10- 2 BER. This indicates the feasibility of increased capacity through higher frequency reuse, perhaps even including the possibility of frequency reuse within a cell. Thus, we have demonstrated that adaptive arrays can increase both the range and capacity of TDMA cellular systems.
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ACKNOWLEDGEMENTS
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REFERENCES
[1] J. H. Winters, "Signal Acquisition and Tracking with Adaptive Arrays in the Digital Mobile Radio System IS-54 with Flat Fading," IEEE Trans. on Vehicular Technology. November 1993. [2] G. E. Bottomley and K. Jamal, "Adaptive arrays and MLSE equalization," Proc. of the Vehicular Technology Conference, Chicago, Il., June 25-28, 1995, pp, 50-54. [3] K. J. Molnar and G. E. Bottomley, "D-AMPS performance in PCS bands with array processing," Proc. of the Vehicular Technology Conference, Atlanta, GA, April 28 - May I, 1996, pp.1496-1500. [4] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays, John Wiley and Sons, New York, 1980. [5] D. J. Goodman, "Trends in cellular and cordless communications," IEEE Communications Magazine, vol. 29. June 1991, pp. 31-40. [6] 1. C. Nash, Compact Numerical Methods for Computers, 2nd edition, Adam Hilger, Bristol and New York, p. 121. [7] A. Haimovich and X. Wu , "Eigenanalysis based array processing for mobile communications," in Proceedings of the
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Fig. 7 BER versus SIN for the enhanced OMI algorithm with an equal-power interferer - comparison of experimental and computer simulation results. Finally, based on the results of Section II, we note that we should be able to achieve even better performance in our experimental system (when interference is present) by using
1994 Princeton Conference on Information. Science, and Systems, Princeton, NJ, March 1994, pp. 203-208.
684
EricssonIMannesmann GSM Field-Trials with Adaptive Antennas Soren Anderson, Ulf Forssen and Jonas Karlsson Ericsson Radio Systems AB, 8-164 80 Stockholm, Sweden
Tom Witzschel, Peter Fischer and Andreas Krug Mannesmann Mobilfunk GmbH, Am Seestern 1, D-405 34 DUsseldorf, Germany
Abstract -This paper describes Ericsson's GSM testbed for evaluation of adaptive antenna technology and joint field-trials made by Ericsson and Mannesmann Mobilfunk. The implementation principles and algorithms are briefly described together with the field-trial setup. The field-trials was carried out in Dusseldorf, Germany, by Mannesmann Mobilfunk and Ericsson. The paper also presents results from the field-tests.
1.
system level simulations, already carried out at Ericsson using more idealized assumptions, can be found in [4]. From Mannesmann Mobilfunks point of view, the application of adaptive antennas in the 900 MHz frequency band must be focused on finding a solution for capacity improvements. The application area, as currently can be envisaged, could be the macro-cell environment in urban andlor suburban areas. For a new 1800 MHz operator, adaptive antennas can be an appropriate choice of technology from a coverage improvement point of view.
BACKGROUND
The deployment of TDMA based cellular systems is a worldwide success with, for example, the GSM standard in operation in Europe, China, USA, Australia, South Africa and large parts of Asia. Today, more than 100 countries use digital TDMA technology (January, 1997). As the number of users increase, more capacity is needed. This is in particular the case for the 800/900 MHz band where the bandwidth is limited. The avaliable bandwidth at 1800/1900 MHz is larger but the path loss is, on the other hand, higher. Adaptive antennas have generated great interest in recent years and at present, several manufacturers and operators are separately or jointly performing field-tests to gain more detailed knowledge of the potential of the technology. This paper describes a testbed developed by Ericsson in order to evaluate performance (for both uplink and downlink) and to gain implementation knowledge. The testbed, which is fully GSM compatible, has been jointly evaluated by Ericsson and Mannesmann Mobilfunk GmbH.
3. THE TESTBED A simple block diagram of the testbed is shown in Figure 1. The testbed, which was built by Ericsson Microwave in Molndal, implements the air interface part of a GSM base station for one carrier and is designed to work in the DeS 1800 band. The testbed consists of four main parts, antennas, radios, baseband processing and a user interface. In order to facilitate relative measurements, the testbed consists of both a conventional base station and a base station using the adaptive antenna. The reference (or conventional) arrangement consists of a sector antenna with a two-branch receive diversity (Maximum Ratio Combining, MRC) and one transmit branch.
2. AN OPERATORS VIEW ON ADAPTIVE ANTENNAS
The continuously growing number of subscribers in the German 02 (Mannesmann Mobilfunk) network implies also a requirement of continuous adaptation of the network capacity. There are several ways to increase the capacity, e.g., frequency hopping, micro-cell applications and introduction of a half-rate codec. However, the application of adaptive array antenna systems will be an important step in the system-evolution process of current mobile radio network technology. One of the main goals of the evaluation of antenna array technology is to examine which the useful application environments can be for introducing the technology in an existing network. The results from our measurements, that have been carried out on link-level basis, are in principle confirmations of the expected link-level ell improvements. The real potential of the capacity improvement in a dynamic network with adaptive antennas must be analyzed in further investigations including more sophisticated, or realistic, system simulations and preferably also measurements on a system level. Some results from
Radio Hardware (RXlTX Cabinets)
SPARe5 for interface- and DSP control
DSP Cluster
Hard disc for raw data logging
Figure 1. Basic structure of the testbed.
Reprinted from IEEE 46th Vehicular Technology Conference, pp. 1587-1591, May 1996.
685
The antennas, array and sector, are Ericsson designed, microstrip antennas with dual orthogonal polarization. The array antenna gain is 22 dBi and the size is 125x50 em. The antenna is used in duplex operation and has a bandwidth of roughly 100/0, thus covering the DCS1800 band. The gain of the sector antenna is 16 dBi. The radio parts, 10 receive and 9 transmit radios, are built using redesigned RBS205 parts, Ericsson's commercial DCS 1800 radio base station. Calibration circuits are used to control phase and amplitude deviations between the parallel circuits. The circuits also use common synthezisers that are distributed out to the transceivers. The baseband part was designed to supply a high computational power in order to comply with the high data rate in GSM, while still providing enough flexibility so that various algorithms could be loaded and evaluated. It consists of 21 DSP's, TI floating point C40, connected in a cluster. The user interface is implemented on a SPARe5, which is also connected to a hard disc in order to facilitate both on line evaluation of statistics and storage of large amounts of raw data.
of a mobile radio network. Geographically, the areas are located in typical urban and suburban environments and both in flat rural and hilly rural terrain in and outside the city of Dusseldorf, Germany, The herein presented measurements were collected at test sites covering urban and suburban parts of Dusseldorf. The array antenna was installed on a balcony on the tenth floor of an office building (antenna height approximately 40 m). The surrounding buildings in the covered area were at least 10m lower. The base station test equipment was installed near the antenna (feeder cable length approximately 20 m). The measurements on the other sites included in the measurement campaign involved installation of the antennas in 30m high towers at sites used in the Mannesmann GSM network. Certain parameters, like distance from the mobiles to the base station system, separation angle between the two mobile stations, the output power of the base station and the mobile stations are very important for facilitating a relevant investigation of system performance. Therefore, a set of small measurement areas ("squares") with a diameter of approximately 100300 m were defined. The distance between the squares and the array antenna position was typically 100m to 1 Ian for the first set of squares and some 2.5-4.5 Ian for the second set of squares. The main advantage of this way to select the measurement areas is the resulting independence of any navigation information during the measurements. The adaptive antenna testbed is used together with two test mobiles. These special mobiles can transmit a predefined GSM signal in the uplink and measure BER-, RXLEV- and RXQUAL values in the downlink. The base station is able to measure the signal quality, e.g. BER, for sector and array antenna in parallel. Raw data, symbol sampled I and Q, can also be collected in real time for all ten receive chains. A predefined GSM signal can also be transmitted in the transmission part of the base station. The following test setups were used: 1. Uplink elN measurement One MS drives in a certain square and transmits. The BS measures signal strength and quality by means of using different algorithms. 2. Downlink elN measurement One MS drives around in a certain square and transmits. The BS receives and estimates directional information and transmits in downlink using different downlink algorithms. The MS also measures signal strength and quality. 3. Uplink ell measurement Two MS's drive in different squares simultaneously and transmit on the same channel. The BS here estimates directional information for one or two MS's and measures signal strength and quality using different algorithms. 4. Downlink ell measurement Two MS's drive in different squares at the same time. One MS is transmitting. The BS estimates directional information for the active MS and measures signal strength and quality and transmits towards the estimated direction. Both mobiles measure signal strength and quality.
4. IMPLEMENTED ALGORITHMS A. Uplink Receiving Algorithms The uplink uses MLSE equalizer combining, eight receving branches form the input to an eight-dimensional estimation circuit which reduces data to a one-dimensional soft-valued bit stream that can be used in the convolutional decoder. Two versions have been implemented, one version assuming white spatial disturbance and the other assuming colored spatial disturbance. The two algorithms are denoted Maximum Ratio Combining (MRC) and Interference Rejection Combining (IRe), respectively, see further [1]-[3] for a detailed treatment. Some simulation results for a two-branch receiver are shown in Figure 2 in order to illustrate the differences. Note that the IRe performance is, of course, heavily dependent on interference synchronization, propagation channel etc. B. Downlink Algorithms
The downlink algorithms consist of two parts, one for Direction-of-Arrival (DOA) estimation and one for computation of downlink beamfonning weights using the results obtained from the DOA-estimation. The DOA-estimation block uses synchronously sampled uplink data (1- and Q-data from each of the eight receive branches) as inputs. Beamforming weights, computed using the DOA-estimates, are used by the radio hardware to form the downlink antenna diagram. Different algorithms has been implemented and both fixed beam selection and fully adaptive schemes are evaluated.
5. TEST SETUP A. Planning and Preparation of the Field Trials
B. Evaluation Methodology
Five different test sites were chosen for the investigation of adaptive antennas in representative propagation environments
The evaluation of the adaptive array antenna system is based
686
C. Uplink ell Performance
on the described measurement cases in different measurement areas. Measurements are carried out both for the uplink and the downlink CIN- (carrier to noise) and C/I- (carrier to interferer) environments. The possible coverage improvement compared to a standard sector antenna is evaluated in the elN measurement environment. The amount of interference reduction that is possible to achieve with array antenna technology compared to a sector antenna is evaluated using two mobile stations (carrier and interferer) in the CII measurement environment. Uplink performance is measured as both mobiles transmit on the same channel. The base station measures quality, e.g. BER, for sector- and array antenna in parallel. Quality improvements in BER can then be depicted versus ell at the sector antenna. The downlink quality improvement is, of course, more difficult to measure explicitly. In the testbed, it works in the following way for the ell tests. There are two cases: one where the mobiles measure power transmitted from the sector antenna, and one where the mobiles measure power transmitted using the array antenna. The difference in received signal strenghts at the two mobiles is computed and compared with the difference when the sector antenna transmits in the same measurement situation.
The measurement case considered here consists of two measurement squares which are essentially co-located in both angle and distance, both located about 1.6 km from the base station in an urban area. From Figure 7, we see that the MRC array antenna algorithm performs some 7 dB better than the sector antenna MRC algorithm, while Figure 8 demonstrates a further improvement of some 8 dB for the array antenna IRe algorithm relative the array antenna MRC algorithm. The results indicate that the scattering around the MS and the MI is significant, otherwise the performance gains (array vs sector) would only be around 0 dB for the MRC algoritm and some 9 dB for the IRe algorithm. Hence, the algorithms exploit the propagation environment in a very efficient manner. D. Downlink Cil Performance
Any potential path-loss difference between the two measurement squares (some 30 degrees apart, 1 km from the base station) is examined in Figure 9. For the sector antenna, the relative signal level (MS versus MI) follows the corresponding sector antenna gain quite well, we can indeed find the expected 3 dB (median) difference, which is due to the varying sector antenna diagram levels towards the two considered directions. In Figure 9, we also compare the signal level received at the MS relative that received at the MI for fixed and steered beam downlink transmission, respectively. We find that the median interference suppression, compared with that obtained using sector antenna transmission, is significant (14 and 18 dB, respectively). This reduction is well in line with the beamforming diagrams that can be generated using the measurements of the array that has been made earlier (at a measurement range). Other measurements that have been carried out, with both shorter and longer distances and varying relative angular differences between the MS and the MI, have demonstrated similarly good agreements with measured beamfonning diagrams.
6. TEST REsULTS The MS speed was approximately 30 km/h in all cases. There were no line-of-sight conditions present. Initial remarks: The numbers that can be seen in some of the below "BER-graphs" are measures of the amount of data the corresponding BER value was computed from. Furthermore, both EblNO and ell are measured at one of the receive branches of the sector antenna. A. Uplink CIN Performance
In Figures 3 and 4, the BER performance in a elN environment (suburban area) is depicted for the MRC and IRe algorithms, respectively. The improvement of the array antenna configuration relative the sector arrangement is 4-5.5 dB for both algorithms. This should then be compared with the array antenna gain over the sector antenna, which is some 6 dB. The improvements are as expected, taking into account the particulars of the implementation of the considered algorithms. Similar results have also been found from measurements collected in areas of the type hilly terrain.
E. MS Tracking Performance
Figure 10 depicts the results from a tracking test performed in an urban area (close to downtown DUsseldorf). As can be seen, there is a very good agreement between the estimated and the true direction towards the MS. Observe also the reflection over the river Rhein (located between the BS and the MS in this particular measurement case) at 34 degrees. 7. SUMMARY AND FUTURE WORK
B. Downlink CIN Performance
This paper describes Ericssons GSM testbed for adaptive antennas and some of the results from field-tests performed with the testbed. Our measurements indicate that the higher gain of the array (6 dB in our case), relative a sector antenna of the same height, to a large extent can be translated into a system performance gain. The measurement campaign totally consists of five sites with various characteristics in terms of propagation environments and installation requirements.
In Figure 5, an example of the DOA-estimation behavior for the case of a single MS, located about 1 Ian from the base station in a suburban area, is given. Figure 6 demonstrates the gain, relative to sector antenna transmission, of two narrow beam downlink transmission alternatives. The scenario considered here involves a MS angularly located between two fixed beams, which explains why the fully adaptive downlink beamforming approach performs 1 dB better. Tests performed in hilly terrain areas demonstrate similar behavior, whereas one in a more demanding urban area can find lower gains due to the angularly wider local scattering around the MS.
ACKNOWLEDGMENTS
The development of the testbed described herein involved a large number of people at Ericsson in Molndal and Kista. 687
Amongst others, the authors are greatly indebted to: Fredrik Ovesjo, Hans Mahler, Per Frossling, Johnny Widen, Mattias Gustavsson, Mathias Lindeborg, Ludwig Wallander, Jonas Sandstedt and Sven Petersson. The field tests involved Tom Witzschel, Peter Fischer and Andreas Krug, all at Mannesmann, as well as a number of students, temporarily employed by Mannesmann.
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REFERENcES [1]
"Adaptive Arrays and MLSE Equalization", G. Bottomley, K. Jamal, In proceedings oflEEE VTC'95, pp. 50-54.
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[2] "Interference Rejection Combining for GSM", 1. Karlsson, 1. Heinegard, In proceedings of ICUPC '96, Cambridge, USA. 1
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[3] "D-AMPS Performance in PeS Bands with Array Processing" , K. Molnar, Third Stanford Workshop on Smart Antennas in Mobile Wireless Communications, July 1996.
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[4] "Antenna Arrays for TDMA Personal Communication Systems", U. Forssen et. al, In proceedings of ICUPC'95, Tokyo, Japan , pp. 382-386 . [5] "Adaptive Antenna Arrays for GSM900IDCS 1800", U. Porssen et. al, In proceedings oflEEE VTC '94, Stockholm, Sweden , pp. 605-609.
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Figure 7. BERvs C/I at the sector antenna for the MRC array algorithm. The sector antenna arrangement employs a two-branch MRC algorithm. The MS and the MI are co-located both in angle towards the BSand in distance to the BS. The measurement area is located in an urban propagation environment.
Figure 9. Cumulative distribution functions for signal levels received at MS and MI located at two measurement squares. Geographically, the two measurement areas are separated by some 30 degrees in angle at 1 to 1.5 km distance from the BS. Depicted here are the relative levels (tv1S vs MI) when transmission is made with the sector antenna, and with the array antenna using fixed- and steered beams, respectively. The MS antenna is located vertically on the roof of the measurement van. The measurement area is located in a suburban propagation environment.
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Figure 10. Results from a tracking test made in an urban area. A reflection (over the river Rhein) can be found towards 34 degrees at one particular part of the test route. MS speed was approximately 30 km/h, and the distance to the MS varied between 2.5-3.5 km over the route.
Figure 8. BERvs C/I at the sector antenna for the IRC array algorithm The sector antenna arrangement employs a two-branch MRC algorithm. The MS and the MI are co-located both in angle towards the BSand in distance to the BS. The measurement area is located in an urban propagation environment.
689
PRELIMINARY MEASUREMENT RESULTS FROM AN ADAPTIVE ANTENNA ARRAY TESTBED FOR GSMlUMTS Preben E. Mogensen, Klaus I. Pedersen, Poul Leth-Espensen, Bernard Fleury *, Frank Frederiksen, Kim Olesen, Steen Leth Larsen
Center for Personkommunikation, Aalborg University, Fr. Bajers Vej 7 A, DK-9220Aalborg 0 F..<\X: +4598 15 1583, Email: [email protected] . Abstract: This paper presents preliminary measurement results for an 8 element adaptive antenna array in an urban environment. The results show that in most cases the received signal consists of a single narrowly scattered component impinging from the direction towards the mobile. The angular spread (AS) is found to be no larger than 5 degrees for the test measurements. The correlation coefficient between slow fading and AS is -0.56, indicating that the AS increases when the local signal strength fades. The paper also shows results for a direction of arrival (DoA) algorithm, which is designed for downlink beamsteering in a GSM related system. The results indicate that an averaging window size of one SACCD frame (0.48 s) is a wise choice for DoA estimation in a non-FH network.
horizontal angle of 1200 • This creates an unnecessary high level of interference to/from other cells reusing the same frequency channel. By adaptively steering a beam in the direction of the desired mobile, a reduced interference level is achieved which in turn results in a capacity gain. More advanced algorithms may also place nulls in the direction of potentially strong interfering signals [1]. The potential gain of deploying adaptive base station antenna arrays is obviously related to their geometry as well as the employed processing algorithm. Moreover, multipath spreading in the mobile environment, and features/limitations in the mobile communications system are also important issues to consider when analysing quality, capacity enhancement and cost efficiency of deploying adaptive base station antenna arrays . The work presented is performed as a part of the EU funded TSUNAMI II research project. The results mainly focus on angular spreading and direction of arrival (DoA) estimation.
I. INTRODUCTION Th e use of adaptive antenna arrays in mobile communication systems is an emerging technology area. An adaptive antenna array basically consists of a number of antenna elements combined via an amplitude and phase control network to dynamically form a desired antenna beam . The phase and amplitude weights (beamforming network) may be implemented in either RF-circuitry, or real-time digital signal processing hardware, or as a hybrid of these two methods.
II. BRIEF DESCRIPTION OF THE TESTBED AND THE MEASUREMENTS
The conceptual situation illustrating the deployment of adaptive antenna arrays in mobile communications is depicted in Fig. 1. A sectorized base station configuration typically covers a
A. Testbed Description The TSUNAMI II Stand-alone Testbed I consists of a transmitter (mobile unit) and an adaptive antenna array receiver (base station unit). The carrier frequency is within the DCS1800 frequency band. The frame format is a modified GSM multiframe TDMA structure with a bandwidth of approximately 200 kHz. In TNO a 63 bit PN-sequence is transmitted for the purpose of channel sounding, and in TNt the GSM TCHIFS data format, is being transmitted . The mobile test unit is equipped with a GPS positioning system. The base station has 9 parallel RF receiving units. Eight of them are connected to an 8 element linear antenna array, while the ninth receiver is connected to 'a sector reference antenna. The adaptive array processing is performed by two digital signal processors (DSP's). The adaptive antenna array consists of vertical polarised dipole antenna elements mounted in front of a ground plane. The horizontal element spacing is half a wavelength. A more detailed description of the testbed can be found in [3,4].
* B. Fleury is with the Communication Technology Labo rato ry ofthe Swiss Federal Instit ute of Technology, Switzerland. He is currently on sabbatical leave at CPK . Thi s stay is partly supported by the EU Hum an Capital and Mobility Programme, Contract No. ERBCHBGCf940550 and the "Bundesamt fuer Bildung und Wissenschaft, Switzerland, Contract No. 96.0261.
IThe Stand-alone Testbed is secondary to the TSUNAMI II National Host Field Trial Equipment, see [10J
Adaptive radiation pattern
. . Mobile station
Antenna array
Fig. 1: Beamsteering by using an adaptive base station antenna array:
Reprinted from IEEE Vehicular Technology Conference, pp . 1592-1596, 1997.
690
B. Measurement Description The measurements were performed in downtown Aalborg city, a typical European city characterised by irregular street layout and mostly 3-5 story buildings with only a few higher buildings. The adaptive antenna array receiver was installed on a 41meter high roof of a power plant. Four test routes were measured, see Fig. 2:
the coupling between the antennas is discarded in the derivation of Eq. 1. Moreover, broadside corresponds to ein = 0° . Assuming that the propagation environment is uncorrelated scattering [7] the measured PAS , Pm (8), is expressed as a function of the true PAS , p, (8 ) , according to
Pm(8) = flw(s,e int p,(S ;n)dS in .
(3)
An example of a measured PAS is depicted in Fig. 3 (solid line). The PAS is averaged over 0.48 second in order to remove the fast fading component. As a reference, the figure also contains the theoretical PAS that results from Eq. 3 when Pr(8)
llcm
is a delta pulse (only one incident wave) located at the angle where the measured PAS takes its maximum. The main lobe of the two curves are almost identical meaning that the received power is concentrated within a very narrow angle . The higher sidelobe level for the measured PAS is mainly caused by mismatches in the antenna array, and is not a result from angular spreading. --Measured PAS ------.Theoretical PAS
-5
Fig. 2: Simplified city map showing the location and direction of the adaptive base station antenna array, as well as the 4 test routes.
-10 .
TIl. ANGULAR SPREAD
-20
When estimating the potential gain of deploying adaptive antenna technology in cellular systems, it is essential to have a realistic model describing the angular spreading behavior of the propagation environment. In this paper we focus on angular spread (AS) as a measure of the width of the power angular spectrum (PAS). The former quantity is defined to be the root of the second central moment of the latter functions [5].
-25
-3~90
(1)
In this expression, N is the number of elements and p(S ) is P
o ;
-50
-30
- 10
10
30
50
70
90
B. Derivation of the AS Estimator Provided the propagation environment is uncorrelated scattering, then the channel delay spread is equal to the delay spread of the measured power delay spectrum minus that of the system impulse response [8]. This relation allows to compute an estimate of the true delay spread directly from that of the measured power delay spectrum. Unfortunately, such a simple relat ionship does not exist between the true AS and the measured PAS. The reason is that the integral in Eq . 3 is not a convolution. The derivation of the proposed AS estimator basically relies on the same approach as followed for estimating the delay spread . A function
their field pattern which is approximated by [3]
{cos (8 );-90 < S ~ 90 .
,
-70
Fig. 3: Measured PAS (solid line) and theoretical PAS assuming that only one wave is impinging from the direction were the measured PAS reaches its maximum (dashed line).
l-exp ftt(sin8 ;n -sin8)
p(8) =
I · I \ \
Incidence Angle [deg.]
A. Relationship Between True and Measured PAS 's The complex angular response of a uniform antenna array with half a wavelength element spacing, combined with a conventional beamfonner [6] to a wave with unit amplitude , impinging from the direction 8 in is given by
_ l-exp[jN1t(sin8 in - sinS)] W(8,8;n ) [ ] p(8).
.
(2)
otherwise
The factor p determines the directivity of the antenna. It
cr ; =g(em'cr~)
was found that selecting p = 0.625 yields a good match between P(8) and the measured field pattern of the elements. Notice that
(4)
is obtained that allows to compute an estimate of the true AS,
ot ' 691
from the measured mean incidence direction,
em'
and the
AS,(j m' of the measured PAS. The function has been numerically evaluated assuming that the true PAS is a wrapped Gaussian function. The estimated AS is then chosen to be the value of g evaluated at the estimated mean incidence angle and AS. This method yields accurate results provided that the two _following conditions are satisfied: (i) The function g actually depends on the shape of the true PAS. Thus, the true AS must be small enough so that g is less sensitive to the shape of the PAS. (ii) The signal-to-noise-ratio is high so that the impact of noise on the measured PAS is negligible. For the computation of the AS we only consider the angular window corresponding to the main lobe of the PAS. This limitation is introduced because the estimated AS is very sensitive to the magnitude of the sidelobes in the PAS. As long as the signal power is concentrated in a narrow angular region, this sub-windowing only has a minor impact on the results. The estimated power level, mean incidence direction, and AS are reported in Fig. 4 as a function of time for measurement route 2. It can be observed that the estimated AS lies within a range of 0° to 5°. Moreover, the mean incidence direction and the AS seems to be uncorrelated. This was observed for all four measurement routes.
fades. To test this hypothesis we compute the slow fading pattern by subtracting an estimate of the path loss component from the received power. Fig. 5 shows a scatter plot of the AS versus the slow fading pattern. It is observed that as the signal fades the AS tends to increase and vice versa. The regression line computed for that scatter plot is also reported in Fig. 5. Its slope is equal to -0.IOdegidB, and the correlation coefficient is -0.56. 4.0 3.5 ";3.0
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Fig. 5: Scatter plot showing the AS versus slow fading pattern. so 50
100
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Fig. 6 Dynamic angular: power spectrum during a time snapshot of0.55 seconds. Notice the signal in the direction ofthe mobile, at approx. 20°, fades in a combination with a high side lobe level at ± 90°, at the relative time of approx. 0.3 s.
Fig. 4 Behavior of the estimated power level, mean incidence angle, and AS versus time for measurement route 2.
D. Situations with Large AS As already stated, the method used for estimating the AS only yields accurate results when this quantity is sufficiently small. This condition seems to be mostly satisfied along the selected routes as the main part of the received power remains concentrated in a small angular region. However, as illustrated in Fig. 6 situations also occasionally occur where the received power is considerably spread in direction. This happens when
C. Dependency ofthe AS on Slow Fading It is expected that a dependency exists between power fluctuations due to slow fading and AS. When the power fades it is indeed likely that the strong signal component at the mean incidence direction fades, and the total received signal is thereafter mostly contributed by scattered waves. If this statement is true, then the AS should increase as the signal 692
the component impinging from the direction towards the mobile station vanishes. This observation stresses the necessity to derive a more accurate estimate of the AS with unrestricted validity.
could be performed instead. In case DTX is on, each frame contains 21 bursts which are used for estimating the DoA. Estimated impulse responses obtained from the measurements along the four routes have been used to test the performance of the DoA algorithm. The estimated DoA is compared to that calculated from the GPS information. These results are reported in Fig. 8 - Fig. 11. In Fig. 9 DTX is on and a block average over 21 bursts is performed.
IV. DOA ESTIMATION In parallel to the testbed development, array processing algorithms for GSM related systems have been developed [2,5]. The proposed downlink beamsteering techniques rely on an uplink DoA estimate. The DoA algorithm uses the estimated 8dimensional impulse response which is obtained by correlating each received burst at the antenna outputs with the PN training sequence. For each antenna, the estimated impulse response is 11 symbols long. The DoA algorithm finds the strongest among 22 fixed beams with directions in the set
80 - -- Estimated DoA 60 ···..•.. ·CalculatedD oA
~
e =[-72.7,-59.4,-49.8 ,-41.9,-35.0,-28.5,-22.6,-16.6,-11.0,
C>
-c
-5.5,0.0,5.5.11.0,16.6,22.6,28.5,35.0,41 .9,49.9,59.4,72 .7,88.2] The power evaluation for each of the 22 directions is carried out based on the steering vector w(e )
= [1, exp(- jn sinCe )),... ,exp(- j(N -1)1t sin(e)) f
0
c:
- 20
-40
(5)
-60
-80
and the estimated impulse response vector as follows
o
(6)
100
150 Time [sec]
200
250
Fig. 8: Estimated DoA when DTX is off using a moving average window of 21 bursts. The DoA calculated from the GPS information is the dotted line.
In this expression B is the number of bursts over which averaging is performed and ht,b denotes the estimated impulse response vector for delay f in burst b. The logarithm of the power over 3 taps is calculated for ke [1...9]. The direction and delay tap corresponding to the maximum power is then expressed as (Bo ,ko) =argee9.ke[I•...9] maxP(B.k) .
50
80 60
--- - Estimated DoA CalculatedDoA
40
(7 )
Calculating (Bo,ko) necessitates an exhaustive search over 9
c. CD
20
"C
delay taps and 22 directions. The value B, which determines the averaging period, can be varied. Simulation results presented in [2) have been obtained using B=2I. This value was selected because in discontinuous transmission (DTX) mode it coincides with the number of received bursts during a period of almost one SACCH frame period. The burst structure when DTX is off or on is illustrated below.
';;;' 0 C> c:
«
- 20
- 40 -60
-80
DTX/No DTX (21 bursts)
o
50
100
150 Time [sec]
200
250
Fig. 9: Estimated DoA using DTX on a block of impulse responses. It can be seen from Fig. 8 that the DoA algorithm occasionally estimates a direction quite different from that of the mobile. Inspection of the PAS at these points reveals that this function exhibits a deep fade in the direction of the mobile in combination with a component off that direction. This result indicates that averaging over 21 bursts is insufficient in the NonFH case. Applying a moving average window of 104 bursts (one
(b)
Fig. 7: Burst structure when (a) DTX is on and the SACCH is active and (b) where DTX is off.
The DoA algorithm proposed in [2] is based on a moving average over the bursts. An average over a block of 104 bursts
693
SACCH) instead, the estimated DoA is always close to the direction of the mobile station. Using DTX on the same route and averaging over a fixed block of impulse responses (21 out of 104) yields the result depicted in Fig. 9. The DoA estimation algorithm shows almost identical performances when DTX is on (see Fig. 9) and off. Results for test route 1 and 3 are reported in Fig. 10 and Fig. 11 respectively. In both cases, a moving average over 21 bursts is performed and DTX is off. The large angular deviation in Fig. 11 between GPS information and the DoA estimate in the range 350-500 s is due to poor GPS information and not a DoA estimation error.
period (0.48 s) is sufficient for the algorithm to track the temporal changes of the DoA.
VI. ACKNOWLEDGEMENTS The work presented is partly financed by the EU, ACTS programme as part of the TSUNAMI II project, Technology in Smart Antennas for Universal Advanced Mobile Infrastructure (Part 2). Nokia Tele Communications has also kindly cosponsored the work. Celwave and Analog Devices have sponsored the antenna array and DSP boards respectively. 80 - - - Estimated DoA : 60 ...... 'Calculated DoA , ..
80 60
---- Estimated DoA Calculated DoA ..
40 :
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.
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c. c: c(
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350
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250 300 Time [sec]
.. ~
350
400
450
500
Fig. 11: Estimated DoA for route 3. The GPS position was lost on some parts ofthis route. and therefore differences occur.
400
Fig. 10: Estimated DoA on route 1. It can be seen that the algorithm looses track when the mobile is behind the antenna (see maps).
VII. REFERENCES [1]
It is observed from Fig. 10 that there is a good agreement between the estimated and calculated DoA's except when the true DoAs is close to -9Odeg. In this case the algorithm looses track since the mobile moves "behind" the antenna. The direct path is lost. The sudden large deviation between the estimated and calculated DoA arise because the magnitude of the angular response of the array (see Eq. 1.) takes large values for directions near +90 and -90 deg.
v.
...
[2J
[3] [4]
CONCLUSION AND FUTURE WORK
[5]
Preliminary results are presented which have been derived based on first measurements conducted in an urban area employing an adaptive 8 element antenna array testbed. A method for estimating small angular spreads is proposed. It is found that the angular spread is confined within 0-5deg. A correlation of -0.56 between the angular spread and slow fading is also observed. Based on the measurement data the performance of a DoA algorithm designed for GSM related systems is verified in a real interference-free environment. The results show that the algorithm is capable of tracking the DoA towards the mobile station. An averaging time of approximately one SACCH frame
[6] [7] [8J
[9]
[10]
694
Zetterberg. Per. "A comparison of two systems for down link communication with antenna arrays at the base station. 1995. submitted to IEEE Trans. on Vehicular Technology. Mogensen. Preben, P. Zetterberg, H. Dam. P. Le\h-Espensen, F. Frederiksen. " Algorithms and antenna array recommendations (Part I)", TSUNAMI 2 Technical report AC020/AUClA1.2IDRlP/005/al . 27 Oct. 1996. Mogensen. Preben, F. Frederiksen and H. Dam. "A DSP and dataacquisition architecture for an adaptive antenna array testbed". Proc. of DSP ' 96 Scandinavia. pp. 99-106. Copenhagen. June 18-19. 19%. Mogensen. Preben E.• Frank Frederiksen. Henrik Dam. Kim Olesen. and Sten Leth Larsen." TSUANMI 11 Stand-alone Testbed", Proc. of ACTS Mobile Summit, Granada, Spain, Nov. 1996, pp. 517-527. Eggers. Patrick. "Angular Dispersive Mobile Radio Environment Sensed by Highly Directive Basestation Antennas," Proc. Personal. Indoor and Mobile Radio Communications (PIMRC·95). September 1995. pp. 522526 . Krim, Hamid, M. Viberg, "T wo Decades of Array Signal Processing Research."IEEE Signal Processing Magazine, July 1996. pp. 67-94 . Bello. P.• "Characterization of Randomly Time-Variant Linear Channels," IEEE Trans. on Comm. Syst. , vol. CS-ll. no. 12, Dec. 1963. pp. 360-393. Zetterberg, Per. and Poul Leth-Espensen, "A Downlink Beam Steering Technique for GSMlDCS 1800IPCS 1900". IEEE Proc. of Personal. Indoor and Mobile Radio Communications . Taipei. Taiwan. Oct. 1996. pp. 535539 . Zetterberg, Per. Poul Leth Espensen, and Preben E. Mogensen, "Propagation Model. Direction of Arrival and Uplink Combining for Use in Mobile Communications", Proc. of ACTS Mobile Summit. Granada. Spain. Nov. 19%. pp. 500-509. Clarkson, 1.. "An Advanced Antenna System - Unplugged". Proc. of ACTS Mobile Summit. Granada, Spain, Nov. 1996. pp 302-307 .
Performance Evaluation of a Cellular Base Station Multibeam Antenna Yingjie Li, Member, IEEE, Martin J. Feuerstein, Member, IEEE, and Douglas O. Reudink
Abstract- Experimental test results are used to determine the performance that can be achieved from a multibeam antenna array, with fixed-beam azimuths, relative to a traditional dualdiversity three-sector antenna configuration. The performance tradeoffs between the hysterisis level, switching time, and gain improvement for a multibeam antenna are also examined. The multibeam antenna uses selection combining to switch the signals from the two strongest directional beams to the base station's main and diversity receivers. To assess the impact of beamwidth on overall system performance, the following two multibeam antennas were tested: a 12-beam 30° beamwidth array and a 24-beam 15° beamwidth array. Both multibeam antennas were field-tested in typical cellular base station sites located in heavy urban and light urban environments. Altogether, the system performance is evaluated by investigating three fundamental aspects of multibeam antenna behavior. First, the relative powers of the signals measured in each directional beam of the multibeam antenna are characterized. Then, beam separation statistics for the strongest two signals are examined. Gain improvements achievable with a multibeam antenna compared to the traditional sector configuration are determined in the second phase of the analysis. Results indicate that in excess of 5 dB of gain enhancement can be achieved with a 24-beam base station antenna in a cellular mobile radio environment. Finally, the effects of hysterisis level and switching time are characterized based on gain reductions relative to a reference case with no hysterisis and a 0.5-s switching decision time. Useful approximations are developed for the gain effects associated with varying hysterisis levels and switching times. The resulting design curves and empirical rules allow engineers to quantify multibeam antenna performance while making appropriate tradeoffs for parameter selection.
A
1.
INTRODUCTION
s THE 900-MHz cellular subscriber population continues to grow at a rapid pace, service providers are forced to find new methods of enhancing the coverage and capacity of their networks. One approach is to simply build additional, smaller cell sites to increase capacity or to fill coverage holes (i.e., cell splitting). Unfortunately, building new cellular base stations is a time-consuming, expensive process. In many high-
Manuscript received July 12, 1995~ revised October 27, 1995. This work was supported by U S WEST NewVector Group Inc. Y. Li was with U S WEST NewVector Group Inc., Bellevue. WA 98008 USA. She is now with Lucent Technologies Bell Laboratories, Whippany, NJ 07981 USA. M. Feuerstein was with U S WEST NewVector Group Inc., Bellevue. WA 98008 USA. He is now with the Radio Performance Group at Lucent Technologies Bell Laboratories, Whippany, NJ 07981 USA. D. Reudink was with U S WEST NewVector Group Inc., Bellevue, WA 98008 USA. He is now with MetaWave Communications Corporation, Redmond, WA 98052 USA. Publisher Item Identifier S 0018-9545(97)01323-6.
traffic areas, cell sizes are already about as small as can be accommodated given real-world handoff delays. Sectorization is another method of increasing capacity, but with associated tradeoffs in trunking efficiency. Once again, in many cellular markets, sectorization has been utilized ~o its practical limit. Most cellular service providers are choosing to adopt digital modulation techniques to cope with increasing traffic demands and to accommodate the drive to reduce per-subscriber infrastructure costs. However, an existing customer base of analog-only subscriber radios must still be serviced, especially during the transition period to a digital air interface. As analog radio channels are converted to digital service, the blocking rates on the remaining analog channels inevitably will increase. The situation may well worsen in the future because, at least in the United States, analog advanced mobile phone service (AMPS) cellular service is the only universal standard available for customer roaming from one market to another. Thus, the transition to digital cellular air interfaces may, for a time, actually exacerbate the capacity problems of an already strained analog system. In the 1.9-GHz frequency band, new personal communications service (peS) entrants are faced with the daunting task of building a complete wireless network from scratch. pes operators must quickly achieve coverage and capacity parity with today's cellular carriers in a frequency band where propagation path loss is significantly higher. To create such a network and to simultaneously minimize the number of required cell sites, system operators must take advantage of every possible improvement in link margin. For both 900-MHz and 1.9-GHz wireless systems, one economical approach to the problem of increasing capacity and coverage is to use multibeam adaptive base station antennas. By using adaptive control to keep a narrow beam pointed in the direction of each subscriber served by the cell, the effective gain and carrier-to-interference ratio can be dramatically improved compared to a typical sector configuration [1]-[5]. In addition, for established systems, such a technique can make use of existing cell sites, thus eliminating the costly step of deploying new sites. In this paper, the gain improvement of a low-complexity multibeam antenna system is investigated. The multibeam antenna uses 12 or 24 narrow beams, each with fixed pointing directions. Selection combining [6]-[8] is used to switch the strongest two directional beams for a given subscriber to the base station's main and diversity receivers. In the proposed switched-beam method, the complexity associated
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 46, No.1, pp. 1-9, February 1997.
695
' --I
9{>~~f 850 MHz
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Fig. I. Functional block diagram of mobile transmitter and base station data-acquisit ion receiver system.
with adaptively scanning the beam-pointing azimuth (e.g., by varying complex weights in a beam-forming network [1]-[5]) is avoided by switching between fixed-beam azimuths [6]. The beam-switching approach allows the multibeam antenna and switch matrix to be easily integrated with existing cell site receivers as an applique. Multibeam antenna performance is measured using experimental field tests from existing cellular sites located in heavy urban and light urban environments. The paper examines only reverse link improvements on the base station receive path; however, similar concepts can be used to provide improvements to the forward link base station transmit path as well [1], [3], [4]. Subsequent sections of the paper are organized as follows. Section II describes the experimental setup and the two multibeam antennas that were tested. In Section III, the field test environments and test methodologies are presented. The experimental results for both multibeam antennas are discussed in Section IV. Finally, Section V summarizes the key results and draws relevant conclusions . Acknowledgments are noted in Section VI.
II. EXPERIMENTAL SETUP The experimental test setup included a data-acquisition receiver system located at the cellular base station site in conjunction with an 8S0-MHz continuous wave transmitter placed in a roving test vehicle. Functional block diagrams of the test transmitter and receiver systems are shown in Fig. I. The base station data-acquisition system was developed to measure received signal strength simultaneously from each beam of the multibeam antenna array and also from each antenna of a dual-diversity three-sector configuration. The data-acquisition system was capable of digitally sampling up to 30 receive input ports at a sampling frequency of 1 kHz per port. The receive chain for each sampling port consisted of the following: a bandpass filter (for image frequency rejection) and a directional coupler (for injection of a calibration signal) feeding a measurement receiver with a 3D-kHz intermediate frequency bandwidth. The receive signal-strength indicator (RSSl) outputs from each of the 30 measurement receivers were fed to a 3D-channel 12-b analogto-digital (AID) conversion card. The AID card was located
696
in an MSDOS computer, which also contained a large hard disk for data storage. All field data from drive tests were stored in binary files as raw A/D samples; later post processing introduced the effects of calibration offsets. averaging, and other signal processing. The drive test vehicle contained a signal generator, power amplifier, and a mobile whip antenna. The test vehicle also included a global positioning system (GPS) receiver connected to a logging computer that recorded time versus position information for later analysis. During the course of the experimental work, two multibeam antenna arrays were field tested. The first multibeam antenna had 12 beams, each with approximately 30° beamwidth. The antenna physically consisted of three panels with four beams per panel. The three panels were oriented to provide nonoverlapping 120° azimuth coverage (i.e., the three panels in concert provided complete 360° coverage in azimuth). The actual beamwidths and antenna gains for the four beams on each panel were 35° (14.1 dBd), 30° (15.3 dBd), 30° (15.3 dBd), and 35° (14.1 dBd), with the variations due to inevitable beam broadening off the boresight of the panel. Front-to-back ratios for the beams ranged from 20-35 dB. The second multibeam antenna was a 24-beam array with approximately 15° beamwidth per beam. Again, the antenna physically consisted of three panels, except in this case, each panel had eight individual beams. The actual beamwidths for the eight beams on each panel were 23° (15.2 dBd), 18° (17.7 dBd), 15° (18.7 dBd), 14° (18.4 dBd), 14° (18.4 dBd), 15° (18.7 dBd), 18° (17.7 dBd), and 23° (15.2 dBd). Front-to-hack ratios for the beams ranged between 20-39 dB. Typical worstcase sidelobe levels for both the 12- and 24-beam arrays were down approximately 13-20 dB from the main peak of each beam. The reference antenna system for each of the tests was a traditional dual-diversity three-sector configuration. Each of the three sector faces consisted of two 92° beamwidth (11dBd gain) antennas separated by approximately 4.5 m for spatial diversity. During drive testing, received signals were simultaneously recorded from each of the six sector antennas (three sectors x two antennas/sector). III. DESCRIPTION OF TESTS To investigate the influence of the propagation environment on multibeam antenna performance, two locations were tested: a heavy urban site and a light urban site. The heavy urban cell site was located on the roof of a seven-story office building on the fringe of the downtown core of Seattle, Washington. The cell site was situated near a dense urban environment with a number of tall office buildings in the vicinity. Streets in the downtown core were lined with tall buildings, creating the typical urban canyon prevalent in many modern cities. A major commuting highway passed close to the cell site. The light urban cell site was located on the roof of an II-story office building in the central business district of Bellevue, Washington (a suburb of Seattle). The business district consisted of a cluster of approximately a dozen office buildings, ranging from 10 to 25 stories, interspersed with
commercial business and shopping areas. Locations further from the center of the business district were mixed with typical residential neighborhoods. The cell site was located close to the intersection of two main commuting highways. Four separate measurement campaigns were conducted: 12and 2~4-beam antennas in heavy and light urban environments. During each measurement campaign, the test vehicle was driven along selected drive routes that extended to the edge of the measurement system coverage range. Drive routes were selected to provide a thorough mix of speeds and conditions, as might be experienced by a representative cellular subscriber. Emphasis was placed on highway routes commonly used for commuting. Attempts were made to uniformly cover all azimuth directions around the cell sites. In this manner, an extensive database of measurements was collected for each of the antenna configurations.
IV. EXPERIMENTAL RESULTS To derive the results presented in this paper, the raw signal-strength samples were post processed as follows. First, redundant samples obtained when the measurement vehicle was stationary for extended periods of time (e.g., street lights, stop signs, etc.) were removed. Second, signal-strength readings near the noise floor of the measurement system were removed to eliminate the influence of limited receiver dynamic range. Third, the raw I-kHz samples were averaged over 500ms intervals to remove the effects of fast fading. The resulting local mean received power measurements were then used to provide estimates of the relative performance differences between the dual-diversity sector antennas compared to the multibeam arrays. A. Relative Power Differences and Beam Separations The received power, normalized to the strongest beam, versus rank, based on received power for the 24-beam antenna, is shown in Fig. 2. For comparison purposes, results for both Bellevue (light urban) and Seattle (heavy urban) are shown. Each curve in the figure represents an average of the differences for all the measurements in that environment. As expected, there is a significant difference in received power from the strongest to the weakest beam. For the light urban environment, the total range was 20 dB, while for the heavy urban, the range was 17 dB. As a point of reference, the worst-case sidelobe and backlobe levels for the antennas were approximately 13-20 dB down from the main beam. The reduction in signal range for the heavy urban compared to the light urban was due to the additional reflection and shadowing present in the more dense urban setting. In Fig. 3~ the cumulative distribution function (CDF) of the received power difference between the first and second strongest beams is shown for the 24-beam antenna. For the heavy urban case, 90% of the measurements had a received power difference between first and second strongest beams of 7.5 dB or less. For the light urban case, the difference was 10 dB or less for 90% probability. As expected from Fig. 2, the difference was smaller for the heavy urban, compared to
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the light urban, due to the additional scattering present in the dense urban environment. A plot of probability versus beam separation between the first and second strongest beams is shown in Fig. 4. In the figure, a separation of one indicates that the first and second strongest beams were adjacent to one another. A separation of two means the first and second strongest were separated by one beam, and so on. Obviously, for the 24-beam antenna, the farthest the first and second strongest beams could be separated was by 12 beams. From the figure, the probability of having the first and second strongest beams adjacent (i.e., beam separation of one) is 72% and 55% for the light urban and heavy urban environments, respectively. This illustrates an important
characteristic of the multibeam antenna performance: there is a high probability that the first and second strongest signals will occur on physically adjacent antenna beams. Given that the directions of the strongest two beams tend to be concentrated over a narrow angular extent, typically two-three beamwidths across, it is of particular interest to examine the correlations between the signals from these beams. For the multibeam antenna, the correlation coefficients between the first and second strongest beams are shown in Table I, as calculated using the Pearson product moment formula. When interpreting the correlation coefficients from Table I, it is important to be cognizant of two facts. First, the correlations are calculated based on the SOO-ms averages, not the raw 1ms samples; therefore, the tabulated numbers do not represent correlations between the fast fading on the two strongest beams . Second, the calculated correlations include the effects of selection combining to obtain the strongest signals; thus, the correlations are not between two specific beams, but rather every 0.5 s, the best two average signals are selected. From Table 1, the strongest two signals are more highly correlated in the heavy urban environment, where building shadowing effects and multipath are predominant. For the heavy urban environment, the correlation coefficient for the 24-beam antenna exceeded the l2-beam antenna. In the light urban case, the 12- and 24-beam antennas had approximately the same correlation coefficients. As a relative comparison, the correlations calculated betwee n the signals from specific pairs of physically adjacent beams (i.e., by not performing
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selection combining) are much lower than the correlations between the two strongest signals. It is also quite fascinating to note that the correlations between the two strongest signals from the six sector antennas tend to be slightly higher than the coefficients shown for the multibeam antennas. Of course, in a typical sectored cell site, the main and diversity receivers are fed signals from the pair of antennas assigned to that specific sector with no guarantee that they are 'indeed the two strongest signals.
fit have means and standard deviations that agree to within 0.2 dB. The correlation coefficient between the experimental histogram and the Gaussian model is 0.998, indicating the goodness of the model fit. The log-normal model provided excellent agreement with the experiment data from 12- and 24-beam antennas in both heavy and light urban environments. Fig. 7 is the CDF of gain relative to the dual-diversity sector configuration for the 24- and l2-beam antennas in the Seattle (heavy urban) environment. The mean gains, using linear averaging, for the 12- and 24-beam antennas are 2.95 and 5.23 dB, respectively. The difference of 2.28 dB illustrates the performance advantage of the 15° beamwidth compared to the 30° beamwidth. As might be anticipated, the measured gain difference of 2.28 dB is close to the actual gain differences between the individual 15° and 30° beamwidth antennas. The standard deviation for the 24-beam antenna gain exceeded that of the 12-beam antenna by 2.31 dB, showing the greater performance variability associated with the narrower beamwidth antenna system.
B. Gain Improvement Relative to Sector Antenna
One significant aspect of multibeam antenna performance is the gain improvement that can be experienced compared to a dual-diversity sector configuration. In this section, the received signal strength from the strongest multibeam antenna is compared with the signal strength from the strongest of the six sector antennas (three sectors x two diversity antennas per sector), once again using 500-ms averaged data. A histogram of relative gain for the 24-beam antenna in Seattle (heavy urban) and Bellevue (light urban) is shown in Fig. 5. The mean gains, using linear averaging, are 5.23 and 5.35 dB for the heavy and light urban environments, respectively. This gain improvement translates to potential extended coverage range for cells employing multibeam antennas. Such an approach could result in a lower cell density and, hence, lower network deployment costs. For example, with a propagation path loss exponent of 3.5, an additional gain of 5 dB theoretically results in a 38% extension in coverage range. The relative gains are well modeled as being log-normally distributed (i.e., Gaussian distribution for decibel values). One typical example is shown in Fig. 6 for the 24-beam antenna in the heavy urban environment, where the gain histogram is plotted in decibels along with a Gaussian least mean squared error (LMSE) curve fit. Based on statistics derived from the decibel values, the experimental data and Gaussian best-
C. Effects of Switching TIme and Hysterisis Level
There are several reasons for adding hysterisis or increasing switching time in the beam selection process. Since the multibeam antenna system uses predetection switch combining to feed the base station's main and diversity antenna ports, it is possible for phase changes to occur at each switching event [8]. For an analog FM system, these phase transients at the discriminator output can result in audible clicks in the recovered baseband audio signal [8]. Therefore , it is desirable to avoid unnecessary switching or ping-ponging between beams. Reducing the amount of switching also lessens the computational burden on the processor controlling the beam selection. Finally, slowing the switching speed eases the problem of handoff and interference coordination issues for a widespread deployment of multibeam antenna sites.
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Fig. 8 illustrates the effect of hysterisis by plotting the probability of switching to a new beam versus hysterisis level. The probability of switching to a new beam is determined based on actual switch counts determined with a O.5-s decision rate. In other words, a switching decision was made every 0.5 s; the probability of switching was calculated as the number of times the antenna was actually switched, divided by the total number of opportunities to switch. Results are plotted for the 24-beam antenna in both light urban and heavy urban environments . The first observation from Fig. 8 is that the probability of switching to a new beam is always higher in the heavy urban environment compared to the light urban case. This is
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logical given the increased shadowing and reflections present in the dense urban environment. One would expect the building blockages and street comer effects to be more pronounced in the heavy urban case, causing more frequent switching between beams. For the O-dB case (i.e., no hysterisis), the probability of switching to a new beam is 24% and 13% for the heavy urban and light urban cases, respectively. Adding 3 dB of hysterisis reduces these probabilities to 6% and 3% for the heavy and light urban cases, thus avoiding excessive switching. Adding hysterisis to the switching decision process reduces ping-ponging and increases the average dwell time on a given beam. Increasing the switch decision time also increases the average dwell time on a particular beam. However, both the increasing hysterisis level and increasing switch decision time cause a reduction in the effective gain of the multibeam antenna, when compared against the optimum case. The reduction in gain is a consequence of the fact that the receiver will remain connected for longer periods of time to an antenna that is not actually the strongest beam. The tradeoffs between hysterisis level, switching decision time, and gain are shown graphically in Fig. 9. In the figure, gain is calculated relative to the reference case, which is no hysterisis and a 0.5-s switching decision time. With no hysterisis, increasing the switching decision time from 0.5-6 s results in a gain reduction of approximately 2 dB. With a O.5-s switch time, increasing hysterisis level to 5 dB results in about I-dB gain reduction . Combining as-dB hysterisis level with a 6-s switching time results in a 2.5-dB penalty in reduced gain. Fig. 10 is a 2-dimensional (2-D) plot containing several slices of the surface shown in Fig. 9. The dramatic impact of switching time on the effective gain is illustrated. From the slopes of the curves, each I-s increase in switching decision time results in approximately a I/3-dB gain reduction, which holds for the 24-beam antenna in the heavy urban case. The influence of the propagation environment is illustrated in Fig. 11, where gain reduction is plotted for the light urban
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and heavy urban settings for the case of 2-dB hysterisis. By examining the slopes of the two curves, one can obser ve that the gain reduction is a stronger function of switching time for the heavy urban environment compared to the light urban. This makes logical sense given the larger degree of building shadowing and street comer effects for the dense urban setting. The result is also in complete agreement with Fig. 8, which shows the increased likelihood of switching at each opportunity in the heavy urban environment. In Fig. 12, a set of six curves is plotted to simultaneously illustrate the impacts of hysterisis level, switching time, and
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propagation environment. From the similar slopes of all the curves, one can conclude that the sensitivity of gain reduction to changes in the hysterisis level is independent of the environment type and the switching time ; hence , as a rule of thumb, each decibel increase in hysteris is level results in approximately a l/lO-dB gain reduction . which hold s true for the 24-beam antenna. Another meaningful observation is that the difference in gain reduction between the two environmental types is only significant for switching times greater than about I s. In other words, if the switching time is held to less than approximately 1 s, there is little additional degradation when moving from the more benign light urban environment to the harsher heavy urban setting. As a final comparison, in Fig. 13 the 12- and 24-beam antennas are compared by plotting gain reduction versus switching time. Based on the slopes of the two curves , the performance difference s between the two antennas can be noted . As might be expected, the gain reduction for the 24beam antenna is a stronger function of switching time when compared to the 12-beam antenna . The larger reduction in performance for the 24-beam antenna is expected given the 15° beamwidth relative to the 30° beamw idth of the 12-beam antenna. One would expect to have to switch faster for the narrower beamwidth array as the mobile move s. V . C ONCL USION S
A low-complexity multibeam antenna, based on switching between beams with fixed boresight orientations, has been evaluated in this paper. Switching between fixed beams dramatically reduces the complexity of the associated signal processing hardware and also allows the antenna system to be readily integrated with existing cell site receivers. The reverse link performance of both 12- and 24-beam arrays has been field tested in light and heavy urban propagation environments.
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Based on an extensive experimental measurement database, three fundamental aspects of the performance of cellular base station multibeam antennas have been investigated : I) relative power differences and beam separations for the individual directional beams; 2) gain improvements achievable relative to a traditional three-sector configuration; and 3) the influence of switching time and hysterisis level on the effective gain improvement. Based on experimental observations in heavy urban environments, the relative power difference between the strongest and weakest beams is approximately 3 dB lower than for a light urban environment. Such an observation is consistent with the extra multipath reflection and shadowing that are prevalent in a dense urban setting. This interpretation is also borne out by examining the received power differences between the first and second strongest beams, where the difference is significantly less for the heavy urban than for the light urban case. A calculation of the probability of beam separation shows that, with a high probability, the first and second strongest beams will be located in physically adjacent antenna beams. In fact, in over 80% of the measurements for the light urban environment, the first and second strongest beams were either adjacent or separated by only one beam. Based on the calculated correlation coefficients, the strongest two signals are more highly correlated in the heavy urban environment , where building shadowing effects and multipath are prevalent. The gain improvement obtainable from a multibeam antenna relative to a dual-diversity sector configuration has been examined. Experimental results in heavy and light urban environments show that linear average gain improvements of approximately 2.9 and 5.2 dB can be achieved from 12and 24-beam antennas, respectively. CDF's and probability density functions of gain relative to a dual-diversity sector configuration have been presented. Based on goodness-of-fit measures, the probability density functions for gain improve-
ment in decibels are well modeled as Gaussian. The design curves presented in this paper can be used to estimate the coverage improvement that can be achieved with 12- and 24-beam antennas in representative urban environments. Experimental results have been used to characterize the tradeoff between hysterisis level, switching time. and gain for a multibeam base station antenna. As expected, increasing hysterisis level results in less ping-ponging between antennas ; however, the effective gain of the multibeam antenna is correspondingly reduced. As a rule of thumb for the 24-beam antenna, each decibel increase in hysterisis level results in a I/lO-dB reduction in effective gain for the multibeam antenna, compared to an ideal no-hysterisis case, due to the suboptimum switching operation associated with hysterisis [8). Increasing the switching decision time is one method of reducing the burden on the computer that controls antenna assignment; however, increasing the switching time also reduces the effective gain of the multibeam antenna. As an approximation for the 24-beam antenna, each I-s increase in switching time results in a 113-dB reduction in effective gain, compared to a switching time of 0.5 s, for the multibeam antenna in the heavy urban environment. Once again, the effective gain reduction is due to the fact that the receiver remains connected for longer periods of time to antenna beams that do not contain the strongest signal. As expected. due to the more complex propagation environment, the gain of the multibeam antenna is a stronger function of switching time in the heavy urban environment than in the light urban case. Even though the probability of switching to a new beam is higher in the dense urban environment, if the switching is accomplished rapidly enough. then the effective gain will not be markedly reduced. Results have shown that the difference in gain performance between the light and heavy urban environments is only significant for switching times greater than approximately I s. Because of the smaller beamwidth , the gain of the 24-beam antenna is more sensitive to the switching time than the 12-beam antenna. As a final point, several relevant topics, other than gain enhancement effects, should be mentioned when assessing the overall performance of multibeam antennas. The ability of multibeam antennas to combat fast fading through angular diversity influences the effective coverage area of the cell [2], [7], [8]. The correlation of fast fading on the two strongest beams from a multibeam antenna should be contrasted against that of a typical spatial diversity antenna pair, thereby determining the relative effect of the angular diversity benefit [9]. For capacity-constrained and interferencelimited circumstances, the issues of interference reduction [i.e., carrier-to-interference (CII) improvement) and, hence, frequency reuse efficiency are significant concerns [I], [2], [4). These topics represent important aspects of multibeam antenna performance that are not within the scope of this paper. ACKNOWLEDGMENT
The authors would like to thank D. Jones, P. Perini, M. Harrison, D. Ellingson, and J. O'Connor for performing the experimental tests described in this paper.
702
REFERENCES
[1] S. C. Swales, M. A. Beach, DJ. Edwards,and 1. P. McGeehan, "The performance enhancement of multibeam adaptive base-station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Techno!., vol. 39,pp.56-67,Feb.1990. [2] S. P. Stapleton and G. S. Quon, "A cellular base station phased array antenna system," in IEEE Veh. Technol. Conf, 1993, pp. 93-96. [3] M. Goldburg and R. H. Roy, "The impacts of SDMA (spatial division multiple access) on pes system design," in Int. Con]. Unio. Personal Comm., 1994, pp. 242-246. [4] P. Zetterberg and B. Ottersten, "The spectrum efficiency of a basestation antenna array system for spatially selective transmission," in IEEE Veh. Technol. Conf, 1994, pp. 1517-1521. [5] G. Y. Delisle and A. T. Denidni, "Experimental investigations of phased array characteristics for pes applications," in Int. Conf Universal Personal Comm., 1993, pp. 49-53. [6] T. Aubrey and P. White," "A comparison of switched pattern diversity antennas," in IEEE Veh. Technol. Conj., 1993, pp. 89-92. [7] D. G. Brennan, "Linear diversity combining techniques," Proc. IRE, vol. 47, pp. 1075-1102, June 1959. [8] W. C. Jakes, Ed., Microwave Mobile Communications. New York: Wiley, 1974. [9] 1. R. Pierce and S. Stein, "Multiple diversity with nonindependent fading," Proc. IRE, vol. 48, pp. 89-104, 1960.
703
Space division multiple access (SOMA) field trials. Part 1: Tracking and BER performance G.Tsoulos J.McGeehan M.Beach
Indexing terms: Digitalbeamforming, Adaptive antennas, SDMA
suppress interference [3--6], and beam steering to focus energy towards desired users [7-10]. By exploiting the spatial domain via an adaptive antenna, the operational benefits to the network operator can be summarised as follows [5]: • Capacity enhancement • Coverage extension ('smart' link budget balancing) • Ability to support value added services (e.g. high data rates, user location, etc.) • Increased immunity to 'near-far' problems • More efficient handover • Ability to support hierarchical cell structures
Abstract: An adaptive antenna testbed for mobile communication applications is briefly described and results from field trials presented. The goal is to provide an experimental demonstration of both transmit and receive digital beamforming supporting SOMA user access. Trials are presented for a typical urban environment with different combinations of user positions and the ability of the employed adaptive algorithm to establish the link and track the channels is investigated alongside the link BER performance. The tracking performance of the adaptive algorithm used for SOMA is also tested for an indoor environment against maximum ratio combining and a fixed grid of beams.
1
2
SDMA testbed
The experimental testbed that was used in the field trials was that developed under the RACE TSUNAMI project [11]. The DECf radio standard was selected as the operational wireless bearer since it could be readily integrated with the adaptive antenna platform and, furthermore, DECT can be operated in an isolated radio cell mode, thus allowing networking aspects (e.g. handover) to be addressed at a later phase. Some key characteristics of the OECT system [12] are: • Frequency band: 1880-1900MHz • Number of carriers: 10 • Carrier spacing: 1.728MHz • Peak transmit power: 250mW • Carrier multiplexing: TOMA - 24 slots per frame • Frame length: lOms • TOD with two slots (up-down links) on the same carrier • Gross bit rate: 1.152 Mbit/s • Net channel rates: 32kbiUs traffic (B-field) and 6.4kbit/s control/signalling (A-field), per slot The testbed hardware consisted of an eight-channel system employing a patch antenna array which could be deployed in various configurations and eight independent linear up and down conversion chains which transform the signals to quadrature baseband (see Fig. 1). The baseband system provides two independent bidirectional wideband beamfonner outputs to the DECT radio system. The digital beamforming devices were two DBFII08 chips [13], each providing 32 million complex operations per second processing rate and 8 bit complex data, 11 bit complex weighting coefficients. The two DECT radios were modified such that they only operated on a single fixed frequency and timeslot
Introduction
Over the last few years the demand for service provision via the wireless communication bearer has risen beyond all expectations. The extraordinary fact that some half a billion subscribers to mobile networks are predicted by the year 2000 worldwide [1], introduces the most demanding technological challenge: the need to increase the spectrum efficiency of wireless networks. Filtering in the space domain can separate spectrally and temporally overlapping signals from multiple mobile units, and hence the spatial dimension can be exploited as a hybrid multiple access technique complementing FDMA, TOMA and COMA. One such approach is usually referred to as space division multiple access (SDMA) and enables multiple users within the same radio cell to be accommodated on the same frequency and time slot. Realisation of this technique is accomplished using an adaptive antenna array which is effectively an antenna system capable of modifying its temporal, spectral and spatial response by means of amplitude and phase weighting and internal feedback control. Numerous approaches using adaptive antennas have been considered to exploit the spatial domain, for example null steering to isolate cochannel users [2], optimum combining to reduce multipath fading and ©IEE,I998 lEE Proceedings online no. 19981782 Paper first received 29th April and in revised form 14th November 1997 The authors are with the Centre for Communications Research, University of Bristol, Merchant Venturers Building, Woodland Road, BristolBS8 1UB, UK
Reprinted with permission fcom lEE Proceedings of Radar, Sonar, and Navigation, G. Tsoulos, J. McGeehan, and M. Beach, "Space Division Multiple Access (SDMA) Field Trials Part 1: Tracking and BER Performance," Vol. 145, No.1, pp. 73-78, February 1998. © 1998 by Institute of Electrical Engineers.
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and were also capable of supporting both speech and channel quality assessment measurements (received signal strength and bit error rates). The DECT subsystem contains two digital basestation controllers synchronised together as master and slave and are running in synchronism with the digital beamformers . To compensate for mismatches and component drift within the multiple analogue signal paths , a calibration system was employed. Calibration is required for DBF systems where each channel contains a complete transceiver, each of which must be amplitude and phase matched over the entire signal bandwidth, as explained in greater detail in the companion paper [14]. Two SDMA channels were supported on two independent DSPs, thus enabling the trial system to establish two links within the same timeslot and frequency channel of DECT air interface, through the spatial domain. In addition to real-time beamforming operation speech mode, the beamforming system also had 16 Mbytes DRAM for data logging for postprocessing applications . A PC-based controller with VME-bus interface was used to initiate the beamforming algorithms, enable the measurements, and finally to read the stored data out of the system. The system can work in two different modes: o The diversity or nonadaptive mode, where classical diversity algorithms like switched diversity, selection diversity and maximum ratio combining are employed. This approach is conceptually much simpler than a fully adaptive method and is usually efficient for simple scenarios [11]. o The adaptive mode, where the SDMA technique is used with the help of the MUSIC algorithm [15]. Each iteration of the adaptive algorithm consists of three steps: (i) Estimate the number of signals and directions of arrival (DOA) using the MUSIC algorithm . The update rate is 4Oms, i.e. the computation is spread over four DECT frames, while the covariance matrix is cal-
culated by taking 100 snapshots from one of the four frames. The MUSIC algorithm estimates the number of sources N, by counting the eigenvalues N:ig above some user defined threshold Th'eig which depends on the environment. It then computes the MUSIC spectrum and finds the peaks. It counts the number of peaks NsMUS above another user defined threshold Th'M US which again depends on the environment and if NsMUS < N/g then N, = NsMUS (else N, = N/g). During the field trials the Th'MUS was kept high (e.g. 25dB), so that most of the MUSIC peaks are considered and the estimate of the number of signals is governed by the eigenvalue threshold Th'eig, which was chosen to be around 18dB. (The values for the parameters of the SDMA algorithm were chosen after initial tests were performed for each different scenario.) (ii) Apply the DOAs to a tracking algorithm [16]. When the tracking algorithm finds a source with DOA which remains within an angular range (typical values range from 80 for rural and 150 for urban environments) for a number of successive iterations (6 for the results shown here; by increasing this value to 15 or 20 one effectively reduces the probability to receive weak signals or sudden disturbances but also increases the delay in recognising the signal of a new user), it is assigned to a track. The first track to be established in this way is assumed to be for user A or the master handset , and the second is assumed to be for user B, or the slave handset. If other sources then appear they are treated as additional interferers . When track for user A is established, tracking for user B starts . When the track for user B is also established, the two tracks are used for the look directions of the beams for user A and user B. The tracking algorithm uses a Kalman filter to process the raw super-resolution DOA estimates. If the sources being tracked are temporarily lost the tracking algorithm will use the last estimates of position and velocity to predict its current position . If the source reappears within the predefined angular window of the predicted position, the algorithm will pick up the
705
track again. The tracking algorithm can handle crossing sources if the velocities are sufficiently different. (iii) Synthesise beams for the users. Calculation of the weights for the two orthogonal beams (W A.1> W B.J can be done by solving the following system of equations:
arrival associated with the beamformer. The angle of arrival associated with the steering vector, which produces the maximum inner product with the calculated weight vector, is taken to be the notional angle of arrival of the maximum ratio combiner when an angular comparison is required. After demodulation the signal is compared with a known reference sequence to calculate the bit per symbol errors that have occurred. The reference sequence is a 320 bit sequence in the Bfield of the data field of the DECT burst.
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w{fw Al. 0 (1) where ( )8 represents complex conjugate transpose. Note that the Tx weights are always the complex conjugate of the Rx weights, i.e. adaptive retransmission is used. During the call setup procedure for establishing the DECT links to the two handsets for SDMA operation the initial weight vectors for the two beamformers are loaded for both transmit and receive modes and form two initial beams over which the two users communicate with the base station. Hence, during acquisition the azimuthal position of each user is aligned with the orientation of each initial beam . The MUSIC algorithm is waiting for the appearance of the first strong signal, which it will assume is user A (synchronisation must be first for user A). The tracking algorithm assigns this signal to track 1, and then performs the same steps for user B. The second signal is assigned to track 2 by the tracking algorithm. After the beamformer has acquired both users, two orthogonal beams are computed, and are set up in the two DBFs. Since the adaptive algorithms used to perform optimisation operations will be required to cope with nonstationary environments, one issue which needs attention is the required processing update rate to perform the adaptive beamforming operation. This depends both on the algorithm used and the rate of change of the environment. An indoor picocellular enviroriment is characterised by slowly moving users and high scattering. The fading envelopes at the different antenna elements will be uncorrelated if they are spaced at least half a wavelength apart at the carrier frequency and the low mobile speed results in a relatively long coherence time for the channel. Thus , it is feasible for an adaptive antenna to track the fading envelopes of the signals at each of the antenna elements. Such a scheme suppresses unwanted signals and also minimises the effects of fading , and requires an algorithm update rate which is fast compared to the channel coherence time. On the other hand, a macrocellular environment is characterised by high mobile speeds and very low scattering at the basestation. Since in such environments it is difficult to obtain spatial diversity at the basestation unless large arrays are used (more than lOA. interelement spacing (17)), if a small array is used (e.g. eight elements with 0.5A. spacing), the fading will be more or less constant across the array. Furthermore, if the adaptive processing tracks the mobile signal while simultaneously forming nulls in the direction of cochannel interference sources, then in this case the required update rate is not governed by the coherence time of the channel but rather by the angular velocities of the signal sources. 3
3. 1 Outdoor urban environment
The results are from the trials performed in Bristol during early 1996. The area is a typical outdoor urban environment around the engineering building of the University of Bristol (Fig. 2). The base station antenna is a "}J2 linear deployment at a height of approximately 30 m above the ground.
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3.1.1 SOMA for two stati onary users: User A is at - 300 (or point 2), while user B is at - -3 50 (or point 1) and both users are stationary. The beamformer was able to spatially separate and support the users throughout the whole experiment with a HER which was much less than 10-3. Fig. 3a is a plot of the cumulative distribut ion function (CDF) of the power from each antenna element and the output of the beamformer, for the period of the experiment. More than IOdB improvement in power has been achieved with the adaptive antenn a against the power achieved over a single element. From Fig. 3a the antenn a element with the best CDF curve is selected and in Fig. 3b the power received from that element is plotted along with the power at the output of the beamforme r. 60
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3.1.3 SOMA with two crossing users: In this scenario the two users walk towards each other, user A stops in the middle of the route while user B keeps on walking and crosses user A. User A starts from point 2 and user B from point I (see Fig. 2). From Fig. 7 it can be seen that the beamformer can track the two users apart from the section where they overlap each other. Here the tracking algorithm can not spatially separate the users and hence the system produces an erroneous response for the second user, as it can be seen from the tracked DOA for user B for the period from 30 to 4S seconds. The explanation for this effect is as follows: The tracking algorithm has an estimate of the position and the velocity of each source, thus, the next
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3.1.2 SOMA for two users moving towards each other without crossing: Here, the users start from points 1 and 2 shown in Fig. 2. During the test they initially move at walking pace towards each other up to a point where their angular separation is almost 150 as seen from the base station, here they stop and return to their original positions. In Fig. 4 the tracked DOAs are given, illustrating the ability of the spatial
707
position can be extrapolated from this data. If the raw DOA estimate is outside the defined window (15°), this estimate is ignored and the algorithm attempts to predict where the source is from the last estimate of the source velocity. This works perfectly for user A, but the track for user B is lost and appears to have a track towards 60°, probably confused by some of the spurious raw DOAs produced by the MUSIC algorithm . Meanwhile, the response from the MUSIC algorithm in terms of the raw DOAs for user B, recovers as the sources move apart (45s). However, the tracking algorithm is still indicating that user B is now somewhere between 60° and 20° and so it ignores these new estimates. At t = 50s the raw DOAs are within the threshold range of the slave track, so it interprets them as new estimates of the position of user B and the tracking recovers. Clearly, it is not a requirement for the adapt ive antenna to be able to separate spatially overlapping users. Ways to cope with such problems could be to make use of the handover mechanism to a different frequency or time slot, or to employ a polarisation sensitive adaptive antenna [18]. Due to the fact that the tracking algorithm is confused about which handset is which during the user crossing, the 'number of the bad slots has increased dramatically during crossing for both the up and down links. 60
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3.2 Tests for an indoor environment Fig. II shows the map of the area of the indoor tests. The adaptive antenna was mounted at a height of 1.5m and the user followed the route shown. Due to the fact that in an indoor environment coherent multipath rays have much wider angular spreading (as opposed to the very small angular spreading around a dominant direction in large cell environments), the MUSIC algorithm is expected to have poor performance [19]. This can be seen from Fig, 12 where a lot of spurious response has
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been produced . The maximum ratio combiner (Fig. 13) along with the approach of a fixed set of beams (Fig. 14), seem to produce better results, although the high sidelobe levels of the antenna pattern can cause erroneous response. NLOS
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In non-line-of-sight (NLOS) points (after t = ISs) there are a lot of multipath rays which reach the base station with similar strength but with rapidly and widely varying angles of arrival. These rays cause additional problems to a direction-finding-based beamforming algorithm. Clearly, super-resolution algorithms like the MUSIC algorithm are not suitable for an indoor environment and also generally for environments with highly correlated multipath. Even with super-resolution algorithms that can cope with coherent multipath, it is questionable whether such an approach would be able to acquire the correct position of a user if the scattering of the environment were more than the free-space antenna beamwidth. An optimum combining approach would be more suitable for such environments but particularly for an indoor scenario, its advantage over
Conclusions
An experimental demonstration of both transmit and receive digital beamforming supporting SOMA user access has been presented. Trials in a typical urban environment clearly demonstrated the potential gains of adaptive antenna technology and the viability of the SOMA technique. It was shown that it is possible with the adaptive antenna system to establish the required links and track the channels with an acceptable BER performance (better than 10-3) , for a variety of operational scenarios; however, no attempt has been made here to optimise either the system architecture or control algorithms. 5
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much more simple non or semi-adaptive approaches, like the grid of beams approach, still remains to be proved.
References
ZYSMAN. G.: 'Wireless networks', Sc i. Am. J.• 1995,10, pp. 5255 2 MARCUS, M., and DAS, S.: 'The potential use of adaptive antennas to increase land mobile frequency reuse'. Proceedings of the lEE 2nd international conference on Radio spectrum conservation techniques, Birmingham, UK, September 1983, pp. 11J-117 3 WINTERS, 1.: 'Optimum combining in digital mobile radio with co -channel interference', IEEE Trans .• 1984, Vf-33, pp. 144-155 4 NAGUIB, A.. PAULRAJ, A., and KAILATH, T.: 'Capacity improvement with base station antenna arrays in cellular CDMA', IEEE Trans . Veh. Technol., 1994,43, pp. 691-698 5 TSOULOS , G., ATHANASIADOU, G., BEACH, M., and SWALES, S.: 'Adaptive antennas for microcellular and mixed cell environments with DS-CDMA', Kluwer Wirel. Pers. Commun. J., 1998, WPC-7, 6 TSOULOS. G., BEACH, M.. and SWALES, S.: 'DS-CDMA capacity enhancement with adaptive antennas', Electron . Lett.• 1995,31, pp. 1319-1320 7 SWALES, S., BEACH , M., EDWARDS, D., and McGEEHAN, 1.: 'The performance enhancement of multi-beam adaptive base station antennas for ceUular land mobile radio systems', IEEE Trans. Veh. Technol ., 1990,39, pp. 56-67 8 TSOULOS , G., and BEACH , M.: 'Sensitivity analysis of capacity enhancement with adaptive multibeam antennas for DCSI800', Electron . Lett.• 1996, pp. 1745-1746 9 THOMPSON,1., MULGREW, B., and GRANT, P.: 'Bearing estimat ion techniques for improved reception of spread spectrum signals'. Proceedings of international symposium on Spread spectrum techniques and applications, ISSSTA'94, Oulou, Finland, July 1994, pp. 544-548 10 ZETIERBERG, P., and OTIERSTEN, B.: 'The spectrum efficiency of a base station antenna array system for spatially selective transmission' , IEEE Trans. Veh. Technol .. 1995,44, pp. 651660 II TSUNAMI project final report , R2108IERAJWP1.3IMRlP/0961 b2,1996 12 WERBUS, V., VELOSO, A., and VILLANUEVA, A.: 'DECfcordless functionality in new generation Alcatel PABXs', Electr . Commun. J., 1993, pp. 172-180 13 ' DBF Il08: digital beamfonning device'. Report 0361.82.01.92 ERA TEchnology Ltd., Leatherhead, Surrey, KT22 7SA 14 TSOULOS , G., McGEEHAN, 1., and BEACH . M.: 'Space division multiple access (SDMA) field trials. Part 2: Calibration and linearity issues', lEE Proc.- Radar Sonar Navig.• 1998, 145, (I), pp. 79~4 15 SCHMIDT, R.: 'A signal subspace approach to emitter location and spectral estimation', IEEE Trans., 1986, AP-34, pp. 276-280 16 PEREZ-NEIRA, A., and LAGUNAS, M.: 'High performance DOA trackers derived from parallel low resolution detectors '. Proceedings of the 8th IEEE signal processing workshop on Statistical signal and array processing , SPWSSAP-96, Corfu , Greece , 1996, pp. 558-561 17 LEE, W.: 'Effects on correlation between two mobile radio base stat ion antennas ', IEEE Trans .• 1973, COM--21, pp. 1214-1224 18 COMPTON, R.: '00 the performance of a polarisation sensitive adaptive array' , IEEE Trans ., 1981, AP~, pp. 71&-725 19 SHAN , T., WAX, M., and KAILATH, T.: 'On spatial smoothing for direction-of-arrival estimation of coherent signals', IEEE Trans .. 1985, ASSP-33, pp. 806--811
709
Space division multiple access (SOMA) field trials. Part 2: Calibration and linearity issues G.Tsoulos J.McGeehan M.Beach Indexing terms: Digital beomforming, Adaptive antennas, SDMA
2
Abstract: In Part 1 an experimental demonstration of both transmit and receive beamforming supporting SDMA user access was given. In Part 2 some implementation issues critical to the performance of an adaptive antenna system are studied from a practical point of view. Results for the sensitivity of the system to calibration errors and the linearity of the transmit power amplifiers are now presented from measurements made during the field trials.
Channel error correction and calibration
A relatively simple mathematical model can be employed to investigate the sensitivity of the system to calibration and linearity offsets, outlined as follows [1]. Assume that the input signal at the nth antenna element is represented by x~n(t) = Re(ej(w+O)t) (1) where Re(x) is the real part of the signal x, OJ is the angular frequency of the carrier, and n is the angular frequency of the signal. After down conversion to digital baseband I, Q, the signal can be expressed as
x~1Lt(t)
1
Introduction
Calibration is required for digital beamforming systems where each channel contains a separate transceiver, each of which must be amplitude and phase matched over the entire signal bandwidth. Sources within a digital beamforming system which are likely to give rise to distortion products include: gain compression, intermodulation products, internally generated noise, spurious effects from mixers, amplitude and phase mismatch across the signal bandwidth, I and Q imbalances, quantisation noise, DC offsets, saturation effects, sampling jitter and spectral aliasing from the AID and D/A conversion, time delay mismatches. Distortion introduced by any of these mechanisms will effectively preclude precise pattern control and significantly reduce the overall benefit that adaptive antennas can offer to mobile networks, e.g. in terms of enhanced capacity. In the following Sections the techniques employed in the RACE TSUNAMI testbed to tackle these problems are introduced. First, the calibration method is discussed and experimental results for the sensitivity of the adaptive antenna system to the calibration process presented. A technique which eliminates the nonlinearity products within the digital beamforming transceiver, i.e. the power amplifiers, is introduced and experimental results with and without linear amplifiers presented. ©IE~ 1998 lEE Proceedings onlineno. 19981781 Paper first received 29thApriland in revised form 14thNovember 1997 The authors are with the Centre for Communications Research, University of Bristol, Merchant Venturers Building, Woodland Road, BristolBS8 1UB, UK.
= (1 + c~)ejOt + c:~c
+
E 00
+ c:~Qe-jOt
c~ e j mn t
(2)
Iml>! where EnA is a complex constant representing the amplitude error differences between the receiver modules, EnDC is a complex constant representing the DC offset caused by the ADC and the low pass filter, e"IQ is a complex constant representing the I, Q imbalance caused by differences in the amplitude and quadrature phase between the I and Q channels, and e"H is a complex constant representing the higher-order harmonics caused by device nonlinearities. Note that eqn. 2 describes the signal from the nth array element; this is why it does not include any phase component due to interelement distance. Using spectral analysis with the Fourier transform, the individual signal components can be isolated
= x~ut(w)lw=o c~c = x~ut(w)lw=o c~Q = x~ut(w)lw=_n = x~ut(W)lw=3n
1 + c~
c:
(3)
(4) (5)
(6) where in eqn. 6 only the third-order harmonic is considered. Estimation of the channel errors involves injection of a known pilot tone in each of the channels and measuring the output signals. The differences in amplitude and phase between these signals and the original stimulus constitute the channel errors. These errors can be represented as a diagonal matrix, the inverse of which is the error calibration matrix. There are two basic approaches for the calibration to be combined with the DBF design. The first is called direct calibration, and provides calibrated signals to both the DBF and the
Reprinted with permission from lEE Proceedings on Radar, Sonar and Navigation, G. Tsoulos, J. McGeehan, and M. Beach, "Space Division Multiple Access (SOMA) Field Trials Part 2: Calibration and Linearity Issues" Vol. 145, No.1, pp. 79-84, February 1998. © 1998 by Institute of Electrical Engineers.
710
demanding. Here the quality of the calibration will be evident from the noise floor of the MUSIC spectrum, which is an indicator of the null depth that is achieved in the field.
weight controller. An alternative approach is calibrating only the steering vector of the weight controller. This approach is based on the observation that the only difference between a calibrated and an uncalibrated output, is that for the former the steering vector p must be multiplied by the inverse of the calibration matrix (C)
y(k) == xT(k)R-1(k)p
-10
m
.----t.
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cal ibr at.iori
C5
Ycal(k) == (Cx(k))T(CxT(k)x(k)C)-lp {:} Ycal(k) == xT(k)R-1(k)C-1p (7) where x(k) is the observed signal vector element at instant t = k'I'; T, is the sampling interval, R-I is the inverse covariance matrix of the observed s; -nal, y and y cal are the normal and the calibrated output signals respectively and [ ]T denotes transposition. An important factor to be considered for the calibration process is if the error correction is frequency dependent (wideband) or frequency independent (narrowband). Frequency dependent calibration errors originate from the incorporation of multiple independent receivers in each of the element channels and consideration of the channel bandwidth versus carrier frequency. In [1] it was shown that for an ideal array with 256 elements designed to produce a null with -85dB depth for radar applications, there is a 2-3dB reduction in the null depth between the 0 and 1% bandwidth cases, for amplitude errors ranging from 0 to 2dB and phase errors ranging from 0° to 200 dB. With 10% bandwidth the null depth reduction is between 23 and 26dB when compared with the 0% case. There are two general approaches for wideband calibration [1]: Frequency domain correction: Each element signal is transformed to the frequency domain, then multiplied with the calibration correction factors and finally transformed back to the time domain with an inverse Fourier transform. The advantage of this approach is that excellent compensation can be achieved. The disadvantage is that it needs two FFT operations in each channel. Time domain correction: Here the calibration can be performed with one FIR filter for each channel. The role of the filter is to equalise the frequency response of each channel. With existing systems for mobile communications (e.g. GSM, DCS1800, IS-95), the used bandwidth is less than 2% of the central frequency, hence frequency dependent errors will not have major impact to the performance of the system, as opposed to radar or future third generation wideband mobile communication applications. Thus a single fixed-frequency calibration method can be employed. The accuracy required from the calibration process is mainly driven by the effects of the element mismatches on the adaptive signal processing. For algorithms such as selection diversity which rely on power measurement, it is only necessary to ensure that any gain mismatches introduced in the receive chains do not bias the power measurements made across the array. For algorithms employing fixed receive beams the distortion of the beam pattern resulting from the element mismatch must be considered. For super-resolution algorithms, such as the MUSIC algorithm, which are known to be susceptible to array mismatch errors [2], the calibration requirements are significantly more
t:
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Fig.1 Theoretical calibration mismatch performance for two-element antenna array
To obtain an approximation of the calibration accuracy required, a simple model of two antenna elements where the second element introduces amplitude and phase mismatches, can be considered. Fig. I shows the variation of the residual cancellation error due to the amplitude and phase mismatch. Ideally, the output should be zero (i.e. a perfect null steered towards an interferer) which occurs when there is no amplitude or phase mismatch. A maximum phase mismatch of I ° and amplitude mismatch of 0.1 dB must be maintained if a -30dB null depth level [3] is required to secure sufficient interference cancellation for mobile communication applications.
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Fig.2 Receiver calibration outline
a Beamforming mode
b Calibration mode
3
Receiver and transmitter calibration tests
As outlined in Fig. 2, calibration of the receivers in the RACE TSUNAMI testbed was implemented offline (not continuously), using an external signal generator to produce a carrier tone which was split eight ways and directly injected into the multiple receiver ports. Using this approach the tone need not be phase locked to the receiver local oscillators, since it is only the relative phase and amplitude of the signals in each channel that needs to be determined, Also, the amplitude and phase relationship of the 1:8 power splitter was calibrated in the laboratory before the system level results could be obtained and it is assumed that these parameters do not vary significantly with time. Calibration was performed before the trials and at regular intervals during the trials to account for thermal drift in the receiver and transmitter subsystems, as well as connector wear and cable rerouting necessary for the numerous field deployments. In summary, this process was achieved by measuring the transfer functions of all eight paths and applying a (complex) correction coeffi-
711
cient to each path to make each path have the same nominal characteristics. For transmitter calibration, the calibrated receivers were connected directly to the transmitter outputs and the transmitted vector was compared with the known excitation. 3
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Fig. 3 shows the variation in amplitude and phase distortion of the signals in the transmit path observed over several hours of operation in room temperature and numerous days (-165 tests). The gross fixed offset varies over almost 2dB and 180° across the different elements (see mean values). The amplitude and phase distortion of each array element are given relative to the first array element. Also, precalibration, the worst dynamic offset about the mean is 4.1dB in terms of amplitude and 35° in terms of phase variations. As shown for the receive path, this dynamic variation can be attributed largely to temperature variation in the system. Figs. 4 and 5 show similar analysis but for the receive path. Here measurements were taken for a temperature range from 14 to 27°C and also for a fixed room temperature of 24°C repeated on different days. It can be seen that the mean distortion varies over a range of almost 1.76dB and 180° across the array prior to calibration. Also, the dynamic calibration offset varies over a range of almost 1.1dB and 20° about the mean, respectively. Furthermore, it can be seen that even with the same room temperature the amplitude and phase mismatches can change significantly from day to day (0.575dB and 19° for amplitude and phase variation, respectively). This underlines the necessity for regular and possibly continuous calibration to constantly ensure low levels of amplitude and phase errors in the transceiver chains. In addition, the TSUNAMI system contains a ganged AGC providing some 40dB of control and hence the calibration must be repeated for each gain setting since the element mismatch varies at different gain settings. Fig. 6 shows that the amplitude and phase distortion vary over a range of 0.23dB and 2.7°, respectively, with different gain settings. The gain is adjustable over 40dB range in
Amplitude b Phase Mean values are plotted. Maximum and minimum values for each element are indicated by bars separated by dashed lines
Q
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712
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Fig.6 Mismatch for differ ent gain settings and for each antenna element a Amplitude b Phase Mean values are plotted . Maximum and minimum values for each element are indicated by bars separated by dashed lines
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is limited within a window of 0.057dB and 0.58°, respectively, (Figs. 7 and 8). This level of performance can be translated to an approximate cancellation level of -35dB from Fig. 1. Further, Fig. 9 shows the I, Q diagram produced from the unmodulated signals of the sixth antenna element. The offset ellipse is the uncalibrated output, the inner noisy circle is the calibrated output but with calibration factors calculated in a previous day, and the outer centred to zero circle is the calibrated output. With the old calibration factors the ellipse is corrected and centred to zero, but only after the new calibration the system appears to have reached
-0.4 Ll--72--3~;"";""-4~-~5'---~6--~-~-7 8 b element
Fig.7 Rx calibration as function of temperature a Amplitude mismatch after calibration b Phase mismatch after calibration
After calibration, the mean amplitude and phase variation is limited to 0.02dB and 0.08° across the array, while the worst case dynamic variation for the elements
713
an acceptable level of purity. The only problem that was noticed during the experiments was that although all the circles should have the same radius, there are small deviations as it can be seen from Fig. 9. This is because the array elements can still introduce small imbalances which the calibration loop employed in this testbed cannot correct. 4
Linearity issues for adaptive antennas
The spurious-free dynamic range of the transceivers also plays an important role in the performance of the adaptive antenna and it is usually specified in terms of the maximum level of intermodulation products (IMP) that can be tolerated. The creation of the IMPs is a complex nonlinear process. For two signals at frequencies!1 and!2 present at the input to a device or system, the output frequency spectrum can be represented as the sum of discrete frequencies
E E A(i,j) . (i· /1 ± j · /2) 00
spectrum ~
00
(8)
i=1 j=1
where the amplitude of each intermodulation product is given by A(i, J). The majority of the IMPs are at frequencies outside the bandwidth of the desired user. Especially when a signal without constant envelope passes through nonlinear devices, the signal is spread, which effectively means that adjacent channel interference and eventually capacity reduction is caused. These effects will be even worse when multiple carrier base station architectures are employed alongside adaptive antenna signal processing to support handover in the frequency domain whenever users can not be maintained on SDMA mode. The use of digital baseband beam forming techniques poses high linearity demands on both the RF/IF upand down-conversion chain. This is because the weights of the beam former are carefully calculated and constructed at digital baseband and distortions in the up or down conversion chains can alter the antenna beam pattern [1]. Some of the fixed mismatches between channels can be calibrated out using calibration techniques, as already discussed, but nonlinearity effects in the transceiver chain cause intermodulation distortion which cannot be calibrated out in any practical way. In [4] an example for the impact of inband intermodulation distortion on sidelobe levels, null depth, main beam width and change in null direction, was given for an eight-element linear array with Dolph-Tchebyscheff weighting. It was shown that as the level of nonlinearities increases, the sidelobe level also increases and there is considerable shift of the nulls. These changes have serious system implications as the degree of interference suppression will be severelyimpaired. The intermodulation products will increase the level of spurious radiation and this will ultimately reduce the performance of the adaptive antenna. Generally, there are two solutions to the problem • use of backed-ofTclass-A amplifiers. This solution is not power efficient, and furthermore the required levels of IMD for such systems cannot be easily achieved. • use linearisation techniques [5, 6]. RF power amplifier linearisation is a classic problem to wireless system designers. The use of feedback or feedforward from a nonlinear system to provide a correction signal is a well established technique for
714
improving the linearity of amplifiers. The feedforward linearisation technique [5] was considered to be the most suitable approach for adaptive antenna applications mainly because of the high degree of linearity attainable, broad frequency bandwidth and unconditional stability. The feedforward amplifier operates by comparing the distorted main amplifier output-with an undistorted reference signal; the error signal generated is suitably combined with the main amplifier output such that the distortions are cancelled. Eight amplifiers based on this architecture were built for the TSUNAMI testbed system. Each module includes an IQ up converter and fully adaptive control (using a TMS320C50 DSP card) of the amplitude and phase control in the main amplifier signal path. When the signal passes through the main amplifier, the nonlinearity of the amplifier generates IMPs at about -27 dBc while at the output of the operating feedforward amplifier the IMPs are at about -65dBc. The use of linear power amplifier technology will also support the deployment of multiple carrier base station architectures alongside adaptive antennas, with capability of supporting IMD levels better than -80 dBc using multiple adaptive loops. This is important, as mentioned, since the SDMA operation cannot always be maintained owing to the spatial location of the users, thus handover to the frequency domain will be necessary. desired user
interferers
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Fig. 10
-40
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0
20
40
60
angle, deg Difference of measured radiation patterns with and without lin-
ear power amplifiers
In Fig. 10 the difference of the measured radiation patterns with and without power amplifier linearisation is shown. The environment where these measurements were performed was an open field test area. The testbed transmits with a radiation pattern which is the response to a simulated scenario where the desired user is approximately at 32° and there are interfering users at -3, -32, -37, -43, -45, and -50°. Power measurements were performed every 5° along the circumference of a circle with radius -20m which had the adaptive antenna array at its centre. Without linear power amplifiers the sidelobe level is generally increased and in particular for the nulls there is a decrease which ranges from 5 to 12dB. 5
Discussion
In the pursuit for schemesthat will efficiently utilise the spectrum, attention has turned into spatial filtering methods using advanced antenna techniques, adaptive or smart antennas. An experimental demonstration of
both transmit and receive beamforming supporting SDMA user access was performed in Part 1 of this paper [7] where it was demonstrated that it is possible with the adaptive antenna system to establish the required links and track the channels with an acceptable BER performance (better than 10-3) , for a variety of operational scenarios. In Part 2, some implementation issues critical to the performance of an adaptive antenna system have been investigated. Results showed that the amplitude and phase distortion of the transceiver paths vary over time and for different gain settings and indicated the need for regular calibration. After calibration the amplitude and phase errors were reduced to 0.057dB and 0.58°, respectively. The last values can theoretically give an approximate cancellation level of -35dB. Finally, the sensitivity of the system to the linearity of the transmit power amplifiers was discussed. It was shown that the produced radiation patterns are affected from the intermodulation distortion introduced by the power amplifiers and that this problem can be combated with the use of linear power amplifiers. To further develop adaptive antenna technologies for third-generation systems, a follow-on project ACTS TSUNAMI II is under way. Here, within the family of the GSM derivatives, it was decided to use the DCS 1800 standard because arguably it is closer to third generation than GSM, but most importantly, it utilises very similar frequency bands to UMTS/ IMT2000 proposals. The TSUNAMI II field trial will exercise the adaptive antenna system to fully identify the performance of the adaptive antenna relative to the performance of the existing trisectored DCS 1800 base
stations. Of particular interest are parameters which impact on cell sizing, fading protection and, most importantly, spectral efficiency gains using adaptive antenna technology. These will be assessed both analytically and via field trial experimentation. 6
Acknowledgments
The authors thank the CEC for funding the RACE TSUNAMI project and the partners of RACE TSUNAMI for their contributions to this activity. Special thanks are due to R. Arnott of ERA Technology and R. Wilkinson and C. Simmonds of Bristol University. Finally, the authors thank the reviewers for their useful comments. 7
2
4
5 6
7
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References LANGSTON, J., SHASHIKANT, S., HINNMAN, K., KEISNER, K., and GARCIA, D., Design definition for a digital beamfonning processor, Rome Air Development Centre, NY, 1988 SHAN, T., WAX, M., and KAILATH, T.: 'On spatial smoothing for direction-of-arrival estimation of coherent signals', IEEE Trans., 1985, ~-33, pp. 806-811 TSUNAMI project final report, R2108IERAlWP1.3IMRIPI0961 b2, 1996 XUE, H., BEACH, M., and MCGEEHAN, J.: 'Non linearity effects on adaptive antennas'. April Proceedings of the 9th international conference on Antennas and propagation, ICAP'95, Eindhoven, Netherlands, 1995, pp. 352-355 KENNINGTON, P., and BENNET, D.: 'Linear distortion correction using a feed-forward system', IEEE Trans. Veh. Techno/., 1996,45, pp. 74-81 TSUNAMI, I., Baseline technology, R2108/ERAlWP5.IIDS/l1 0221b1, 1994 TSOULOS, G., McGEEHAN, J., and BEACH, M.: 'Space division multiple access (SOMA) field trials. Part 1: Tracking and BER performance', lEE Proc.-Radar Sonar Navig., 1998, 145, (I), pp. 73-78
Chapter 6 Applications and Planning Issues
T
HIS book concludes with six papers that deal with more general issues related to adaptive antennas including specific applications for user location, indoor wireless high data rate networks, planning issues, and novel techniques that have attracted a great deal of attention recently, such as space-time processing and multiple input, multiple output systems, that seem promising to open new directions for this technology in the future. The first paper demonstrates how very high data rates (in excess of 1 Gbps) can be achieved in an indoor environment with the use of antenna arrays at the transmitter and receiver ends. The work from Stanford University studies the performance of linear and circular base station antenna arrays with different topologies, angular spread, and number of elements in order to extend the coverage of a net-
work. The next work considers the use of adaptive antennas in order to solve resource planning problems. In particular, improvement of the reuse distance is considered in conjunction with the fractional loading technique in the context of improving spectrum efficiency and reducing the effort for network planning. A recent mandate by the Federal Communications Commission that requires all cellular communications service providers to provide user location services for emergency calls has triggered much of the work in this area. Caffery and Stuber consider the application of time and angle of arrival methods to achieve this goal in CDMA systems. Finally, the last two papers of the book present work on multiple input, multiple output systems and processing methods to achieve high data rate communication.
717
High Data Rate Indoor Wireless Communications Using Antenna Arrays Michael J. Gans, Reinaldo A. Valenzuela, Jack H. Winters, and Manny J. Carloni
AT&T Bell Laboratories Holmdel, New Jersey 07733 USA Abstract
typically 60 dB (relative to 1 meter, averaged over the multipath fading), while the nns delay spread is typically on the order of 100 ns (2,3]. This rms delay spread limits the maximum data rate to about 1 Mbps. Current proposals consider equalization or multicarrier processing to increase this data rate to 20 Mbps, but the circuitry is near the complexity limit for an economical system and maintaining reasonable outage probability with a 60 dB propagation loss (relative to 1 meter) may be difficult to achieve. Here we consider the use of phased arrays (tested in our experiment by using directive antennas) to increase the power margin and decrease the delay spread of the signal at the receiver, thereby permitting data rates in excess of 1 Gbps. Note that if the multipath in a building generated signals at the receiver that were uniformly distributed in power and delay spread with respect to angle-of-arrival, antenna arrays would not be useful. : However, results using the propagation-prediction techniques of [1] for in-building propagation over an entire floor of the Crawford Hill Building show that this is not the case. In particular, our results show that arrays at the transmitter and receiver with 25 0 beamwidths can isolate one ray, with high probability, and thereby achieve nearly the full gain of the antennas and eliminate delay spread. To support this conclusion, we present experimental results for 622 Mbps at 19 GHz from several locations within the Crawford Hill building, using manually-scanned directive (15 0 beamwidth) hom antennas. We have investigated ways of economically fabricating antenna arrays with these beamwidths, which, based on our results, would make entire floor coverage at high data rates economically feasible. In Section 2 we discuss the data rate limitations due to power margin and delay spread. The effect of antenna arrays is studied in Section 3, using both propagationprediction results and experimental data.
In this paper, we consider the feasibility of indoor wireless communications at very high data rates (up to multi-Gbps). In particular we wish to use one base station to cover the entire floor of an office building, which may have in excess of 60 dB propagation loss relative to 1 meter. This feasibility depends on two factors: received signal power margin and delay spread. Based on results using the propagationprediction techniques of [1], supported by experimental results up to 622 Mbps, we conclude that neither multicarriers, equalization, nor antenna arrays with less than 1600 elements at one end of a communication link are economical methods for increasing the data rate substantially above 10 to 20 Mbps for multiple room indoor wireless coverage. However, based on the propagation-prediction techniques of [1] and verified by our experimental measurements using directive antennas (15 0 beamwidth) at both ends of a link between the center of the Crawford Hill building to an end laboratory, we have shown that high-speed ubiquitous communication is feasible. Using antenna arrays with 50 to 200 elements at both the transmitter and receiver, we expect to obtain entire floor coverage at data rates in excess of 1 Gbps.
1. Introduction In this paper, we consider the feasibility of indoor wireless communications at very high data rates (up to multi-Gbps). In particular we wish to use one base station to cover an entire floor of an office building. This feasibility depends on two factors: received signal power margin and delay spread. Previous measurements have shown that the maximum propagation loss for a single floor in several office buildings, including the Crawford Hill building, is
Reprinted from 6th International Symposium on Personal, Indoor and Mobile Radio Communications, pp. 1040-1046, 1995.
719
2. Data Rate Limitations Second Floor Averaged Propagation Loss 4 feet into Offices, Crawford Hill
Consider first the received signal power margin. The margin is given by
(Dm MMSUred by OL RSR & RAV at18 GHz.2I10194)
10
r----------------
(1)
o Eb
where N
o
is the energy per bit to noise density ratio at
,,
the receiver and -
is the ratio required to achieve a
,
e,
P rec -=-No N
-0
.,.,
(2)
where Prec is the received signal power given by [6, p. 490]
,
\
\
\ ----- ----,- --------------------------------
.E -30
...o
Fixed at -50.1' dB beyond 30 met....
\
m
given bit error rate (BER). Now,
,,
, --------------,----------------- -------------------------
-20
req
Rcvd Power dBm Propaglltlon Loss ----
---------'""~--------------- -34.34Lgt(r)up to 30 meters range
"
Eb
No
-10
,,
-
-~--------------------------------
""
""
"",
,,
-40
------------~------------------------
-50
---------------\\-;'\~,,."'i--
,
'~\\
"
-60
-----------------------------------------
(3) -70 - - - - - - - - - - - - -
1
10
100
Range in Meters
N is the noise power given by N =kTB·NF
--.J
(4)
Figure 1 and Pa is the power out of the transmit amplifier, Lcr is the loss of cable to the transmit antenna, G, is the transmit antenna gain, A is the wavelength, Lp is the propagation loss relative to 1 meter free space, Gr is the receive antenna gain, L CR is the loss of cable from the receive antenna, k is Boltzmann's constant (l.38xlO-2o mWIH:zJ°K), T is the system noise temperature, B is the bandwidth, and NF is the noise figure of the receiver. In (4), we assume that the signal bandwidth is equal to the data rate. Let us consider typical values for the following parameters: Pa=23 dBm, LCT=l dB, LcR=l dB, T 290 OK, and NF - 6 dB. To operate with data rates up to Gbps, the carrier frequency must be in the range of 19 GHz (or higher), or A - 3xl08/1.9xl010 m. Now, the prediction techniques of [1], along with experimental measurements [1], have shown that in several buildings, including Crawford Hill (which has sheetrock interior walls), with a suitably-placed base station, the maximum propagation loss, Lp , on one floor is equal to 60 dB. In particular, Figure 1 shows propagation loss
Propagation loss versus range for the Crawford Hill building.
measurements of the Crawford Hill building made by R. S. Roman, O. Landron, and R. A. Valenzuela at 18 GHz using omnidirectional antennas. These results show the loss, averaged over the multipath fading by moving the antenna over an area with a radius of several wavelengths, for transmission from the center of the main hallway to 4 feet inside each room (i.e., no line-of-sight). The maximum loss is seen to be less than 60 dB. Thus, with a 60 dB loss, from (2)
E
Nb
o
= 71 dB - IOloglO(B) + Gt + Gr
(5)
where Gt and Gr are in dB. If we assume that a BER of
10~8 is ~Uire~, IWith coherent detection of binary phase
shift keying
0-
720
-
N
req
12 dB [6, p. 380], and the margin
receive antenna gain is cancelled by the loss of power from the signal outside the beamwidth. Similarly, transmit antenna gain would not increase the margin. However, results using the propagation-prediction techniques of [1], for transmission to users up to 4 rooms away, show that even though the signal received by a user can arrive via hundreds of rays at different angle-ofarrivals, about 50% of the signal energy, i.e., total power from all rays, which is approximated by the multipathaveraging in the measurements mentioned before, is usually concentrated in one ray. Thus, directive antennas should provide an increase in multipath-averaged received signal power over isotropic antermas within 3 dB of their directive antenna gain. For an antenna with a beamwidth in azimuth and elevation (assumed equal) of e in degrees, the gain for small eis given by (see also [7])
is given by (from (1) and (5)) Margin
= 59 dB
- lOloglO(B) + G, + Gr.
(6)
Thus, with isotropic antennas (Gt=Gr=O dB), the maximum data rate is about 800 kbps. (Note that to increase the data rate limitation due to power margin, coding could be used to permit a higher raw BER. For example, with BER = 10-2 , and coding to reduce this to 10- 8 , an additional 4.3 dB margin can be obtained. Thus, from (6), the maximum data rate would be 2 Mbps.) This is the maximum data rate considering the loss averaged over the multipath fading, however. Multiple paths from various directions produce fades in signal strength which vary with distance at wavelength intervals. In practice, additional margin (with a correspondingly lower data rate) must be considered because of this fading. For example, with a single receive antenna and Rayleigh fading (note, however, that our results in Section 3 indicate that the fading is Rician in our building), 10 dB of additional margin is required for 90% availability and 20 dB is required for 99% availability, which lowers the data rate limit to 80 and 8 kbps, respectively, for full, single-floor, building coverage. Of course, at millimeter wavelengths a user would only have to move the receive antenna by a fraction of an inch to move out of a fade, and therefore might not need this margin. However, even though the user antenna may be stationary, the environment changes, which makes this method not practical, and may mean that even a 99% availability would be unacceptable due to frequent, but short, outages. Diversity can be used to greatly reduce this additional margin, though, with two receive antennas cutting the margin required for a given availability in half (in dB). Next consider delay spread. For many buildings, the rms delay spread is on the order of 30 to 250 ns [2,3]. Since without equalization, a BER of less than 10-8 requires an rms delay spread less than about 10% of the symbol period, this rrns delay spread results in a data rate limitation of about 1 Mbps. Note that this is similar to the limitation due to power margin (without the multipath fade margin), and thus both factors need to be reduced to operate at very high data rates.
G :::: 10loglO [
J 360}2 1 1 1-81t J
.
(7)
Note that this beamwidth and gain can,~_~El~!!l~Y an array of M antennas, with the gain, \G=10Iog 10M. For example, from (6) (which assumes L p ':: 60 dB), to~obtain enough receive power to support 155 Mbps (with a ray with power 3 dB less than the total received signal power) requires an antenna gain G=26 dB, or, from (7), a 400-element (e= 10 0 ) base station array with omnidirectional antennas at the users. Note that the required gain is given by the product of the gain of the receive and transmit antennas. Thus, we could also use a 100-element (8=20 0 ) base station antenna with a fourelement handset, or a 20-element (6=45°) array at both ends. Thus, for example, antenna arrays with 15° beamwidths (183 elements) at the transmitter and receiver should support up to 10 Gbps (if delay is not an issue). Note that these results do not consider additional fade margin due to multipath fading as in the omnidirectional antenna case. We do not consider the multipath fading to be a concern with directive antennas for two reasons. First, an isolated ray should have no fading, since multiple rays are required for fading and the single ray should remain relatively constant in amplitude over many wavelengths. However, although the prediction techniques of [1] may generate only a single ray, the environment may actually generate multiple rays that are closely spaced, rather than a single ray, which could result in fading, albeit with longer fading intervals than with omnidirectional antennas. This fading would typically be Rician, though, with a large K, which greatly reduces the required fade margin. Second, the prediction technique of [1] shows that, at least with the Crawford Hill building, there are typically seven isolated rays, with
3. Antenna Arrays First, consider the use of higher gain antennas to increase the margin. Note that if the multipath causes the received signal to be uniformly distributed in power with respect to angle-of-arrival, increased receive antenna gain does not increase the margin. In this case the increase in
721
low delay spread and sufficient power. In this case, we obtain seventh order diversity. which reduces the required fading margin even further. Therefore. we expect the required fade margin to be substantially less than with omnidirectional antennas and will therefore not consider it. Next. consider antenna arrays to reduce the delay spread problem (see also [4], which also considers directive antennas at both ends of the link to reduce delay spread. but for line-of-sight systems). Since the data rate limitation due to margin (without an additional fade margin) and delay spread are about the same and arrays are needed to increase the maximum data rate due to the margin limitation. we would hope that an M-element array would increase the data rate limitation due to delay spread by the same factor as that due to power margin. Similar to our above results. if the distribution of received power and delay spread was uniform with respect to the angle-of-arrival. directive antennas will not increase the data rate limitation due to delay spread. However. as stated above. the prediction techniques of [1] show that the power of the received signal is not uniformly distributed in angle-of-arrival. However, the range of signal delays can remain large even for small beamwidths as shown below. An example of this problem is shown in Figure 2, where with omnidirectional transmit and directive receive antennas the delay spread is large even when the receive beamwidth is small. As a result. higher directivity reduces the total power of the weaker rays with respect to the strongest ray within each beam. but not necessarily the delay spread of the signals in the beam . Therefore, the maximum data rate remains below 1 Mbps until the power of the weaker rays becomes small enough. At this point the delay spread limitation is essentially removed. Thus. as the antenna directivity is increased. the data rate is limited to 1 Mbps by the delay spread until the directivity exceeds some value at which point the data
rate dramatically increases to the power margin limitation. Furthermore. as discussed above. additional margin due to multipath fading is no longer needed. Note that under these conditions. neither multicarrier techniques nor equalization can significantly increase the data rate. Until the beamwidth is sufficiently narrow . the number of carriers or the length of the equalizer must increase linearly with the data rate. independent of the antenna directivity. These techniques become very complex and expensive for data rates greater than 20 Mbps. The critical parameter is therefore the directivity required for high data rates. Consider first an omnidirectional transmit and directive receive antenna. as in Figure 2. Using [1]. Figure 3 shows the beamwidth required at the receiver versus the number of equalizer taps (see below) for 90% coverage at a 1 Gbps data rate. These results were generated for the model of the Crawford Hill building (see Figure 4). using ray tracing to determine all rays received with up to 3 reflections. To show the most optimistic results for the
25
!
20
'0 15
~
10
5
Eye p8lllIIly • 20 dB
21.-.:Dons 100 150 200 250 Beamwidlh (0egrMs)
Figure 3
300
350
The beamwidth required versus the number of equalizer taps for 90 % coverage with omnidirectional transmit and directive receive antennas.
~~'IIIII[I[lliDJ • I [/I J I 1
14
118m
~
R-I46
Figure 2
An example case of omnidirectional transmit and directive receive antennas where the delay spread is large even when the receive beamwidth is small.
Figure 4
722
The bottom floor of the Crawford Hill building.
effect of equalization and arrays, we only considered delay spread and used a coverage requirement of a 20 dB eye penalty (nearly closed eye) as given by a 1/.9 ratio of the power of the strongest ray (plus other rays within ±.5 of the symbol period) to the sum of the powers of the other rays (with delays outside of ±.5 of the symbol period) within the beamwidth. Furthermore, we considered an N-tap decision feedback equalizer that eliminated the rays in either the strongest N precursor or N postcursor symbol intervals. Figure 3 shows that. even under these overly optimistic conditions, coverage within a distance of four rooms requires 5° beamwidths, and equalization does not significantly reduce the required beamwidth. Thus, a 1650-element array is needed. which appears to be impractical with today's technology. Therefore, consider using directive antennas at both the transmitter and receiver. Note that the example given in Figure 2 would benefit greatly from this strategy. On the other hand, the example given in Figure 5 shows that even with directive antennas at both ends a small be~width may not always reduce the delay spread. However, we would hope that these cases are rare. Therefore, to further improve performance we consider searching over all rays to find the beam direction with the lowest BER due to thermal noise and delay spread. Thus, we could choose a beam with a ray with lower power than the strongest ray, but with less delay spread. To illustrate these results. and determine the critical antenna size. consider the data rate limitation given by directive beams for one floor of Crawford Hill (see Figure 4) with a 10-8 BER requirement. Specifically, for each ray, using the prediction techniques of [1] with rays with up to 3 reflections, we determined if the receive
E
_b
No
..c._
Figure 5
An example case of directive transmit and receive antennas where the delay spread is large even when the beamwidth is small.
1000"'"
0.8
t
- -
_';:1"
0.6
~ 0.4 0.2
400
15
Figure 6
20
25
30
Beamwidlh (Degrees)
35
40
45
Availability versus beamwidth for several
data rates for transmission to the edge of
the coverage area using the propagationprediction techniques of [1] for the Crawford Hill building with rays with up to 3 reflections.
was
greater than 12 dB and, for all the rays within the beamwidth, the rms delay spread was less than 10% of the symbol period. For a given receiver location, an outage occurs if no ray can be found that meets these requirements . For each beamwidth, we chose 60 locations, at the edge of the coverage region, and determined the availability at a given data rate. Figure 6 shows our results for the availability versus beamwidth (with data points taken at 2.5° intervals) for several data rates. These results show that for data rates greater than 20 Mbps, the availability depends primarily on the beamwidth. Availability greater than 90% requires a beamwidth less than 30° (:::50 elements) for 45 Mbps, but 1 Gbps requires only a 25° beamwidth. Thus. if the beamwidth is narrow enough to isolate at least one ray for 45 Mbps operation, data rates up to 1 Gbps and higher are also feasible. For a 13° beamwidth (244 elements), the maximum data rate exceeds 1 Gbps with 100% availability. Thus, in all 60 locations 13° antennas find
an isolated ray with enough power to support Gbps data rates. In fact, our results show that about 7 isolated rays (in different 13° beams) with sufficient power are usually found for each location. To support our conclusions, we performed the following experiment, which is summarized in Table 1. Using the LuckyNet [5] setup, we transmitted up to 622 Mbps at 19 GHz within Crawford Hill. For Table I, radio link factors are given in [5-7]. The propagation loss is determined as 3 dB less than free space (assuming half the total power in one beam, as above) minus 3.4 power law excess loss (from Figure 1). The transmitter was located in the hallway near the library and reception area on the first floor and the receiver was positioned about 12 feet inside room RI46 at the end of the corridor (see Figure 4). Although this is the short end of the corridor
723
(about half the length from the reception are to the other end of the corridor), propagation measurements (see Figure 1) show that the propagation loss (L p ) is similar to that for the longer end, i.e., an average of about 50 dB. The transmit and receive antennas were 15 0 beam width hom antennas, which could be manually scanned. BER measurements were made at a combination of 6 locations by moving the antenna height or lateral position within a few feet at both ends of the link. At each location, both antennas were manually scanned to try to jointly find the best transmit and receive angles. Note that there are over 33,000 possible transmit/receive angle combinations with 15° beamwidths, and therefore it was not practical to exhaustively search for the best angle. However, since the transmitter was located down a long corridor, we assumed that pointing the transmit antenna toward Rl46 would be most likely to give the good performance, With this general direction for the transmit antenna, the receive antenna was manually scanned to find a reasonable BER, and the transmit angle was then adjusted slightly to try to improve this performance. We
found that good receive angles could not be determined a
priori, e.g., pointing at the door did not always result in a satisfactory BER. The strongest receive signal had a
propagation loss of 51 dB, compared to the predicted propagation loss with omnidirectional antennas of 50 dB. This is in agreement with our expected result of the strongest ray containing about half of the total receive power. The BER results for 6 locations are shown in Table 1 and range from 3xlO-8 to 10-3 • Note that even a 10-3 BER is acceptable since with coding the error rate could easily be reduced below 10- 8 • Table 1 also shows the variation in BER with bit rate at one location. Except for the highest data rate measured, 622 Mbps, we did not have a clock recovery circuit and used a coaxial line to feed the clock to the receiver. Data rates were adjusted slightly to synchronize the coaxial-line-fed clock to that of the signal received by radio. The BER is nearly constant for data rates greater than 210 Mbps, which implies an irreducible BER (albeit, low BER, ~10-7) that is independent of the data rate, i.e., the received signal consisted of one strong ray with much weaker rays with delay spreads in excess of 5 nsec. Thus, with sufficient receive power, data rates in excess of 1 Gbps should be possible. This experiment only presents anecdotal results at a few closely-spaced locations to support our conclusions. The propagation loss was 10 dB less than is required for full floor coverage (with a maximum 60 dB propagation loss), but we did not exhaustively search all transmit/receive angles. Even a computer-controlled exhaustive search with directive homs would take many hours for each location, so experimental measurement of availability awaits the construction of phased array antennas.
1. Transmitter Radio Frequency Power Amplifier Output Power Transmitter Cable Losses OmDidirectional Antenna Gain Directional Antenna Gain
19.00GHz +23dBm 1 dB 4.5 dB 22 dB
2. Signal: Data Rates, Mbls BPSK, Coherent:, Piiol Aided
622.08, 340,210,110.50& 10 2'·1 PRBS NRZ
3. Receiver: Noise Figure Bandwidths, MHz Reqd. Output SNR, for 10-IBEK, dB Antenna Gains & Cable Loss
6dB Approximately equal to Bit Rate 12 dB Same as for the Tnnsmitter
4. Propagation: r-J.4 Below 1 meter Free Space, Half of Power in Main Ray. Assumes obstructed paths. W,
4. Conclusions
A)2 ('; )34 (G G) where '0 = 1 meter. = W, (4x,o T'
Economically-fabricated antenna arrays with about 100 elements which determine and track the optimum combination of transmit and receive beams, along with networking issues, are complex problems that require further research. However, assuming such technology were available, we make the following conclusions. Based on the propagation-prediction techniques of [1], supported by experimental measurements, neither multicarriers, equalization, nor antenna arrays at one end of a communication link are economical methods for increasing the data rate substantially above 20 Mbps for multiple room indoor wireless coverage. However, based on propagation-prediction techniques and verified by our experimental measurements using directive antennas at both ends of a link between the center of the Crawford
Expected Omni Revd Power@l34ft. -8S.5dBm Expected Orelni Revd Power@l34ft -SOJldBm
s. Margin @ 622Mbls, (neglecting intersymbol Omin:Exp.-Reqd for 10- 3 BER, dB Direta:Exp.-Reqd for 10-1 BER, dB
interferences due to delay spread): -17.7 +17.3
6. Error Rate Measurements:
6 dearly spaced locations @ 622.08 Mbls gave sweet spots of 3*10-1, 6*10-1, 1*10-7,2*10-4,3*10-4, & 1*10-3 BEL At One location the BER varied with bit Rate as shown: 621.06 Mbls 6*10-5 BEK 340 Mbls 1*10-7 BER 210 MBls 1*10-1 BER 110 MBls 0 BER SO Mbs 0 BER 10 Mbs 0 BER
Table 1
Indoor Wireless Error Rate Measurement
724
Hill building to an end laboratory, we have shown that high-speed ubiquitous communication is possible. Using antenna arrays with 50 to 200 elements at both the transmitter and receiver, we can expect to obtain entire f oor coverage at data rates in excess of 1 Gbps.
[2]
A. A. M. Saleh and R. A. Valenzuela, "A Statistical Model for Indoor Radio Propagation," IEEE J. Selected Areas Commun., Vol. SAC-5, pp. 128-137, Feb. 1987.
(3]
D. M. 1. Devasirvatham, "Tune delay spread measurements of wideband radio signals within a building," Electronic Letters, Vol. 20, pp. 950-951, Nov. 8, 1984.
[4]
P. F. Driessen, "High-speed wireless LANs with directional antennas," Proc. of VTC'94, Stockholm, Sweden, June 7-10,1994.
[5]
M. J. Gans, T. S. Chu, P. W. Wolniansky, and M. J. Carloni, "A 2.5 Gigabit 23~Mile Radio Link for LuckyNet," Proc. of GLOBECOM'91, pp. 1065-1068, Dec. 2-5, 1991.
[6]
H. Taub and D. L. Schilling, Principles of Communication Systems, McGraw-Hill, New York, 1971.
[7]
J. D. Kraus, Antennas, 2nd edition, McGraw-Hill, 1988.
Acknowledgements We gratefully acknowledge R. S. Roman and O. Landron for providing the experimental results presented in Figure 1, and A. A. M. Saleh for proposing the highspeed data measurements.
References [1]
R. A. Valenzuela, "A ray tracing approach to predicting indoor wireless transmission," Proc. of VTC'93, pp. 214-218, Secaucus, NJ, May 18-20, 1993.
725
On Optimizing Base Station Antenna Array Topology for Coverage Extension in Cellular Radio Networks JEN- WEI LIANG, AROGYASWAMI
J.
PAULRAJ Information Systems Laboratory, Stanford University Stanford, CA 94305
Abstract
sults from two factors: array gain and diversity gain. In [5], the authors show that using maximal ratio combining maximizes the array gain, and the average array output SNR is given by
Use of higher frequencies (1.8 GHz) for the US upper tier PCS cellular service and the FCC regulations on the network build out have resulted in significant interest in improving coverage of cellular networks. Networks whose coverage is limited imply that thermal noise is the limiting factor. Also, since the for.ward link (base station to mobile) has higher power than the reverse link, cell coverage is usually limited by the reverse link. This coverage can be extended by improving the reverse link budget. Use of receive antenna arrays for boosting array gain on the reverse link is therefore of great interest [1, 2]. When receive antenna arrays are used at the base station, several conflicting choices affect system performance and cost. Some of these aspects are: the number of antenna elements (and channels) improves coverage but also increases system cost; the maximum span of the array increases diversity but must be limited for convenient deployment on a tower; large inter-element spacing can increase diversity but cause grating lobes at the same time. These conflicting requirements mean that a careful design of the array topology can minimize the cost. In this paper, we study performance of linear and circular base station antenna arrays with different topologies, angle spread, and the number of elements. We compare alternate topologies using maximal ratio combining for narrowband systems such as AMPS and IS-54.
I
(1) where M is the number of antennas and SN R; is the element SNR. Diversity gain comes from various sources such as polarization, multipath in spread spectrum systems, interleaving and coding, and spatial decorrelation. In this paper, we study how the base station antenna topology affects spatial diversity and hence changes the system performance. Maximum diversity gain can be achieved from uncorrelated fading at each antenna. Large inter-element spacing decorrelates signal fading but it also creates grating lobes which cause direction ambiguity and power transmission to the undesired directions. These conflicting requirements reveal that a careful design of the base station antenna array topology can effectively maximize the spatial diversity at minimum cost.
II
Signal Model
We consider transmission from a single mobile to a base station. The base station receiver performance is only limited by thermal noise and no co-channel interference is present. We assume Rayleigh fading and fourth power loss, and do not consider slow fading such as shadowing. CDMA systems can benefit from multipath diversity provided by resolved paths received at the RAKE receiver [6]. However, in this paper we assume that delay spread is much smaller than a symbol period and therefore the channel undergoes flat fading. As mentioned earlier, other diversity dimensions such as polarization, multipath, are neglected. Signals transmitted from the mobile are assumed to be scattered by 20 scatterers around the mobile,
Introduction
Recent studies [3, 4] have shown that antenna diversity can substantially improve the performance of most wireless communication systems. In this paper, we are interested in using base station antenna arrays to extend the coverage of cellular networks. The improvement in performance using antenna array re-
Reprinted from IEEE 45th Vehicular Technology Conference, Vol. 1, pp. 866-870, July 1995.
726
... ." .." " ."l~}
\
-:·-0·...
o
.......................
0
b d ;.......0-...
_c~~>· ry , ..............
i-J2'i
1
(A ······/ ·.,··········· ···O\ \. -,
5A
0 .... ...........0.
(a)
(b)
(c)
(d)
0:
j
....
- :SCATfERS Figure 1: A constant angle scatterer model.
~
0 ·-Q-·0··0·-0-·0
(e)
-------.-Et~==========~ ir
~
0 ·-Q-'0··········-Q-·O··0
I
7.5A
I
(I)
Figure 3: Different topologies under consideration .
'(
LOCAL SCATIERERS
BA SE STATION DOMINANT SCATTERERS
Figure 2: A simple scattering scenario.
We now conside r the com bining schemes for the bas e st at ion ant enna array. The traditional beamforming assumes planar impinging wavefront , and signals are cophased and weighted to maximize SNR at th e array output . The array SNR for the planar beamformer is given by
(2) all of equal power . In Fig. 1, we illust rat e a constant angle scatterer model where the signal angle spread () is independent of the distance between scatterers and the base station . This model is applicable when there are dominant scatterers confining the scattering angle as shown in Fig . 2. The phase angle of scatterers have uniform distribution in·{O:"~;'l and scatters ar e randomly distributed on a const ant radius scattering arc centered at the base station . This is a simplified version of a scattering 'disc' model, but it is adequate to illust rate the topology tradeoff studied in this paper . Fig . 3 shows 6 different base station antenna topologies we have studied , which are (a)single circular array with >../2 and (b)large inter-element spacing , (c)single circular array plus one sensor , (d)dou ble circular arrays , (e)single linear array, and (f)double linear arrays. Different antenna patterns ar e used in circular and linear arrays: a cardioid pattern pointing away from the cent er for circular arr ays and an omnidirectional pattern for linear arrays .
where A1 is the number of antennas , Xi is the received signal at antenna i , Wi (¢) is the expected complex antenna gain for a signal arriving from angle ¢ , and O"~ is the noise power. However, the planar wavefront assumption is not valid in a cellular environment beca use of th e presence of angle spread in multipaths. On the other hand , maximal ratio combining exploits th e fact that th e impinging signals have arbitrary wavefront, and signals are com bined to maximize the output SINR. The array output SNR, ignoring the interference , is given by
(3) To study the coverage extension, , we set up the range, angle spread , and mobile power for scatterers and combine signals as the array output for each sna pshots. We obtain the BER of the i t h snapshot
727
SINGLE BLOOM CIRCULAR ARRAY
r:~I~
2.8
2.6 2.4
, 10
12
14
NUMBER OF ANTENNAS
16
18
20
22
Figure 4: Coverage extension for a single circular array. from the AWGN BER curve, and it is given by
BER;
= 2erfc(VSNR;). 1
(4)
We average over snapshot BERs to determine the BER for a Rayleigh fading channel, which is given by 1 N
BER= N LBER;,
(5)
i=l
where N is the number of snapshots. With the scattering angle and the target BER 10- 2 fixed, we increase the number of antennas and BER will improve. We then increase the range until the BER exceeds 10- 2 and thus obtain the coverage extension. All our results are normalized with respect to the range obtained from using a single omnidirectional antenna.
Simulation Results
Figure 4 shows the maximum coverage extension for a single circular array as a function of the number of antennas and angle spread. Using 12 antennas at 0 degree angle spread can provide 50% coverage extension. We should note that when we use cardioid antenna patterns in a circular array, approximately only about half of the elements participate in receiving signals. As we increase the angle spread, signals at the sensors become more uncorrelated and larger coverage improvement can be obtained from the increased diversity gain. However, diversity gain is saturated when signals at the sensors are totally uncorrelated.
10 20 Number of Antennas Scattering Angle=30
30
i:l~ z
0
10 20 Number of Antennas Scattenng Angle=40
30
10 20 Number of Antennas Scattering Angle=60
30
1:1/ I [I~I 1:1/ I i:1/
1.8
III
0
z
2.2
1.2
Scattering Angle=20
Scattering Angle=l 0
z
0
10 20 Number of Antennas Scattering Angle=50
30
z
0
z
0
10 20 Number of Antennas
30
z
0
10
20
Number of Antennas
Figure 5: Coverage extension comparison between single and double circular arrays. Therefore, as the angle spread becomes larger, the corresponding range increase is less pronounced. The comparison between single and double circular arrays appears in Fig. 5. Simply dividing sensors into 2 circular arrays with 5A apart, as shown in Fig. 3(a) and (d), we can further increased the coverage from 60% to 100% using 10 antennas and 10 degrees angle spread. However, when the angle spread increases substantially, the difference between single and double circular arrays becomes less obvious and both topologies perform in a similar way. Diversity is mostly provided by angle spread and hence the topology of the base station antennas has less influence. Figure 6 has one additional result for one large circular array as shown in Fig. 3(b). We find that a large circular array has slightly better coverage than double circular arrays when the angle spread is large. However, with 6 elements and 5-X diameter in a circular array, the inter-element spacing is 2.5A and grating lobes become a serious problem for transmission and direction-finding, i. e. , grating lobes cause ambiguity in the angular position of the desired mobile and it also causes interference by transmitting power in the undesired directions. In Fig. 7, we compare a single circular array with one circular plus one sensor as shown in Fig. 3(a) and (c). Placing one sensor 7.5-X apart, receive antennas acquire more diversity from this separated lone sensor and thus increase the coverage from 70% to 100%
728
30
r:~12:10 z
0
10 20 Number of Antennas Scattering Angle=30
?20 r:~1 L I [I ;;;;20 O
1:1 30
z
0
z
0
10 20 Number of Antennas Scattering Angle=40
30
z
0
10 20 Number of Antennas Scattering Angle=30
30
z
0
z
0
10 20 Number of Antennas Scattering Angle=40
1:17 1:17 11:17 [17 z
0
~ 31
10
20
Number of Antennas scat:tering Angle=50
t:~ z
0
10
20
Number of Antennas
I
30
~ 31
30
10
20
Number of Antennas
0
10
20
Number of Antennas
z
0
30
10
20
Number of Antennas
I
30
10
20
Number of Antennas
t7 t.7 ~ 31
Scattenng Angle=60
t:~ z
30
z
0
~3
Scattering Angle=50
10 20 Number of Antennas
30
z
1
0
10 20 Number of Antennas
Figure 7: Comparison between single circular and single circular+ 1.
for 10 antenna elements and 10 degrees angle spread. Thus, it is reasonable to use the 5-element circular array in Fig. 3(c) for transmission to avoid large grating lobes and use all 6 sensors for reception to obtain more diversity. We also find that when the number of antennas is large, the difference of the range increase between 2 curves becomes smaller. The comparison between a single linear array and double linear arrays is shown in Fig. 8. With the distance between the center of two linear arrays fixed (7.5A in Fig. 3(f)), both topologies achieve the same coverage extension at a large number of antennas.
when the system lacks diversity, for instance, there are few antennas and small angle spread. In a practical cellular system, only a limited number of antennas can be deployed and hence topology plays a more important role in the base station antenna design.
References [1] B. Khalaj, A. J. Paulraj, and T. Kailath, "Antenna arrays for CDMA systems with multipath," in MILCOM'93, pp. 624-628, 1993.
[2] J. H. Winters, "Presentation from First Workshop
on Smart Antennas in Wireless Mobile Communications," tech. rep., Stanford University, June 1994.
Concluding Remarks
In the previous section, we showed the coverage extension results for six different base station antenna topologies. Beyond the array gain described in Eq. (1), using multiple antennas at the base station provides different diversity gain with different topologies. We studied three diversity sources in this paper: the number of antennas, angle spread, and topology. The results show these three sources interact with one another. Topology becomes less important when large angle spread or a large number of antennas are present in the system. A careful design of the base station antenna topology can effectively provide more diversity and hence extend the coverage for a cellular network, especially
[3] J. H. Winters, J. Salz, and R. D. Gitlin, "The
impact of antenna diversity on the capacity of wireless communication systems," IEEE Trans. Commun., vol. COM-42, pp. 1740-1751, February 1994.
[4] A. Jalali and P. Mermelstein, "Effects of multipath and antenna diversity on the uplink capacity of a CDMA wireless system," in GLOBECOM'93, vol. 3, pp. 1660-1664, 1993.
[5] J. William C. Jakes, Microwave Mobile Communications. John Wiley & Sons, 1974.
729
I
30
Scattering Angle~o
Figure 6: Comparison between small circular, large circular, and double circular arrays.
IV
30
30
f~I;10 I z
0
&4
1:1 z
0
10 20 Number of Antemas Scattering Angle=30
i~l~ z 0
10
20
Number of Antennas Scattering Angle=40
~. f~l~
30
I
10
20
Number of Antennas Scattering Angle=50
f~l~
z 0
30
Scattering Angle=20
10
20
Number of Antennas
30
30
z
0
10
20
Number of Antennas
i~17 z 0
10 20 Number of Antennas
30
30
Figure 8: Comparison between single and double linear arrays.
[6] J. S. Lehnert and M. B. Pursely, "Multipath diversity reception of spread-spectrum multiple-
access communications," IEEE Trans. Commun., vol. COM-35(11), pp. 1189-1198, November 1987.
[7] P. Balaban and J. Salz, "Optimum diversity combining and equalization in digital transmission with applications to cellular mobile radio," IEEE Trans. Commun., vol. COM-40, pp. 885-894, May 1992.
[8] D. Parsons, The Mobile Radio Propagation Channel. John Wiley & Sons, 1992.
730
USAGE OF ADAPTIVE ARRAYS TO SOLVE RESOURCE PLANNING PROBLEMS M. Frullone*, P. Grazioso*,
c. Passerini", G. Riva*
* Fondazione Ugo Bordoni,I DEIS - Universita' di Bologna
Villa Griffone - Pontecchio Marconi, 1-40044, Bologna Tel.: +39 51 846854; Fax: +39 51 845758 e-mail: [email protected] Abstract - The usage of adaptive antenna arrays in cellular systems is currently being investigated by many researchers. In this paper we show by means of simulation that adaptive antenna arrays allow to reduce the reuse distance, and hence to increase spectrum efficiency. Furthermore, it is shown that, by adopting a fractional loading factor, it is possible to adopt a cluster size equal to 1, which avoids the need of frequency planning altogether, and allows a further improvement in spectrum efficiency.
II ADAPTIVE ARRAY In the following we assume to use a planar circular array of
N=8 elements situated at the base station. The elements are
supposed to be half-wavelength dipoles, and the radius of the array is half-wavelength. It has been supposed that no mutual coupling exists between array elements: it can be demonstrated that the presence of the mutual coupling does not affect the behaviour of the adaptive array under certain, widely satisfied, conditions.
I INTRODUCTION
s;
signals Sk,i(t) (O
The usage of adaptive arrays in cellular systems has been recently supported in the research community [1-2]. The advantage of adaptive antenna arrays is their capability to filter out co-channel interference by means of a proper null bemforming, thus allowing the adoption of lower reuse distances. Another technique which is currently investigated is fractional loading [3]. It consists in assigning to each base station more bandwidth than strictly needed from a traffic point of view. This poses some interference problems, because the reuse distance becomes actually smaller. Therefore one must take advantage of a lower channel occupancy, by means of a proper technique, to guarantee a satisfactory protection ratio. Some examples of these techniques are: intra-cell handover or opportune channel selection algorithms when bad channel quality is detected; frequency-hopping techniques, whose performance become better when channel occupancy is lowered; adaptive arrays at base station, as reduced number of cochannel interferers, even if with higher power, can be coped with more easily through a proper exploitation of the available degree of freedom. The adoption of these techniques povides often advantageous results and allows an increase in system capacity. Here we focus on the combining of adaptive arrays and fractional loading; we will show, by means of a Monte-Carlo simulation, that by combining these two techniques, it can be possible to adopt a cluster size equal to one, i.e. to avoid the need of frequency planning.
is
Nu
xi = L[Sk,l(td Sk,2(td ... Sk,N (td]. At time k=l
ti, si
"
is the reference digital signal, that will be used by the receiver to perform the adaptation process. The sampling times ti are chosen opportunely. In the following we assume that the sampling period Tb is equal to the inverse of the bit-rate of the digital system employing the adaptive receiver. We will denote with Rn the estimate of the input crosscorrelation matrix and with Pn the estimate of the correlation between the reference signal and the input vector.
1~ * T R n =~x. x n l l
(1)
i=l
Pn =
n
LX; Si
(2)
i=l
The accuracy of the estimates depends on the number of samples and on the channel variability. We will focus on the chosen number of samples later on. We have supposed that during the sampling interval the channel variability is negligible. With these assumptions, the weight vector W n defined as:
Reprinted from Proceedings of the 46th Vehicular Technology Conference, Vol. 1, pp. 527-530, April 1996.
731
GSM standard and the studied system will be pointed out.
Performance have been calculated for just one of the frequencies available for the system. Thus the associated wavelength defines the array radius. We assume that the performance are weakly dependent on this reference frequency (that is, the total band allocated is narrow). The service area has been supposed characterised by a uniform distribution of traffic. The radio coverage is achieved using a Base Station (BS) endowed with an adaptive antenna array (for sake of brevity this kind of BS will be called "adaptive BS" in the rest of this paper, in contrast with "nonadaptive BS", i.e. a BS with conventional antennas). The purpose is to reduce the eel by means of the interference cancellation properties of the adaptive receiver and verify the possibility to obtain a system that doesn't require planning because of an opportune choice of system parameters. The BSs are located in the centre of the cells. The adopted cluster-size (CS) fixes the reuse distance of the channel and the number of channels available per cell
(3)
minimises the mean square error en =
2
n
L
1/ S i -
W~ xi II ·
i=l This approach in solving the problem of minimising the mean square error between a reference signal and a vector of revealed signals is referenced to as Direct sample-Matrix Inversion (DMI) technique [4]. The invertibility of the matrix R n is guaranteed if uncorrelated noise is superimposed to the
received signals. This condition is usually satisfied in real systems. To develop the simulation of the behaviour of an adaptive array receiver the noise has been assumed as Gaussian and zero-mean. The noise power has been chosen as comparable with the actual noise of a cellular network radio base station. Under these hypothesis, the number of samples required to obtain a solution with an accuracy of less than 3 dB has been proved to be n = (2N - 3) = 13 [4].
R = Co , where Co is the total number of available channels
For the simulation of the communication system some additional assumptions have been done:
CS
The determination of the cluster size is a crucial point in the dimensioning of a cellular system, and we will briefly discuss two alternative policies.
1) . the correlation between Sk,j' and Sk,j" is a complex number whose modulus is 1 and whose phase is evaluated as a function of the direction of arrival of the wave and the relative positions of the j' and j" antennas;
III.A - Dimensioning using a blocking probability
2) the correlation between two samples of the same signal is zero (i.e. the channel is non-dispersive in time); 3) the sources of the signal can be regarded as points; 4) the cross-correlation between two different signal tends to zero as the number of samples tends to infinity; 5) the noise is uncorrelated with each sample; The first point is reasonable if the distance between elements is small; the third and the fourth point allow to reduce the computational burden of the simulation (assuming that the user can be distinguished by a color code); the second point can not be satisfied in real communication systems especially if microcellular technology is involved. However for macrocellular systems, this assumption can be considered reasonable. Because of these assumptions, the results obtained should be regarded as an upper bound of the performance that can be obtained introducing adaptive arrays in actual communication systems.
~II THE
This method is the most usual, and is performed in a similar fashion to the dimensioning rules used in fixed networks. The grade of service is defined by the call blocking probability, Pb (in this paper mobility management issues are not dealt with, hence the probability of hand-over failure is not considered). In this case, the signal-to-interference ratio (SIR), used as the quality parameter of the system, determines the actual cluster size. Once the maximum call-blocking probability is imposed, the mean number of active users M can be dimensioned solving the equality Ph = BE(R,M), where BE is the Erlang B formula. As R is depending on the CS (the greater CS, the lower R), also the offered traffic per cell, M, will decrease when the cluster size increases. Finally, the channel occupancy probability (i.e. the probability that a channel is in use) is
M R
Po = -(I-Ph)· In the following numerical examples, we have imposed a tolerable blocking probability Ph = 1%.
MOBILE COMMUNICATION SYSTEM
We have analysed a TDMA/FDMA mobile radio communication system. For this class of systems, the resource planning is an essential procedure to avoid the Co-Channel Interference (Cel). System performance in term of capacity are bounded by the CCI. A GSM-like system is assumed, i.e. a great number of the parameters needed for its definition are similar to the GSM ones. Wherever it will be necessary, differences between the
IIJ.E - Dimensioning using a fractional loadingfactor An alternative way to perform system dimensioning is based on the fractional loading concept. Let us consider a system dimensioned using blocking probability, as explained in section III.A, with cluster size CS and offered traffic M. We now adopt a lower cluster size, allowing each base station to
732
use a higher number of channels than strictly required to serve the traffic, M . In the limiting case, CS is 1, i.e. the whole bandwidth is available at each base station . In this case, as the number of channels available in every cell is very large, the call blocking probability is negligible , and actually reduces to o in all practical cases. Therefore, the system capacity is now limited by interference; as a matter of fact, the adopted CS is not able to guarantee the required SIR and have to resort to some other technique, as mentioned in section 1, in order to match the interference constraints. The average number of active users in each cell can be expressed as M = F CO' where F is a number between 0 and 1. Obviously , if CS = 1, the channel occupancy Po is equal to F. The capability to provide higher fractional loading factor, F, is then dependent on the ability to properly reduce CCI (e.g. using adaptive arrays).
V RESULTS In this section we will report some results obtained for non-adaptive and adaptive base stations respectively . The first ones are reported as reference to show the benefits obtained by using adaptive base stations.
A. Results for the non-adaptive BS
The chosen transmitted power provides the CDF of SNR (signal to noise ratio), shown in fig. 1, in the case of nonadaptive BS and CS 12. The SNR is above 10 dB in about 99% of cases, thus meeting the imposed constraint.
=
0, 4
IV COMMUNICATION SYSTEM SIMULATIONS
0,35
II
0 ,3
The steady-state of the communication system has been simulated . It is supposed that, in each time-slot, n bits of the transmitted burst are devoted to the adaptation . Snapshots have been generated with a random number of users . For each snapshot, 50 frames, during which the short-term fading varies, have been considered. Cells have been supposed hexagonal, and users are placed randomly inside them. The performance are calculated referring to an active channel in a target cell. The reference signal, s i of the user in the target cell is assumed to be known. Interferers belonging to the cochannel cells in the nearest 3 tiers have the probability Po (channel occupancy) of being present. Po is calculated assuming a number Co =124 of available channels. Then, the probability of k users contemporarily active in the 36 interfering cells of the 3 nearest tiers is Pk
p rOblS N M +
I:----~
!
0.25
i
0, 2
!
0,15
: I
0, 1
! i
0,05
°
I
i
°
,
I
I i
I I i
I
t.> 10
j/
-
/
/
1/
/
/
I
! I
SN R
15
25
20
30
Fig 1. CDF of SNR for CS = 12 (non-adaptive BSs) The interference analysis is reported in figure (2), as a function of the cluster size . Omnidirectional BS antennas are assumed, and the minimum cluster size that satisfies the constraint on SIR is 12. This is a reasonable value compared with the CS = 3x3 of the actual GSM where the BSs are not situated in the centre of the served cell.
=Ck6)p~ .(I _ Po )36- k .
The radio links are affected by: 1) path -loss, depending on the distance, with a = 3.5 inverse power-law; 2) long-term fading, log-normally distributed with standard deviation of 6 dB; 3) short-term fading, Rayleigh distributed, uncorrelated between two subsequent frames. The active users (both in the target cell and in the cochannel cells) are regarded as the sources of the signals
0,4
0,35 0,3
ProbISIR<x}
_______._L. !
.•L. I
I
.
.~::-:!:-...-+
0,25 0.2 0, 15 0, 1
Sk -i (r), as defined in section II; therefore 15 N u 5 37 For each snapshot, Rayleigh fading is averaged over the 50 frames. This is different from GSM quality measure, which is averaged over a time period of 100 frames. The Cumulative Distribution Functions of the SIR and SNR are evaluated over 3000 snapshots . In this study it is assumed that a SIR less than 10 dB is tolerable in 5 % of the cases.
0,05
°·10
_._._+_ -5
_+"L-~+
o
10
«:
SIR 15
20
Pig. 2. CDP of SIR for different CSs (non-adaptive BSs) With the adopted numerical values (Co = 124, Pb =0.01), it is straightforward to evaluate the system capacity by a
733
simple inversion of Erlang B formula: we obtain that the carried traffic is about 4.5 Erl/cell. As final remark of this subsection, it is worth noting that the use of the fractional loading technique at non-adaptive BS doesn't produce any performance improvement.
Of course, a mixed solution involving adaptive arrays , intermediate CS and fractional loading could be put into practice. However, carried traffic results .are worse than the CS= I solution. VI. CONCLUSIONS
B. Results for the adaptive BS
In this paper we have discussed the usage of adaptive antenna arrays to solve frequency planning problems and to improve spectrum efficiency. The analysis was carried out by means of a Monte-Carlo simulation; we have shown that the adoption of adaptive antenna arrays allows a drastic reduction in the minimum cluster size which fulfils the requirements on SIR. In the numeric example here developed, the minimum cluster size could be reduced from 12 to 4; this, along with the higher trunking efficiency , led to an increase of almost 500% in the carried traffic. We have also shown that using a fractional loading factor it is possible to adopt a cluster size equal to 1, which avoids the need of frequency planning altogether, and allows a further improvement in spectrum efficiency . These guidelines can be applied in planning advanced cellular systems, even based on already existing air interfaces such as GSM.
In this section we will report the SIR results for the adaptive BS antennas. The curves in fig. 3 refer to the dimensioning criterion based on blocking probability, above introduced (see section lILA) and are directly comparable to those shown in fig. 2. We note that, by adopting adaptive BSs, the cluster size can be as low as 4. In this case, the carried traffic is about 21 Erllcell, with an improvement in spectrum efficiency of almost 500% with respect to the one obtained with non-adaptive BS antennas. 0,4
0,3S 0.3
0,2S 0,2 O,IS 0,1
REFERENCES
O,OS
°
-10
-S
°
10
IS
[1] T. Ohgane, T. Shimura, N. Matsuzawa and H. Sasaoka, "An Implementation of a CMA Adaptive Array for High Speed GMSK Transmission in Mobile Communications", IEEE Trans on Veh , Tech., vol. VT-42, No.3, August 1993, pp.282-288. [2] O. Norklit and J. Bach Andersen, "Mobile Radio Environments and Adaptive Arrays", PIMRC '94, The Hague, September 1994, pp.725-728 . [3] M. Frullone, C. Passerini, P. Grazioso and G. Riva, "Advanced frequency planning criteria for second generation cellular radio systems", ICf '96, Instambul, April 1996. [4] I.S . Reed, J.D. Mallett and L.E. Brennan, "Rapid Convergence Rate in Adaptive Arrays", IEEE Trans. on Aerospace and Electronic Systems, vol. AES- 10, No.6, November 1974, pp. 853-863
20
Fig. 3. CDF of SIR for different CSs (adaptive BSs) Finally, we have analysed the effect of dimensioning the system adopting CS = 1 and using a fractional loading factor (see fig. 4). We obtain that a fractional loading factor F = 0.3 is achievable. This means that every cell uses in average about one third of the available 124 channels, that is 37 channels; in other words, the carried traffic is now about 37 Erl/cell, with a further improvement of 75% with respect to the previous case. 0,4 0,3S
Prob(SIR <xj
0,3
0,2S 0,2 O,IS 0,1 O,OS
°·10
.._-- ---- .r':·'"!;.~ .~ .. ~?-r ·5
°
10
15
20
Fig. 4. CDF of SIR for CS = 1 and fractional loading
734
Subscriber Location in CDMA Cellular Networks James Caffery, Jr., Student Member, IEEE, and Gordon L. Stuber, Senior Member, IEEE
Abstract~ubscriberradio location techniques are investigated for code-division multiple-access (CDMA) cellular networks. Two methods are considered for radio location: measured times of arrival (ToA) and angles of arrival (AoA). The ToA measurements are obtained from the code tracking loop in the CDMA receiver, and the AoA measurements at a base station (BS) are assumed to be made with an antenna array. The performance of the two methods is evaluated for both ranging and two-dimensional (2-D) location, while varying the propagation conditions and the number of BS's used for the location estimate.
Index Terms-Cellular CDMA, position location.
O
I. INTRODUCTION
VER THE past decade, considerable attention has been given to vehicle location technology, and numerous applications have been proposed. Recently, in a few testbed areas, rental cars outfitted with location devices and map displays have aided visitors in unfamiliar territory [1]. Taxi and delivery drivers have utilized location technology in Tokyo to navigate the myriad of streets. Fleet operators use location technology to improve product delivery times and to improve the efficiency of the fleet management process. Emergency and police dispatchers have also utilized location technology to locate dispatch vehicles and emergencies for improved response times, In cellular telephone networks, location technology could be used for radio resource and mobility management [2], [3]. For example, a service provider who may have multiple agreements with personal communication services (Pf'S's), cellular, or satellite carriers, could offer its customers the ability to choose a carrier that best suits their needs at a given time and location [4]. Also, the Federal Communication Commission has recently released an order, to be implemented in two phases, requiring cellular service providers to provide a mechanism for generating subscriber location estimates for Enhanced-vl l (E-911) services [5]. A further application of location technology is in the rapidly emerging field of intelligent transportation systems (ITS's), which are designed to enhance highway safety, system operating efficiency, environmental quality, and energy utilization in transportation [6], [7]. Each of the above applications requires a method for determining and relaying the location of vehicular and pedestrian mobile stations (MS' s).
Manuscript received April 19. 1996; revised March 3, 1997. This work was supported by GTE Mobilnet. Portions of this paper were presented at the 5th IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), Toronto, Ont., Canada. September 1995. The authors are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. Publisher Item Identifier S 0018-9545(98)02477-3.
Automatic vehicle location (AVL) techniques have been studied thoroughly in the literature for the purpose of vehicle location. AVL systems entail the acquisition of information about the location of MS' s operating in an area, and all require the processing of that information to form location estimates. There are three basic AVL methods: dead reckoning, proximity systems, and radio location [8]. Dead reckoning computes the direction and distance of travel from a known starting position [8]. In proximity systems, the nearness of an MS to fixed detection devices is used to determine its position. The devices can be anything from magnetic sensors to conventional radio transmitters and receivers. Radio location systems attempt to locate an MS by measuring the radio signals traveling between the MS and a set of fixed stations (FS' s). The signal measurements are first used to determine the length or direction of the radio path, and then the MS position is derived from known geometric relationships [8]. Radio location can be implemented in one of two ways-either the MS transmits a signal which the FS's use to determine its location or the FS' s transmit signals that the MS' s use to calculate their own positions [e.g., the global positioning system (GPS)]. There are several fundamental approaches for implementing a radio location system including those based on signal-strength [9]-[12], angle of arrival (AoA) [13], and time of arrival (ToA) [3], [14], (15]. It is important to note that line-of-sight (LOS) propagation is necessary for accurate location estimates. Many of the existing location technologies use dead reckoning, radio location with GPS, or hybrids which require specialized subscriber equipment, the cost of which can severely limit their availability to the average consumer, With these technologies, the MS formulates the location estimate which may be relayed to a central site. Another approach for providing location services is to use the cellular telephone networks. A method has been proposed in [1] which incorporates the cellular network into the location process. However, this service requires a GPS receiver in the MS to determine the location, and the cellular network is only used to relay the location information to a central site. Only one previous work has examined a subscriber location technique that relies solely on the cellular network, that is based on signal attenuation measurements [12]. This paper examines the feasibility and performance of radio location techniques in code-division multiple-access (CDMA) cellular networks. CDMA is the chosen access scheme, since it appears to be the leading candidate for third generation cellular networks. The cellular network is used as the sole means to locate the MS' s, and the location estimates are determined through reception of signals that are transmitted by the MS at
Reprinted from IEEE Transactions on Vehicular Technology, Vol. 47, No.2, pp. 406-415, May 1998.
735
a set of base stations (BS's). This approach has the advantage of requiring no modifications to the subscriber equipment. Specifically, radio location methods based on AoA and ToA are studied. We concern ourselves with performance in terms of absolute accuracy with no concern given to the rate of location updates that can be achieved. The remainder of this paper is organized as follows. Section II outlines the methods employed for the AoA and ToA techniques that will be used for the performance evaluation. The propagation models for macrocellular and microcellular systems are discussed in Section. III, followed by simulation results in Section IV. A discussion of some practical issues for subscriber location is given in Section V, followed by some concluding remarks in Section VI.
BS
II. RADIO LOCATION SYSTEM
Fig. 1. MS-BS geometry assuming a ring of scatterers for macrocells.
A. Angle of Arrival AoA techniques estimate the location of an MS by using directive antennas or antenna arrays to measure the AoA at several BS's of a signal that is transmitted by the MS [13], [16]. Simple geometric relationships are then used to form the location estimate, based on the AoA measurements and the known positions of the BS' s. With the AoA method, a position fix requires a minimum of two BS's in a 2-D plane. In this paper, we consider the error due to multipath propagation, but do not consider angle estimation errors. Multipath propagation, in the form of scattering near and around the MS and BS, will affect the measured AoA. For macrocells, scattering objects are primarily within a small distance of the MS since the BS's are usually located well above surrounding objects [17], [18]. This results in reception of signals from all directions at an MS while the BS receives signals from a small azimuthal spread. For microcells, it has been suggested that the BS' s be placed below rooftop level (lamppost height) in order to confine the signal coverage to a small area [18]. As a result, the BS becomes surrounded by local scatterers and signals can arrive at the BS from a much broader range of angles. Consequently, the AoA approach, which may be used for macrocells, is impractical for microcells. Gans [14] and Jakes [17] have modeled the macrocellular propagation environment as a ring of scatterers about the MS, with the BS well outside the ring. Fig. 1 illustrates this geometry, where the primary scatterers are assumed to be on a ring of radius a about the MS. The distance between the BS and MS, d, is assumed to be much greater than a. We assume that the MS uses an omnidirectional antenna, so that
1 p(,) = 27r'
o :S ,
< 27r.
(1)
The distribution of the AoA at the BS, 0, is given by
d,
dO p(O) = 2p(,).
(2)
From the geometry of Fig. 1, we find that [14] 2
d-y ~ [ (~) - ({3 - B)2
] -(1/2)
dB.
(3)
r(t)
1\
Timedelayestimate, 't Fig. 2. The DLL used for time-based subscriber location.
Therefore, p(0) is
p(0) == {
K [
(~/ -
({3 - B)2] -(1/2) ,
0,
{3 - BM 5: B $ {3 + BM otherwise
(4)
where
OM == arctan( a/d) 1
K= 2 arcsin (dO ). aM
Note that for d ~ a, a small angle approximation can be invoked, with the result that OM ~ aid and K ~ l/tr. The model p(8) provides the AoA distribution for signals arriving at a BS. Our model goes one step further by assuming that a measured AoA at a BS also has the distribution p( 0). Since the measured angles are not equal to the true angles to the MS, the lines of position from the BS's will not intersect at the same point. This problem is resolved by deriving the location estimate from the centroid of the set of points defined by the intersecting lines of position. With three BS' s, for example, the lines of position intersect at
736
three points: (Xl: Yl), (X2' Y2), and (X3' Y3). The location estimate (x: y) is obtained by averaging the coordinates of the points of intersection. i.e., ;r == (Xl + X2 + x3)/3 and Y == (Yl + Y2 + Y3)/3.
B. Time of Arrival Many popular radio location techniques are based upon the measurement of the arrival times of a signal transmitted by an MS at several BS"s. These methods determine the distance between an MS and BS by measuring the time a signal takes to travel from the BS, to the MS, and back again. Geometrically, this provides a circle, centered at the BS, on which the MS must lie. Using at least three BS' s to resolve ambiguities in two dimensions. the intersection of circles provides the MS"s position.' This method is often called the ToA method and has the disadvantage that it requires the MS to act as a transponder in which processing delays and non-LOS propagation can introduce error. To overcome these limitations, time difference measurements rather than absolute time measurements can be used. Since the hyperbola is a curve corresponding to a constant time difference of arrival (TDoA) for two BS's. the time differences define hyperbolas, with foci at the BS' s, on which the MS must lie. The intersection of hyperbolas provides the location of the MS. This method is often called the TDoA method. Methods for obtaining the ToA or TDoA estimates include phase ranging [19], pulse ranging [3], [19], and spread-spectrum techniques [20], [21]. Since the cellular system being considered is CDMA, methods for determining the ToA's from the spread-spectrum signal are of interest. The two methods for determining time delays in spread-spectrum communications systems are coarse timing acquisition with a sliding correlator or matched filter and fine timing acquisition with a delay-locked loop (DLL) or taudither loop (TDL) [22]. Previous subscriber location studies have used coarse timing acquisition to obtain the ToA estimates [20], [21]. Since the DLL finely tracks the time delay, it is better suited for a location system. The DLL is an essential part of time estimation used for GPS and provides reasonable accuracy over the satellite-earth propagation channel. Here, the DLL-based location system will be investigated for its performance in cellular propagation environments. The DLL shown in Fig. 2 allows fine synchronization of the local spreading code with the incoming code. It operates by correlating the received signal with the early and late spreading codes c(t - f- + ~Tc) and c(t - f- - ~Tc), respectively, where f is an estimate of the delay between the local and incoming codes. The code phase error signal e(t) is obtained by squaring and differencing the correlator outputs. The squaring operations serve to remove the effects of data modulation and carrier phase shift. The loop is closed by applying e(t) to a low-pass filter, whose output is used to drive the voltagecontrolled clock (VCe) and correct the code phase error of the locally generated code. The parameter ~, 0 < ~ < 1, is called the early-late discriminator offset. The output of the vec provides the ToA estimate f. 1 In
C. Time-Based Location Algorithm
Two approaches are generally used to calculate the location of an MS from ToA or TDoA estimates. One approach uses a geometric interpretation to calculate the intersection of circles or hyperbolas, depending on whether ToA or TDoA is used. This approach becomes difficult if the hyperbolas or circles do not intersect at a point due to time measurement errors. A second approach calculates the position using a nonlinear least-squares (NL-LS) solution [3], [19], [23], which is a more statistically justifiable approach. The algorithm assumes that the MS, located at (xo, Yo), transmits its sequence at time TO. The N BS receivers located at coordinates (Xl, Yl), (X2, Y2), "', and (XN, YN) receive the sequence at times T1, T2, ... , TN· As a performance measure, we consider the function [19]
j"i(X) = C(Ti - T) - J(x t
x)2 + (Yi - y)2
-
(5)
where c is the speed of light, and x == (x, Y, T)T. This function is formed for each BS receiver, i == 1, ... , N, and all the Ii(x) could be made zero with the proper choice of x ; y, and T. However, the measured values of the arrival times t. are generally in error due to multipath and other impairments, and non-LOS propagation introduces errors into the range estimates that are derived from the arrival times. 1) Unconstrained NL-LS Approach: To obtain the location estimate from the raw ToA data, the following function is formed:
F(x) ==
'""' Qi ? f,;?(x) c:
(6)
i=l
where the O:i' S are weights reflecting the reliability of the signal received at BS i. The location estimate is determined by minimizing the function F(x). A simple approach for solving the nonlinear least squares problem in (6) is the steepest descent method, where successive location estimates are updated according to the recursion
where J.L is a constant (scalar or diagonal matrix), (Xk, Yk, Tk)T, V'x == d/dx, and
8FI 8x
\JxF(Xk) == \7 xF (X) lxk
=
737
==
Xk
8FI by Yk
(8)
bPI
N
2f- ( ) 2'""' ~Qi t Xk i=l
8T
Tk
x, - Xk J(Xi - Xk)2 + (Yi - Yk)2
N
2LO:; h(Xk) i=l
J(Xi -
Yi - Yk + (Yi - Yk)2
Xk)2
N
-2cLfi(Xk) i=l
general, locating an MS in n dimensions requires n + 1 measurements.
Xk
(9)
Since T is small (microseconds) compared to x and y (meters), the scalar step size J..L should be small enough to allow T to converge to a solution. Consequently, J.l. is chosen to be the diagonal matrix J.-L
==
/-Lx [
0
o
0 00]
J..Ly 0
BSI
(10)
J-L-r
where J-Lx, J.Ly ~ J..L-r. The recursion in (7) continues until lI\7x F (xk)1I is smaller than some prescribed tolerance f. One drawback of the steepest descent method is its slow convergence. Other algorithms have been investigated [19], [23], which form the solution to (6) by linearizing Ii(x) with a Taylor series expansion about Xk and keeping only the first order terms, i.e.,
BS2
(11)
where 6 == (6x , by, b-r)T = x and solving
Xk.
Substituting (11) into (6) (12)
for 6, the vector
Xk
is updated by (13)
This new estimate is substituted back into (11) and the process 8yl+cI6-r1< E, where f is a prescribed is reiterated untilI8 x l+1 tolerance. When the MS is either close to the BS' s or near the perimeter of the area defined by the polygon with the BS's as its vertices, then the linear approximation approach has convergence problems [3], [19]. For microcells, the MS is always within a short distance of the serving BS, so this method is not appropriate. The convergence problem arises from the approximation of fi(X) with the linear terms of the Taylor series expansion. Other objective functions F(x) can be formed replacing, for example, fl(x) with Ifi(X)I. However, these methods usually do not perform as well as minimizing the sum of squares [3]. 2) Constrained NL-LS Approach: It may be possible to improve the time-based location algorithm due to the fact that the range error is always positive [24]. This is because the ToA estimates are always greater than the true ToA values due to multipath propagation and other impairments. Also, the range estimates derived from the ToA estimates are greater than the true ranges due to non-LOS propagation. Therefore, the true location of the MS must lie inside the circles of radius Ti == C(Ti - r), i = 1, .", N, about the NBS's, since the MS cannot lie farther from a BS than its corresponding range estimate (Fig. 3). Mathematically, this implies r;
= c(ri - r) 2 J(Xi - x)2 + (Yi - y)2
(14)
where (x, y) is the position of the MS. Since the unconstrained NL-LS algorithm does not take this restriction into account, a constrained NL-LS approach can be used to force the estimate at each iteration to satisfy (14). However, the LS solution is complicated by the nonlinear functionals Ii(x) as well as the
Fig. 3. The location of the MS is constrained to the intersection area (shaded region) of circles of radius C(Ti - T) centered at each BS.
nonlinear inequality constraints of (14). Note that (14) implies that
J(Xi - x)2 + (Yi - y)2 - C(Ti - r) ~
o.
(15)
We recognize from (5) that the left side of the inequality in (15) is simply 9i (x) = - Ii (x). Hence, the restrictions Ii(x) 2 0 are formed, where the area within the constraint boundaries is known as the feasible region. There are many approaches to forming numerical solutions for NL-LS problems with nonlinear inequality constraints of the form gi(X) :::; 0 [25]. One simple, yet effective, method uses penalty functions to modify the objective function F (x) and form a solution using an unconstrained approach as in the previous section. The penalty functions provide a large penalty to the objective function when one or more of the constraints are violated. The objective function in (6) is modified to include the penalty functions 9i(X) as follows [25]:
F(x) ==
N
L
i=l
N
at fl(x) - P L[9i (X)]- 1
(16)
i=l
where P is positive for minimization. As any constraint is approached during the search, the penalty term forces F toward infinity, thus forming a natural optimum within the feasible region. This approach requires that the initial guess be placed within the feasible region. A method for doing this is described in [25]. The search procedure can be viewed as the optimization of a sequence of surfaces which tend toward the true value of the objective function. Initially, an unconstrained search method is used to provide an artificial optimum x 1 with a large value of P == PI- The next stage is initialized with the previous estimate x, and uses a smaller P = P2 to provide a better approximation to the true optimum. In this
738
COST207
where nk(dB) is the local mean envelope (or square envelope) level (in decibels) that is experienced at location k, ~ is a parameter that controls the spatial decorrelation of the shadowing, and {'T/k} is a zero-mean discrete-time Gaussian random process with autocorrelation ¢"P7 (n) == (j2 8(n ). The autocorrelation of nk(dB) is given by
TABLE I SIx-TAP REDUCED TYPICAL URBAN POWER DELAY PROFILE
COST 207 Model Fractional Power 0.189 0.379 0.239 1.6 0.095 2.3 0.061 5.0 0.037
Delay-To (J.lS) 0.0 0.2 0.5
(19)
way, the solution approaches the constraints more closely, if the optimum happens to lie close to one of the constraints. The penalty constraints become smaller at each stage, forming a monotonic-decreasing sequence PI > P2 > '.', and the sequence of artificial optima x. , X2, ... tends toward the true optimum. The search continues until several iterations fail to produce a change in the objective function. This formulation essentially replaces a constrained optimization by a sequence of unconstrained optimizations. III. PROPAGATION MODELS A three-stage model is used for the radio propagation environment, that includes multipath-fading, shadowing, and path loss. The particular models used in this paper for macrocellular and microcellular propagation environments are now described.
where as is called the shadow standard deviation. Typical values of the shadow standard deviation range from 5 to 12 dB in macrocells [17], [18], [28]. If we assume that the local mean is sampled every T s, then the autocorrelation can be expressed as
where ED determines the correlation between two points separated by a spatial distance D and v is the velocity of the MS. The simulations in the sequel assume a shadow decorrelation of 0.1 at a distance of 30 m. Several empirical path loss models have been presented in the literature, one of the most useful being Hata's model [28], which expresses the path loss in terms of the carrier frequency, BS height, MS antenna height, and the type of environment (urban, suburban, or rural). Rata's model for medium or small city urban areas is used in the sequel with a carrier frequency of f == 850 MHz, BS antenna height of 100 m, and an MS antenna height of 2.5 m.
A. Macrocells
For wideband spread-spectrum systems, the channel can be modeled by the M -tap tapped delay line
h(t) ==
M
L
ui, (t)8(t
-
Ti)
( 17)
i=O
where the {Ti} are the tap delays and the {W·t } are the tap gains, assumed here to be complex Gaussian random processes. For numerical convenience, the tap delays can be chosen to be an integer multiple of some small delay T, i.e., Ti == krr, i == 1,,'" M. The first tap delay TO is determined from the MS-BS geometry of Fig. 1 by calculating the distance traveled by a signal transmitted from the MS in a random direction according to p('"Y) and reflected from the ring of scatterers to the BS. The remaining delays are chosen according to the six-tap reduced typical urban delay profile defined in COST207 [26] (see Table I). The model deviates slightly from the COST207 model by assuming a classical Doppler spectrum for all taps, i.e., in the simulations the taps gains are all generated by using Jakes' method [17]. Shadow fades have been described from measurements as being lognormally distributed with a standard deviation that depends on the frequency and the environment [18]. Gudmundson [27] has suggested a simple Markovian model to describe variations in the local mean envelope (or squared envelope) level due to shadow variations. With this model nk+1(dB)
== ~rlk(dB)
+ (1 -
~)TJk
(18)
B. Microcells
The wideband channel and shadowing models discussed above can also be used to model microcellular propagation. However, the power delay profiles are different, and the standard deviation of shadowing in microcells typically ranges from 4 to 13 dB. Further differences in the propagation models for microcells and macrocells are discussed in the following. Microcellular path loss is often described by a two-slope characteristic, where the area mean == E[n] is given by [29]
n
n = 10 loglO [da(l: d/g)b ]dBm
(21)
where A is a constant, d is the radio path length, 9 is the break point (that ranges from 150 to 300 m), and a and b determine the slopes before and after the break point. In the simulations, we assume 9 == 150 m and a == b == 2. An important consideration for microcells is the corner effect, which occurs in microcellular scenarios when an MS rounds a street corner. To account for this effect, LOS propagation is assumed to the MS until it rounds the comer. The non-LOS propagation after rounding a street comer is modeled by assuming LOS propagation from an imaginary transmitter that is located at the street comer having a transmit power equal to the received power at the street corner from the serving BS. The area mean (in dBm) is given by (22), at the bottom of the next page, where de is the distance between the serving BS and the comer.
739
was used for the loop filter [i.e., F( s) = 1]. For the vee, the output time delay estimate and input waveform are related by
f(t)
.8S1
Fig. 4. Manhattan street microcell deployment.
Due to the site-specific nature of the microcellular propagation environment, techniques such as ray tracing have been developed. In this study , ray tracing concepts are used to calculate the propagation delays for the wideband channel model. A Manhattan street microcell BS deployment is assumed as shown in Fig. 4. When the MS is LOS with a BS, a four-path model is used, consisting of a direct path, a roadreflected path, and two wall-reflected paths. The taps of the wide band channel model are generated using Jakes ' method, appropriately modified for Rician fading. When the MS is non LOS with a BS, i.e., around the comer, a different approach is taken to determine the propagation delays . Since the literature prov ides no results that describe the power delay profile for an MS that is around a comer from a BS, a simplistic model is chosen. A four-path non-LOS propagation model is used that includes two paths that arrive from diffractions at the building comers in the street intersection and two remaining paths whose delays are generated by adding random delays to the first two paths . All paths are assumed to be Rayleigh faded. The model chosen here is inconsequential, because the extra time delay for non-LOS BS's introduces a large amount of error into the location algorithm. Hence, accurate modeling of multipath propagation on non-LOS streets is not necessary; only a means of introducing the excess propagation delay around the street comer is needed .
IV. SIMULATIONS The location techniques described in Section II were simulated in the macrocellular and microcellular environments described in Section III to determine their performance. The spreading code used was an m sequence of length 127 and chip rate Te- 1 = 1.2288 Mcps . In the DLL, an all-pass filter
= KveeTe
it
u(x) dx
(23)
where K vee is the gain of the vee, T; is the chip period, u(t) is the output of the loop filter, and the vee is assumed to begin .operating at time t = O. A simple accumulator models the operation of the vee in the computer simulations with the constant K veeTe = 0.003. Note that there is a limitation in the accuracy that can be achieved when simulating the DLL on a computer. As a result, we limit the resolution of the DLL to 1/120 of a chip to limit the simulation time. Consequently, the ranging resolution is limited to approximately 2 m, which causes all range estimates to be in error even in the absence of propagation impairments. However, with such a fine resolution, propagation impairments will be the predominant source of location error .
A. Range Estimation Ranging measures the I-D distance between an MS and BS. Only the time-based method is employed for ranging since AoA ranging does not make sense . For macrocells, our ranging results assume that the first path to arrive from the COST 207 model is a LOS path . Consequently, the ranging results for macrocells are very optimistic by disregarding the extra propagation caused by non-LOS propagation when a direct path does not exist. For microcells, the Manhattan street microcell deployment in Fig. 4 is assumed. 1) Effect of Standard Deviation of Shadowing, as : Fig. 5(a) shows the effect of the shadow standard deviation on the mean and standard deviation of the range estimation error with an early-late discriminator offset ~ = 1/2 and a chip-energy-tonoise ratio E e / No = 10 dB. The mean ranging error increases by approximately 10 m as a s increases from 4 to 12 dB. The standard deviation of the ranging error also increases due to the increased variability of the shadowing process. 2) Effect of Ec/N o: Fig.5(b) shows the effect of E e/No on the ranging error with ~ = 1/2 and as = 6 dB . The increase in ranging error for decreas ing E e/ No is expected in any system . The effect is not as pronounced in the microcellular environment due to the smaller delay spreads. 3) Effect of d : The effect of multipath on the tracking ability of the DLL can be explained by observing the distortion that multipath causes on the correlation function of the spreading code, which has a triangular shape for a rectangular chip-shaping pulse. Fig. 6 shows an example of a distorted loop S curve for the case of two multipath components, the second having half the power of the first and delayed by T e/2. Observe that the tracking error introduced by multipath propagation is reduced by using smaller ~. However, the
(22)
60.0 ~
4° l' -~ _~
:g 30 ~ ~~ l .~20
,
0--- - -- - ---0-- ----- --
~
~
~...: 40 .0 ~,i " o
0- - -
-.- - - - - - - - - ..
e ---------.-- -
g 50.0 ~
-- -------1
Ot-t-t-O
f
e
§
I
1
Macrocell
+--+ Microcell (LOS)
gf
.-
1
10
gf
I r 30 .0 ~
~
~ Macrocell
+--+ Microcell (LOS)
"
"
I
'0"
6
8
10
Standard deviation of shadowing, cr, (dB)
------
Il.l
~
Fig. 5. Effect of (a) shadow standard deviation (E r /N o (dashed lines) of the ranging error.
6
12
= 10
dB) and (b) Eo/ N u (a.
=
~
j
i1
- 1.0
L
Fig. 6.
- ' - -_
- 1.0
15.0
dB) on the mean (solid lines) and standard deviation
- - L -_
0.0
_
---'_~ 1.0
Error in delay estimate (chips)
V ARIOVS V ALVES OF LI. . V ALVES ARE IN METERS
t. 1/2 1/4 1/8
I
~
- 2.0 - 2.0
=6
FOR
g
o
1
-.J
M EAN R ANGING ERROR AND S TANDARD D EVIAT ION
~
..J
10.0
1
TABLE II
=1/2
6 = 118 6 1120
1.0
5.0
E/No (dB)
6 = 1/4
.g 'C '"
-- - - ---
-'--__
' -_ _......L_ _- - '
0.0
..
· ·0·------0
(b)
(a)
2.0
' . 0 ..
20.0 ~~_......Ac
o '------'---~---'------'--- ~ 4
'0-
2.0
Distortion of the S curve due to multipath for different values of LI. .
minimum size of Do is limited by hardware considerations (such as the clock rate) and the precorrelation bandwidth in the DLL. Band limiting tends to round the autocorrelation peak which limits the discrimination between the early and late correlation when using small Do [30]. Simulation results for various Do are presented in Table II with a, = 6 dB and Ec/No = 5 dB. The results show that the ranging error mean and standard deviation can be signifi cantly reduced by using a smaller Do . B. Two-Dimensional Location
Two-dimensional (2-D) location estimates the MS location by using several BS' s. Here, we focus on the accuracy of the location estimates as a function of the number of BS' s. This is an important consideration, since using more BS' s means more processing and an increased load on the network. Assuming a transmit power of I dBW (the maximum for Class III IS95 MS's) , a noise power of - 100 dBm was added to each BS receiver for the time-based method. The macrocell and microcell deployment scenarios are as follows.
Microcell Mean Std. Dev 15.6 15.5 10.8 12.1 4.5 6.1
Macrocell Mean Std. Dev 30.8 23.6 25.3 19.1 16.9 18.9
J) Mac rocells: We assume a distance of 6000 m between BS' s, i.e., the cell radius is 3000 m. Assuming known BS positions , the MS is randomly placed among the BS' s and the nearest BS' s are used for the location process. For macrocell s, ToA and AoA approaches are compared when using two-fi ve BS' s in the location process as a function of the scattering radius a about the MS. The simulations examined both the unconstrained and constrained location algorithms of Section II-C, which almost always converged with a < 1 for each BS. The mean and standard deviation of the location error for the ToA method using the unconstrained NL-LS algorithm are shown in Fig. 7(a). For a given scattering radius, the mean and standard deviation decreases when more BS' s are used. As expected, a larger scattering radius increases the location error due to nonLOS propagation. Recall that LOS propagation is necessary for accurate ranging and location estimates. The mean and standard deviation of the location error for the ToA method with the constrained NL-LS algorithm are shown in Fig. 7(b). Unlike the unconstrained NL-LS case, the performance is not improved significantly when more BS' s are used. Table III compares the performance of the unconstrained and constrained NL-LS methods, where both algorithms are initialized with the same location estimate. The mean location error is reduced up to 30% by using the constrained NL-LS algorithm. The constrained NL-LS with three BS' s perform s nearly as well as the unconstrained NL-LS with five BS's. This implies that the constrained NL-LS algorithm can result in less network loading.
741
800
S
'B
r~3 BS
_ _ 4BS
600
'" "CI
600 ,-... 500
S '-'
6-----6 5 BS
-r-----.-------,
_ _ 4BS
"'-----6 5 BS
"0
§
<1l
... 300
400
9
]
-
0--0 3 BS
I
'B '" 400
:::
~
r - - -- , - - --
200
o
I o
~:::
_ --_:~
0" " " ,
.S: 200
,,~"""" f :····:····"
Cii u
o
"".''':
~
-~----'------"---~--'---~ 200 400 600 800
100 O l.-------'-----'------'---~----'
o
LI
Radius of scatterers about mobile, m
200
400
600
Radius of scatterers about mobile, a (m)
(a)
800
(b)
Fig. 7. Two-dimensional location in macrocells for the ToA method using the (a) unconstrained and (b) constrained NL-LS algorithms. Solid lines denote the mean error, and dashed lines denote the standard deviation: o , = 6 dB. TABLE III
REDUCTION IN THE MEAN LocATION ERROR OF THE CONSTRAINED NL-LS ALGORITHM COMPARED TO THE UNCONSTRAINED NL-LS ALGORITHM FOR THE TOA METHOD IN MACROCELLS. V ALVES ARE IN METERS
'--'2BS O-~ 3 B S
...---.4BS D----D 5 BS
#BSs 0=100 0=300 0=500 0=700 10.0 SO.5 125.7 213.2 3 4.8 29.0 75.7 4 145.0 6.1 23.1 53.4 5 98.8
600
Results for the AoA method are shown in Fig. 8. Once again, a larger number of BS' s decreases the mean and standard deviation of the location error. However, diminishing returns are obtained by increasing the number of BS' s. For a = 100 m and a = 300 m, the unconstrained ToA method outperforms the AoA method for the same number of BS' s. For larger scattering radii, the AoA method steadily improves and performs slightly better than the unconstrained ToA method at a radius of a = 700 m for the same number of BS's. In all cases, the constrained ToA method performs the best. 2) Microcells : As the MS rounds the comer from BSO to BS1 in Fig. 4, its location is estimated using a combination of BS's . For this particular deployment , we assume LOS propagation between the MS and two BS's (four at an intersection) and non-LOS propagation to the other two BS's. Only the ToA method is used since the AoA method is unreliable due to the relatively large AoA spreads in microcells. The number of BS's used to derive the location estimate ranges from two to four. With 2-BS location, BSO and BS2 are used until the MS rounds the comer after which BS1 and BS3 are used. With 3BS location, BSO, BS I, and BS2 are used until the MS rounds the comer after which BSO, BSl, and BS3 are used. Fig. 9 plots the results for the microcell deployment in Fig. 4, and indicates that the location accuracy is improved with more BS's. Fig. 9 shows the effect of the NL-LS weighting factors a on the location performance. The LOS BS's weights are a = 1, whereas the non-LOS BS weights were varied from 0.2 to 1.0. Fig. 9(a) shows that a smaller a can significantly reduce the mean and standard deviation of the location
400
I
200 .
o
o
--'-
200
-'--
400
'--~ _--J
600
Radius of scatterers about mobile, a (m)
800
Fig. 8. Two-dimensional location in macrocells using the AoA method. Solid lines denote the mean error, and dashed lines denote the standard deviation.
error with the unconstrained NL-LS algorithm, especially for 3-BS location . For the constrained NL-LS algorithm, there is no significant improvement for a < 0.6, as shown in Fig. 9(b). It is interesting to note that 4-BS location is much better than 3-BS location, even though an additional non-LOS BS is used. This is because the two non-LOS BS's tend to "cancel" one another's effects as a result of the symmetry of the BS layout. Two-BS location was also considered, using the two LOS BS's with a = 1 for both BS. The mean location error was 6.9 m with a standard deviation of2.0 m for both the unconstrained and constrained NL-LS algorithms. Although three BS's are required for 2-D location, the constraint that the MS must lie on a line between the two LOS BS's provides the additional information needed for 2-BS location. This topographical constraint provides more accurate location information than the additional information from a non-LOS ToA measurement and, therefore, 2-BS location performs better than 3- or 4-BS location. Note that the feasibility of this approach depends on the BS topography.
742
I
250
100 ,--- - .---,,.--
~... ~
"'d
200
-
--,.--
-
-,--
-----. 3 BS <>---04 BS
-----.3 BS <>---0 4 BS
,,-f
;
til
1 150 8'"' ~ 100 =
60
~
40
Q
.~
as
...:l
50 •
,~
- - -
_.---------.----
20 , u uum. um m ~ /
o_- - - -- - - - -O-- -- -- - - -
0_ - - -
0.4
0.6
0.8
1.0
NLOS Least Squares weights, a
/
o ? ---- --- -- ~ -- ------~-- - - ---- -
O '<---~-'--------'-~--'--~----'
0.2
------~
0.2
0.4
0.6
0.8
NLOS Least Squares weights, a
1.0
(b)
(a)
Fig. 9. Two-dimensional location in microcells using the (a) unconstrained and (b) constrained NL-LS algorithms for various values of the non-LOS BS weights, Q . Solid lines denote the mean error, and dashed lines denote the standard deviation .
• aso
•
~110
Approx. cell _".: bOundaries
./ .....
.
"
....
~
. cell I
.
~
Desired mobile
•
Interfering mobile in cell 0
~
Interfering mobile in cell l
......... .."s..•
~
~
~ " " ;'~ ~....vr
as}
•
• aS2 Fig. 10. CDMA multiuser interference scenario.
V. DISCUSSION
The implementation of an effective cellular radio location system, requires a method for identifying the set of BS' s to be used for deriving the location estimate. This is especially important for microcells since the use of non-LOS BS's can introduce large errors in the location estimates as it does in macrocells. The link quality measurements used for handoff initiation may be used to determine the appropriate set of BS's. The appropriate BS's can be determined , for example, by received signal strength measurements . In order to achieve high accuracy, LOS propagation is necessary between the MS and BS's used in the location
process . This may be impossible to achieve in macrocells, but may be attainable in microcells . The additional propagation time to a non-LOS BS leads to a positive range estimate error. Recently, a statistical method has been suggested to compensate for non-LOS induced errors by biasing the range estimates [31]. This paper has not addressed the effect of CDMA multiuser interference caused by the nonorthogonality of the user spreading codes. Multiuser interference along with unequal received power levels from different users leads to the near-far effect, which is mitigated in CDMA systems by using power control techniques. This paper has only considered the case of a single
743
MS with no interferers. Since the location system requires the use of several BS's, multiuser interference may pose severe problems. For example, consider the microcell deployment in Fig. 10, where the target MS whose location is desired (solid black oval) is being served by BSO. To determine its location, BSO, BS1, and BS2 are used. If power control is used, then the signals from all MS's served by BSO (black and light grey) will arrive at BSO with approximately the same power. The same is true for the MS's being served by BSl (medium grey) and BS2 (not shown). To derive the location estimate, BS1 and BS2 must detect the signal being transmitted by the target MS (black oval). However, the signal from the target MS may experience severe multiuser interference from MS's being served by BSl and BS2, since the target MS is not power controlled by those BS's. This multiuser interference will affect the tracking capability of the code tracking loop that is used to obtain the ToA information.
VI. CONCLUDING REMARKS
Subscriber location in CDMA cellular networks has been investigated for both macrocellular and microcellular deployments. For range estimation, it was seen that location error and standard deviation increases with increasing shadow standard deviation and decreasing Ec/No. Under the condition of moderate delay spread, the tracking performance of the DLL can be improved by using a smaller early-late discriminator offset ~. For 2-D location, the location error is reduced by increasing the number of BS's used in the location process. For macrocells, the unconstrained NL-LS ToA method outperforms the AoA method for a small scattering radius, while the AoA method performs slightly better for a large scattering radius. In all cases, the constrained NL-LS ToA method performed best. It should be noted that our AoA-based location results depend on the method chosen for generating the AoA' s, namely, reflection from a ring of scatterers about the MS. A different method for generating the AoA distribution p( 0) may produce different results. For microcells, the AoA method is inappropriate so only the ToA method was used. Again, using more BS's decreases the error and using small weights for the non-LOS BS's provides good error reduction. The ToA method relies on an NL-LS solution. This was obtained here by using the steepest descent method, chosen for its simplicity. Better methods for locating the minimum of a least squares surface could be employed, such as the Levenberg-Marquardt method, to improve the convergence time and possibly the accuracy of the location estimates. REFERENCES [1] R. Jurgen, ''The electronic motorist," IEEE Spectrum, vol. 32, pp. 37-48, Mar. 1995. [2] I. Paton, E. Crompton, 1. Gardner, and J. Noras, "Terminal self-location in mobile radio systems," in 6th Int. Conf Mobile Radio and Personal Communications, 199], pp. 203-207.
[3] H. Hashemi, "Pulse ranging radiolocation technique and its application to channel assignment in digital cellular radio," in IEEE Veh. Technol. Conf., 199], pp. 675-680. (4] A. Giordano, M. Chan, and H. Habal, "A novel location-based service and architecture," in IEEE Personal Indoor Mobile Radio Conf., 1995, pp. 853-857. [5] "Revision of the commissions rules to ensure compatibility with enhanced 911 emergency calling systems, RM-8143," CC Docket 94-102, FCC. Washington, DC, July 26, ]996. (6] W. Collier and R. Weiland, "Smart cars, smart highways," IEEE Spectrum, vol. 31, pp. 27-33, Apr. 1994. (7] R. Jurgen, "Smart cars and highways go global," IEEE Spectrum, vol. 28, pp. 26-36, May 1991. [8] S. Riter and J. McCoy, "Vehicle location-An overview," IEEE Trans. Veh. Technol., vol. VT-26. pp. 7-11, Feb. 1977. [9] W. Figel, N. Shepherd, and W. Trammell, "Vehicle location by a signal attenuation method," IEEE Trans. Veh. Technol., vol. VT-18, pp. 105-110, Nov. 1969. [10] M. Hata and T. Nagatsu, "Mobile location using signal strength measurements in a cellular system," IEEE Trans. Veh. Technol., vol. VT-29, pp. 245-251, May 1980. [11] G. Ott, "Vehicle location in cellular mobile radio systems," IEEE Trans. Veh. Technol., vol. VT-26, pp. 43-46, Feb. 1977. [12] H.-L. Song, "Automatic vehicle location in cellular communications systems," IEEE Trans. Veh. Technol., vol. 43, pp. 902-908, Nov. 1994. (13] S. Sakagami et. al., "Vehicle position estimates by multibeam antennas in multipath environments," IEEE Trans. Veh. Techno I., vol. 41, pp. 63-67, Feb. 1992. [14] M. Gans, "A power-spectral theory of propagation in the mobile-radio environment," IEEE Trans. Veh. Technol., vol. VT-21, pp. 27-38, Feb. 1972. [15] M. Feuerstein and T. Pratt, "A local area position location system," in 5th Int. Conf. Mobile Radio and Personal Communications, 1989, pp. 79-83. [16] S. Anderson, M. Millnert, M. Viberg, and B. Wahlberg, "An adaptive array for mobile communication systems," IEEE Trans. Veh. Technol., vol. 40, pp. 230-236, Feb. 1991. [17] W. C. Jakes, Microwave Mobile Communications. New York: IEEE Press, 1994. [18] J. Parsons. The Mobile Radio Propagation Channel. New York: Halsted, 1992. [19] G. Turin, W. Jewell, and T. Johnston, "Simulation of urban vehiclemonitoring systems," IEEE Trans. Veh. Technol., vol. VT-21, pp. 9-16, Feb. 1972. [20] P. Goud, A. Sesay, and M. Fattouche, "A spread spectrum radiolocation technique and its application to cellular radio," in IEEE Pacific Rim Conf Communications. Computers and Signal Processing, 1991, pp. 661-664. .2lj J. Caffery, Jr. and G. Stuber, "Vehicle location and tracking for IVHS in CDMA microcells," in IEEE Personal Indoor Mobile Radio Conf., 1994, pp. 1227-1231. [22] R. Ziemer and R. Peterson, Digital Communications and Spread Spectrum Systems. New York: Macmillan, 1985. [23] W. Foy, "Position-location solutions by Taylor-series estimation," IEEE Trans. Aerosp. Electron. Syst .. vol. AES-12, pp. 187-193, Mar. 1976. [24] G. Morley and W. Grover, "Improved location estimation with pulseranging in presence of shadowing and multipath excess-delay effects," Electron. Lett., vol. 31, pp. 1609-1610, Aug. 1995. [25] G. Beveridge and R. Schechter, Optimization: Theory and Practice. New York: McGraw-Hill, 1970. [26] "Proposal on channel transfer functions to be used in GSM late 1986," COST 207 TD(86)51-REV 3, Sept. 1986. [27] M. Gudmundson, "Analysis of handover algorithms," in IEEE Veh. Technol. Conf., 1991, pp. 537-541. [28] G. Stuber, Principles of Mobile Communication. Norwell, MA: Kluwer, 1996. [29] P. Harley, "Short distance attenuation measurements at 900 MHz and 1.8 GHz using low antenna heights for microcells," IEEE J. Select. Areas Commun., vol. 7, pp. 5-13, Jan. 1989. [30] A. van Dierendonck, P. Fenton, and T. Ford, "Theory and performance of narrow correlator spacing in a GPS receiver," 1. Instil. Navigation, vol. 39, pp. 265-283, 1992. [31] M. Wylie and J. Holtzmann, "The non-line of sight problem in mobile location estimation," in IEEE Int. Conf. Universal Personal Communications, 1996. pp. 827-831.
744
~~sactions Letters~~~~~~-_~~~~~~ On the Capacity Formula for Multiple Input-Multiple Output Wireless Channels: A Geometric Interpretation P. F. Driessen, Senior Memher, IEEE, and G. J. Foschini Abstract- The capacity of multiple input., multiple output (MIMO) wireless channels is computed for Ricean channels. The novelty is a geometrical (ray-tracing) interpretation of the MIMO channel capacity formula to find array geometries which greatly enhance channel capacity compared to single input-single output (SISO) systems.
y
1+ 2
0
Index Terms-Array signal processing, channel capacity, land mobile radio cellular systems.
1+ 1
A
.
<.
·····O:::~·:·_,:·::::···:::··························· ..........
a
I. INTRODUCTION
MULTIPLE input-multiple output (MIMO) wireless communications channel with a matrix transfer function of independent complex Gaussian random variables has an information-theoretic capacity which grows linearly with the number of antenna array elements n, for fixed power and bandwidth [1]. For line-of-sight (LOS) channels, we use ray tracing to construct a matrix transfer function (channel response) explicitly for some example environments and find array geometries which result in channel matrices with close to ti nonnegligible eigenvalues. with corresponding high capacity. The LOS matrix channel response will change as the receiver is moved. so that a capacity distribution is obtained from the ensemble of sample matrix elements at different receiver locations. Also, a Rayleigh matrix may be added to the LOS channel matrix to form a matrix of Ricean scalars, from which a capacity distribution is obtained. In either case, we define an outage threshold :z; (say 0.01), and define C.r to be that capacity for which Prob{ C > ex} == 1 - x, In what follows, the MIMO channel capacity formula is used to compute the capacity for three example geometries. plus one Ricean example, with a discussion of the results.
II. CAPACITY CALCULATIONS AND ARRAY GEOMETRY The capacity in b/s/Hz of a MIMO wireless system with n r transmit antennas and n R receive antennas with an average received SNR p (independent of nT) at each receive antenna Paper approved by N. C. Beaulieu. the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received February 18, 1997; revised May 29, 1998; and August 11, 1998. P. F. Driessen is with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC V8W 3P6, Canada (e-mail: [email protected]). G. J. Foschini is with Bell Laboratories, Lucent Technologies, Hohndel, NJ 07733-0400 USA. Publisher Item Identifier S 0090-6778(99)01929-7.
R
T
a
0········ I
z
-1
=
Fig.!. Images in street canyon-top view (/ 25 m. /\ T(.r.y ..r ) (l.15.0) rn, R(,f.Y . r ) (1.:) . .30) to (l.G.-!O)
=
=
=
1/.1 m,
was obtained in [1] as C == lO?;2(det[In R + (plnT)H H*]), where the normalized channel matrix H contains complex scalars with unity average power loss, and H* is the complex conjugate transpose of H. The capacity is expressed in b/s/Hz in the narrow-band limit with no frequency dependence. Normalization is achieved by dividing out the free space power loss and setting the parameter o to the desired SNR. 1 This result assumes that H is unknown to the receiver but nn and p are known [I], [2]. The transmitted data has been demultiplexed into substreams which are separately independently coded and modulated on each antenna [6], so that the transmission from each antenna is different. The practicality of such MIMO wireless systems, using space-time coding with no bandwidth penalty, is illustrated in [3 l-l 5] and [7].
A. Line-of-Sight Channels We first consider an environment with only free space nonfading LOS propagation and T and R arrays of nT == I This avoids the need to compute absolute propagation loss and then select the transmitted power to obtain the desired SNR.
Reprinted from IEEE Transactions on Communications, Vol. 47, No.2, pp. 173-176, February 1999.
745
50 45
.,
40 35
.-<
u >. u
....
30
.~
25
'" :: u
n=8 , f = 900 MHz TR spacing larnbda / 2 , T: street wi d t h 2 5m
>.
u u
....
20
'"
15
'0". u
(y , z) =( 1 5 , O)m
T
10
O'---_.---L_ _-..L_ _--'-_ _- ' -_ _-'--_----'_ _-..L_ _---L_
a
20
_
...l-_--l
40 60 80 100 1 20 140 160 180 R location index 0 : (y . z ) =( 5 . 30) m. 200: (y , z)=(6 , 40) m
200
Fig. 2. Capacity versus R location in street canyon T and R height .r = 1 rn, p = 20 dB. >. = 1/3 m.
0 .9 0 .8
-;;;
....''""
u
0.7
'"
0 .6
A
0 .5
.Q
'"
>.
....u u
'"
0.4
"
0.3
0.
.''":: c,
0. 2 0 .1 0
,
10
log(l+n rho)
Fig. 3. Capacity on Ricean channels. n
12
14 16 18 capacity (bits/cycl e)
20
r
22
nl og(l+rho )
= 3.
nn = n antennas. The base and subscriber ends of the link are designated as T and R, respectively, but reciprocity applies. For LOS propagation, and a narrow-band channel at fixed carrier frequency Ie = c] A; ray-tracing from T to R yields the channel transfer function matrix H = H LOS with complex scalar entries
where Ti , R; are coordinate vectors for the ith element of
T , R. H ik is normalized by the distance between the reference locations T I , R I , so that H I •I = 1 and the absolute attenuation need not be calculated. If the antennas are spaced less than A/2 apart at both T and R , H ik = ei 6•k ~ ei 6 for fixed () for all i , k, (H H*)ik ~ n , and C = !og2(1 + n p) = C log, so that Clog increases logarithmically with n . For this case, H = H LOS = H I is of rank 1, and the capacity gain is essentially due to the n -fold array gain in p. For arrays of n more widely spaced antennas at T and/or R, the complex scalars Hik all have magnitude near one but different phases (}ik. For (}ik so that H H* = nln l H =
746
......../ .
3 .?~ts
.., ...... daisy chain to spac e-time processor
eell slte et
center 01 cell
cell .ltea.t ed geo! cell •
cell . ite••t ce nter 01cell covlHlIgaarea 7ceUreuse
~ cell aitea at edga 01 cell cover age area
A = channel group
120 degree .ectora at ea ch cell alte
Fig. 4.
Ce ll sites at center and edge of coverage area.
H LO S = H " is of rank II , and C = n. log2 (l + p ) = G lin . so that G lin increases linearl y with n, An example is Bi k = ~[ (i - -o) - (k - /.;o)f where for II. = 2 and i o = /.;0 = 1 . ll . H lll a x = ( j ~ ) , and the corresponding array geometry is two linear arrays broadside to each other. In what follows. we show three mor e examples of geo metric arran gements for which HLos ~ One such arrangement is an n.-element T array spread along an arc at angles (/Jk - 1 for l: = 1. . . . . u, and a linear R arra y oriented broadside to the center of the arc . From ( I ). for arc radius D » A, and interelement spacing Z'" H i k = ex p(j 2 '-~= r [U - 1) - ":;- 1] sillfPI.'_ Jl which corresponds to the
n;
autocorrelation R:u U ~ /.;) = 2~Re .r~6 e "· (i- I.:)sin.l d/3 from [8. eq. (A- l3) 1 specialized for the n. discrete angle s of arri val (Pk - l ' G ~ G lin when the an gle subtended by the arc 2.6 = ¢"- 1-¢o is consistent with the beam width 2.6 = AI-Z,. at which Ru U - k) = O. The radiation pattern of the R array with all elements in phase E(¢ ) = T1sin( '~""j;)) , where ":;Ill "), _ j 2\
and a sim ilar R arra y on a circle of radiu s D,. S A at the center of the T arra y. From (1 ), H ;k = ex p (j 2"f " cos[U - k) 2,;]). For D,. A/2, the T elements are not in the nulls of the R array. but the elements of H H* for which i - k is odd are zero. and the off-diagonal even elements are approximately O.3n. Nonetheless, the capacity approaches G lin, con sistent with the observation [8] that small correlation « 0.3) has negligible effect on performance . Further calculations confirm that the capacity is robust in the presence of rotation or lateral movement of R or perturbations in the placement of the T elements. The third example is an urban street geometry with two parallel reflectors representing the building walls separated by the street width a (Fig. I). l ±nt represents an image due to m. specular reflections from the wall s. and 10 is the " ground reflection" image not visible in the figure. For thi s street geometry. the elements of H may be written
For z; = A/ 2. and the 11. array elements of T placed along the arc at ¢ k - l exactl y in the nulls of R, H i k = exp(j2r~ (-i - -io)(k - /';0 ) ) where i o = /.;0 = n! I , for which H H* = 11.1" so that H = H n and G = Glin. The second example arrangement is an n-element T array with elements spread evenly around a circle of radius D » A.
=
"( = 21fz,. sin ¢/ A.
747
ITl
Hi k
ex p( - j27f IT; -
- Rl i
+
f
IT; -
k= - m
Rk l/A)
Rkl
!l f - Rkl/A) II; - Rkl
rl.: ex p (- j 2:
(2)
r
where is the amplitude reflection coefficient.? m is the maximum number of reflections considered, and we have assumed isotropic array elements. For eight-element linear arrays with >../2 spacing oriented perpendicular to the street, Fig. 2 shows how the capacity increases as more images are added (and the angular spread of rays increases) and approaches Glin with seven images (Iml :S 3) plus T. Furthermore, the received signal envelope looks increasingly Rayleigh-like as more images are added.
B. Ricean Channels Next we consider the capacity for Ricean channels, where the deterministic component HLo S is fixed as either HI or H n . We follow the simulation methods of [1] using the normalized Ricean channel matrix H == (aHLos + bHRayleigh) with a 2 + b2 == 1 and Ricean K-factor K == a 2 / b2 . The results for n == 3 with p == 100 (20 dB) (Fig. 3) quantify how for closely spaced (~>..) array elements at both T and R, and no reflectors or scatterers such that H LOS ~ HI, the capacity decreases with increasing K toward C == Clog. However, for array geometries such as the above examples, where HLo S ~ H n , the capacity increases with increasing K toward C == Glin . For K == O. C corresponds to that obtained in [1].
cellular system can be combined to form one "edge-excited" (inward-facing) cell (Fig. 4) to enhance capacity for R not close to a base station. This is reminiscent of soft handoff in CDMA systems where multiple base stations serve one mobile, except that in this case, each base station carries a different substream of the transmitted data. The results of the third example suggest that in the absence of reflectors, we may use n antennas at each of n sites, thus replicating the effect of images of the n-element T array. The ray-tracing channel model for H == H LOS + H Rayleigh described here may be useful for the performance evaluation of MIMO wireless systems which use spatial diversity through space-time coding to exploit the available capacity with no bandwidth penalty (e.g. [2], [5]-[7]). REFERENCES [ 11 G. J. Foschini and M. J. Garis, "On limits of wireless comrnunicanon in a fading environment when using multiple antennas," Wireless Personal
Commun., vol. 6. no. 3. pp. 311-335. Mar. 1998. [21 G. J. Foschini, "Layered space-time architecture for wireless commu-
[3] [41
III.
DISCUSSION
Capacities approaching Glin == ti 10g2 ( 1+ p) can be achieved for MIMO channels in an LOS (non-Rayleigh) environment by spreading out the elements of T either explicitly (by placing one element of T at each of ti sites), or implicitly (by adding reflectors which create images of T). The results of the second example suggest that three sectors in a conventional general, the r are different for each image. since they depend on the angles of incidence and reflection, and the surface characteristics. Here we assume has the same constant value 0.6 for all reflections. except the ground reflection [0 which is set to -1. This approximation is sufficient to tllustrate the capacity gain. 2 In
r
748
[5] [6]
[7]
[8]
nication in a fading environment when using multi-element antennas." Bell Labs Tech. J .• vol. 2. no. 2. pp. 41-59. Autumn 1996.V. Weerackody, "Diversity for the direct-sequence spread spectrum system using multiple transmit antennas." in IEEE Int. Conf, Communications. Geneva. Switzerland. May 1993. pp. 1775-1779. N. Seshadri and J. H. Winters, "Two signalling schemes for improving the error performance of frequency-division-duplex (FDD) transmission systems using transmitter antenna diversity." lnt.T. Wireless lnformatton Networks. vol. 1. no. I. pp. 49-60. 1994. D. Agrawal. V. Tarokh. A. Naguib. and N. Seshadri, "Space-time coded OFDM for high rate wireless communication over wideband channels:' in Proc. IEEE Vehicular Technology Conf.. 1999, pp. 2232-2236. D. Agrawal, V. Tarokh. A. Naguib. N. Seshadri. and A. R. ':alderbank. "Space-time codes for high data rate wireless communication: Performance criteria and code construction:' IEEE Trans. Inform. Theory. vol. 44, pp. 744-765, Mar. 1998. D. Agrawal. V. Tarokh. A. Naguib. N. Seshadri. and A. R. Calderbank, "Space-time codes for high data rate wireless communication: Practical considerations," IEEE Trans. Commun .• to be published. J. Salz and J. H. Winters. "Effect of fading correlation on adaptive arrays in digital mobile radio." IEEE Trans. Veh. Technol.. vol. 43, pp. 1049-1057. Nov. 1994.
Optimum Space-Time Processors with Dispersive Interference: Unified Analysis and Required Filter Span Sirikiat Lek Ariyavisitakul, Senior Member, IEEE, Jack H. Winters, Fellow, IEEE, and Inkyu Lee, Member, IEEE Abstract- In this paper, we consider optimum space-time equalizers with unknown dispersive interference, consisting of a linear equalizer that both spatially and temporally whitens the interference and noise, followed by a decision-feedback equalizer or maximum-likelihood sequence estimator. We first present a unified analysis of the optimum space-time equalizer, and then show that, for typical fading channels with a given signal-to-noise ratio (SNR), near-optimum performance can be achieved with a finite-length equalizer. Expressions are given for the required filter span as a function of the dispersion length, number of cochannel interferers, number of antennas, and SNR, which are useful in the design of practical near-optimum space-time equalizers. Index Terms- Equalization, interference suppression, multipath channels, space-time processing.
I
I. INTRODUCTION
N WIRELESS communication systems, cochannel interference (CCI) and intersymbol interference (lSI) are major impairments that limit the capacity and data rate. These problems can be mitigated by spatial-temporal (S- T) processing, i.e., temporal equalization with multiple antennas [1]-[11]. In typical wireless systems where the cochannel interferers are unknown at the receiver, optimum S-T equalizers, either in a minimum mean square error (MMSE) or maximum signal-to-interference-plus-noise ratio (SINR) sense, consist of a whitening filter, i.e., an equalizer that whitens the CCl both spatially and temporally, followed by a decision-feedback equalizer (DFE) or maximum-likelihood sequence estimator (MLSE) [12]. However, under some channel conditions with dispersive CeI, the whitening filter requires an infinite span to achieve near-optimum performance, even with reasonable signal-tonoise ratios (SNR's). For typical fading channels, though, such channel conditions occur only occasionally, and the required filter span for near-optimum performance is finite in most cases. Since the filter span, specifically both the causal and anticausal portions, determines the required memory of the Paper approved by K. B. Letaief, the Editor for Wireless Systems of the IEEE Communications Society. Manuscript received May 20, 1998; revised October 20, 1998. This paper was presented in part at the IEEE Intemaional Conference on Communications, Vancouver, BC, Canada, June 1999. S. L. Ariyavisitakul is with Home Wireless Networks, Norcross, GA 30071 USA (e-mail: [email protected]). J. H. Winters is with AT&T Labs-Research, Red Bank, NJ 07701 USA. I. Lee is with Bell Labs, Lucent Technologies, Murray Hill, NJ 07974 USA. Publisher Item Identifier S 0090-6778(99)05236-8.
DFE and MLSE, these filter spans determine the required complexity of near-optimum S-T processors. In this paper, we first present a unified analysis of the optimum infinite-length S-T processor, considering three receiver types: 1) MMSE linear equalizer (LE); 2) MMSE-DFE; and 3) MLSE. The unified analysis includes both previously published results [12], [17]-[ 19] and additional new material. The objective here is to provide a consistent and comprehensive framework for expressing all these results in a form that is descriptive of the functions and properties of individual filter elements. We then present filter length analyzes for all three receivers by analyzing the z-transform expressions. We show that, with fading channels, the filter spans of these receivers can be truncated such that the average effect of the truncation is small compared to the effect of thermal noise. We then determine the required filter span to achieve near-optimum receiver performance. These expressions for the required filter span as a function of the dispersion length, number of cochannel interferers, and SNR are useful in the design of practical near-optimum space-time equalizers. Using computer simulation, we study the effect of thermal noise on the required filter span for specific fading channels. In Section II, the system model and notation is defined. The unified analysis is presented in Section III, and in Section IV the finite filter span analysis is presented. Section V shows numerical results. A summary and conclusions are given in Section VI. II. SYSTEM MODEL We consider a system where L + 1 cochannel signals are transmitted over independently fading multipath channels to an M -branch diversity receiver. The time-domain complex baseband expression of the received signal on the jth antenna is
Tj(t) ==
L
L L 00
xnthij(t - nT) + nj(t)
(1)
i=O n=-oo
where {Xni} is the transmitted data sequence from the ith source, with the desired source being indexed by i == 0; h ij (t) is the overall impulse response of the transmission link between the ith source and the jth antenna; T is the symbol period; and nj (t) is the additive white Gaussian noise at the jth antenna. The data {Xni} are independently identically,
Reprinted from IEEE Transactions on Communications, Vol. 47, No.7, pp. 1073-1083, July 1999.
749
Fig. 1. A space-time DFE receiver.
Fig. 2.
distributed complex variables with zero mean and unit symbol energy and are uncorrelated between sources. The frequency-domain expression of the above received signal is
Our analysis also includes the use of z-transfonns. The ztransform of a sampled sequence of a continuous-time function g(t) is G(z) ~ ~kgkz-k, where gk == T· g(kT) [we multiply g(kT) by T so that {gk} and g(t) have the same average energy per symbol interval]. The relation between z-transform and Fourier transform is given by the following using the Poisson sum formula [13]:
RJ(f) ==
L
L
X·i(f)Hij(f)
i=O
+ Nj(f)
(2)
where RJ(f), ..: ri(f) , Hil(f)~ and Nj(f) are the Fourier transforms of rj(t)~ {Xnl}~ h'ij(t), and nl(t)~ respectively. Since the data have unit symbol energy E[I~Y'i(f)lr~- == 1 for If I < 1/(2T) where E[·] denotes expectation. The noise at each antenna has two-sided power spectrum density No. The general space-time receiver using a DFE is shown in Fig. 1 (an LE receiver model can be obtained by setting the feedback filter response to zero). It consists of a linear feedforward filter, W J (f), j == O. 1 ~ ... , M -1, on each branch, a combiner, symbol-rate sampler, slicer, and synchronous linear feedback filter B(f). The feedforward filters {W) (f)} are shown as continuous-time filters, but they can be implemented in practice using fractionally-spaced tapped delay lines. The input to the feedback filter is the decided data {x nO} for the desired source. We assume correct decisions (x na == xnO) throughout this study. The input to the slicer (i.e., the space-time processor output) is denoted by sequence {Yn}, with its Fourier transform Y(f) given by Y(J)
=
1\1-1
L L 00
1=0 rn=-oo
G(ei 2rrf T ) =
2
== E[lYn - x no/ ] == T
j
1/ 2T
-(1/2T)
9ke-j2trkfT
=
f
G(f - ; )
'rn=-cx:,)
(5)
where G(f) is the Fourier transform of g(t). and j == It is easy to show that
Tj·l/2T
G'(f)df
-(1/2T)
= ~ 1 G(z) 21rJ
.r
R.
dz z
(6)
where G' (f) ~ 2::=-00 G(f - (miT)) is the Fourier transform of the sequence {9k}. The contour of the integration on the right side of (6) is the unit circle. For convenience, we omit the tilde sign from our z-transform notation throughout the rest of this paper, e.g., G(z) will be written simply as G (z). Furthermore, if G(f) is the Fourier transform of a symbolspaced sequence (instead of a continuous-time function), then
T
j
l / 2T
-(lj2T)
G(f)df=~ 21rJ
f G(z)~. d Z
(7)
III. UNIFIED INFINITE-LENGTH THEORY
(3)
E
f
k=-oc
w, (J - '; )RJ (f - ;) - B(J)Xo(J).
The summation with respect to m in the above equation is a result of spectrum folding due to symbol-rate sampling. Based on the MMSE criterion, the filters are optimized by minimizing the mean square error (MSE)
A space-time MLSE receiver.
A. Optimum Filter Expressions for DFE and LE Receivers The MMSE solution for the feedforward filters {W) (f) } with unconstrained length can be derived by using (2)-(4) and setting the derivatives {aE I aWj (f - (miT))} to zero. This yields
E[lY(f) - X O(f )/2]df·
(4)
where
Fig. 2 shows a space-time receiver using an MLSE. Here, the goal of optimization is to maximize the signal power (without suppressing lSI) to CCI plus noise power ratio, while whitening the CCI and noise components of the .input {Yn} to the MLSE. 750
w~
[Wo(f
-~)
WM-1(f
-~) ...
WO(f
+~)
WM-l (f
+~)]T
(9)
H,
~
[fI (f - ~)
Hi, M -1
iO
HiO(.f+~)
(f - ~) ... ~)r
H"M-1(f+
Rs ~ H~H6
~
L
(10)
H:H,;
+ iVoI.
R s is the correlation matrix of the desired signal, R 1 + 1V
is the correlation matrix of the interference plus noise, I is the identity matrix, and the superscripts * and T denote complex conjugate and transpose, respectively. We assume that the desired and CCI sources are strictly band limited to f == ±.J' / (2T ) (JI is a positive integer), and therefore J == (.I' -1) /2 when J' is odd, and .J == JI/2 when .I' is even (e.g., JI == 1 and .J == 0 when there is no excess bandwidth). Note in (9) and (10) that excess bandwidth provides additional diversity which can be exploited when there is sufficient transmit power outside the Nyquist band. e.g .. in a spread spectrum system [14] (or see also [171). Using the matrix inversion lemma [12. Appendix OJ, it can be shown that
+ R1+,v ] - IH*0
== 1
+
J+N
0
HTR-1 H*' 0
/+.V
()
(1 1 ) -I
Therefore, (8) becomes
W ==
R- 1 H* ~+ lV_ 1 0 * (1 + B (f) ). 1 + H o R 1 + 1yH o
['(1) =
WtR s W WtRI+~VW
(15)
superscript t denotes conjugate transpose W is the output signal power density, and WtR] +lV W is the output interference-plus-noise density at frequency f). Substituting (14) into (15) yields where
.wtn,
( 17)
Equation (17) gives the form of the MMSE solution, well known in array processing [12] (except for the consideration of spectrum folding and feedback filtering). This equation indicates that the optimum feedforward filter consists of a space-time filter R~NHo~ which performs spatial prewhitening (R ~: {; is the whitening filter of CCI and noise) and matching to the desired channel, followed by a temporal filter (1 + B(f)) / (1 + f(f)), which can be regarded as a post-whitening filter under some zero-forcing condition, as described below,
( 18)
+ r(f) jO (1 + B(z))(l + B*(z-l)) 1
.r
dz z
1 + r (z )
1 + I'(e) == SoG(z)G*(z-l)
( 19)
(20)
where the constant So is given by
So ==
e(ln(l+r(f)))
and
(-) ~ T
j
.l / 2T
. -(1/2T)
(21 )
[.J df
(22)
and G(,~) is canonical, meaning that it is causal (Yk == () for I: < 0). monic (gO == 1). and minimum phase (all of its poles are inside the unit circle, and all of its zeros are on or inside the unit circle). Using the Schwarz inequality, it can be shown that the MSE in (18) is minimized when
1 + 8(f) == G(f):
1 + B(z) == G(z).
(23)
Substituting (20)-(23) into (17) and (18), we obtain
*
-1
1
W OF E =RI+NHO SoG*(f)
(24)
and 1 _ e-(ln(l+r(!))) EDFE - - -
(25)
So
Using (16), (20), and (24), the CCI plus noise power density at frequency f is
- 1 H*0 H 0T R I+iV W RI+NW SZIG(f)1 2 t
( 16)
-1 * 1 + B(f) W = RI+NHO 1 + f(f) .
. df
1 + r(f) == SoIG(f)/2;
the
Thus, we can rewrite the optimum feedforward filter solution as
/1 + B(f)/2
where B(z) and I'(z ) are the z-transfonn equivalents of B(f) and r (f). Using spectral factorization theory [19J, 1 + r (f) and 1 + r (z) can be written as
( 14)
Furthermore, we can define the signal-to-interference-plusnoise power density ratio T'( f) at frequency f as
l/ 2T
. -(1/2T)
21rj
i=l
[R s
== T
== _1_
( 12)
R- 1 H*
j O
f
( 11)
L
R]+lY
The optimum feedback filter B ( f) can be determined through spectrum factorization. Substituting (17) into (3), and using (4) and (7), we obtain
r(f) 5 0 (1 + r(f)) .
_
(26)
Note that as f(f) --t DC for If I :S 1/(2T)~ the Cel plus noise power density becomes a constant 1/50 over f. Under this 'condition, we can regard 1/ SoG*(f) in (24) [or 1 + B(f)/l + f(f) in (17)] as a post-whitening filter. As will be relevant later, we can write 1 + B (z) also as
1 + B(z) ==
e[1 + f(z)]
(27)
where C[·] denotes canonical factor. The corresponding Fourier transform is given as
1 + B(f) == C[l + f(f)]·
(28)
Using the above expression, we can write (17) as
751
W
OF E
-1
* C[l
= RI+NHO
+ r(f)]
1 + f(f)
.
(29)
Fig. 3.
An equivalent model of the space-time DFE receiver in Fig. 1.
The optimum LE is obtained by setting B(f) to zero in (17) and (18)
where P is an (L + 1) x (L + 1) correlation matrix whose (a, b)th element Pub is given by
(30)
(34)
and
Furthermore, we can write
Do(f) == VTpTU
(31)
where U rows, and
B. An Alternative Solution for DFE and LE Receivers The optimum space-time filter solution in (17) is based on a general model which does not make any prior assumptions regarding the filter structure. Without loss of optimality, an analytical receiver model suggested by many in the literature (e.g., [15]-[18]) assumes the use of a bank of matched filters {Hij (f)}, each corresponding to the signal source i on diversity branch i, which, after diversity combining, is followed by a bank of T -spaced transversal filters {Vi (f) }, each corresponding to the signal source i (see Fig. 3). This analytical model leads to a different form of solutions which are important to our filter length analysis. The following derivation is similar to the LE receiver derivation in [18], but here we also provide the solution for the DFE. In Fig. 3, the Fourier transform of the input to the slicer can be written as
L Di(f)Xi(f) + N(f) - B(f)Xo(f)
L
vtr-tr-v.
(36)
The MSE for this receiver is given by E
==
EISI
+ ECCI + Enoise
(37)
where EISI
fCCI
== (IDo(f) - (1 + B(f))2)
(38)
=
(39)
(f; IDi(fW)
and cnoise
== (Novtr-v,
= (NoytptV).
(40)
Using (35) and (36), the MSE becomes
(32)
== (ytptpv
- 2Re[(1
i=O
D==PY
IDi (f )12 ==
+1
i=O
E
where Do(f) and Di(f) are the overall channel and feedforward filter responses for the desired signal and the ith interference, and N(f) is the noise at the combined output of the feedforward filters. Let D == [Do(f) ... DL(f)]T and Y == [Vo(f) VL(f)]T. We then have the following relationship:
... , O]T is a column vector with L
O~
L
L
Y(f) ==
== [I,
(35)
+ Nov'r-tv + 11 + B(f)1 2
+ B*(f))yTpTU]).
The MMSE solution for Y
(aE/a~(f)) = 0 for i = 0 to
V = (P
(41)
is obtained by solving
L. We then obtain
+ NoI)-lU(l + B(f)).
(42)
It can be shown that this receiver achieves the same MMSE as (25) (or (31) in the case of an LE receiver) and that
(33)
752
1
1
l+r(f) = No UT(P+NoI)-lU'
(43)
which satisfies
wtR1+NW == 1{3(f)121\lf(f)12H6RI~NH~
IC[f(f)]1 2 f (f )
{I\ }
{ynJ
+t=n~ ~~
-
If(f) /2
1
31
= constant where the overall filter response W is given by
Fig. 4. An equivalent model of the space-time MLSE receiver in Fig. 2.
Again, 1 + B(z) is the canonical factor of 1 Accordingly, we can rewrite (42) as
+
W == W'\lf(f) == R -1 H* C[f(f))
f(z).
I+IV
For an LE receiver
WMLS E
(45)
Fig. 4 shows an equivalent model of the MLSE receiver in Fig. 2. The front-end filters are now represented by spatial filters {W;(f)}~ which maximize SINR of their combined output, followed by a post-whitening filter 'I!(f). Let WI denote the vector of {W; (1)} similar to (9). The signal-tointerference-plus-noise power density ratio f(f) is then [ef. ( 15)]
w'tn-w
= W/tR I+iV. WI·
VVLSE=
(46)
*
opt.
WoptH O
(49)
(3(f)f(f)
(54)
.
1 + f(f) C[f(f)] C[l + f(f)]' f(f)·
(55)
(P+NoI)
-1
.
U(l+f(j))
C[f(f)]
f(j)·
(56)
(57)
(50)
Our filter length analysis is based on counting the number of zeros and poles in the z-transform expression of the optimum space-time filter. For all three receivers (LE, DFE, and MLSE), there are two forms of optimum filter solutions:
we can find a post-whitening filter
W(f) = C[f(f)]
f(f)·
IV. FILTER LENGTH ANALYSIS
Equation (48) has the same form as (17). As a result, we obtain the same expression for r(f) as (16). By factoring f(f) as
r(f) == SlIC[f(f)] 12
C[f(f)]
Thus, the desired signal at the output of the front-end filter has a canonical impulse response; this is the known desired property of the input to an MLSE [21] (in addition to the noise being white).
where
W'
+ f(f)]
H6 W == Do(f) == C[f(f)]·
(48)
,I+N
1 + f(f)
C[l
W DFE .
In (54), when f(f) ~ 1 such that 1 + f(f) ~ f(f), then WrvILSE ~ W DF E , i.e., the optimum front-end filter for an MLSE receiver is equivalent to the optimum feedforward filter of a DFE receiver. This is usually the case when there is no eel and the input SNR is sufficiently high [20]. However, it is generally not true in the presence of strong eel's. Also, using (53) and (56), it can be shown that
The maximum f(f) is, therefore, given by the maximum eigenvalue of RI~NRs. Let W~Pt be the eigenvector corresponding to this maximum eigenvalue, and substitute WI == W~pt into (46). We obtain
opt
(53)
Accordingly
(47)
W'R
=
V l\ILSE = V DFE
The optimum W' is obtained by solving (ar(f)j8Wj(f)) == 0, for j == 0 ~ 1, ... , M - 1; this gives the relationship
(3(f) ==
.
This relationship is extendable to the case where we use the analytical feedforward filter model in Fig. 3 to represent the front-end filter of the MLSE receiver. Thus, we can also write
C. Optimum Linear Filtering for MLSE
.
f(f)
0
Comparing (53) to (29), we find that
(44)
f(j)
(52)
(51)
753
1) one based on the general model (with a linear filter on each branch); 2) one based on the analytical model (with a bank of matched filters on each branch, followed by common filters).
The filter length determined by each solution is valid under different assumptions. The filter length based on the general model is valid when M < L + 1, i.e., when the number of interferers is equal to or greater than the order of diversity due to multiple antennas and excess bandwidth for convenience, we write the overall order of diversity as M, instead of M~. J The filter length based on the analytical model is valid when M ~ L + 1. The reason for these different conditions will become apparent later. Since the general analytical approach for determining the filter length is the same for both solution forms, we only provide details for one of them below. We choose to work on the analytical model case because it is slightly more complicated than the other case, and because the condition under which it is valid (the order of diversity exceeding the number of dominant interferers) is where the most interference suppression is achieved, i.e., an array with M antennas can null up to M - 1 interferers [23]. We begin by working on the MMSE solution for the DFE receiver. The z-transfonn equivalent of (42) is
+ N oI )- l U (l + B(z)) ~ Q- 1 U (1 + B(z))
V == (P
(58)
where V ~ [Vo(z) ... VL(z)]T and Q ~ P + Nol. The element Pab of matrix P is the z-transfonn of the sampled sequence of 1\1-1
~
Loo oc
haj(T)hbAr - t) dr.
(59)
Thus, each element qab of matrix Q has a two-sided response such that it includes both a causal factor and an anticausal factor of equal length. If we assume that all channels {h i j (t) } have a finite memory of K symbol periods (h i j ( t) == 0 for t < 0 and t > KT), then the causal and anticausal factors of qab will be polynomials of order K. Using the matrix identity [22]
Q-l
= Qadj IQI
(60)
where Qadj is called the adjugate matrix of matrix Q, and the (a, b)th element Qab of the transpose matrix of Qadi is called the cofactor corresponding to the (a, b)th element qab of matrix Q [Qab = (-l)a+bIQabl, where Q ab is obtained by deleting the ath row and bth column of Q], we can rewrite (58) as
v = ~ljU(l + B(z)) [Qoo where 1
+ B(z)
QOl
IQI
QOL]T (1 + B(z))
Q01
...
QOL]T
(IQI/O[lQID. O[Qoo]
Vo(z
(63)
)
Qoo/C[Qoo]
= IQI/O[lQIJ .
(64)
Thus, this filter is anticausal (cf. [17]). We now focus on each term in (63). Defining a permutation a as a one-to-one mapping a: (0, 1, ... , L) ---+ (aQ, 0"1, ... , a L), the determinant of Q is [22] (65)
where sgnl c ) == +1 or -1 depending on whether the number of exchanges in permutation (1 is even or odd, and the summation is taken over all (L + I)! permutations (1. Note that the product of two polynomials of order (J, and b results in a polynomial of order a + b, while the sum of them gives a polynomial of order max] (J" b]. Since each element £jab of matrix Q includes a causal factor and an anticausaI factor, each of order K, IQI will, in general, have a causal factor and anticausal factor, each of order K(L + 1). Similarly, Q ab will, in general, have a causal factor and anticausal factor, each of order K L. Accordingly, IQI/C[lQIJ will be anticausal (and maximum phase) with order K(L + 1)~ and C[Qoo] will be causal (and minimum phase) with order K L. Combining these results together, the causal part of each filter ~ (z) (for 'i > 0) will have K L zeros and K L poles, and its anticausal part will have K L zeros and K(L + 1) poles. Since each front-end matched filter hij ( - t) is anticausal with length K (we can always set the synchronization timing such that h i j (t) is a causal function), the overall feedforward filter on each branch will have a causal part with K L zeros and K L poles and an anticausal part with K (L + 1) zero and K(L + 1) poles. In general, a pole filter has an infinite impulse response. Nevertheless, we can always truncate a pole filter which is causal and minimum phase, or anticausal and maximum phase, such that the effect of truncation is small compared to the background noise. Thus, the lengths in units of T of the causal and anticausal parts (denoted as C and A, respectively) of the optimum feedforward filter can be given as
C == KL(l + a) A = K(L + 1)(1 + a).
(62)
(66)
Here, a determines the truncated length of a pole filter 1 (1 - €z)-l or (1 )-1. Note that C = 0 when L == 0; this agrees with the known result that, in the absence of CCI, the optimum feedforward filter of a DPE is anticausal. Note that the required length of the feedback filter is K + C, since the optimum feedback filter completely cancels the postcursors of the desired signal.
ez-
= 0 [NoUT1Q-lU]
C[\Qt]
[Qoo
Note that the common filter for the desired signal is
(61)
== e[l + f(z)]
= C[Qoo]·
V=
DFE (M ~ L + 1):
is given by [see (43)]
1 + B(z)
Thus, we obtain
754
Similarly, we can estimate the filter length for an LE receiver using (6]), with B (z) set to zero; this gives LE (Ai 2: L
C == K(L A == K(L
+ 1): + 1)(1 + (~) + 1)(1 + o ).
K
(67)
Note that the causal length of the LE receiver is greater (by Kc'i) than that of the feedforward filter of the DFE receiver. The above results are valid under the condition that the MMSE sol ution in (61) is compact, meaning that there is no cancellation of highest order terms in the summation in (65) for all determinants. By working on specific examples, we found (6 I) to be compact when 11/[ ~ L + 1. Otherwise, the MMSE solution based on the general filter model [given in ( 17)] is compact. Using the same analytical approach as above, we estimate the filter length for the case M < L + 1 as LE and DFE (J\;1
<
L
+ 1):
+ n) - K ~4 == K M (1 + n).
C == Kl'vl(l
(68)
The filter length results are the same for both LE and DFE receivers in this case. We now focus on the MLSE receiver. Using (63) and the relationship in (55), we obtain
[(200
(201
(20L)T
V ~ILSE == - - - - - - - - - - - - - - - ((IQI - lVo()oo)/C[IQI- lVo(2oo)) . C[(200]"
(69)
Comparing (69) with (63), we see that individual terms in the two equations have the same highest order and, thus. the two filters have the same length. This is also true when we compare the filter lengths using the general filter model. We, therefore, conclude that the optimum front-end filter for the MLSE receiver has the same length as the optimum feedforward filter of the DFE receiver; namely MLSE (!vI ~ L
+ 1):
C==KL(l+o:) ~4 ==
K(L
+ 1)(1 + o )
section. Nevertheless, the general relationship regarding how the required length of the whitening filter varies with the dispersion length, number of interferers, and order of diversity, should remain unchanged for any given SNR. V. NUMERICAL RESULTS
We now study the effect of thermal noise on the required filter span of DFE and LE receivers. As discussed earlier, we expect the required filter span of an MLSE receiver to be the same as that of a DFE receiver (although this needs to be proven for all given SNR's). For the purpose of illustration, we assume a single-carrier system using quaternary phaseshift keying with Nyquist filtering. We also assume a multiray delay profile for all the channels, where all the rays are of equal power, independently Rayleigh faded, and uniformly spaced by an interval T (the symbol period). The fading is assumed to be independent for different signal sources and diversity branches. We choose this channel model for several reasons: first, we define the rays to be T -spaced so that, with Nyquist filtering, the channel memory is strictly restricted to K. for a given K + I-ray delay profile (see the discussion at the end of this section regarding cases where the channel memory is not strictly restricted to K). Furthermore, since the frequency response of aT-spaced multiray channel is periodic, with period 1IT. there is no need to consider spectrum folding in our numerical computations. Accordingly, we assume the use of T -spaced equalizers in all the finite-length performance results. Finally, we assume the rays to be equal power so that the results obtained give a worst case (i.e., more conservative) estimate of the required filter length for a given channel memory compared to channel models with unequal delayed paths. Fig. 5(a)-(c) show the infinite-length performance of MMSE-DFE and MMSE-LE receivers compared to the matched filter (MF) bound. The performance is given in terms of the average bit error rate (BER) as a function of the average input SNR, where the average is over Rayleigh fading. The fading of individual channels was generated by Monte Carlo simulation. For a given set of channel realizations, the BER was computed as
rY:)
r; == ~2 erfc ( V2
(70)
and
MLSE (!VI
< L + 1):
C == KJvl(l + a) - K ~4 == K I'll (1 + 0:).
(72)
where erfc(-) is the complementary error function [13], and the output SNR "Yo is given by [19] 1 !O ==-- -l~
(71 )
Note that the above analysis does not take into account the effect of thermal noise, i.e., strictly speaking, the filter length expressions given above are valid only when the input SNR approaches infinity. As the SNR decreases, we expect the required length of the whitening filter V or RI~l\TS(z) (5 (z) denotes the temporal filter, e.g., S (z) == 1/( 1 + I'( z)) for an LE receiver) to decrease and eventually approach zero when thermal noise dominates both Cel and lSI. Although analytical results are not available, it is possible to study this effect through numerical examples, as shown in the next
755
for DFE
EDFE
10
== -
1
ELE
-
l~
for LE
(73) (74)
and
'10 == (f(f))
(75)
and EDFE, ELE, and f(1) are given by (25), (31), and (16), respectively. Note that we only consider the case of M ~ L + 1, i.e., the receiver has a sufficient number of antennas to suppress all
M
= 2, L = 1, SIR = 0 dB
MMSE-DFE . - - . MMSE-LE MFBound
-...
10-1
W
10-3
Q)
ttl 2 a: 10-
... 0
-
CD
10.5
10-4
L......~...J...._-....L~~........ ~-L.....:.......>...'----"-..J
o
5
10
20
15
25
5
30
(a)
=
...
...e
4·Ray(K=3) 6-Ray (K= 5)
w 10-4
'--_-'-_.J.LI..:u..:-.........L.-~-.L~~"'___...J
o
5
10
15
20
25
30
SNR (dB)
5
(b)
__-=,.- 2-Ray (K= 1)
...e W
-
CD
10-2
4-Ray(K=3) 6-Ray (K= 5)
10.3 10-4 10.5
0
5
10
15
20
25
--
30
SNR (dB) (c)
Fig. 5. Infinite-length performance of space-time MMSE-LE, compared to the MF bound. (a) L 1, M = 4. (c) L 3, AJ = 4 , L
=
=
15
20
25
Fig. 7. Effect of channel dispersion length I\ on the required filter length of a space-time MMSE-DFE. L = 1. M = 2. and "7 = 18 dB.
MMSE-DFE . - - - MMSE-LE MFBound
Q)
10
Filter Length A and C
M = 4, L = 3, SIR = - 4.8 dB -
ttl
=
MMSE-DFE .-- . MMSE·LE MFBound
2·Ray (K= 1)
-
25
Fig. 6. Effect of SNR on the required filter length of a space-time O. Four-ray channel (l{ 3) . MMSE-DFE. L
M= 4, L = 1, SIR = 0 dB
a: ...
20
15
Filter Length A
SNR (dB)
10.5
10
=
MMSE-DFE and I , AI 2. (b)
=
dominant interferers (the remaining interference can be treated as Gaussian noise if its total energy is sufficiently low). We
assume that each of the L interferers has the same power as the desired signal. Thus, the average signal-to-interference ratio is o dB for L = 1 and -4.8 dB for L = 3, Fig. 5(a)-(c) show that the infinite-length DFE performs to within 1-2 dB of the MF bound in all cases, while for M = L + 1 [Fig. 5(a) and (c)], the performance of the infinite-length LE is worse than that the MF bound by up to 6 dB, at BER around 10- 4 . Fig. 6-9 provide results for finite-length MMSE-DFE and MMSE-LE receivers. In all of these figures, we plot the average BER as a function of the length (the number of symbol-spaced taps) of the causal and/or anticausal portion of the filter on each diversity branch. The total filter length is C + A + 1, where C and A are the length of the causal and anticausal portions, respectively, as defined earlier. In all of the figures, the BER is shown to decrease with the filter length until it reaches an asymptotic value (a "floor"). The arrows on the right side of each curve show the corresponding infinite-length BER (due to an artifact of the computation, the infinite-length BER's do not necessarily match the BER floors exactly) . The triangle symbol on each curve indicates the required filter
756
I
TA BLE
RE QUtRED FILTER L ENGTH RESULTS IN FIG . 6, C OMPARED WITH PREDICTED LE NGTHS B ASED ON (77) (SHOWN
10,1
a:
10. 2
= O. .\1 =
L
IN P ARENTHESES).
1.
1{
AND
=3
y in dB
3
6
9
12
IS
18
A
4 (3.9 )
5 (4.8)
6 (5 .7)
7 (6.6)
8 (7.5)
8 (8.4)
ttl
~
0
-
10.3
~ ~
ill
iii
10.4
10.6
7,
C OMPARED
WITil P REDICTED L ENGTHS B ASED ON (77) (S HOWN tN
10. 5
L = 1. .\ 1 = 2 .
P.~ RENTHES ES ).
0
20
15
10
5
Fig. 8. Effect of the number of interfere rs L on the required filter length of a space- time MMSE·DFE. .\f -I. J,' :3. and -x9 dB .
=
=
=
length to achieve "near-optimum" performance . where "nearoptim um" is defined here as being within 5% of the BER floor. We assume in all the DFE results (Figs. 6-8) that the feedback filter is sufficiently long such that it completely cancels the postcursor lSI of the desired signal. Fig. 6 shows the effect of the average input SNR "1 on the required filter length of the DFE receiver, assumin g a fourray channel model (K = J) with no CCI (L = 0). In this case, the optim um feedforward filter is aruicaus al [see (66)], so the performance is given only as a function of A . The results for IvI = 1 show that the required filter length to achieve near-optimum performance increases by one for every 3-dB increase in SNR in most cases (the required length stays unchanged when increases from 15 to 18 dB). We will discuss this relationshi p in more detail later. The example results for M = 2 and M = 4 show that the required filter length does not change with the number of diver sity antennas . Fig. 7 shows results with different dispersion lengths. assuming L = 1, M = 2, and "1 = 18 dB. In this case, the performance is given as a function of the lengths of both the causa l and anticausal portion s of the filter; the results for each portion are obtained by assuming a sufficient length for the other portion . These results show an approximately proportional relationship between the required filter length and the channel dispersion length :
c ~ 1.8K ~4 . 6 K ,
for L = 1, At! = 2 and "1 = 18 dB . (76)
Fig. 8 shows results for different values of L. assuming a four-ray channe l model (K = 3) with M = 4 and "1 = 9 dB. The average SIR is 0, -3, and -4.8 dB for L = 1. L = 2, and L = 3, respectively. The results show that the required filter length grows linearl y with the number of interferers. So far, all the simulation results generally agree with the analytical results in (66), i.e., the required filter length to achieve near-o ptimum performance grows linearly with K and L , but it does not depe nd on M . In additio n, we found
757
= 18
AND"
dB
K= I
K=3
K=5
c
2 (1.8 )
5 (SA )
9 (9)
A
5 (4.6)
14 (i3.8)
23 (23)
25
Filter Length A and C
A
II
TABLE
RE l.IUIRED F ILTER L ENGTH R ESULTS IN F IG.
TA B LE
III
R EQUIRED FI LTER L ENGTH R ESULTS tN FI G.
8,
CO MPARED
WITH P REDICTED L ENGTHS B ASED ON (77 ) ( S HOWN IN P ARENTHESES ).
JJ =
4.
h'
= 3.
WD ~
= 9
dB
L=O
L=I
[= 2
[= 3
c
0 (0)
3 (2.7)
5 (5.4)
8 (8.1)
A
6 (5.7)
8 (8A )
10 (11.1 )
12 (13 .8)
in this section that the required filter length grows almost linearl y with the average input SNR in decibels . Combi ning these results together. and taking into account the fact that the antica usal part of the filter always includes a matched filter of length K . we obtain the followi ng empirical formulae for the required filter length: DFE:
C
~
K £ 1>("1 )
A
~
K
+ K (L + 1)1>("1)
(77)
where (78)
and "1 is in decibels. The good agreement between the filter length s predicted by (77) and the simulatio n results are shown in Tables I-III. Although not shown here, we also found good agreement when testing the empirical formula e against simulation results with other sets of parameter values . Despite its empirical nature, (77) has meaningful analytical justifications. First, it gives the same form of expression for C as the analy tical result in (66), except for the dependence on the SNR [which is also expected of 0: in (66)]. Second, when "1 -; cc such that 1>("1) » 1, the expression for A in (77) becomes A -; K (L + 1)4>("1); thus, we also obtain the same form of expression for A as (66) . As discussed earlie r, (66) is also valid when "1 -; co . Finally, (77) gives the length A as
of a linear filter on each antenna branch, followed by a DFE or MLSE . In this analysis, we derived explicit expressions for the linear filter [e.g., (29), (44), and (54)], which are novel to the best of our knowledge. Using z-transform analysis, we also derived expressions for the linear filter length showing that the required span is proportional to the channel dispersion length and the number of interferers. We then used computer simulation to derive empirical expressions for the required filter span which show that the span is also proportional to the input SNR in decibels. The derived empirical expressions for the required span are in good agreement with simulation results with Rayleigh fading and a uniform-delay spread profile. These expressions are useful in the design of practical near-optimum space-time equalizers.
10-1
-
10-2
Q)
c:
10. 3
~
0
~ ~
W
iIi
10-4 10.5 10.6
o
5
10
15
20
25
30
35
Filter Length A and C Fig. 9. Required filter length for space-time MMSE-LE. L and "'i = 18 dB.
= 1. JI = 2.
the sum of the matched filter length K and the length of the anticausal portion of the whitening filter K (L + 1)¢>(;:y) which decreases with decreasing input SNR; this agrees with the intuition that the length of the whitening filter should approach zero when thermal noise dominates both CCI and lSI. As for the LE receiver , the analytical results in (67) show that it has the same anticausal length A as that of the DFE receiver, and its causal length is given by C = A - K . Thus. we simply modify (77) as LE:
C = K(L + l) ¢>(;:Y) A = K + K(L + l)(p(;:y) .
ACKNOWLEDGMENT
The authors would like to thank M. V. Clark. D. D. Falconer. Y. Li, and L. J. Greenstein for useful discussion s and suggestions.
(79)
The required filter length results in Fig. 9 agree well with the lengths predicted by the above empirical expressions. Equations (77) and (79) give useful empirical expressions for predicting the required filter span of space-time DFE and LE receivers for a given SNR. The empirical function ¢>(;:y) is given in (78) only for a specific channel model (Rayleigh fading and a uniform delay spread profile). This function can be easily determined for other channel environments , by studying only the single -antenna, no CCI performance, similar to the way we determined ¢(;:y) from the results in Fig. 6 and Table 1. Note that even if the channel memory is not strictly limited to K , in practice we can truncate the memory such that the energy outside the truncation length is below some given value. The above empirical approach should still be valid in that case as long as K is defined consistently throughout, because K ¢(;:y) should be independent of the definition of K. Finally, note that K is typically defined by the system specifications . Our uniform delay profile results give the required filter length for a given K that will meet the system requirements for any delay profile. VI. CONCLUSION In this paper, we studied optimum space-time equalization of dispersive fading channels with CCI. We first presented a unified analysis of optimum space-time equalizers, consisting
758
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759
Author Index
A Abend, K., 596 Acar, L., 359 Agee, B. G., 236 Amin, M. C., 62 Anderson, S., 685 Ariyavisitakul, S. L., 749
B Beach, M. A., 387, 466, 704, 710 Berkowitz, R. S., 596 Brookner, E., 221
c Caffery, J. C. Jr, 735 Carloni, M. J., 719 Cantoni, A., 607 Capon, J., 146 Cardieri, P., 20 Cardoso, J.-F., 78 Chen, T., 51 Compton, R. T. Jr, 314, 324, 359, 418 Cupo, R. L., 680
o Djuric, P. M., 56 Driessen, P. F., 745 Dudgeon, D. E., 553 DuFort, E. C., 623
E Edwards, D. J., 3887 Eggers, P. C. F., 33 Er, M.-H., 607 Ertel, R. B., 20
F Feuerstein, M. J., 695 Fessler, J. A., 272 Fischer, P., 685 Fleury, B., 690 Forssen, D., 685 Foschini, G. J., 745 Franks, R. E., 663 Frederiksen, F., 690 Friendlander, B., 406 Frost, O. L., 157 Frullone, M., 731 Fuhl, J., 481
G Gabriel, W. F., 195 Gans, M. J., 525, 719 Gardner, W. A., 236 Gerlach, D., 463 Giannakis, G., 67 Godara, L. C., 95 Goldberg, J., 58 Golden, G. D., 680 Goode, B. B., 3 Grant, P. M., 500 Grazioso, P., 731 Griffiths, L. J., 3 Gupta, I. J., 332
H Haardt, M., 286 Hansen, R. C., 614 Hargrave, P. J., 560 Hatori, M., 399 Herd, J. S., 603 Hero, A., 54, 272 Howell, J. M., 221
761
Imai, H., 399 Itoh, K., 371
J Jablon, N. K., 213, 579 Jeffs, B., 44 Jensen, M., 44 Johnson, D., 167
K Kailath, T., 178, 184, 224, 261, 444 Karlsson, J., 685 Kohno, R., 399 Krim, H., 64 Krolik, J., 81 Krug, A., 685 Ksienski, A. A., 332
L Larsen, S. L., 690 Lee, I., 749 Leth-Espensen, P., 690 Li, J., 510 Li, Y., 695 Liang, J.-W., 726 Liberti, J. C. r-., 452
M Mailloux, R. J., 649 Mantey, P. E., 3 Marshall, D., 655 Martin, C. C., 680 Mathews, B. D., 569 Matsuzawa, N., 673 McGeehan, J. P., 387, 704, 710
Author Index
McWhirter, J. G., 560 Messer, H., 58 Mogensen, P. E., 690 Molisch, A. F., 481 Mulgrew, B., 500
N Naguib, A. F., 444, 486 Nehorai, A., 73 Nossek, J. A., 286
o O'Donnel, T. H., 630 Ogawa, Y., 371 Ohgane, T., 673 Ohmiya, M., 371 Olesen, K., 690 Oprea, A. M., 33 Ottersten, B., 250, 471
p Papadias, C. B., 297 Passerini, C., 731 Pasupathy, S., 399 Paultraj, A., 297, 444, 463,486, 726 Pedersen, K. J., 690 Pesquet, J.-C., 64
R Rappaport, T. S., 20, 452 Reed, J. H., 20
Reudink, D.O., 695 Rice, M., 44 Riva, G., 731 Roy, R., 224
s Salz, J., 435, 519 Sasaoka, H., 673 Schell, S. V., 236 Schmidt, R. 0., 190,663 Schreiber, R., 205 Shan, T.-J., 178 Sherman, K. L., 680 Shimura, T., 673 Silverstein, S. D., 636 Simmers, J. A., 630 Sollenberger, N. R., 680 Southall, H. L., 630, 649 Sowerby, K. W., 20 Spencer, Q., 44 Steyskal, H., 603 Stoica, P., 510 Stuber, G., 735 Swales, S. C., 387, 466 Swami, A., 70 Swindlehurst, A. L., 75
T Thompson, D. J., 60 Thompson, J. S., 500 Toftgard, J., 33 Tong, L., 80
762
Tsoulos, G. V., 1, 49, 301, 303, 466, 551,661,704,710,717 Tugnait, J. K., 72
v Valenzuela, R. A., 719 Vanderveen, M. C., 297 Vaughan, R. G., 351 Viberg, M., 250
w Ward, C. R., 560 Ward, J., 418 Wax, M., 178, 184 Weiss, A. J., 406 Widrow, B., 3 Winters, J. H., 339, 378, 427, 435, 519, 525,680,719,749 Witzschel, T., 685 Wolniansky, P. W., 680
x Xu, G., 261
y Yuen, S. M., 596
z Zatman, M., 655 Zetterberg, P., 471, 535 Zheng, D., 510
Subject Index
A Adaptive-adaptive processing, 51, 221 Adaptive processing, 553 Adaptive algorithms CMA, 119, 673 constraint LMS, 115 constraint MUSIC, 123 COLD, 510 ESPRIT, 124, 224 JADE, 297 LMS, 8, 113, 347 MEM, 121 MUSIC, 123, 190, 663 MLM, 122 MVDR,121 Normalized LMS, 115 RLS, 118 SAGE, 272 SCORE. 236 SMI, 112 unitary ESPRIT, 286 WSF, 125 performance, 127 implementation, 205 Adaptive transmission with feedback, 463 Akaike information criteria (AIC), 127 Angle of arrival, 95 Angular spread, 691 Applebaum arrays, 337 Array imperfections, 579 Array manifold, 226, 569 Array processors, 607 Array topology, 726
B Beamforming, 95 beam-space, 105
broadband, 106 conventional, 100 delay-sum, 110 downlink, 306, 535 digital, 110,555,560 frequency domain, 109 networks. 623 Benefits of adaptive antennas, 306 capacity, 311, 339, 387, 444, 452, 466, 481 coverage, 309, 525 fading reduction, 371 handover, 307 power control, 307 signal quality, 307 transmit power, 310 BER, 519 Broadband signal processing, 14 Buttler matrix, 649
c Calibration, 232, 636, 711 CDMA, 444, 452, 466, 486, 500 Channel impulse response, 21 Channel models, 20 angle of arrival indoor, 44 Bad Urban, 27 Extended Saleh-Valenzuella, 28 Extended Tap-Delay-Line, 28 Gaussian Wide Sense Stationary, 26 Geometrically Based Single-Bounce, 24 Lee's propagation model, 23 Typical Urban, 27 Cochannel interference, 341 CLOSEST method, 124 Cluster phenomenon, 28 Coherence bandwidth, 37
763
Coherent signals, 178 Constraint optimization, 157 Convergence rate, 113 Correlation matrix, 99 Cross-polarization, 38 Cyclostationarity, 261
o Degrees of freedom, 103, 355 Delay lines, 107 Delay spread, 21, 308 Detection probability, 307 Digital beamforming, 560 Diagonal loading, 681 Diversity, 33
E Eigenstructure method, 112, 122 Eigenvalues, 100 Eigenvectors, 100 Emergency calls, 301 Equalization, 67 Errors amplitude, phase, DC, IQ, 710 correlated arrivals, 129 element failure and element position, 133 pointing, 132 weight, 133 Expectation, 99 Experiments CMA, 673 DMI,680 MRC, IRC, 686 multibeam, 695 SDMA, 704,710
Subject Index
F Fading correlation, 435 Forward-backward averaging, 655 Frequency correlation function, 34 Frequency hopping, 359
G
Generalized sidelobe canceller, 213 Gram-Schmidt orthogonalization, 598
H Hermitian matrix, 99
Indoor, 378, 719 Information theoretic criteria, 124, 184 Interference canceller, 399
K Kalman filter, 705
L
Least square minimization, 562 Linear prediction method, 121, 170 Linearly constraint algorithm, 157 Linearity issues, 714
M Macrocell, 21 Matrix Inversion Lemma, 102 Measurements spatial signal measurements, 29 diversity, 34 polarization, 38
Microcell, 21 Min-norm method, 123 Misadjustment, 11 Multibeam systems, 387 Multipath fading reduction, 371 Multiple input-multiple output systems, 311, 745 Multiresolution analysis, 64 Mutual coupling, 332, 406, 603
N Network planning, 731 Neural beamforming, 630 Neutral network approach, 120 Non-linearities, 569
o Optimum combining, 304, 339, 351, 519,680
p Phased array, 525 Polarization effects, 325 Polarization sensitive adaptive antenna, 324 Power delay profile, 34 Pre-processing techniques, 126 Prewhitening, 354
s Scatterers, 23 SDMA, 304, 481, 704, 710 SFIR, 304 Signal acquisition, 427 Signal subspace, 226 Signal tracking, 427 Slotted ALOHA,418 Smart antennas, 303 Space-time processing, 749 Space diversity, 80 Spatial complex correlation coefficient, 24 Spatial smoothing, 178 Spectral diversity, 80 Spectral estimation methods, 120 Spectrum analysis, 146, 167, 195 Spectrum efficiency, 394, 471 Statistical signal processing, 51 Steady state response, 217 Steering vector, 99 Subspace fitting, 250 Switched beamforming, 304, 695 Systolic array, 566, 596
T TDMA/SDMA, 481
Transpose matrix, 99 TSUNAMI, 690, 704, 710
a Quantization error, 133 Quiescent radiation pattern, 105
R Ray tracing, 29 Reference signal, 316, 348, 374 Robust beamforming, 134 Rotman lenses, 614
764
u User location, 735
v Van Atta, 3
w Wiener filtering, 104, 214
