RADAR PRECISION AND RESOLUTION
G. J. A. Bird, C.Eng .. M.I.E.R.E. EMI Electronics Limited, Wells. Somerset
PENTECH PRESS
LONDON
© G. 1. A. Rird. 1974
First Published 1974 Pentech Press Limited: London
8 John Street, wei N 2HY
ISBN 07273 1801 2
Printed in England by The Whitefriars Press Ltd., London and Tonbridge
PREFACE
This book was w r i t t e n t o h e l p practISIng enginee r s, and s t u d e n t s o f r a d a r t h e o ry, t o u n d e r s t a n d h ow unce r t a i n t y fun c t i o n t ec hn i q u e s c an be used t o analy se the p e r fo rmanc e o f rada r system s . It is comm o n fo r mode r n systems t o employ sophist ic a t ed m o dul a t i o n, a n d t h e i r chara c t e ristics are n o t always apparen t from a n i n t u i t i ve t r e a t me n t . Engi ne e r s must h ave c o n fidence i n m a t h em a t ic al t e c h n iques (and kn owledge o f any lim i t a t ions) if they are to apply t h em t o new p rob l e m s . It is the a u t ho r ' s opinion that t h i s c o n fidence is best i n s p i re d b y s t ressing b a s i c principles and by giving t h e proofs o f all mathem a t i c al resu l t s, the reby establ ishing t h e c o n d i t i o n s nece ssa ry for t h e i r vali d i t y . This philosophy has b e e n adopt e d i n w r i t i ng t h e book. To facilit a t e a fi rst reading, t h e c o n c l u sions a n d discussion usually precede the de t ailed m a t h emat i c s; t h i s a l so m akes it easy to u se the book for reference purpose s . The book h as b e e n d i v i d e d i n t o t wo part s . Chapt e rs 1 t o 5 cover t h e r a d a r t h e o r y and i nclude nume rous r e fe r e nces t o Ch aptErs 6 and 7 which p rovide a firm fo u n d a t i o n of t h e unde rlying t ra nsfo r m theory . It is sugge s t e d t hat the reade r may l i k e to c o n c e n t r a t e i n i t i ally on Cha p t e r 1, fol lowe d by S e c t ions 4.1, 4.2, 4.6 and Chap t e r 5. This -procedure will acq u a in t h i m w i t h t he significance of the u n c e rt a i n t y fun c t io n a n d h o w i t c an be u s e d t o divi d e r a d a r receivers i n t o t w o cl asses-m a t c h ed fil t e r receive rs and Four i e r t r a nsform receivers . The th e ory is c onsolidat e d in Chapt e r 5 b y ap ply ing the re sult s t o two practical m o d u l ati o n wave forms; the reader should then be in a po s i tion to understand the more advanced resu l t s given i n the established
literature . The second part of the b oo k provides the m athem a ti cal b a c k gro u n d to Chapt ers 1 to 5 . Chapter 6 de als with t h e rel a t i o n s hip be tween Lapl ace and Fourier transforms . The discrete Fourier transform (OFT) is also discussed and it is shown how two OFT theorems lead to the fast Fourier transform ( F FT) process. Chapter 7 covers the Hilbert transform and complex analytic signals. The author has always found 'engineering' treatments of this lat ter sub ject t o be particularly unconvincing-it is hoped that the reader will find the present treatment t o b e much more satisfactory. The author has fou nd Part 2 to be a useful reference source when working in other fields of signal processing. The summary of the main notation in Appendix 5 is also useful in this respect.
Preface The fi rst d raft of this book was written as an inte rnal report for the Wells laboratories of EMI Elect ronics L t d . The mate rial has also been used as the b asi s of a series of aft e r-hours lectures t o the au thor's coll eagues. The au thor would like to th ank EMI Electronics L t d. for permission t o publ ish and a l so his coll eagues for m a ny helpful di scussion s. G. J. A. Bi rd
CONTENTS
Preface
PART l. RA D A R THEORY
I.
THE RADAR UNCERTAINTY FUNCTION
1.1 1.2 1.3
1.4
1.5
2.
T HE
Limitations
and
characteristics
of
the
uncertainty
function The Lm.s. error criterion
3
5
The area of uncertainty of a received signal Resolution and precision The relationship between the
MATHEMATICAL
T,
wand
x,
TREATMENT
6
y planes
OF
7
THE
UNCERTAINTY FUNCTION
2.1 2.2
The mathematical properties of
2.3
The effect of repetition upon the uncertainty function
13
2.4
The effect of linear FM upon the uncertainty function
19
Proofs of the properties of
20
2.5 3.
4.
Derivation of the uncertainty function
X(T,
w)
X(T, w)
9 II
WORKED UNCERTAINTY FUNCTION EXAMPLES 3.1
The rectangular pulse with constant carrier frequency
30
3.2
The rectangular pulse with linear FM
34
3.3
The rectangular pulse with triangular FM
37
SIGNAL PROCESSING METHODS 4.1
The matched filter receiver
41
4.2
The Fourier transform receiver
46
4.3
The effect of finite processing time
51
4.4
The matched filter concept
59
The
62
4.5
mathematical treatment of the matched filter
receiver 4.6
Practical Fourier transform calculators
64
4.7
The mathematical treatment of the Fourier transform
65
receiver
5.
SOME EXAMPLES OF SIGNAL PROCESSING METHODS 5 .1
The rectangular pulse with constant carrier frequency
68
5 .2
The rectangular pulse with sawtooth FM
76
Contents PA R T 2. M ATH E M A TIC A L BA CKGR O U N D
6.
LAPLACE AND FOURIER TRANSFORMS 6.1
The 2-sided Laplace transform
6.2
The significance of the strip of convergence
6.3
The I-sided Laplace transform
6.4
The Fourier transform
6.5
The physical interpretation of Laplace and Fourier transforms
6.6
Fourier transform symmetries
6.7
The Fourier transform of a conjugate function
90
6.8
Limiting cases of the Fourier transform
6.9
The delta function
92 92
94
6.10 Fourier transform notation 6.II Using the p-multiplied Laplace transform notation 6.12 The discrete Fourier transform (OFT)
6.13 Evaluation of the Fourier transform by means of the DFT 6.14 T he fast Fourier transform (FFT) process 6.15 Proofs of the properties of the OFT and FFT
7.
85 86 87 88 89
99 99
100 103 106 108
HILBERT TRANSFORMS AND COMPLEX ANALYTIC
SIGNALS 7.1
Summary of the main results of Chapter 7
114
7.2
The Hilbert transform
116
7.3
Envelope and phase functions
7.4
The
complex
122
7.5
The results of multiplying real or complex analytic
125
7.6
Filtering either real or complex analytic signals
129
7.7
The com pression of a Doppler shifted pulse
131
exponential
approximation
121 to
the
analytic signal signals
Appendix I Complex conjugate terminology
137
Appendix:2
138
Parseval's theorem
Appendix 3
Fresnel integrals and their relationship to linear FM
Appendix 4 A short table of Hilbert transforms
141
Appendix 5
A summary of the main notation
141
138
References
145
Index
146
Chapter 1.
THE RADAR UNCERTAINTY FUNCTION
The unce rtainty functio n , int ro duced by Woodwa rd [ I ] , is a function of two variables represen t ing signal de l ay and Doppler shift. I t may be regarded as a 'figu re of m e rit' by which various waveforms and processing methods can be compare d . This chapter gives a qualitative treatment of the uncertainty func tion and its propertie s .
1.1 LIM ITATIONS A N D CHARA CTERISTICS OF THE U N C ERTAINTY FU N CTION
The return sign al from a radar target is a mo dified version of the t ransmitted sign al . The modifications are due to the paramet e rs of the target which can , in p ri nciple , b e deduced by comparing the re turned signal with the transmit ted signa l . The unc e rtainty function resu l t s from a co nside rat ion of t h e signal retu rned from a single point targe t . An example of such a t a rget would be a small insect flying at constant veloc ity t oward an 'upward looking' radar. The radar retu rn is assumed to differ from the transmi t ted sign al in only two way s , namely: (I) A time del ay (x), p roportional to the radial range of the targe t . ( 2) A constant f reque ncy shift (y) of the whole signal spectru m , p roportional to t h e t arget radial velocity . T h e variable y is c alled the Doppler offset; it is posit ive for targe t s travelling towards the radar. Strictly speaki ng , the assumption of a constant Doppler offse t is an app roximation which is only valid when the target velocity is small com p ared with the veloCity of propagation of the transmit ted signal . The true effect is a distortion of the signal spect ru m , due to an offset which varies with frequency. I t is normally realistic t o assume a constant Doppler offset in the case of elect ro-m agnetic propagat ion in air , but caution must b e exe rcised i n such applications a s sonar [ 2 ] . With the ab ove assumpt ions, the target m ay b e regar de d , at any one time, as a point in the x, y p l ane . Use of the uncertainty fu nction allows one t o define an a rea of unce rtainty in the x , y p l ane , inside
The radar uncertainty function
4
convenient mathematical p rope rt ies. F o r the present d i scussion, 8, will be d e fined th rough the followi ng relationship:
J U{t-x, wo+y)-f{t-x+r, wo +y +w)}2 d t
00
8,2 =
(1.1)
I t is shown in S e c t i o n 2.1 t h a t th e above d e finition leads t o
00
8,2 = 2 wh e r e
r L/(I)]2 df-Ix(r, w)1 cos [ (wo+y+ w)r- Arg{x(r, w)}] x(r,w)= (a(t)a*(r+T)e - i w tdt _'00
(1.2)
( 1.3)
and
(1) a(t) is
a comple x baseband funct ion which is
transmitter modulation, as discussedin
(2) r, ware di
target co-ordinates (see Fig.
(:i)
,1IJ(r)12 df
=
derived
from the
Section 2.1.
1.2).
F i( .e.
In
to as 'the ambiguity function'. Also, x(r, w) and Ix(r, wW c an be found called by either n a me . A full mathematical treatme nt o f x(r, w) i s given i n Chapter 2. Only two the established literature it is s o me time s referred
0), its value at t h e origin : at wh ich po int it is equal to 21-.'. independent of signale ner gy, F. (2) x(r, w) normally changeswith r muc h more slowly than does cosl(wo+.\' +w)r I h ecause Wo is a hi ghcarrier ( I ) x(r, w)is never greater than X(O.
Equation
�
21:'
1.2 =
I
can be re-written in the more cllnvenient form:
-
I 21:'
1).:(1, w)1 cosl(wu +\' +w)r
.
Arg{x(r, w )}]
( 1 .4 )
A graphical interpretation of Equa tion show s a typ ical variation of g 2/21:' along a line in the r, w plane pa rall el to the r a x i s . 1-.'trepresents before the s i gnal can b e said o t different . As Wo is a high carrier frequency, 8,2/21:' is a rapidly fl uc t ua ti n g function of r, a nd always lies between the solid lines in Fig. 1.1. Si n c e pr ec i sion with ambiguities s e p arat e d by distances of th6 01 d e r of the tr ansmi t ted Signal wavelength
5
The radar uncertainty function
2
1
1
2E l x (1,W)1
-- - ------
1
�
--- - ------- -
' 2E I x (1,W)1
--- -
-----
./ Normalised �
threshold
- - -- - ---------
I n s u fficie n t diffe rence
o
1, 1----.-
Fig. 1. 1. A graphical interpretation of Equation 1.4
(m i n i mum value) cu rve i s chose n as a test o f whe t h e r t h e t h re sh o l d h a s been exceeded. T h u s , t h e t a rget range co-o r d i n a t e c a n b e said t o l i e i n t h e range s 0 t o 7), o r 72 to 73. T h e e xam p l e u s e d h a s b o th range-u nc e r t a i n t y and am b i gu i t y . Figu re 1.1 sh ows t h a t ach i evable prec ision c a n be inc reased b y red uc ing th e ra t i o c�/E . S i n c e E t i s u l t im a t e l y set by system n oise , t h i s i m p l ie s a n increase i n t h e t ransm i t t e d e n e rgy , E. 1.3
THE A R E A OF UNCE R T A INTY OF A R ECEIV ED S I G NA L
Figure 1.1 shows that Equ at ion 1.4 can be re-w r i t t e n to read: A rec eiv ed si gnal m ay be l oc a t e d wh ere
IX(7, w)1 � U'
I -E(/2E �
(1.5)
E( in Equat ion 1.5 represe n t s the (arb i tra ry) t h re sh o l d l eve l which t h e mean sq u a re e rror must exc eed b e fore i t i s p o s s i b l e t o m easu re t h e paramet e rs o f t h e ret urned signal. Si nce system n oise has not been considered , t h e i mpl i cat ion is that Et excee ds the syst e m noise level by a s ign ificant amount. Table
1:."/1:.'(
Contour of
define� the
(ratio)
I\(T, wll/21:.· has log,o"
which
of uncertainty
2.5 dB
0.4 dB
10
20
IX(T. w)I/2E
area
6 dB
2
100
1.1
0.04 dB
been
converted to dB
by
the formula:
6
The radar uncertainty function
Equa t i o n 1.5 c a n b e u se d t o c a l cu l a t e c o n t o u r l e v e l s o f I X(T , w) I /2£ i n side wh ich t h e t a rget para m e t e rs of a signal o f a given e n e rgy l i e . T h e a re a e nc l o se d b y such a co n t ou r i s t e rm e d t h e 'a r e a o f u n c e r t a i n t y ' . Exa m p l e s for various signal-t o-t h resho l d e n e rgy ratios a re given i n Tab l e 1.1.
1.4
R ES O L UTION AND P R ECIS ION
Re solut i o n i s use d h e re t o i n d icat e th e abi l i t y t o separa t e t h e re t u rn s from sev e r a l target s (as o p p osed t o 'p rec i si o n' w h i c h w a s used above t o i n d i c a t e t h e p e rfo r m a nce w i t h respect t o a s i ngl e t a rge t). I f the requi re d p re c i s i o n has bee n d e fi n e d a s a n area i n t h e T , w pl ane, and t h a t area i s e ncl ose d wi t h i n a m uch l a rge r a rea o f u n c e r t a i n t y (fo r a given sign al-t o-th reshol d rat io) i t c a n b e s t a t e d t h a t t h e giv e n signal d oe s no t mee t t h e spec ificat i o n. N o amo u n t of c leve r sign a l processing w i l l al t e r t h i s fact (assu m i ng, o f cour se, t h a t t h e signa l-e n e rgy t o t h resho l d-level ca n n o t b e i nc rease d ) . Th e unc e rt a i n t y fu n c t ion c a n n o t b e used d i r ec t l y t o m ake a sim i l a r stat e m e n t rega r d i ng reso lut i o n . If t h e un ce rt a i n t y areas co r respon d i ng t o t wo t a rge t s are p l o t t e d i n t h e x, v pl ane a n d t he y ove rl ap, reso lut i o n wi l l b e di ffic u l t o r impo ssib l e. If t h e t wo signal s were muc h h i gh e r t h an t h e rec e ive r t h resh o l d level t h e co r respo n d ing unce rt a i n t y a r e as wou l d b e s m a l l a n d m igh t no t overl ap. However, o n e sign al co u l d s t i l l b e m u c h st r o nge r t h an t h e o t h e r a n d i t s response s i d e l o b e s coul d compl e t e l y swa m p t h e weake r sign a l . One coul d o n l y b e su re t hat t h i s woul d n o t h ap p e n i f t h e Ixl = 0 con t ours sur ro u n d i ng each target co-o r d i nat e di d n o t i n t e rsec t. Th i s l a t ter c r i t e r i o n is not very use ful fo r the t y p e s of signal wh ich are usual l y s t ud i ed by means of the unce r t a i n t y fu nct i o n , and does n o t h e l p i n t h e case o f a sign al i n cl u t t e r. I n tui t ively, o n e m ight t h i n k t h a t if t h e a r e a o f u nc e r t a i n t y o f t he weakest signal al one was m a rk e d o ff i n t h e x, y plane t h e t est fo r resolut i o n i n t h e p r esence of a sign al 1/ dB s t ronge r w o u l d b e wh e t h e r t he -1/ dB c o n t our o f Ixl/2c·, su rrou n di n g t h e st ro nge r s ig n a l co-o r d i nates, e nc l o sed t h e a rea o f un c e rt a i n t y o f t h e w e ake r sign al . Al t h o ugh t h is argu m e n t has val i d i t y i n t h e c ase o f rece ivers w h i c h gi ve an out put wave fo r m in t h e shape of a cut t h rough X(T, w), it d o e s n o t a p p l y t o cases w h e re th is i s not so. I n pa r t i c u l a r , s u c h a l t e rn a t ive cases may we l l l e a d t o b e t t e r reso lut i o n. A t rue assessm e n t o f t h e p reci sion and reso l u t io n p r op e r t i e s o f a give n rad a r sy s t e m can o n l y be m a d e b y st u dy i ng t h e shape o f t h e p rocessed o u t p u t p u l se w h i c h i s u se d t o d e t e r m i n e t h e val u e o f t h e delay o r D o p pl e r p a ra m e t e r b e i ng me asu re d . I f a single figure i s
7
The radar uncertainty function
required as a me asu re o f p rec ision o r reso l u t io n , it can be ob tained b y measu ring t h e width of t h i s output pulse a t som e arb i t r a ry l evel. In this b ook t he 'precisio n ' o f a system will be defi n e d as the width of its outp u t pulse measu re d be tween its half-am pl i tu de (i.e . -6 dB) point s . Table 1 . 1 shows t h at this implies a sign al-t o-th reshold energy ratio of unity . The 'n dB reso lut ion' of a system will be d e fined as the width of its ou t p u t p ul se me asured between its widest -n dB p o in t s .
The ab ove definitions apply to an actual syst e m , the inherent p recision o f a transmitted signal will be defined as the -6dB width o f t h e a p p ropriate c u t t h rough IX(7, w)l. 7 and ware rel at ed t o range and veloc i t y by the formulae
2r 7= -
( 1 .6 )
c
2wov w= --
( 1 .7)
c
where r and v are range and ve locity (wi th respect t o the ta rget ) , c is the velocity of p rop agation and Wo is t he carrie r frequency . Thus i f, fo r a given system, the d e l ay p recision is give n by 71, the corre spo nding range precision w i ll b e (C71 /2). The velocity precision, correspo n d i ng t o a Dop p l e r p recision of Wi> will be (cwI/2wo) a n d it shou l d be no t ed t hat this c an be improved by inc reasi ng Wo.
1.5 THE REL A TIONSHIP BETWEEN THE
7, w
AND x, y P L ANES
The co-ordi nat es x, y define a point with respect to the receive r , while the co-ordinates 7, W define the same point with respect t o th e t arget , Fig . 1.2. Y
Y,
/compari50n point
r�---:L _____
, I I I I I I I I I-----I....-.I ____
__
Target
Fig. 1.2. Relationship between x, y and
i,
W
8
The radar uncertainty function Y
W
y
,
--D \...W
-------+--�---.x x,
Fig. 1 . 3. T ran slation of X(T, w) to x, y plane
Since i t fol l o w s t h at
x = (Xl -7), X(7, w) = X [-(x-xd, (y- ydl
The above relat i o n ship is illust r a t e d by Fig. 1.3.
Chapter 2 THE MATHEMATICAL TREATMENT OF THE UNCERTAINTY FUNCTION
This chapter is used to show how the r.m.s. error criterion leads to the function
x(r,w)= f a(t)a*(t+r)e-jwtdt. 00
_ 00
Also, some mathematical properties of x(r,
2.1
w)
are studied.
DERIVATION OF THE UNCERTAINTY FUNCTION
It is assumed that a real signal, f(t), is transmitted such that
faCt) = aCt)ejwot is the corresp onding complex analytic signal (Chapter
(2. 1 ) 7).
Equation
implies that the real transmitted signal is given by
j(t) = Re{fa(t)}
2. 1
(2.2)
Defining
aCt) = la(t)1
ejp(t)
(2.3)
leads to
f(t) = la(t)1 cos[wot+
(t)]
(2.4)
A subtle point worth n oting is that with the above definition of f(t),
la(t)1
and
(t)
are not arbitrary functions. Let f(t) be written:
f(t) = Re { b(t) e jwo t }
where
bet) = Ib(t)1
that is
where
Ib(t)l, OCt)
ej(J(t)
f(t) = Ib(t)l cos [wot + O (t)]
(2.5)
are defined as the real, arbitrary, functions applied t o
the amplitude and phase modulation terminals, respectively, o f the transmitter.
9
10
The mathematical treatment of the uncertainty function Unless the carrier frequ e ncy W(j i s sufficiently high, it does n ot
follow that
Equ ation
aCt) is equ al to bet).
2.5
Rather,
u si n g the re lationship :
aCt)
has to be calculated from
aCt) e j w o t = fa Ct) = f(t)+ittt)
(2.6)
A ful l discussion o f this point is given in Section 7 .4 . T� e dist in ction
b e twe e n the complex an aly t i c signal an d the exp on e n t ial approxima tion h a s t o be m ade in t he case o f son a r. Some impo rtant d i ffe rences b e tw e en radar an d sonar are de t a ile d b y Krame r [2] . Fo r t u n a t ely , for p ractical rad a r applicat i on s, i t is n o t ne cessary t o use Equ a t i on 2.6 as aCt) c a n u su ally b e conside red equal to b(t ). The r.m .s. e r ro r c rite rion has b e e n discuss e d in S ect i o n 1 .2. The r.m. s . diffe re n ce , &, between tw o re al sign a l s f(t) and get), is defin e d t h rough t h e rela t ionship
J (f(t)-g(t)J2 dt
00
&2 =
(2.7)
An i m p o r tan t p rope r ty o f complex a nalyt i c signals (p rove d i n Se c t i on 7.2.5) i s that Eq u at i o n 2.7 can b e repl ace d by
&2
J lf�(t) - ga(tW dt 00
=
�
(2.8)
Using the r ela t ion sh i p s
f�(t) = f(t)+ jf(t) &(t) = g(t)+ i(t)
Equat i o n 2.8 c a n b e expanded t o give 28,2 =
J lt�(t)12 dt+ J l!;ra(tW dt - 2 J Re{J�(t)ga*(t)} dt 00
00
00
Eq uat i on 2.9 foll ows s ince
Re{J�(t )ga*(t)} = f(t)g(t)+Rt )i(t)
To p r ove Equat ion 1.2 it i s necessary t o p u t
j{t)=f[t-x, wo+y]
g(t) = f[t-x+ T, Wo+l' + w] Hence
j�(t) = aCt-x)
ej(wo+y)( t -x)
gael) = aU -x + T)
ej(wo + y+w)(t - X+T)
(2.9)
The mathematical treatment of the uncertain ty function Substitution of the above in Equ ation
give s
2.9
J la(t-xW dt + J la(t-x + TW dt -2 J Re{a(t-x)a*(t-x + T) e - jwt e jw x e - j(wo+y+w)T} d t 00
11
00
2&2 =
00
(2. 1 0)
B y u sing the relationship between the spec tra of real and comple x analytic signals (Section 7.1), an d b y the app l ication of Parseval's the orem (Appendix 2), it ca n be shown that
J laCt-x W dt = J la(t 00
00
_00
_00
J [fa (t ) 12 dt = 2 J Lf(t)]2 dt 00
-
x
+ T ) 12 dt =
_00
Al s o , by changing the dum m y var ia b l e Equa tion 2. 1 0 b e come s
{
2 Re e j WX e - j(w o +Y + W)T
Thus, w ith the definition
00
_I a
t to (t + x), the last integral i n
(t) a * ( t + T) e -j(t+x)w dt
f a( t) a *(t + T ) e
}
00
X(T, w ) =
-
(2. 1 1 )
j w t dt
Equ a tion 2 1 0 m ay b e w ri tte n in the form .
J [f(t)]2 dt - Re { e j(w o 00
&2 = 2
-
+ Y + W)T
X( T ,
w)}
(2 . 1 2 )
Equ ation 2. 1 2 i s also equ ivalen t to
J Lf(t)]2 dt-lx(T, w)1 cos[( wo + y + w ) T-Arg X( T , w)] 00
&2 = 2
(2. 1 3)
Th e di s c u ssi on in Se ct ion 1.2 uses Equat i on 2. 1 3 t o estab l ish th e physi cal s i gn ifican ce of X(T, w). 2.2 THE M A THEM A TI C A L PR OPE R TI ES OF
X(T,
w) i s defined by
X(T , w )
J a( t)a * ( l + T ) e -j w' dr 00
X( T , W)=
(2.14 )
The most important properties of X ( T , w) are stated below; proofs a re
gie v n St utt
[3, 4] and Siebert [5].
X( T . w ) = ,�{a(t) a*( r + T )}
(:2.15)
12
The mathematical treatment of the uncertainty [unction
�
XCi, w) = 2
_
r 00
A *Ux)A [j(w + x)] e-jxT dx
(r.16)
X*(i, w) = e-jwT X( -I, - w)
(2. 1 7)
IX(i, w)1 = IX(-i, -w)1
(2. 1 8)
I X(i, w)1 can be regarded a s a 3 -dimensi on a l solid , placed u po n the i , w plane. Tw o-dimensional fun c tions can b e ob t ained by t aking cuts t h rough IX (i, W)I. The symmetry indicate d in Equa tion 2. 1 8 means t hat t he resu l ting fun ctions will be eve n , if the cu t s pass through the origin o f t he i, w plane.
O
f la(t)12 dt � IX(i, w)1 00
X ( , 0) =
(2 . 1 9)
_00
E quation 2 . 1 9 means tha t IX(i, w)1 can never be greater t han i t s value at t he origin . Note t h a t sin ce aCt) w a s de rived from faCt)
f
00
J [f( t)p dt; 00
X(O, 0) is equal to twice t h e ene rgy of f(t). ]2 = 2� fT'X(i, wW di d w [X(O, 0) _00
i.e .
la ( t ) 1 2 dt = 2
( 2. 20)
_00
Equ ation 2 . 2 0 exp resse s the fa ct that the total volu me of t he I X Ci , W)12 sol i d is constan t , rega rdle ss of the for m of aCt). This means that any steps take n to c o n ce n t r a t e W)12 i n a n a rrow spike at t h e origin m u st also resu l t in a l arge spread of low e r leve l s o f t he fun ction. A change i n t h e form of a t) gives the resul t s set o u t in Table 2. 1
IXCi,
C
Table 2.1 a(t)
AUw)
X(T, w)
t
aCt) ejb 2
cJbw2 A Uw)
a(cxt)
e-jbT2 X[T, w + 2bT]
ej bw2 x[T-2bw, w] 1 - X[CH, w/a] lal
I;j 1
A Uaw)
X[T/a, aw]
The significance of Equation 2.21 is discussed in Section 2.4.
Rep e t i t i on o f a b asic w avefo r m , i.e. c hanging aCt) t o n-J i
L
=O
Cia ( t- ik)
(2.21 ) (2.22) (2.23 ) (2.24)
The mathematical treatment of the uncertainty function
13
leads to an uncertainty function given by
n-I n-I-m m C:C,+me-jwlk L e-jw kX(T+mk.w) L m=1 1=0 n-I-m n-I + L X(T- mk. w) L C,C:+m e-jw1k
where C,
m =0
is a real or complex m u ltiplier (commonly unity) a nd 11 the
number of p uls es
S e ct ion 2.3 .
.
The significance of Equation 2.15 is discussed in
The composite function aCt) + bet),
leads to
X(T, w) = Xaa(T, w) + Xbb(T, w) + Xab( T, W)
+
where
XuvCT. w) = or,
by
(:�.25)
1=0
inspec t ion
XLiI.( T.
I
u(t)l'*(t + T)
w) = ;;; f II( t)l' * (1
ejwT Xa*b( - T, -w) e-jwr dt
+
(2.2h) (2.27) (2.2R)
T)} dt
2.3 THE EFFECT OF REPETITION UPON THE UNCERTAINTY FUNCTION
A general by
case o f repetition of a basic w3veform. a(t). 11 times i� given
h(t)
=
1/
L
1
i 0
C/a(t
Ik)
By letting C, take on comple x values. i.c.
C/
= a
/ 1.'1>1
ucceeding pulses can hc givcn differcnt amplitlldcs and initi:t1 ph :l\c<;.
The resultant lIncertainty function is given hy Equation 2.2:'. T h l'
proof of Equation 2.15 is givcn in Sectilln Three
important
casc
2.5.').
�Irisc wh erc thc alllplitudcs ()f
pulses remain Clln t31lt and either
\lICCl'l'll!ll�
(1) The nwdlilatioll is applied b\ 'ptlll{ a C()ntlllll()U�I� ·nll1l1ll1� carrier oscilbtor. i.c.
.lU)=
1/ /
L
I
0
U
IJ..)
C11Iwot+c)(f
dll
The mathematical treatmen t of the uncertain ty function
14
(2) The osc ill at o r is rest arted (or e ach 'pul se' , and succeeding r. f. pulse s a re identical, i.e. n -I
= L
f (t)
i=O
la(t - ik)1 c os [ wo(t - ik)+¢(t - ik)]
(3) The oscillat or i s resta rte d fo r e ach 'pu l se' , but the initial phase is random , i . e. f (t) =
n-I
L
i=O
la(t-ik ) 1 cos [ w ot +¢(t -ik )+¢; ]
I t is show n in Sec t i o n 2.5 .10 that the c o mple te X fu nction for case
(I) b eco m e s
(n ') f
m= - (n -I)
s i n [ ( n- Iml)e]
e- i(n-I-m)8
sin e
w h e re e
x [r- m k, w ]
wk 2
-
=
For case (2), t he X fun ct i on i s given b y a similar expression (n
� L..
si n [ (n- Iml)e] [ eiwokm e-i(n-I-m)8 X r-m k, w ] .
1)
m =- (n -I)
s i ne
For c a se (3) the X fun ct i o n i s give n by n- I
L
m=1 +
e-iwmkx(r+mk, w) n
L
I
m=l
+e
x(r-mk, w)
_i(I1_1)8
{
L
i= 0
n-I-m
si n[n01
. e �In
n-I-m
L
;=0
}
ei(¢i+m-¢i)e-iwik
ei (¢i-¢i+m) e-iwik
X(r, w).
Ass u m i ng , fo r simplicity, that x(r , w) o f a single p ulse i s contained i n a fi n it e a rea o f the r, w p l a n e ( F i g. 2.1) the e ffe c t o f repe tition is t o g i ve a n ew x(r, w) c on sist ing o f ( 2 n - 1) o f t h e origi nal fu nctions spac e d a l on g the r axi s at in t e rva ls of k (th e repe tition period).
The mathematical treatment of the uncertain ty jUnction w
15
----. ---1---+-
Fig. 2.1. T o tal ar ea of unc er tai n ty of a hypot h etical x(r,
For cases (I) and fac tor
w)
(2) all the multiple functions are we ighted by the sin
[¥ ] . [Wk] 2 (n-Iml)
SIn
whe re Iml takes on values b etween 0 and (n - 1 ) , according to which one of the multiple fun ction s is being considered. For case (3) , only t he function correspon ding to m = 0 is a ffected b y t he weighting factor. Plots of t he we ighting fa ctor a re shown in F igs 2.2 and 2.3. The independent variable is take n as m ( e ffec tively T ) and w , respectively. I
I --� --n
/ /
(
/
/
"
/
t""/ --- --
// I
I I
I
-
/, / 1
i
Jr'I
/
/ I' I "
I
I
l
I
I
:
:
I
I
I
I
:
I I I
r I
:
I 1
I
I
t I
I
,
�
I " I "- '
I I
:
t I
I
I
o
-1
I
I
I I
I
:
I !
"
: :
1
: 1:
I I
I
,
I
- "1 --- -t -- -, - --- ... --
-2
- (n-1)
/
/ /
/ / /
-
"T: I I I
"
',
---
'. "
,
"
- ----
�
I',
I
I J
!
2
'-
',
>
(n-1)
m-
Fig. 2.2.
. [wk 1 -"2 I ml)J . [wk] 2
Sin
(n-
�In
for w = 0 and
m
an i nt eger
16
The mathema tical treatment of the uncertainty function _ _
Fig. 2.3.
��-Iml
. [w2"k ] --. 2 (n -Iml)
Sin
for n-Imlhigh
Sin
T h e fu n c t i o n
f(x)
=
sin (nx) . smx
is discusse d fu lly b y Guillem in [7] w h e re it i s show n that , for large h a s the form -Sin (nX) nx
nl
]
n,
it
r e pe a t e d w he n x e quals in t egra l mUltiples of 'IT. A s n ge ts la rger , the main l obes of the ab ove fun ction b e c o me n a r row e r and talle r , but the rati o of main-lobe to side-l o be ampl i t u de rema ins constan t; it is thi s fac t which le ads t o the w e ll kn ow n Gi b b s phenomenon o f spe c trum theory. It can b e see n t h a t fo r sm all va lu es o f T (an d hence m ) and fo r large t he weight ing fa c t o r has t he fo rm o f narrow ridge s (or b a rs) w h i c h run paralle l to t he T a xis . The in teraction o f the we ighting fac t o r w i t h t he in dividual ' X's' w ill be r e fe rre d t o as t he 'D oppl e r b a r' e ffect. The Do pple r b a r e ffe ct fo r cases (1) and (2) is summ a rised b y Fig. 2.4 w h ich show s t h e area o f u n c e r t a i n t y of a t rain o f the w aveforms w hi c h gave F ig. 2.1. F or the n on-coherent case (3), the concent ra t ion bars on ly ap pl y t o t h e X fun c tion a t the o rigi n . Usin g t h e quan titat ive de fin i t i on o f precision given in Se ction 1.4 means t hat t he Doppler p re cision resu l t i n g fro m t he D oppler bar e ffec t i s given b y
n,
G-:r)nk =
1 .2,
n�1
The mathematical treatment of the uncertainty function
17
The function which is of interest , from t he pOin t of vie w of radar theory, is IX(7, w)1 rather t han X(7, w ) . Figure 2.4 illust rates that , in regions of t he 7, w pla ne wh ere no ove rlap occurs, t he modulus of the combined function is e qu al t o the modulus of t he appr opriat e 'X', times the weigh t ing fac t o r . The mult iple a reas o f unce rtain ty produced b y repetition a r e mathe mati cal expressions of the w e l l k now n 'secon d ( o r higher)-tim e round effe cts' encountered in practi cal radar systems . I n practice , the mult iple areas will a lso be we ighte d by the effe cts of range at tenuation. One factor in the choice of pulse repetition frequency (Ilk) is to remove amb iguit ies and a llow the 'X' function at the origin of Fig. 2.4 t o be the dominan t one . If the latter con di tion i s satisfie d , i t can be stated t hat the Doppler bar effect applies t o cases (I), (2) and (3 ) , i .e . gat e d , pulse d a n d n on-coherent sou rces. The fact t hat the ampl itu de of t he total u ncertaint y function increases as n increases means that , for a given receiver threshold , re pe tition increase s the potentia l precision . Physica ll y , this is be cause the ene rgy of a repetitive train is n times the energy of a single pul se . When plotting the unce rtainty func tion , for d ifferent values of n, it is convenient to remove t he ab ove eff ec t by dividing b y n. =
Area in which f u n ctio n
'15
c o ncent rated
X["[-K, ] w
w
2n K
--
-2K
-K
f
--
2n K
K
2K
"[
Fig. 2. 4. Total area of uncertai nty of a train of the p ulses whi ch g ave Fig. 2. J
18
The mathematical treatment of the uncertainty function
2.3.1 Coded waveforms
The full for m o f Equation 2 .25 has to be used when s tudying de libe rately coded wave trains. In this case n o Doppler bar e ffect w ill occur (the e ffect o f coded repe ti ti on is to produce a n ew 'single-w o rd' w aveform which is m a de up of a numbtr of bits). Of course i f a train of identical words is t ransm itte d , the Doppler b a r e ffect will a ct upon the unce rtainty func tion of a single w ord. When considering the uncertain t y fun ction of a w o r d consisting of L contiguous c o de d bits, occu rring at t = 0, t = d, t = 2d, etc. i t is convenient t o w rite Equ at ion 2.25 in the following form:
X(T, w) =
L-l
L
m=O
Xb(T-md, w)d(m, w)
L-l
+
L
m=l
e-jmwdXb(T+md, w)JIi*(m,
-
w)
whe re Xb (T, w) refers to a single-bit pu lse , and
d(m, w) =
L-l-m
L
i=O
Ci c7+ m e-jiwd
If the bit envelopes h ave d u ra tions which are n o grea ter than d, it fo l l ows that Xb(T, w) = 0 fo r IT I> d. T hu s , in a given portion o f the T, w p l ane, only two displa ce d Xb fun ctions w ill be involve d in the calculation o f X(T, w). The a b ove is illustrated by Fig. 2.5 which show s t h e regions of influ ence of t he individual b i t fu nctions on t h e X func t ion of a fou r-b it w o rd. W
t I IT+ 2d)--'I-I'-X IT) .... I.O-- X IT- 2 d)----.., I I I I I--- x 11 +3d) . : . X I'"(+d ) --r--- x IT -d) ----X-I.I-.- IT 3d) ------.1
I
I I--X
:
I
I I
I
II
b
i
I
b
I I I
I
:
b
:
b
--
I
i
I
b
I
I
b
I
I I I
:
b
I I I
I
I
�----+_----�----�----_4------�----�----�----� 0 12d :IL-l)d :Ld
: -4d I I I I I
-3d
1-2d I I
I-d
I I
I I
:
I I
:
Id II I
: I
I I
I
I I
I I
:
:
I
__
'"(
I I
:
I
Fig. 2.5. R egi ons in w h ich the displaced bit x' s ex ist for a 4·bit w ord wit h b it dur at ion �d
It should be parti cu l a r l y noted th at along t he lines parall e l to the w axis where T = ±md{m = 0, 1, . . . (L - 1 )}, Xb(T - md, w) i s the only non-zero Xb fu n ction.
The mathematical treatment of the uncertainty function
19
The function represented b y .s# (m, w ) is determined by the code and is indepen dent of the bit shape. For the special case of Ci real ( e. g. 00 , 1800 phase coding) .s#(m, w ) is equal to d*(m, - w ) . A fu rther discussion of the effects of signal repetition and coding is given in Section 4. 3 .
2.4 T HE EFFECT OF L I N EA R FM UPO N T HE U N C ER T A I N TY FUN CT ION
If linear f.m. is added to a c o mp lex modulation functi on a C t), t he resul t is t o give a new m odulation function of t he form aC t) eib f , i.e. an extra qua drati c phase te rm. Table 2.1 show s that this results in the uncertain ty function changing from X(T , w) t o
b e-i r2 X[T, w+2bT]. The above result means tha t the modulus of the new uncertainty fun ction is a she ared version of the modulu s of the original uncertainty func tion . A characteristic which formerly occurred at the co-ordinate s TI, WI is tr ansferred t o w+2bT:=Wt
That i s , to the new co-ordi na tes Til (WI - 2bTI)' The effect is illustrated in F i g . 2.6. When evaluating formulae numerically, some confusion can resu l t over t he choice o f t he parame ter b. I t i s convenie n t , in algeb raic manipulation , to use t he si ngle le tter b; however , when using the results it is m ore useful t o t hin k in terms of the physical 'fre quency sweep' . Interpreting instan t an e ous f re quency a s w· :=
.
I
dcp dt
-
the freque ncy swe ep , in t ime t, cause d by the qu adratic phase term w i ll be 2bt. Thus, for a n enve lope of durati on d, t he quadratic ph ase term will lead to a t o t a l fre quency sweep 6
= 2 bd.
(2.29)
I t follows t hat , when using results base d upon t he linear f.m. formul ae , b should be give n the va lu e b=
6
2d.
(2.30)
20
The mathematical treatmen t of the uncertainty function w
--�------+---�r--1
w
Ix [1,
W +
]I
2b1
Fig. 2. 6. T h e sh earing effect of linear FM u pon t h e ar ea of uncertainty
A term often used in the litera t u re is ' disp e rsion factor'. Using the ab ove notati on , the dispersi on fa ctor is equal t o the dimen sio nless p roduct 6 d. F o r the comm on pra c t i ca l sit u ation o f a fixed envelope d u ration , an increase in the dispe rsio n fa cto r means an in crease in the total frequency swe ep 6 .
2.S PROOFS OF THE PROPERTIES O F
x (r , w )
Th is se ction contains the pr oofs o f the mathematic al properties o f x (r , w) w hich were sta t e d i n Section 2 . 2 . 2.S. 1 Proof of Eq uation 2. 1 S
Equation 2 . 15 foll ow s, b y i nspe ct i o n , fro m t he definitions o f x(r , w ) a n d the F ou ri e r t ra n sfo r m .
The mathematical treatment of the uncertainty function
21
2.5.2 Proof o f Equation 2.1 6
From the results of Sec ti on 6.7
ff{a*(t)} =A*(-jw). Hence
ff{a*(t+r)} = eiWT A*(-jw). Exp anding Equation 2. 15 b y me ans o f the convolu tion t heorem gives 00
1 x(r, w) = 21T
f .
eJXTA*(-jx)A [jew-x)] dx.
_ 00
Changing t he dummy variable to
f
-x
give s
00
1 21T
x(r,w)= whi ch is Equation 2 . 16.
A*Ux)A[j(w+x)] e-iXTdx,
2.5.3 Proof of E quation 2. 1 7
Since E qu ation 2. 14 leads to
[x y z] * = x*y*z*,
J a*(t)a(t+r) e iWT dt 00
x*(r, w) =
_00
Chan ging the dummy va riable to (t - r) gives
J a*(t- r)a(t) e iW (T-T) dt 00
x*(r, w) =
_00
which be comes
J a(t)a*(t-r) e iWT dt. 00
x*(r, w) = e - i uJ T
Th e ab ove integral is equal, by inspection, to x( -r ,-w), t hus proving e quation 2. 1 7 .
2.5.4 Proof of Equation 2.18
Fr om Equation 2 . 1 7
Ix*(r, w)1
=
Ix(-r,-w)l.
The mathematical treatmen t of the uncertain ty function
22
Also , since IX *( 7, w)l = IX(7, w)l , Equ a t i on 2.18 follow s .
2.5.5 Proof of Equation 2.19
The e quality in Equat i on 2.19 foll ow s b y i n spec t i on , reme m b e r i ng that
�12 =xx*.
T he i n e q uali t y i n E qu a t i on 2.19 can be ob t ai n e d from t he Sc hw a rz inequal i t y [6]. Sin ce
I
�
I_L f(x)g(x )
dx
2
�_
l
00
lf(xW dx
_[ (x 00
[g
)12 dx
i t fol lows t h a t
The a p p l ic a t i on of Pa rseva l's t he o rem ( Appe n d i x 2) sh ow s t h a t t he tw o r ight-hand i n t egrals a re equal. Since i t has a l r eady been sh own t h a t
J la(f)12 dt = X(O, 0) 00
it fol l ows tha t
which p r ove s E q u a t ion 2.19.
2.5.6 Proof of Equation 2.20
The p roof of E q u a t i o n 2.20 is ob t ai n ed b y r e p l acing I X( 7 , wW by X(7 , w) X *( 7 , w). X(7, w) is expan ded by means o f Eq ua t i o n 2.14, and X *(7 , w) by Equation 2.16. T he result is
JJ IX(7, wW d7 dw 00
fffa(t)a*(t 00
=
+
7) e -jwt df
fA 00
21n
Ux)A * [j( w +x)] e j XT dx d7 d w .
The mathematical treatment of the uncertain ty function The right-han d si de m ay be re-a rranged to give
I 2rr
IIa(t)A(jX) I 00
00
I
00
a*(t+T) e
jXT d
T
_00
A*U(w+x)] e - j
23
w t dw dt dx
Sin ce the inte grals over T and w, i n the a bove expressio n , contain all the T and w fu n ctions they may be evaluate d separately. Now
J a*(t +T) ejxT dT J a*(T) ej (T - t)x dT 00
00
=
=
Also
A
e -jxt *Ux)
_00
J A*U(w+x)] e - jw t dw J A*Uw) e-j(w-x)tdw 00
_
00
=
00
=
2rr e jx
ta*(t).
The ab ove results follow from the de finitions of the F ourier transform and its inve rse . Substitution in the expressio n for
JJ IX(T, WWdTdw 00
gives
H IX(T, wW dTdw J a(t)a*( t) dt J A(jx)A*Qx)dx 00
00
=
00
Parse val's the ore m sh ows the la st integral to be equal to 2rr times the middle integral. It has already be e n show n that
f a(t)a*(t) dt 00
=
_00
Hence
II
f
00
_00
la(tW dt
=
X(O, 0)
00
2Irr
IX( T, wW dTdw = [x(O,O)f
which is Equati on 2.20.
2.S. 7 Proof of Equations 2.2 1 and 2.22
De fining
f b(t)b*(t+T) e -j wt dt 00
Xl(T, w)
=
_ 00
The mathematical treatment of the uncertain ty function
24
and subs t i t uti ng
gi ves
J a(t ) ej b r a * (t + r) e -j ( f + T )2 b e -j w f d t ""
x \ (r, w) =
J a(t ) a * ( t + r ) e - jbT2 e - j (w + 2 bT )t dt. 00
= Hence
p r o v i ng E q u a t ion 2 . 2 1. S i m i l ar l y t a k i n g
gives
f
00
=
2rr
T - 2 b W )X dx A * U X ) A [j ( W + X ) ] ejb w 2 e -j( .
H e n ce
p r o v i n g E q u a t i on 2. 2 2 . 2 . 5 . 8 Proof of Equ a t ion s 2.23 an d 2 . 2 4 If
b e l ) = a( M )
J a ( al )a * l ( l + r )a j e - j w t dt 00
x \ ( r, w ) = t hen
J
00
XI ( r, w) =
lal
a( l ) a * ( l + ar ) e - j w f/Ci. d t
The mathematical trea tment of the uncertain ty function
25
which is Equ ation 2 . 23 . Also, i f BGw) =AGaw) t hen
J
00
1 x \ ( r, w) = 2 T{
A*GQX)A [ja( w+x)] e - jx T dx
changing t he d ummy vari ab le t o x/a gives
J
00
1 X \ (r, w) = __ 2 T{ l a l =
1
j";I
A*Gx)A [j(aw +x)] e - jx T/a dx
x [r/a, awl
which i s Equation 2.24.
2.5.9 Proof of Equation 2.25
To p rove Equ ation 2.25 let n
b(t) = I Ci a(t - ik) i= O Hence
b(t) = Coa(t)+C\ a(t-k) . . . +Cn a(t-nk) b*(t+r)= C�a*( t+r)+cra*(t+r--k) . . . +Ga*(t+r- nk) givi ng
b(t )b*(t+r) = Co C�a(t)a*(t+r)
+C\ C�a(t-k)a*(t+r) . . .
+
+ Cn C�a(t - nk)a*(t+r) + Co Cfa(t)a*( t+r- k) + C\ Cfa(t - k)a*(t+r- k) . . . + + Cn Cfa(t -n k)a*(t +r-k)
+ Co Q'a(t)a*(t+ r-n k)+ c\ CJa(t - k)a*(t +r- nk) . . . + +Cn C�a(t-nk)a *(t +r- n k)
26
The mathematical treatment of the uncertainty function
Since
j
J
00
��
a(t + x) a * (t + y ) e - j w t dt =
i t follows t h a t
J
-�
a(t ) a * (t -x + y ) e - j w ( t - x) dt
00
X I ( T , W) =
b (t) b *(t + T) e - j w t dt + n + e - i w k c;, Q' X ( T + n k , w)
= Co Cl)X( T , w)
+ Co crX ( T - k, w) + e - j w k C I Cl' X [ T , w ] . . .
+ n + e - j w k Cn C fx [ T + (n - l )k , w ]
+ Co c,iX( T - nk , w) + e - j w k CI G X [ T - (n - l )k . w ] + + e - j w n k Cn C* n X(T W) Hen ce X I ( T , w) = e - j w n k X(T + nk , w ) { c;, Co' } + . . . I
,
+ e - i wk X( T + k , wH cl Co' + C2 ci e - j w k . . . + Cn G I e - j w (n - l )k � n C: c;, ci . . . cl k w + w k e j Co T, X( w e j + + } H C�
+ X ( T - k , w H Co C; + C I G e - j w k . . . + c;, _ I c: e - j w (n - I ) k } . . . + X( T - l I k , w H Co G }
which leads
to
X J T, w ) =
n
L
m =1
+
e - j w m k X( T + mk , w ) n
L
m =0
X( T - m k , w)
n -m
L
;= 0
n -m
L
;= 0
G + m Ge - j w ;k
C G+ m e - j w ;k ;
so that the new 'n' i s e q u a l t o the n u m b e r o f Re p l a c i n g n b y n p u lses give s n-I n- I-m m w w = X I ( T, ) L e j k X( T + mk , w) L C; + m G e - j w;k m=1 1=0 +
11 - 1
L
m=O
X( T - mk, w)
n - I-m
L
;=0
c;c7+m e - j w;k
The math ema tical treatmen t of the uncertain ty function
27
2. 5 . 1 0 Special cases o f Equation 2 . 25
Thre e special case s will n ow be considere d .
(I) G at e d sou r ce .
f(t) = H ence
n-I
L
;= 0
la(t - ik ) 1 cos [ wo t + ¢ ( t - ik ) ]
fa ( t) =
n-I
L
;= 0
a(t - ik ) e i w o t
which c orresponds t o C; = 1 . T h u s t h e coeffic i e n t o f X ( T + mk, w ) is a ge ometric p rogression given b y e -i w m k p + e - i w k + . . . + e -i W k (n - l - m ) }
{
-i W k (n - m ) . - e - J wm k e k e -i w _ l
_ I}
Similarly the coefficien t of X (T - m k , w) is
By defining e =
�k
{
e -i W �(n - m ) - I e -J W k _ 1
}
and u si ng the identity
( I _ e - 2 i x ) = 2j e - i x sin (x)
the above coefficients can be written as follows : X ( T + m k , w ) � e - j(n - I + m w X( T - mk , w) � e j (n - l - m W
{ {
sin [(n - m)8 ] . e SIn
sin [ (� - m )8 ] SI n e
} }
which is expressed compactly by saying t h a t the c o e fficient o f X(T - m k , w ) is W l e -j (n - - m
{
Sin [ (n - I m l ) e ] . SIn e
}
fo r 0 � I m l � n - 1 . The expression for X lT , w) then b e c omes : (n l ) -j w sin [ (n - l m l )8 ] i e (n - l - m X [ T _ m k, W ] . SIn e m = - (n - l )
{
(2) Pulsed source . n
{
-I
f(t) = L
;= 0
}
l a(t - ik) 1 cos [ wo (t - ik ) + ¢(t - ik ) ]
}
28
The mathematical treatment of the uncertainty function Hence
faCt) =
n
-I
L
;= 0
which corresponds t o C;
e - i wo ki a (t _ ik )ei wo t
= e- iwoki. Thus
C·I+ m C'" = \:""_i wo km I
Since the above products are not fu nctions o f i, the coefficients of x(r + m k, w) and x(r - m k, w ) are once more geomet ric prog resssions . The same reasoning as in the gat e d case lea ds to the result
x 1 (r, w) =
i
n
m-- n-
(
l
I)
{eiWokm
e -i (n - l _ m )8
{
�
Sin [ ( . - l m l ) e ] ,.10 e
}
x [ r _ mk' W ]
}
(3) N o n - c o h e re n t sou r c e . This i s t he case for Ci = e jq)i w he r e ¢i t akes on r a n dom values be twe en 0 and 21T r a d i a n s . The co-effi cien t of x(r - m k. w) fo r m = 0 i s a ge ome t ric p r ogression n-I
L
i= 0
Ci c1 e -i wik
=
e-i wik i {
n-I
L
i= 0
si n [ n e n ( = e - n - I )8 si n e
f
wh e re e = wk/ 2 . The c o e ffic ients o f X ( T ± mk, w) for m ± 0 are not geom e t ri c p ro g ressi o n s and c a n n ot be exp ressed i n a c l ose d fo r m .
2.5. 1 1 Proof of Equation 2.26
Eq u a t i o n 2 . 2 6 can be prov e d b y usi n g t h e fu n c t i o n aCt) + b et) in the d e fi n i t i on of X (T , w):
r [ a ( t ) +b(t ) ] [a *(t +r)+b * (t + T)] e 00
X( T , w) =
_00
M u l t ip ly i n g ou t gives
-i wt dt
X(T , w) = Xaa ( T , w)+Xbb ( T ,w)+Xab ( T , w)+Xba ( T, w) w h e re
_L u(t) v*(t+T) e 00
Xu v ( T , w) =
- iw
t
dt
The mathematical treatment of the uncertainty function
29
I t follows that
Xba (r , W) = (b(t)a*(t +r) e -j wt dt 00 Changing the dummy variable to (t - r) gives Xba (r , w) = ej w T (a*(t)b(tr) e-jwt dt 00 _
_
Hence
Substitution of the above in the expression for Equ ation 2.26.
x(r, w)
gives
Chapter 3 WORKED UNCERTAINTY F UNCTION EXAMPLES
This chap ter is use d t o illustrate the m e thod of c alculating and representing u n ce rtainty fun ctions .
3.1
THE RECTANG U LAR PULSE WITH CONSTANT CARRIER FREQ UENCY
F o r t h e rectangular pulse w ith constant carrier frequency t h e b aseband m o du lating fu nction is given by
a(t) = H en ce
a * ( t + 7) =
-t
Thus , for 0 < 7 < d
-t+d t --
n-- 1
a(t ) a * ( t + 7 ) =
o
- "C + d
t .....
An d for -d < 7 < 0
a(t ) a * ( t + 7) = F i nally , for 17 1 > d
n-- 1 -t d t .....
a(t ) a * (t + 7) = 0
F r o m Equ ation 2 . 1 5 H e nce for 0 < 7 < d
X ( 7 , w) = .�{a( t)a * (t + 7) } X( 7 , w ) =
l _ e -j (d - T ) W
jw
----:---
30
Worked uncertainty function examples .
31
� sin l � (d - T)]
(3 . 1 )
or
X(T , W) = e -j( w /2 ) (d - T) For -d < T < 0
. r
1 - e - j (d + T) W . --X( T , w) = el W T -JW
Hence
X (T , w) = e - j ( w /2 ) (d - T)
�
sin
[�
Equations 3 . 1 a n d 3 .2 c a n be comb i ne d to give
X ( T , w)
=
e - j ( w /2 ) (d - T)
X ( T , w) = 0,
ITI
�
l�
sin
]
]
(d - I T I ) ,
>d
]
(3 . 2 )
(d + T )
}
(3.3)
T h e cut a l o n g t he T axis o f t h e uncer t a i n t y fu nct i o n i s given b y IX( T , 0)1 , w here I X( T , 0 ) 1 = d - I T I , ( 3 .4) X (T , 0 ) = 0 , W
The cut a l o ng the Ix(O , w) l , w here
}
axis of t h e uncer t a i nt y fu nc t i o n i s given b y
I X( 0 , w ) 1 = d
'1
l
si n ( uxl / 2 ) ( uxl/2 )
I
(3.5 )
Eq ua t i ons 3 .4 a n d 3 . 5 a re i l l ustrated by F i gs 3 . 1 a nd 3 . 2 . T he fu l l 3-dim ensi onal fu n ct i on o f Equa t i on 3 .3 is ill u s t r a t e d b y t h e c om p u t e r p l o t of F ig. 3 . 3 ; F ig . 3 .4 i s a com p u t e r p l o t sh ow i n g t h e a re a o f u n ce r t ai n t y fo rme d by the - 6 dB c on t ou r of I X (T , w) l . -- -
-1
--- d
o
L/cJ
Fig. 3. 1 . I X ( T , 0 ) 1 for a recla l l � lar p u lse wilh CO I I S la l l 1 carrier jrcqu el/ c \ '
Worked u ncertain ty function examples
32
-4
-2
o
w d/ 2n
2
4
Fig. 3.2. I x (O , w ) 1 for a r ectangular pulse with con st a nt carrier fr equ en cy
T
Fig. 3. 3. Th e un certain ty fun ctio n of a r ect angular pulse wit h const ant carrier fr equency (tr uncat ed at half height )
U si ng F i g . 3 .4 t o ge th e r w i t h the q u a n t i t a t i ve d e fi n i t i on s o f pre cisi o n an d r e s o l u t i on (Sec t i o n 1 .4 ) gives t h e i nh e ren t delay a n d D op pl er p re c I s I ons a s T
-
d
wd
=
I
1 .2 2 rr =
(3 .6 ) (3 .7)
33
Worked uncertainty function examples W dj2 Tt 0.8 0.6 0.4 0. 2 - 0.6
- 0.4
-0.2
0. 2
0.4
0.6
l:jd
-0.2 - 0. 4 - 0.6 - 0.8 Fig. 3.4. Th e - 6 dB area of un cert ainty of a rectangular pulse with cons tant carrier frequency
while the correspon din g figures for resolution a re
J = 2 (for complete resolut ion)
c.::: = 6 3 7 (fo r 60 dB resolution)
(3.8)
( 3 .9)
The relationship between T / LV a nd range/velocity is given in Section 1 .4 (Equations 1 .6 and 1 . 7) ; making the appropriate substitutions leads to the followin g relationships for inheren t range and velocity p recision
r = 0 . 5 cd
(3 . 1 0)
= 0.6
(3 . 1 1 )
v
Note th at pulse .
fo d
c fo d
is equal to the numb er of RF cycl e s contained in the
Worked un certain ty jill1ction examples
34
3 . 2 . T H E R ECT ANG U LAR PU L S E W I T H L IN EA R FM
For t h e r e c t a n gu l a r p u l se w i t h l in e a r F M t h e b ase b a n d m od ul a t i n g
func t i on i s gi ve n
by :
I t fo l l ows , fr o m E q u a t i on 2 . 2 1 t h a t , i f X tCT , w ) a p p l i e s t o t h e p u l se w i t h ou t FM , t h e r e q u i r e d fun c t i on is e- j bT 2
X I f T , w + 2b T ]
Hen c e from Se c t i o n 3. 1 ( Eq ua t i on 3. 3 )
X ( T , w ) = e _ jb; 2
e- i (W +2 bT)(d - T)/ 2
sin
X( T , w)
=
0,
(
2 + W �T
__ _ _
[ � w
2b T
)
]
}
(d - I T I ) ,
(3. 1 2)
Th e c u t s a l o n g t h e u n c e r t a i n t y fu n c t i on axes are giv e n b y
I X( T , 0) 1
=
I X ( T , 0) 1
d
=
0,
I ��� Si n [ b
I
- ITI )]
I
,
/ 2) sin( wd I X (0 , W ) 1 = d (wd/ 2 )
ITI < d
ITI > d
I
'I
J
( 3. 1 3 )
(3. 1 4)
Usi n g t h e re la t i on ship b = 6. /2d (Equ a t i o n 2 . 30) , I X (T , O ) I c a n b e w ri t t en i n t h e fo rm
IX( T, O ) I = 0 ,
IJI < I �I
> 1
}
(3 . 1 5 )
Th e a rgum e n t o f t h e s in e fu n c t i o n i s illu s t rate d by F ig . 3.5 . N u l l s o c c u r i n IX (T , 0)1 w h e n e v e r t h e g ra p h o f F ig . 3 . 5 pa sses t h r o ugh i n t egral m u l t i ples o f n . It c a n b e see n that whenever t h e d ispersion fac t o r d6. is large t here w i ll be a l a rge n u mb e r of n ulls-w i t h c o r re s p o n d i n g p e aks b e tw e e n t h e m . For a l arge d i s pe r s i on fac t o r the l as t p eak w i ll occ u r i n t h e vi c i n i t y o f I T lid = 1 a n d w ill have a n a p p r ox im a t e value o f 2d/(d6. ) . Thus for a d isp e rsi on fa c t o r o f 2000 t h e last peak i n I X( T , 0) 1 w i ll be a b o u t 60 dB d own o n t he m'a i n lobe.
..
Worked uncertain ty function examples
(n+1)
11:
I
11:
o
--
- - -t- - I I I I I I
I
I
I --- - - -t- - - -tI - - -tI I I I I I I I I I I I I I I I
n 11:
2 11:
35
I I I -+----1---+-+ ---+I I I I I I II 1 I I
1 I 1 1 t 1I I I 1 1 I I I 1 1 -
-
- - - -
I I
:
L
- -
I I
+ I
- -
1
+
I
- - -
i
:
1
I I I
I
I I I -+ I
I
-
I
T
I
-
I
: :I :
I
I
I
I � I --
Fig. 3. 5. Plo t of
T: (1 - III I )
A fu r t h e r poi n t t o n o t e is t h a t for small values o f 1 7 1/d , i . e .
� d
=
1 76 1 & 0 1 d6 ""'"
I X(7 , 0) 1 c a n be represe n t e d b y I X(7, 0 ) 1
�
d
I
!
sin(76 / 2 ) , (76 / 2)
1 7 6 1 � 0. 1 d6
(3 . 1 6)
Since t he sin(x )/x fun c tion has i t s n ul l s when x is equal to i ntegral mul t iples of 1T , it follows that it is convenie nt to l abel t he 7 a nd w axes fo r t his fu n c t i on in t erms of 76 21T
and
wd
21T
respec tive l y . I t follows from Equ ations 3. 1 4 a n d 3 . 1 6 t ha t t h e Doppler p recision a n d resolution a re given b y : wd
2 1T
= 1 . 2 (prec ision)
'::: = 6 3 7 (60 dB resolution)
(3. 1 7) (3 . 1 8)
36
Worked uncertainty function examples
Fig. d�
=
40
3. 6.
Th e uncertainty fu nction of a rectangular pulse with linear FM and
(trunca ted a t half heigh t)
wd/2lt 5
� � ----------+5 -+5----------�------ �-r �------� 2.5
. to 2 lt
-2.5
Fig. 3. 7. Th e - fJ
and d �
=
40
-5
dB area of uncertain ty of a rectangular pulse with linear FM
37
Worked uncertainty function examples For large values o f dfl the d elay precision will b e given b y
;�
=
( 3 . 1 9)
1 .2
The complete u n ce rtainty fu nction is ill ustrat e d b y F igs 3 .6 a n d 3.7 . No te t hat there i s strong delay-Doppler c oupling, the consequences of t his a re discusse d i n Sec tions 4. 1 , 4 .2 a n d 5 .2.
3. 3 T H E RECT A NG U L A R PU L S E W IT H TRIA NG U L AR FM
The baseband modulating fu nction to be studied in t his se ction is given by
c ( t ) = aCt) + b et )
where
(3 . 20a)
aCt ) =
[ rL -d 1] e- jbr
( 3 . 2 0b )
bet) =
[ rL1] d
( 3 . 2 0c )
0
l-
0
l-
H e nce the in stantaneous fre quen cy is given by
w( t) =
S/JU2bd -d o
d
The above de finitions maint ain t he relation sh i p b. = 2 bd ; i n t his case b. is t he t otal frequency change (rad/s) during either t he down sw eep or the u p swe e p . T h e results of Section 2 .2 ( Equations 2 .2 6- 2 . 2 8 ) show t h a t the com p lete uncertainty fun ction is the sum of the unce r tainty functions of a(t) a n d b et) an d o f the 'cross un certainty fu nctio ns' , i .e .
x(r , w ) = Xaa ( r , w ) + X bb (r , w ) + Xa b (r, w ) + e j w T X:b ( -r , - w)
(3 . 2 1 )
whe re 00
X uv( r , w ) = f u(t) v *(t + r) e -j w t dt _ 00
If
a{ t ) =
[ D1 J l-
e -j
bt2
(3 . 22 )
38
Worked uncertainty function examples
it foll ows that aCt ) defi ned by E quatio n 3 .2 0(b) is given by
aCt ) = cx( t + d )
(3 2 3 )
3 . 2 3 all ow :xaa ( T , w ) and X bJ.T , w ) Section 3 .2 . The final expressi ons a re
Equations 3.20(c) and from the results of
:xaa (T , w ) = ex p j
[� ] [� (: ) ,
:xaa ( T , w) = 0 , and
X bb ( T , w ) =
(T + d) - b Td
exp
j
W 2b T
sin
W 2b T
[� - ] ; [ (} )
]
W
2b T
w 2b T
to b e found
(d - I TI ) , IT I < d
}
(3 .24)
I TI > d
(T d ) - b d T sin
.
�(2t+d)bd
(d
- ]. IT!)
ITI < d I TI > d
}
(3 . 2 5 )
I t is shown i n Section 3 .3 . 2 that t h e c ross term NJ b (T , w ) is given by
Xa b ( T , W ) = O ,
T
Xa b( T, w ) = exp j ( wT/2) e x p -j (bT2/2 ) exp j ( w2/8b)
J( rr / 4 b )
{ [ C( u I ) + C(U 2 ) ] -j [ S( U I ) +S( U 2 ) ] }, 0 < T < 2d
( 3 . 26 )
T > 2d w here C(u) , S(u ) are F r e sn el integrals (see A p p e n d ix 3 ) arid
UI = U2 =
J( ; ) [ 1 � l J ( : ) [ 1 .- � ] b 2
b 2
+
2 T
2 T .
0
} (3 . 2 7a) }
(3 . 2 7b )
Worked uncertain ty function examples
39
3 . 3 . 1 Interp re tation of resul ts
The a b ove exp ressions l o o k ver y forb i d di n g espec ia l l y w h e n it i s realised t h a t t h e sign i ficant fun c t i on is t he m o d u l u s o f E q u a t i on 3 . 2 1 . Th ey can', h oweve r , be e as il y i n t e r p re t e d . Xa a (r , w ) a n d x b i r , w ) a r e sh e a r e d fu n c t i o ns of t h e fo r m d i scu ssed in Section 3 .2 ; t h e m a i n p a rt o f Xaa (r , w) r u n s b e tw ee n t h e fi rst a n d t hi r d quad ran t s as opposed t o �b(r , w ) w h ich r u n s b e tw e e n t h e second a n d fo u rth qu a d ra n t s . For a la rge d ispersion fact or t h e r e wi ll b e l a rge areas of the r, w plane where t he two fun c t i o n s do n o t in t e r ac t signi ficant i n te ra ct i o n ta kes place a t the o rigin l e a di n g t o a l a rge s i n gle l obe . The F re sn e l cross fu nc t i on s fo rm a pe de s t a l t o t h e ab ove two fun ctions. It w o u l d be ve ry d i ffi c u l t to ca l c u l a t e t h e e x a c t fo rm o f t h i s p e d e s t a l b u t , for t u n a t el y , i t c a n be s h own t o be sma l l c o m p a r e d wi t h the a m p l i t u de of xaa (r , w ) a n d Xbb (r, w) . F or h igh'precision a p pl i ca t i on s, i n t er e s t is c e n t red on the p e r forma nce for mode rat e values o f r . It ca n be seen t h at i n this region the e ffec t i v e m u l t i pl i e r o f xaa (r , w) and �b(r , w) is the i n d i v i d u a l p u l se l e n gt h , d. Pu t t i ng b = an d usi n g Equa t io n sh ows t h at t he m u l t i p l i e r fo r Xab(r , w ) is
�/2d
d�.
3. 2 6
w)
i s sm a l l for This fact s h ows t h at t he r e l a t i v e a m p l i t u d e o f xa b ( r , l a rge v a l u e s o f F u r t h e r i n fo r mat i o n can be o b t a i n ed fr o m t h e a r gu me n t s o f t h e F re sn el i n t egra l s . S ub s t i t u t i n g fo r b , E q u a t i on 3 . 2 7 ( a ) b ec o me s II ,
=
112
=
d �J( d �) [1 :J(��)I I
Hence , along t he r axis
= !..
F o r l a rge v a l u es o f
( F i g . A3 . 1 ) . A lo n g t h e w a x i s
+
2 rr
d�,
WrD.d J , 0 < S < I wd -I , O < !.. < I r� d
/(d �) . 0 < S < I
d '\/
}
2rr
t h e F r es n e l i n t eg r a l s w i l l he e q u a l
(3.28 )
to
OS
40
Worked uncertain ty fun ction examples
For large values of db.. the F resnel in t egrals w i l l b e e qu a l to zero . Al ong t he main ridges w = ±7 b.. /d. hence UI
=
� J(��) [1 ± I ] ,
0<
u2 = ; J(��) [l + 1 ] ,
0<
�<1 �< 1
l e ad i ng t o F re snel i n t e grals e qual t o 0 and (for high db.. ) 0 .5 . I t should b e n o t e d that use o f th e t ri angu la r F M wavefo rm a ll ow s ta rge t range changes t o be d i s t inguish e d fro m vel ocity ch ange s . I f e ither a matche d filter o r F ou ri e r t ran sfo rm rec e i ve r (se e C hap t e r 4) is used t o
t rack a given t a r ge t , t he c o r re c t se t t i n g o f t he ' tu ning' c ont rol will b e where the o u t p u t wave fo rm change s from a d o u b le t o a sin gle p u l se .
3 . 3 .2 Proof o f Equation 3 .26
,
� b( 7 w )
can be ca l c u la t e d in t h e foll ow i ng E qu a t i o n 3 .2 2 , a ft e r a c ha n ge of d u m m y va ri able
�b( 7 .
f a (t - ;) b * (t �) 00
w ) = ej( w T / 2 )
+
_
manner :
e-jw t d
00
fr om
t
Using Equati ons 3 . 2 0(b ) a n d 3 . 2 0(c )
*(t �)I JL- 1 la (t - �\b 2) 2 2
I
+
2
=
-l
t�
l
'
0<7
Also,
Equ at i ons 3 .2 6 and 3.27 fol l ow when t he a b ove in fo rma tion is used with the results o f Appendix 3 ( Equ a t i on A 3 . 6 ).
Chap ter 4
SIG NAL PRO CESSING MET HODS
This c h a p t e r i l l u s t rates radar sign a l processi n g b y desc r i b i n g t w o ' o p t i m u m' receive rs- t h e m a t ch e d fi l t e r rece iver a n d t he F ou rie r t ransform receiver . The fo rme r i s b as i ca l l y a range -m e as u ri ng system w hi c h a lso o b t ai n s D o p p l e r i n format i o n , whi l e t he la t t e r c a n be rega r d e d as a D op ple r m e asu rin g system w h ich a l s o ob t ai n s ran ge in formation . The a b ove re ceivers are op t i m u m , from the p rec i si o n p oi n t o f v i ew , i n the sense that t he t a rge t p arame t e rs are determine d from w ave fo rm s which a re cuts t h rou gh the u n ce r t ai n ty fun c t i on o f t h e t ra n smi t t e d signal . T h e d i scu ssion in Sec t i on 1 .4 show s th at such rece ive rs a re n ot n ecess a ri ly optimum from the p oi n t o f view of resolu t i o n . In p r ac tice re so l u t i o n can be imp roved (at t h e e x pe n se o f p rec ision ) by u si ng modifi c a t i on s o f the ' op ti m u m ' rece ive rs . M o s t prac tica l r a d a r p roce ss in g m e t h ods can b e ide n t i fi e d as s u c h modi fi c a t i o n s .
4. 1 THE M ATCHE D FI LTER R E CEI V ER
The matched fi l t e r c o r respon d i ng t o a given real sign a l f(t ) ca n b e defin e d a s t h e fi l t e r w i t h a n i m p ulse respo nse o f f ( - t ) . I t i s sh ow n i n Sec t i on 4 .4 th at :
( 1 ) A match e d fil t er w i l l p rocess f(t ) t o give a n ou t p u t having a peak value w h i ch i s gre ater t h a n (or as gre at a s ) t he peak ou t p ut fr o m a n y other fil t e r h av i n g a n im p u l se response o f t h e same ene rgy a s t h e ma t ched fil t e r . ( 2 ) I f f(t ) is i n w h i t e noise , t h e fil t e r ma t ched t o f(t ) w ill give t h e optimum o u t pu t peak-sign a l t o n oise ratio . ( 3 ) A fil t e r havi ng p ro p e rt i es ( \ ) a n d (2 ) is physi c a l l y re a l i sa b l e p rovi ded t hat f(t ) is o f fin i te d u ra t i on ( i . e . time-limi t e d ) . The m a t c h e d fil t e r receiver wi ll b e defined as a supe rhet receiver using a matched filter i n its IF sec t i o n . A bl oc k diagra m of su ch a re ceiver is sh own in F ig . 4 . 1 . 41
Signal processing methods
42
[t - x , \ -f
5 . 5. 8.
I--
Mixer
signal
]
W +Y-W V o
Filter R F E n velope is :-'------i m a t c h e d t o 1------. f [t, W , ] 1 I. '----'---' I X Lt - x , y + w o - w v - w lj
I
II
Tu n a b l e local osc i l la t o r
Fig. 4. 1 .
Th e ! n a t ch e d filter rec e i ver
It i s s h o w n i n S ec t i o n 4 . 5 t h at t h e ou t p u t RF e n v e l op e of t h e m a t c h e d fi l t e r rece i v e r i s gi ve n b y
� I X [ ( / - x), ( w o - w , - w \. + y ) ] 1
w h ere I X ( T , w )1 is t h e u n c e rt a i n t y fun c t i o n c o r r espo n d i n g t o
laU ) 1
ej¢(r)
I t is assu m e tl t h a t ( wo - W v ) i s h i gh e n o ugh fo r t h e resu l t s o f . S e c t i o n 7 .4 t o a p p l y . T h us t he R F e n ve l o pe o f t he m a t c h e d fi l t e r rec eive r o u t pu t i s a t im e fu n c t i o n w h i c h h as t J l e s a m e s h a p e a s a c u t ( ta k e n p a r a l l e l t o t h e T a x i s ) t h r o u g h t h e u n ce r t a i n t y fu n c t i on ce n t r e d o n t h e x, v c o -o r d i n a t e s o f a p a r t i c u l a r t a rge t . T h e c o n s t a n t v a l u e o f w fo r w h ic h t he cu t i s t ak e n c a n be a dj u s t e d b y v a ry i n g t h e l oc a l osci l l a t o r fre q ue n c y W v. The e ffe c t s of v a ri a t i o n s i n W v a n d D o p p l e r s h i ft v a rc sh o w n i n F i g . 4 . 2 ; a t yp i c a l o u t p u t s i gn a l is s h o w n i n F i g . 4 . 3 .
- - - - - - - - - -
I I � I
- - - - - - - - - - -
I
The receiver
output
is a c u t a l o n g
t h i s l i n e - l i n e d i s p l ac e m e n t i s de t e r m i n ed b y t h e p r e c i s e va l u e of
Fig. 4. 2.
How
Wv
w v a n d y ur tcrm ille t h e u n cer tain ty jim ction cu t
Signal processing methods
43
Fig. 4. 3. A typical r eceiver au tpu t pulse
Bearing in min d t hat 'x ' rep resents t he time at which a ta rget return is received , Fig. 4.3 appe a rs t o ind icate that the receiver outpu t pulse sta rts b e fore t he r e t u rn signal a r rive s ! This parad ox is resolve d in Section 4 .4 where it is shown that the strict m athemati cal mat ched fil t e r - being non -cau sal-cannot be re alised . T he p r ac tical mat ched filter must include a finite amoun t of del ay - th is delay d oes not affect i t s optimu m properties . Since the m ax im um value o f the uncert ain t y fun ction occurs at the o ri gin of the T , w pla n e , the rece ive r w ould be 'tuned' to a particular targe t b y a dj usting the l o cal oscillator fre qu ency to maximise the echo from that ta rge t . Once the re ceiver w as ' t u ned ' , the targe t delay parame ter could be o b t aine d ( t o within an accuracy set b y som e threshold) b y n o t in g t h e time d i fference b e tw een the t ransmi tted sign a l a n d t h e p e ak. o f the o u t p u t pulse . The ta rget Doppl e r shi ft could be de duc e d from a k n owledge o f the value o f l ocal oscillator frequency W v u sed t o o b t ai n the optimum p u lse shape . The threshold in the Dopp le r measu rem e n t would b e set b y the ability t o r e c ogn ise the optimum pulse sh ape . An e l ab o ra ti o n o f the r e ceiver w oul d be to inc lu de the l ocal oscill ator in a c ont rol loop wh ich w ou l d keep the receiver tuned to a p articular t a rge t o nce it was sel ected and acquired . Such a system would give a fast r e a d out of variations in target p osition (the shi ft of the 'blip' on the face o f an oscill oscope , for examp l e ) , and a slower readout o f va riati ons in targe t . velocit y- th e speed of the vel ocity readout b e in g go ve r n e d b y that o f the local oscill ator cont rol loop . I t would be possib le t o obtain Doppler information faster by using a bank of filters an d appropriate sw i tching . Ce rtain wave forms, a rectangular pulse wit h linear F M , fo r example e xh ibit a cou p l ing between del ay and D opple r . T his m eans that i f t h e a b ove tunin g pr oce dure w ere ca r ried out , t h e p u l s e shape a nd ampli t ude w ould not change a pp reciably , t h e predominant e ffe ct of the t u ning proc edure would be t o give a shi ft in t h e pulse position . T his e ffect merely c onveys, in p ra c t i cal terms, th e p recision information con tained in the uncertainty fu nction of the p articular transmi t ted waveform .
44
Signal processing methods
Del ay-Doppler coupli ng can b e a n a dvant a ge as w e ll as a disadvantage . A s yste m using su ch a w ave fo r m w ill yield a n e ar o p timum o u t p ut , fro m the sign a l detectability p oint o f vie w , even i f t he receiver is b a d ly mis t u n e d . Figu res 4 . 4 and 4 . 5 i ll ustrate the ab ove p oi n t by c o mp a ring t he o u t p u t s fro m a sy stem u sing l i nea r FM (i.e . d elay-Dopple r co upl in g) and one u sing a constant c a rrier p u l se (no delay-Doppler c oupling). --
Co r rectly tuned R x
- - - - - - M i st u n e d R x
Fig. 4. 4.
Ou tpu t of a iiI / car FM m a t ched jilter
Rx
-- C o r r e c t l y t u n e d R x ------ M istuned
Rx
t -
Fig 4. 5. 011 t{J l l t of a
cOllSta l l t jfequ e/l cy pu ls e match ed jilter
Rx
A U p ra c t i c al rad a r sy s t e m s t ra n s mi t t r a i n s o f p u lses , ra t he r than a si n g l e p u l se . I t w as sh ow n i n Se c t ion 2 .3 t ha t t he e ffe c t o f a t rain o f p ulses o n t h e u nc e r t a i n t y fu n c t i o n w a s t o c au se de lay ambiguities a nd t o give a n imp rovem e n t i n D o p p l e r p rec ision .
4. 1 . 1
A m a tc h ed fil ter for re petitive pulses
I t s h ou l d be obv i ous fro m t h e desc rip tion of t h e m a t ch e d fi l t e r receiver
t hat pulse t rai n s will alw a y s lead t o d e lay amb iguities, e .g . ' se c o n d - tim e - rou nd' a m biguity w h e n t h e signal delay is gre a t e r than the p e ri o d of t h e pulse repetition fre quenc y . I m p r ovem ent in D o ppler prec ision , h ow e v e r , w il l o n ly b e obt aine d i f t h e filt e r i n the receiv e r is m at ch e d t o the t rai n of pu lses rather than to a si ngle pul se .
Signal processing methods
45
One method of forming a p ulse t rain wit h inh eren tly high Doppler precisi o n i s t o t ransmit an exac t r e pe ti t i on of a basic waveform at intervals of k seconds. F igure 4 .6 sh ows a method o f c onver ting a filter, G(jw), which i s mat ched t o a single pu lse in t o a filter matched t o such a repe t i t ive train . I n p ut ( n - l ) s e c t i o n d e l ay l i n e
'----
e -jwk
G ( j w)
-
e - i w k r-- . . . . -
e -i w k
�
F i l t e r l a t c hed to a sing l e pulse
�
Output
Fig.
4. 6 . A matched filter for n iden tical pulses
A repeti tive t rain of
n pul ses of the
for m fit ) can be desc ribed b y
fR (t) = f(t)+f( t - k )+ . . . +f [ t - (n - I )k ]
Hence t he sp ectrum of t he t rain is given by
FRUW ) = FU w) [ I + e - iwk +
. . .
+
e - i(n - I )Wk]
The t ransfer fun c t i on o f the required ma tched fil ter i s F;(jw), i .e .
n [ F *U w) ] [ 1 +eiwk + . . . +ei ( - I )Wk ]
I n the above expression F*(j w) is the t ransfe r fun c t ion of a fil ter matched t o a single pu lse (sh own as GOw) in F ig . 4.6) while ei wk is the t ransfer fu nc t i on o f a network whi c h advances a signal in t im e by k seconds. S ince the adva nce netw o rk i s not physically real isable t he fi l t e r is not realisable . H owever , a delay a t the matched fi lter outpu t w i l l not affect its op t i m um prope r t i e s , hence an equivale n t filter which is realisable w o uld have a t ransfer fu n c t i on
1) F* (j w ) l l +e i w k+ . . . +e i (Il - w k 1 e - i (n - I ) w k
That is
i F* ( jw } l l + e- wk+ . . . + e -j (n - I) w k I
leading t o tJle sche me o f F i g . 4.6.
Signal processing methods
46
I t c an b e c learly see n from F ig . 4.6 that the imp rovement in Doppler p recision p re dicted by Secti on 2 .2 (Equati on 2 .2 5 ) i s obt ained , phy sically , b y t he i n teraction o f the n pu lses with each other . T hus although an outpu t will be obtained from t he filter of Fig. 4.6 a fter the fi rst pulse o f the t rai n , the sy stem will n o t achieve i ts full p re cision u n t i l a time o f nk secon ds has elapse d . T he m a t c h e d filte r receive r i s c learly op timum w he n use d t o receive the echo from a sin gle t a rge t . This is because it exploits t he ful l prec isi on in here n t i n th e u ncertain t y fu nction o f the t ransmitted w av e fo rm . T h e receiver w i ll n ot b e optimum when used again st mu l t ipl e t a rge ts an d cl u t t e r , owing t o t h e slow rate of fall-off of t he side l ob e s o f many u n ce rta i n t y func t i o n s . R e s o l u t i on c a n b e im p roved ( a t the e xpense o f p recision) by using a n o n - m a t c h e d fil t e r i n t h e receiver- t h e m a t c h e d fi l t e r c h arac teristics c a n th en be use d as a r e feren ce a gai n s t which im p r ov em ent i n resolu t i o n a n d l oss o f p r e c i si on a n d si gn al-to-n oise r a t i o can b e j u dged . F or some w ave fo r m s t h e ma t c h e d fi l t e r a l t hough ph y si c a ll y re al isa b l e may b e very d i ffi c u lt t o m a n u fa c tu re ; in t h ese cases a s u i t a b l e n on -m a t ch e d fi l t e r m ig h t b e u s e d e v e n w h en i t i s o n l y necessa r y t o achi e v e high p re c I si o n . E x a m p l e s o f s u c h a l t e rn a t ive sch emes a re given i n C h a p t e r 5 . 4 . 2 T H E F O U RI E R T RAN S FO RM REC EI V E R
t ra n s fo rm receiver w i l l b e defi n e d as a sys t e m w hi c h c a l c u la t e s t h e spec t ru m o f t h e p rod u c t o f t h e re t u rned sig n a l a n d a n o ffse t ve r s i o n o f t h e t ra n sm i t t e d s i g n a l A b l oc k d ia g r a m o f s u c h a The
Fourier
.
r e c e i v e r i s s h o w n i n F ig . 4 . 7 .
a D o pp l e r m e asu ring sy s t e m w h ic h o b t a i n s r a n ge i n fo r m a t i on as a ' b o n u s' ; i n t hi s se n se it is t h e o p p o s i t e o f t h e m a t c h e d fi l t e r r e c e i ve r . I t i s sh ow n i n S ec t i on 4 . 7 t h a t t h e m u l t i p l i c a t i o n p e r fo r m e d i n t he F o u r i e r t r a n s fo r m rece ive r l e a d s t o a p a i r o f d i s p l a c e d u n c e rt a i n t y fu n c t i on s fo r e a c h t a rget . T h e r e c e i v e r ou t p u t w a v e fo r m h a s t h e sa me shape as a c u t t a ke n p a r a l l e l t o t h e W a xi s t h r o u g h t h e m od u l u s o f t h e u n ce r t a i n t y fu n c t i ons : t h e si t u a t i o n i s i l l u s t r a t e d h y F i g . 4 . 8 . T h e a m o u n t o f s e p a ra t i o n b e t w een t h e tw o u nc e r t a i n t y fu nc t io n s r e s u l t i n g fro m a g i v e n t a rge t i s d e t e r m i n e d b y t h e freq ue n c y o f t h e fi x e d o ffse t osc i l l a t o r i n F i g. 4 . 7 . I f t h e fix ed o ffse t osc i l la t o r w as o m i t t e d ( o r i f i t s fre q u e n c y WI . w a s n ot h igh e n o u gh ) , t h e s i d e l ob e s f r o l n t h e t w o u n c e r t a i n t y fu n c t i o n s w o u l d o verl a p a n d l e a d t o a loss o f p re c is i o n . O n t h e o t h e r h a n d , i f WI. i s m a d e s u ffi ci e n t ly h igh t h e e ffec t o f t h e s i d e l o be s o f th e l o we r u n ce rt ain t y fu n c t ion ( F ig . 4 . 8 ) w ou l d be The
Fourier
t ra n sform
rec e iv e r
is b a s i ca l l y
Signal processing m ethods
R et u r n s i g na l
L . F. p r o d u c t t e r m s
\
!
S . S . B.
mixer
�
[
!
[
F. T . cal c u lator
]
47
l [<
tx
-t v '
W-Y-WLJ I
( i f wL i s h i g h e n o u g h )
- f t-t ' W - W o L v Va r i a b l e de lay (t ) v __ f t , W - W L o Aux. S . S. B
m ixer
]
Fixed offset osc i l l a t o r
,
cos (w t ) L
�--- [ ] \ \ [
P a r t of tx output
f t,w o
=
a ( t ) c os W t + ¢ ( t o
�
Fig. 4. 7. The Fourier transform receiver W
- -
+ i
- - - -- -
[
8 /
]
X x - t ' W - y - wL
v
I I · I ----+-- T h e r e c e i v e r o u t p u t .IS a c u t a l o n g t h i s l i n e - l i n e d i s p l a c e m e nt i 1 --I I is d e t e r m i n e d b y t h e p r e c i s e
I
i
I
I
v a l u e of t · v
------r---�--_+-- t v IX I I I I
- - -
t I I
Fig.
- - - - - - -
�
]
X [t - x , w + y + w L v
4. 8. Th e effect of W L and tv 011 the receiv er outpu t
negligible a t the + ve fre quen cies w h e re t h e upper u ncertainty fu nc tion is strongest . To operate the rece iver one would adj u s t t he variable del ay c on t rol to obtain a m axim u m output p ulse a m p l i t ude and then read the Doppler offset by noting the frequency at w h ich the p e a k oc curs. If the variab l e delay control was calibra t e d the targe t range c o u ld b e fo und b y realising t h a t the pu lse sha pe would b e op ti m u m w h e n tv = x. I t c an be seen that , since the Four ier t ra ns fo rm re ceiver t akes a c u t t hr ough t h e uncertai n t y fu n c t i on , i t p osse sses a l l t h e precision a n d
48
Signal processing methods
resolution properties of the matched filt er receiver. Specit1cally, it is optimu m for use against a sin gl e targe t and suffers fro m side lobe swamp ing when used against multiple targets a nd clutter. The same advantages and disadvant ages will resul t from the use of transmitter waveforms with and with out delay-D oppler coupli ng as in the case of th e match ed filter receiver . F igures
4 .9
and
4. 1 0
are
an alo gou s t o F ig s 4.4 and 4.5 .
" , \ \ , \ I \ I \ I \ I I \ I \ , \ I I , I , I I I '
, ,-,
,,-, : \; ,� ,
�:
--
Correct t
v ---- - - I n c o r r e c t t
""
sett ing v
sett i n g
.... ,
Fig. 4. 9. Output of a lin ear FM F. T R x -- C o r r e c t t
v ------ I ncorrect t
Fig. 4. 1 0.
sett ing v
settIng
O ut p u t of a con stant frequency pulse F. T Rx
An i n t e re s t i ng c o n se q u en ce 01 d e l a y -D o ppler coupling a rises when t he F o u r i e r t r a n s form rece iver i s u se d w i t h a linear FM w a v e form . I n t h i s case th ere i s oft e n n o n e e d t o u se e it h e r the fixed o ffse t oscillator o r t h e vari a b le de l a y l i n e . T h e re q u i re me n t s are
(I) The t a rge t m us t be s u ffi c ien t ly fa r fr om the t ransmi t t e r . ( 2 ) T h e dispe rsi o n fac t o r d tJ. m u s t b e l a rge . The ab ove s i t u a t i o n a ri se s b e c a u s e the uncert ainty fun ction of a l i n e a r F M sign a l w i t h a h i gh d i s pe r s i o n fac t or is i n t h e fo rm o f a l ong r i d ge i n c l i n e d at an a n gle t o t h e T a x is , (see Sec t i o n 3 . 2 ), and provide s i t s own o ffse t i f the range d e l a y x is la rge e n ough . The effect is i l l u s t ra t e d by F ig. 4 . 1 1 w h i c h i s d raw n fo r y and WI. e q ual t o z e r o . Th e fa ct t ha t t h e a b ove s i t u at i on m a ke s i t impossi b le t o distingu ish b e t ween changin g t a rge t ra n ge a n d changing veloc i ty is a c onseq uen ce of the
Signal processing methods
x t.
d
49
w __ _
___ T h e R x o u t p ut i s a c u t a l o n g t h i s a x i s f o r t y = o
' M a in
r id g e of
X [I X - t) . w]
--+-------��--. t y
/
M a in
r id g e of
X [l t y - X ) . w]
- x t. d
Fig. 4. I I . Uncer tain ty fun ction situation r esu lting fr om the use of lin ear FM in a F. T. R x w ith no offset frequen cy or var ia ble delay
t ransm itted wave form rath er t h an o f t h e receiver . T his effec t is discussed further in Sec tion 5 .2 . A s discussed i n Se c t i ons 2 . 3 and 4. 1 , one o f t he e ffe c t s o f transmitting a train o f p u l ses i s t o imp rove the i n heren t Doppler precision by c oncentra t i ng th e unce r t ai n t y function abo u t m u l t iples of th e p ulse repe tition fre q uen c y . In a m a t c he d fi l t e r receiver add i t ion al .ci rcu itry is r e qu ired to exploit t hi s imp rovement (s ee F ig. 4 . 6 ) . T he e ffective number of p ul ses (usi ng a m a t che d fi l t e r receiver ) i s determ ined b y t h e number o t delay l ine sec tions an d a time o f n k sec onds must elapse be fore t h e ful l prec ision i s achieved . The p recIsIon improveme nt d ue t o repeti tion is ob t ai n ed a u t o matically with the F ou rier t ran s form receiver - the spect rum o f a t rain of pulses is in the form of narrow spi kes and red uces to a 'line spectrum ' for an infini te nu mber of p u l ses. T he sam e t im e p e n a l t y h a s t o be paid fo r t h e imp roveme n t a s i n t he case of t he ma t ched fi l t e r receive r . A true Fou rier t ran sfo rm calc u la t o r w o u ld c a r r y o u t i t s integration over an in fi ni te l y long period , a pract ical ca l cu l a t o r w ou l d have a finite in tegra t i on t im e a nd i t w ou l d be t his t i m e w h i c h w o u l d dete rm ine t h e e ffec t ive number o f p u l ses a n d w h e n t h e fu l l r e s u l t w a s avai lable . Spec i fica l l y , a F o u rie r t ra n s fo rm c a lc u l a t o r w h i c h p r o c es se d m i x e r o u t pu t ba t ches o f d u ra t i o n 1 1 k sec o n d s a c c o r d i n g t o t h e for m u l a '�k
I J( r ) c - iw ' dr
/)
w o u l d resu l t i n a F o u r i er t ra n s fo r m r e c e i v e r h a v i n g t h e s a i l l e prcClsion a s a m a t ch e d fi l t e r r e c e i ve r u s i n g a fi l t e r m a t c h e d t o / I p u l s e s . A t h e o re t i c a l a dva n t a ge o f t h e F o u ri e r t r a n f or m r e c e i v e r i i t s u i t a b i l i t y fo r t h e p roce ss i n g u f a la rge c l a s s o f s i gn a l 3 S u p p os e d t o t h e
50
Signal processing methods
m a t c h e d fi l t e r re c e i v e r w h i c h r eq u i res a d i ffe ren t a n d c o m p l i c a t e d fi l t e r fo r e a c h t y pe o f sign a l . I n p rac t ice t h e a b ove advan t age c a n o n l y b e e x p loi t e d i f a s u i t a b l e m U l t ip l ier a n d calc u l a t o r c a n b e i m p l em e n t e d . A d is a d van t a ge o f t h e F o u r i e r t ra n s fo r m rec e ive r m i gh t b e t h e t i m e t a k e n t o p r o d u c e i t s o u t p u t i n fo r m a t i o n ; t h is d e p e n d s v e ry m u c h on the t y pe o f t ra n s fo r m c a l c u l a t o r use d . A l t h o ugh t h e i n t e gra t i o n t i m e fo r t he F ou rie r t ra n s fo r m c a l c u l at i o n is t h e s a me a s t h e p roc essi ng t im e o f a m a t ch e d fi l t e r , a t r ue F o u r i e r t ra n s fo r m c a lc u l a t i o n h as t o be ca r r i e d ou t at an in fi n i t e n u m b e r of fre q u e n c i e s . I n p ra c t i c e i t w ou l d o n ly b e n e c essa ry t o p e r fo r m t he c alcul at i o n a t a fin i t e n u m b e r of fre q u e n c i e s . T h e fi n i t e i n t e g ra t i on t i m e se ts a l i m i t t o t h e m ax im u m a c h i e va b l e p re c i s i o n , s o t h e r e w o u l d b e n o adva nt age in sp a c i ng t h e c a lc u l a t i o n fre q ue n c ies m u ch c lo se r t h a n t he 'p reci si on fre q u en cy' . A l s o a k n owledge o f the l ik e l y sp r e a d o f t a r g e t p a r a me t er s w o u l d a l l ow t h e calc u l a t ion t o b e per fo r m e d over a fi n i t e r a nge. It fol lows from t h e above t h a t , a l t h ou gh the p rocessi ng t i m e o f a p ra ct i ca l F ou ri e r t ra n sfo rm rec e i v er w o u l d a l w a y s b e s l i gh t l y grea t e r t h a n t h a t o f a c om p ar a b l e m at ch e d fil t e r re c e iv e r , w h e t h e r i t w ou l d b e v e r y m u ch gre at e r depen ds u p o n w h e t h e r t h e c a lc u l at i o n i s c a r ri ed o u t i n ser i a l o r p a r a l l el for m . Pra c t i c a l F ou r i e r t ra n sform sc hemes a r e d i scu ssed b ri e fly i n Se c t i on 4 .6 .
4 . 2. 1 I m p rovemen t of resolution b y weigh ting
As i n t he case o f t h e m a t che d fi l t e r re c eiver , t h e r e so l u t io n p e r fo r m a nc e o f t h e F ou r i e r t ra n s fo r m receiver may n o t b e op t im u m . R e solu t i o n i n th e m a t ch e d fi l t e r rec e iv e r can b e i m p roved b y . frequ e n cy w e igh t i ng ( i . e . t he use o f a n on-m a t c h e d fi l t e r )- t h e c o rr e sp o nd i n g op er a t i o n i n t he F o u ri e r t ra n s fo r m recei v e r is t i m e -w e i gh t ing o f t he p r o d u ct t e r m p r io r t o calcula t i ng i t s F o u ri er t ra n s fo rm . Th e e ffe ct iv e a c t i on o f u n w e i g h t e d F o u ri e r t ransfo ml o r m a t c h e d fil t e r r e c e i v e r s is t o r e p lace e ach p o i n t t arge t ' s p i k e ' b y t h e 3 -d i m e ns i o n a l shape o f t h e u n cer t a i n t y fu nc t i o n . W e igh t i ng h as th e e ffe ct . o f r e p la c i ng e a c h spi k e b y a shape o t h e r t h a n t h e u nc e r t a i nt y fu n ct i o n- h o p e fully a shape w i th im p r oved r e s olu t i o n and n o t t o o b a d ly degrad e d p r e c is i o n prope r t ies . A q u e s ti o n w h i c h a ri s e s is : c a n a F o u r i e r t r a n s fo r m receiver t i m e -we i gh t i n g fu n c t i o n l e a d t o t h e same sh ape as a given m ismat c h e d fIl t er ? The answ e r i s y e s- t h eo re t ica ll y provi d e d t hat t h e t im e origi n o f t h e t i me -w e i gh t i ng fu nc t i o n is s e t b y t h e v a r i a b l e d e l ay c o n t r o l tv' I n m any c ases i t t u rn s o u t t ha t t h e requ i r e d t i m e -w e ighting fun c t i o n is n o t r e a l a n d s o can n o t b e i m p lem en t e d . I t is show n i n Section 4 . 7 (exp ress i o n 4 . 29) t h at t h e sha p e ge n e r a t e d
Signal processing methods
51
b y a match e d fi l t e r receiver u si ng a filter w ith a c o rresponding comp le x a nalytic impulse response o f
a * ( -t)c(t) e j w , t is given by
J a(t - x - z )a * ( - z) c(z ) e - j (wo - w , + y - w v)z dz 00
1-
_ 00
(4 . 1 )
Al so the shape ge nera t e d b y a Fourier transform receiver using a mu l tiplying fu nc tion o f m et - tv) j ust pri o r to c alc u lating the F o u rier t ra nsform is
J a(tv- x - z )a * ( -z ) m( -z ) e - j (wL + y - w )z dz 00
1-
( 4 . 2)
Com parison o f expressions 4 . 1 and 4.2 shows that the c o rrespon ding variables in t h e two receivers are t. Wv and tv, w respectively , as expec t e d . The time w eighting fu nction requ ired t o produce the same shape as a filter having a correspon ding complex a n al y t ic im p u lse response of b et) ej w ,t is met - tv) w here
(4 . 3 )
b ( -t) met) = - a * (t)
A special case o f a time weight ing scheme which does not depend upon the ab ove r esu lt is disc ussed in Section 5 .2 . 1 .
4. 3 THE EFFEC T OF FINITE PROC ESSING TIME
The uncertainty function o f a t rain of n identical pulses can be obtai n e d fr om n-I
n- I
L a(t - ik) L a * (t + T - ik) e -j w t dt i= O i=O
_00 +00
Al th ough the in tegra t i on time runs fro m to Fig, 4. 1 2 sh ows that in the d ominant part of t he -k < T < k) the e ffective integration limits are and
T,
refe'rence to
w plane (i .e .
nk
J for -k < T < O.
-T
In a p ract ical radar system the signal p rocessing is carried out over the d u ration o f a number o f pulses, n. w hic h is v e ry much smal ler than the number actually t ransmitted . Fu rther the processing t ime is not a ffected by the delay T o f the incoming sign a l .
Signal processing methods
52 a(t)
a (t-k)
0
k
" a ( t + -c )
* a (t H - k )
-t
Pig.
4 . / 2.
I
2k
l
[
a t - (n- l )k
( n -l ) k
J
[
" a t +-c - ( n - l ) k
I
]
nk
I
nk -.
Produ ct ill volved in un certain ty fun ction calcu la tion for 0 < T < k
I
a( t )
a
"
2k
k
o
- -c
I
[
a t-(n-l )k
(n-l )k
( t +t )
I
I
f-41·-----
o
I
J
nk
a
Processing Interval
Fig. 4. / 3. S i t u a t ion existing
ill a practical
il-
( t +t - n k )
I I -------.I
rece iver
I
(n + l ) k -t
nk
when 0 <
T
< k
A c o m p a r i s o n b e tw e e n F igs 4 . 1 2 a n d 4 . 1 3 sh ow s t h a t a p r a c t i c a l m a t che d fi l t e r o r F o u ri e r t ra n s fo rm r e c e i v e r w i ll p r od uce a n o u t pu t fu n c t i o n -.J; ( 7 , w) w h i c h is e q u al t o X t (7 , w ) pl u s a n e x t ra i n te g r a t i o n fr o m (n k - 7 ) t o 11k. i .e . -.J; ( 7 , w ) = X t ( 7 . w ) +
J a [ t -- ( n - l )k ] a * [ t + 7 - l l k ] e - i W T d t
v a l i d fo r 0 < 7 < k . I f aC t ) h as a d u r a t i o n d, t he a d d i t i o n a l in t egral i s zer o fo r 0 < 7 < k d, r e su l t i n g i n t h e re d u c t i on o f -.J; (7 , w ) t o X t (7 , w ) fo r t h a t i n t e rv a l . Th u s , fo r l o w d u t y c y c l e pu l se t ra i n s , t he fin i t e p rocessi ng t i m e h as no e ffe c t ove r the d o mi n a n t part o f the 7 , w plane as l o ng as n i s p u t e q u a l t o t he (i n t e g ra l ) n u mb e r o f pu l se s c on t a in e d in t he signal p rocess i n g t i m e . I t c a n a l s o b e se e n t h a t -.J; (7 , w ) w i l l n o t d1 ffer apprecia b l y fro m X t(7 , w ) , w h a t ever t he fo rm o f aCt ) , i f / I is su ffic i e n t l ; h igh fo r t h e c o n t r i iJ u t i o n o f t h e a d d i t i o n a l i n t egral t o b e i n s i g n i fica n t c o mp ar ed wi t h t h a t o f Xt(7 , w ). -.J; (7 , w ) c a n b e e x pe c t e d t o b e sign i fi ca n t ly d i ffe r e n t from X t (7 , w) fo r t h e c a se o f h i gh d u t y cyc l e a n d n s ma l l ; t h i s w i l l n o w b e i n v e s t i ga t e d . By si mp l i fi c a ti o n o f t h e a d d i t i on al i n t e gral , -.J; (7 , w) c a n b e e xp ressed i n t h e fo ll ow i n g fo rm -.J; ( 7 , W ) = X t ( 7 , w ) + e - i 2 (n - I ) 8 X( 7 - k , w ), O<7
Signal processing methods
53
Al so
I/;(T , W ) = X I (T , W ) + X(T + k, w ),
-k < T < O
where X I (T, w) applies to a train of n pulse s , X(T, w) applies t o a single pulse and 8 = 1 wk .
Usi n g t he results o f Sec t i o n 2 . 3 , a n d rememb ering t h a t X( T , w) = 0 for I T I > k, XI(T, w) can b e writt e n in the form
X I (T. w ) =
e
.
- J n ()
sin [ e n - 1 )8 1 x(T + k , w ) Sin e .
sin n 8 - 2 sin [(n - I )8 ] + e -j (n - l ) (} X(T , w) + e -j (n )(} sin 8 sin 8 x
X(T - k, w)
(4 . 4)
valid fo r -k < T < k. Substituting Equation 4 .4 into the fo rmulae fo r I/; (T , w) leads to the expressions sin n8 I/; (T, w ) = e -j ( n l ) (} -'- {X(T, w ) + X(T - k, w )} -
Si n 8
(4 . 5 )
fo r 0 < T < k. Also sin n8 Si n 8
I/; ( T, w ) = e - j (n - l)(} -'- {X(T , W ) + X(T + k, w)}
(4 . 6)
fo r - k < T < O . Th e ab ove expressions fo r I/; (T , w) e xhibit a D opple r b a r e ffec t a s migh t be exp e c t e d an d also verify t h e previous statem e n t s rega rding p ul se t rains with l ow d u t y cyc l e o r large n. For h igh d u t y cyc l e pulse trains the e ffec t o f overlap b e tw e e n X(T , w) and X(T ± k, w) has to b e conside re d ; the c alculation o f I/; (T, w) i s m o re st raight forw a rd t h a n t h at of X I ( T , w) as t h e m u l t i pliers o f the X funct ions are t he same . F o r wavefo rms su ch as linear F M w i t h a h igh d ispe rsi on fac t o r (Sect i on 3 .2), X(T ± k, w) is v e r y small for I T I < k/ 2 a n d b o t h XI(T , w) a nd I/; (T , w) r e d uce t o si n n 8 Sin 8
e - j (n - l )(} -'- X(T, w ),
k k < T <2 2
- -
4.3 . 1 Coded wavefonns
I/; ( T , w) is significantly d i fferent to XI(T , w) when the transmitted signal con sist s o f contiguous coded words. I t is show n b e low tha t
Signal processing methods
54
sub stitu t i on of th e e xpression given in Sec t i o n 2.3 . 1 fo r x(r , w), in Equations 4 .5 and 4 .6 leads t o 8 1/I ( r , w ) = e - j (n - l )
{I L
sin n8 SI n 8
m=O
+ Xb(r -Ld , w )A ( O,
W)
{
A lso
l
X b(r - m d, w )A (m , w )
}
,
0 < r < Ld
8 s i n 11 8 Q Xb(r , w )A (O, w ) 1/I ( r , w ) = e - i (n - l ) -'SI n u
+
where
xi r ,
L-l
m�o Xb(r + ( L - m )d. w ) A (m ,
}
W) ,
- Ld < r < 0
(4.7)
(4 . 8)
w) r e fers to a sin gle b i t pulse and A (m , w )
=
L-l
� ci c7+ m e -i i w d i= O
I t w ill b e n o t e d that t he fun c t i o n A (m, w) d i ffers fr om .s1 (m . w ) o f Sec t i on 2 .3 . 1 in t hat the sum m ation extends ove r t h e full L b i t s rat her t han ove r L - m b i t s . The sign i fi c anc e o f the ab ove resul ts c a n b e s e e n b y compari n g ° ° 1/1 (r , w ) w i t h X \ ( 7 , w ) fo r a 0 , 1 80 phase-coded w avefor m . Typical Ci fo r a I S -b it m axim u m length shi ft register c od e a re
I f the functions are evaluat ed along the 7 ax is a t integral multiples o f t he b i t lengt h , th e resu l t s w il l b e independent o f t h e bit pul se shape (i.e . on ly one displaced b i t X fu nction w ill be involve d at each point a n d i t will h ave u n it amp litude). 1/1 (md. 0) is i ll u s t rat e d b y F ig . 4 . 1 4. It e xhi bi ts the two-level effe c t , ch ar acteristic of maxim u m length codes. T he value o f n a ffe c t s only the amplitude and not t he shape . x\(md. 0) is illustrat ed b y F igs 4 . 1 5 and 4 . 1 6 for n = I a n d 1 0 . The desirable tw o-level e ffe ct is a b sent b u t it c a n be s een tha t the n = 1 0 resu l t is better i n thi s respect than that for n = I . I t should be noted that , w h e n n is smal l , the tw o-l evel e ffect exhib ited by 1/1 (7 , w) i s o n ly p re sen t when processing e x t e n d s over an i n t egral numb e r of w or d s . I f 1 1 is a n on-intege r , the resu l t will l ie som ewhere between the extremes i l l us t rated b y F igs 4 . 1 4 a n d 4 . 1 5 . As 1 1 becomes l a rger, the n e c essi t y for it t o have an in tegra l val u e becomes l ess .
55
Signal processing methods · 15n
I I \ \ \ \ \ \ \ \ \
1 2d
10 d
8d
6d
4d
2d
0 -n
r I I I , I I I I I I , I I I
14d
� - . - . -. - . - . - . -. - . -. -. - .- . - ,
Fig. 4. J 4. 1/1 15
I I \
(md, 0) for
1 5- bit maximu m length 0° . 1 80° phase code
I
\ I \ \
-3
Fig. 4. 1 5.
150
,.
,O
x / md, 0) \ \ I
6d
4d
2d
0
'o
j
/
/"
0
l O d /'12d ..... 14d
8d
,
0
.....
0
/'
.......
0
for a 1 5- bit max im um
length
n =
Fig.
10
4. 1 6.
phase
code:
n =
1
T
\
0 -10
/'
0° . 1 80°
I
I
0
\ \ \ \ \ \ \ \
2d
-- - -
x I (md, 0)
6d
4d .
.
_ e _ .
8d _ _ _ .
12d
lOd
I I I , I I I I I I I 14d
. _ - - e _ _ _ _
,
for a 1 5- bit max imu m le ngth 0° , 1 800
4 . 3 .2 Proof of Equation 4 .5
I t w as s h ow n a b ove t h a t in t h e ra nge 0 <
T
< k
T
ph a se
code
hu t
Signal processing methods
56
Usi ng Equ atio n 4 . 4 a n d remem bering t h a t for 0 <
r
I/; (r , w)
x(r + k , w)
=
0
< k the ab ove b ec o me s =
. si n n e e - J ( n - l ) fJ -'- x(r, w ) e Si n
+ x(r � k, w ) { e - j (n - 2 ) fJ s i n [ ( n Sin e
-
l ) e ] + e - j2 (n - l )e sin
Exp an d i ng the sine fun c t i o n s , t he b racketed exp ressio n b ecom e s
L
H e nce
{ e -j (n - 2 )e e j (n - l ) fJ _ e -j (n - 2 ) fJ e -j (n - I ) e + e -j 2 ( n - l ) fJ e W _ e - j 2 (n - l ) e e -W }
ej fJ
=
2j { l _ e -j 2n fJ }
= e -j (n - l )fJ si n n e
sin n e I/; ( T w ) = e - i ( n - l )fJ -'- { X(T, W ) + X ( T - k , w ) �' Si n e .
w h i c h is E q u a t i on 4 . 5 .
4 . 3 . 3 Proof o f Equatio n 4 .6
I n t h e r a n ge - k < T < 0 Also X(T - k , w )
I/; ( T . w ) = x (r . w ) + X ( T + k , w ) \
=
O . U si n g Eq u a t i o n 4 .4 t he a b ove b e com e s
l ) fJ s i n l 1 e I/; ( T , w ) = e - J ( n -' e X( T . w ) S in "
-
X( T
+
� k , w ) { e - in fJ s i n [( n
sin e
-
l )e 1 + sin
e}
T he expanded b r a c k e t e d ex pression becomes � { e - j n fJ ej (n - l ) fJ _ e - j n fJ e - j ( n - 1 ) fJ e W - - j e � e + '
2J
=
ej e 2j { 1
. . _ e - J2 n fJ } = e - J (n - l ) fJ s i n n e
Hence I/; ( T , w )
=
e - j ( n - I ) f)
w h i ch is E q u a t i o n 4 . 6 .
si n '18 { X ( T . w ) + X( T + k , w ) } S1 11 e
e}
Signal processing methods
57
4 . 3 . 4 Proof o f Equation 4 . 7
T o d e rive Equat i ons 4 . 7 a n d 4 . 8 It I S necessary to s e t t he repetition period k = Ld and then use the fo rmula X ( T, W ) =
L-I
L
m =O
Xb(T - m d, w ) .stI (m , w) +
L- I
L
m= 1
e - i m w d X b( T+md , w ) .stI *(m, - w )
in Equations 4 . 5 an d 4 . 6 . Over t he range 0 < T < L d, �(T + md, w) = 0
hence X(T , w ) become s L -I
L
m=O
X b( T - md, w ) . S'I (m, w )
Sim ilarly X(T - k, w ) = X(T - L d, w ) becomes Xb( T - Ld , w) .>1 ( 0 , w) +
L -I
L
m =1
e-imWd Xb( T + (111 - L )d . w) sl *(m , - w)
Ch anging t h e s u m m a t i o n va r i a b l e t o (L w ri t t e n
+
L -I
L
m=1
- m ) a l l ow s X(T - L d,
w)
to
be
e -i( L - m)wd Xb( T -md . w ) . 0J * (L - m , - w )
Hence X(T , w ) + X(T - L d, w ) b ec o m e s I. - I
+ L Xb( T - l I 1 d . w ) { .d(I I 1 , w ) + e-j( L -m) w d d *(L m= 1
S ince cy' { I 1 / , W)
. d * ( I,
Ill ,
=
I.
-W )
1 - 11/
L
Ci Ci*+ 1 1 1
i=O
=
m
I
L C(Ci ; I .
i=O
C
- ji w d
-
Ill
.
-w ) }
Signal processing methods
58
C h anging the sum m at ion va riable to
(i -
L- I
.s:J * (L - m, - w)
=
L
i= L - m
L + m ) give s
ci c7- L + m e -j (i - L + m ) w d
Si nce t h e se quence is peri odic wi th length L , it follows t ha t
Ci*- L + m = Ci*+m
Hence t he expression for x(r , w ) + x(r - Ld, w ) c an b e w ri t ten
{ Xb( r , w) + X b( r - L d . w )} � ( O . w) +
x
{L
i
I
�
-
-
m
* Ci Ci
+ 1 11
e -ji w d +
i
L
=
-
I
1--
1 11
LL I -
m =1
X b( r - md , w )
ji Ci c:+ m e - wd
}
Substi t u t i o n o f t he a b ove in Equat i on 4 .5 gives Eq uat i on 4 .7 .
4.3 .5 Proof of Equation 4.8 Ov e r t h e r a n ge
- Ld < r < 0 , Xb(T - md, w ) = 0 hence x(r , w) becomes
X b( T . W ) d ( O , w) + C h a n gi n g written
t he
L -I
L e -j I l1 Wd X b( T + md , w ) sy *( m , -w)
m=1
s u m m a t ion v a r i ab l e t o ( L - m ) allow s x(r , w) t o be
I.
- I
Xb( T, w ) ci ( 0 , w ) + L I II 1
C-
j ( I. - m ) w d X ( r + (L - m )d , w ) d *( - Il l , -w) L b
=
X( T + Ld, w ) he comes L -I
L X b ( r + ( L - m ) d , w ) .9I(III , W)
11/
0
X( T , w ) + X(T + Ld, w ) b ecomes
Hence
I. - I
+
=
L
111
=1
Xb( T + ( L - m )d , w) {
s/( m , w ) + e -j (L - m ) wd .91 * (L - m , - w ) }
which c a n b e subst i t uted in Equation 4.6 t o give Equation 4 . 8 .
59
Signal processing methods 4.4 T H E MAT C H E D FILTER CON C E PT
Consi d e r a filte r w i t h a real impu l se response o f g (t) w hich is used t o process a real sign al f(t ) . Assume t ha t t h e filter outpu t h(t) h a s i t s peak value at t = t o . The convolution theorem (Refe re nce 8 , p age 3 9 ) a l l ow s the peak ou t pu t t o be w rit ten in the form
J f{x)g(t o - x) dx 00
h(t o ) =
(4 .9)
It w ill n ow be of i n t e rest to consid e r a second fil te r of impulse resp onse f.1f{tm - t) , where f.1 is a real constan t . Assume that the outpu t o f t his filter is hm (t) and has its peak valu e a t t = tm . Applying the c onvolu tion the orem once m o re , gives
00
00
hm(tm ) =
•
r
f(x) f.1j( tm � tm + x ) dx = f.1
r f (x) dx
(4. 1 0)
_00
The ratio o f t h e pe ak o u t pu t s fr om the tw o filters is given b y
�lf{x)g( to - x) dx L
f.1 _ f ( x ) dx I f the Schwarz inequality [ 6 ] is app l ie d to the integral in the numerat o r , the fol l ow in g relationsh ip re su l t s
(4. 1 1 )
Now
oor
fOO
. g2 ( t o - x ) dx = _ g 2 (X ) dx = 00
(
)
E nergy of the imp U l se re sponse of g( t)
I f the gain factor f.1 i s adj u sted to m ake t h e ene rgies of t he tw o filter impu l se re sponse s equal , Equation 4. 1 1 redu c e s to
Ih ( t o ) 1 �
I hm (tm) I
Thus, i f f.1 i s selected t o give equal fil t e r impulse response energies the peak output of an a rb i t rary filte r , w ith inpu t f(t), can never be gre a t e r th an th at from the fil ter having f.1f(tm - t) as i t s im pulse response. Since
.�{f( t + tm) } =
ejw tm FQ w )
Signal processing methods
60
i t fo l lo w s t h at . � {f(tm
- t)}
=
e - jw tm F( -j w)
The t e nn e -j w tm is the t r an sfe r fu n c t ion of a pe rfe c t delay line (i t s
d e l a y bein g f m ) . A s t he i n c l u sion o f a de lay l i n e will n o t a ffe ct the m agnit u d e of the peak valu e of the fi l t e r ou tpu t , i t is cust omary to refe r t o t h e fi l t e r h a v in g a t ran sfe r fun c tion F( -j w) as th e 'm a t ched fi l t e r ' fo r je t ) . I t fo l l ow s t h a t t h e i m p u l se r e s p o n s e o f t he m a t c h ed fil t e r is f( -t) a n d ( a s j( l ) is ass u m e d t o be re at ) t h a t its t rans fe r fu nct ion is given b y e i t h e r F( -j w) o r F * U w ) . I t s h o u l d be n o t e d t h a t i t i s a lw a y s n e ce ssa ry t o u s e d e l a y i n t h e real isat ion o f m a t c h e d fil t e rs fo r o n e -sided t i me fu n c t i o n s . T o illust rate t h is fac t . l e t
J{ t } =
� t -
i . e . J( I ) is o ne - s i d e d a n d t i m e l i m i t ed t o d sec o n d s . The impulse resp o n se o f t h e c o r respon d i n g m a t c h ed fi l t e r w i l l b e given by R( t ) = j( - t ) , i . e .
g( t } =
� -d
t -
0
Th e a b ove i mp u l se respon se is u n re a l i sa b l e , s i n c e i t i s ov e r b e fo r e t h e i m p u l se i s a p p l i ed � H ow e ve r a r e a l i sab l e i m p u l se respon se w i t h a l l o f th e p rope r t i e s o f t h e m a t c h e d fil t e r w o u l d b e R( r d ) , i .e . -
g( t - d ) =
� o
t -
d
[ t c a n be seen t h a t m a t c h e d fi l t e r s a rc o n l y u n re a l isa b l e in t h e case of non t i m e-l i m i t e d wave fo r m s : such wave fo r m s d o n o t occ u r i n p r a c t i c e , o f c o u rse .
4.4. 1
N o ise pro pert ies of fil t ers
[ f n o ise of c o n s t a n t spe c t ral d e n si t y ( i .e . w hi t e n oise ) is a pplied to the i n p u t o f a fi l t e r , the ou t pu t n o ise pow e r w il l be propo rt ional t o t h e a re a u n d e r t h e m agn i t u d e s q u a r e d response o f t h e fil t e r . Th u s
J IG(j w ) 1 2 dw 00
( O u t p u t n o i se pow e r )
a:
Signal processing methods
61
Parseval's the orem (Appen dix 2 ) show s that the ou tpu t n oise power is also proportional t o the en ergy of the filter impulse response . Thus (Output noise power)
ex
00
J g2(t) d t
_ 00
The above facts lead to t h e concep t o f t h e noise bandwi dth of a filter. The noise bandwidth of a l ow-pass filter is defined as the b an dwidth of a rect angular filt e r having the same d . c . gain , and a m agnitude square d response which encloses the same area as the magnitude squared re sponse of the low pass ftl ter. Since the area under the m agnitude squared re sponse of a unity gain rectangular fi l t e r of b an dwidth B is given by 2B. the fo rmula fo r the noise bandwid th of a filte r of transfer function GO w) is
{
N oise b andwid th (rad /s)
}=
00
J
_ 00
.
/GUwW d w
2 /G(OW
00
= J
2 rr g ( t) 00'-----"-' _
dt
_ _
/G(O) / 2
(4 . 1 2)
The peak signal-to-noise ratio at the output of a fil ter having an impulse respon se get) can be define d as
(�) = ;2(tO)
.1 g2(t) d t
(4 . 1 3)
_ 00
where h(to ) is the peak value of the output due to f(t) at the inpu t . This definition makes (SIN) proportional t o the ratio of peak signal power to noise power. Now consider a matched filter with a peak signal-to-noise ratio o f (SIN)rn ' A ssume that t h e fil te r impul se response is given by J11( - t) a n d t hat J1 h a s been adjusted to equalise t h e impulse response energies, i . e . s o that 00 00 2 g2(t) d t J1 f( - t) d t •
=J
r
I f t he resulting peak ou t pu t signal is
_ 00
hrn (trn ), the output pea k signal
to-noise ratio will be given by
(�)rn = h: trn_)_ = -(
_ _
J1 2
_
J f( - t) d t
Combining t his re sult with Equ ation 4 . 1 3 gives (4 . 1 4)
Signal processing methods
62
As it has al ready b een shown tha t h� (t m )' it follow s that
h2 (to ) can never be greater than
Thu s i f a signal , embedded in white n oise , is applie d t o an arb i t rary filter the o u t pu t peak sign al- t o-noise ratio will never b e greater than the peak sign al - t o-noise ratio a t the ou tpu t o f the matc hed fil t e r .
4. 5 T H E M A T H E M AT I C A L T RE A T M E N T OF T H E
MAT CHED
FI LT E R R E C E I V E R
T h i s se c t ion i s u se d t o i n v e s t iga t e t h e R F e nvelope o f t h e ou t pu t of the rece i ve r d iscussed i n Sec t i o n 4 . 1 ( F i g. 4 . 1 ) . The t ran smitted si gn a l is a s sum e d to be
fi t , wo ) = la( t ) 1 cos [ wo t + ¢(t ) ] a n d t he r e t u rn s ign a l a d elay e d an d D oppl e r sh i fted version o f the t ra n s m i t t ed sign a l , i . e .
f( ( t - x ) . ( W o + Y ) I
The m a i n m i xe r o f t h e rece i v e r is a s s u m e d t o h a v e s i d e b a n d ou t p u t o f
a
sin gle (lowe r )
f(( t - x ) , ( wo + y - wv) I
I t w i l l a l s o be a ss u m e d t h a t ( wo - wv) i s h i g h enough for the e x p o ne n t ia l a p p r o x im a t i o n t o t he c om p l ex a n a ly t i c signal t o apply ( see Se c t i on 7 .4 ) . I t w i l l be s h ow n t h a t t h e o u t pu t R F e nvel o p e o f a n a rb i t rary fi l t e r , h a vi n g a com p le x a n a l y t ic fo rm o f B U ( W - WI ) J is g i v e n b y
Ih a( r ) I = +
J a( z ) b( r - x - z ) e j(wo - w, - w v + Y) Z d z
00
(4 . 1 5 )
I t is a l so show n t h a t t h e i m pu l se a n d freq uency re sponses of t he c o m p l e x a n a l y t ic fi l t e r co rrespon d i n g t o t h e m a t che d fil t e r fo r j(t) a re gi v e n b y
Fo r t h e m a t c h e d fi l t er case t he re ceive r ou t p ut R F envelope is given
by ( 4 . 1 6)
w he r e I X(T , w ) 1 is t he uncertainty fu n c t ion derive d from fit) .
Signal processing methods
63
4 . 5 . 1 Derivation of results
F rom t he above discussion it will be seen that the impulse response of t he arbitrary comp lex analyt ic filt e r is given by b et) ej w , t. Also the complex a nalytic signal corresponding t o the filter inpu t is
aCt - x) ej (wo + Y - w v) ( t - x ) where
aCt) = la C t) I ej1>( t)
Hence , u sing Section 7 .6 (Fig. 7 .7 d) and the convolution theore m , the com p le x analytic form of the rece iver out pu t is given b y
f a( z _ x) eH wo + y - w v) (z - x) b (t _ z ) eH t -z) w ' dz 00
haC t) = t
_ 00
which after a change of dummy variable reduces to
J a(z ) b (t - x - z ) ej ( wo -w , - w v + Y)z dz 00
h/.. t) = t e H t - x ) w ,
(4 . l 7)
The RF envelope is given b y IhaCt) I (see Section 7 .3 ), hence Equa tion 4 . 1 5 follow s . I f t he arbitrary filte r is now replaced by a matched fi l t e r o f impulse response get) it follows from Sect ion 4 .4 that
g(t) = f( -t) C(jw) = F( -jw) = F*(jw)
( 4 . 1 8)
(4 . 1 9)
To fin d the complex a n alytic filter corresponding to C(jw) note that
fi t) = R e {faC t) } = t f!a (t) + fat t) ]
Hence , u sing Section 6 . 7
F(jw) = HFi/(jw) + F;( -jw) } which with Equation 4 . 1 9 gives
C(jw) = HF;Uw) + Fa (-j w)} Sinc e , by definition, Fa(j w) = 0 , for w < 0 , and Fa ( -j w) = 0 , for w > 0 , Section 7.6 (Equation 7 42 ) leads to .
CaUw) = F;Uw) Note particularly that F:(jw) i s complex.
(4 .20)
not equal to FaC -jw), since faCt) is
Signal processing methods
64
Using Section 6.7 (Equation 6 . 1 7 ) with Equation 4 .20 gives
&it) = r:c -t)
(4 .2 1 )
It follow s from Equation 4.2 1 that the outpu t from a matched filter receiver can be obtaine d from Equation 4. 1 7 by replacing b(z) by a*(-z ), hence
f a(z)a* (z+x - t)ej (wo - w . - w v + Y)z dz 00
hi t) = !ej (t - x )w .
(4.22)
The integral in Equ ation 4.22 will be re cognised (Equ a tion 2 . 1 4) as [-(t - x), -Cwo - W I - Wv +y)] . Hence , the outpu t R F envelope is given by X
(4 .23 ) Equat ion 4 .2 3 (Equat ion 2 . 1 8 ).
follows
from
4.22
by
virtue
o f Section 2 .2
4. 6 PRA CTIC A L FOU RIER TR A N SFO R M C A L C UL A TO R S
The calcu lator u se d i n t h e F ou rier t ran sform receive r i s requi re d t o fmd
r1{ t) e - jwt dt
b
a t a selected nu mb e r of valu es of w . The param e t e rs n an d k represe nt t h e number o f processed pu lses and the pulse repe tition frequency p e ri o d , resp ec t ively. Once one bat ch of pulses h as been p rocessed it is required that the in tegrat ors be re se t , ready to pr ocess the ne x t b at ch . The direct m e t h o d o f im plementin g the calcu lat o r w ould b e to expand the a b ove integral to give
J
nk
f( t ) cos( w t) d t j -
j
nk
1{ t ) sin( wt ) dt
Since it is t he modulus (or t he squ are of t he modulus) of t he a bove sum wh ich is require d , a direct imp lemen tation me thod w ould be by the system illustrate d in Fig. 4 . 1 7 . I f p recision requiremen ts dictated calculat ion a t m frequencies it would either be n e ce ssary t o use m parallel channels (ea c h with a different value o f w) , for a fast read ou t , or t 9 t ake m times as long by using one chan nel m time s . T h e system of F ig . 4 . 1 7 could be replace d by any o f the well know n syste ms u se d fo r spec t rum analysis. Owin g t o the rapid a dvance s in in tegrated circuit technique s a part icula rly a t t ractive m e thod m ight be to sam p le the outpu t signal from the re ceiver mu ltiplie r and then fee d
65
Signal processing methods ,...---- x
Square
cos ( w t )
Os c i l l a t o r
In put -
Output
90
·
P ha s e - s h i f t
s i n ( wt )
L..---
x
Square
Fig. 4. 1 7. D irect implementation of F. T. calcu lator
t he samples t o a digital system for calc ulation . A parallel method of calculat ion , making the m ost of finite logi c speeds , w oul d be one of the FFT algorithm s discu sse d in Section 6 . 1 4 . The classic sp ect rum analy sis sche me , using a narrow band I F amp lifier i n conjun ction with a sw ept local oscillator i s esse n t iall y a series scheme sin ce the oscillato r must sw eep slow ly fo r a ccurate results. The equ ivalent parallel scheme would employ a bank of narrow band fil ters each tuned t o a different frequency . The filter bandwi d ths should be approximately equal to l / ( n k) H z .
4 . 7 THE
M ATHE M ATI C A L
TREA T M E N T
OF
THE
TRA N S FO R M REC EI V ER
With reference t o F ig. 4.7 the t ran smitted signal i s de fined as
f i t, wo ) = [a C t) [ cos [ wo t + ¢( t ) ] = Re{a(t) ei wo t} The inputs t o t he main m ixe r a re
Je t - x,
Wo
+ y)
and
j{t - tv, Wo - WIJ
F O U R I ER
66
Signal processing methods
Hence the
product Jt t - x ,
Wo
+ y) f(t - t",
Wo - WL) is equal t o
k7(t - x) 1 k7(t - tv) 1 cos [ (w o +y) (t - x) + ¢(t - x) ]
cos [( Wo - wd {t - tv)+¢(t - t v)]
= t la{t - x) 1 kz(t - tv) 1
{cos [ (y + wL)t + ¢(t - x) - ¢( t - tv) +(w o - wdtv -{Wo+y)x ]
+ cos[(2wo - WL +y )t+¢(t - x) + ¢(t - tv) -Cwo - WL)tv -{ WO+y)x] ) The mixer outpu t i s assumed t o be the first (low fre quency) tenn i . e . x o + y)x L L a la( t - x) l Ia( t - tv)1 e W ( t - ) e - j ¢ ( t - tv) ej (y + w )t � (wo - W )tv e j ( w + +a la( t - x ) 1 kz( t - tv) I e -j ct>(t - x) � ¢ ( t - tv) e -j (y W L ) t e -j (w o - W L )tv ej (w o + y )x
w hich is e q ual t o
)x M a( t - x ) a * ( t - t v) eJ (y + w Ot eJ (w o - wO tv e - j (w o + y + a *( t - x ) a( t - t v) e -j(y + w O t e -j ( wo - WL ) t v ej (w o + Y)X }
(4 . 24)
But 00
J a( t - x) a * ( t - tv) e - j w t d t
.?" { a( t - x) a * ( I - tv) } =
00
J a( l ) a *( 1 + x - tv) e -j(t + x ) w d t
=
_
00
( 4 . 2 5)
= e - jwx X Ix - tv, w ]
Si m i l arly
( 4 . 26 ) wh e r e
J a( t ) a * ( t + T) e - j w t dt 00
X( T , w) = _
00
Using Equations 4 . 2 5 a n d 4 . 2 6 t h e Fourier t ransform o f 4 .2 4 can b e w rit t en ! { e j ( w o - w L ) t v e - j ( w + w o - wOx X [ x - tv , W - Y - W L] +
For
a
e - j ( w + W o + y ) t v e j (wo + y )x X [ tv - x , w + y + wd }
real b a n dpass fun c t ion h ( t ) 9"{h ( t) } = HUw) = IliU w ) I ej8 (j w )
(4 . 2 7)
67
Signal processing methods Hence
h(t)
=
1 2 rr
I
00
I
00
HUw) e jw t dw
_ 00
=
0
2 IHUw) I cos[wt +euw)] df
If h(t) were applied to a physical 'Fou rier transform calcu lator' (e .g. a spectrum analy ser) the outpu t wavefo rm w ould represent the amplitude distribution of the various cosine components, measu red in V/Hz , i.e . 2 IHG w)l. It can be seen , from the ab ove remarks , t hat the outpu t from the F ou rier transform receiver w ill be given by twic e the mo dulus of the exp ression 4.2 7 , when evalu ated for w > O. If W L i s sufficiently high, the contribution from X [ tv - x, w + y + WL ] will be negligible leading to an ou tput waveform given by
4 . 7 . 1 T h e effec t o f weigh ting
The e ffect of t ime weighting in a F ou rie r t ransform rece ive r w ill now be compared with that o f fre quency weightin g in a matched fi lter receive r . To investigate the e ffec t o f mUltiplying the signal at the in pu t o f the Fourie r transform calc ulator (Fig. 4 . 7 ) by the func tion met - tv) , i t is necessary to put m et - tv) as a mul tipl i e r before expre ssion 4 . 2 4 . The resu ltan t exp ression for the Fourier t ransform modulus can be written as
II_Ia(t, - x - z)a*( - z) m( -z )
e
-
j ( W L + y - w ), dz
1
( 4 . 28)
The e ffect of frequency weighting in a matched filte r receive r can be investigated b y assuming that t h e comp lex analytic impulse resp onse corresponding to the receiver filter is a * ( -t)e(t) e j w " rather than a*(-t) ej w1 t. Replacing bet - x -. z ) in Section 4 .5 (Equa tion 4 . 1 5 ) by a *(z + x - t)e(t - x - z ) and subsequen t simplification gives the outpu t RF enve lope as
!/ _f a(t - x - z)a*( - z )c(z ) e -j (wo - wl + y - w v)z / dz
Expressions 4 . 2 8 and 4 . 29 are discussed in Sec t i on 4 . 2 . 1 .
(4 . 29)
Chapter 5 SOME E XAMPLE S O F SIGNAL P ROCE S SI NG M ETHOD S
Exampl es are give n in this chap t e r of the methods used to pro cess two t ypes o f modulated signa l . The resu lting p recision and re solution are d isc ussed in t e rms of the quantitative d e finitions g iven i n Section 1 04 .
5 . 1 THE
RECTA N G U L AR P U LSE
W ITH CONSTA N T
C A R R IER
FREQ U E N C Y
A re c t a ng u l a r p u l s e w i t h consta n t carrier frequency c a n be represe nted by
fi t ) =
r -0-1 cos( wot)
(5 . 1 )
Hence , fo r W o sufficiently h igh ( 5 .2)
I
I
The appro p riate unce rt a i n t y fu nct ion was evaluat e d in Sectio n 3. wh e re it was show n ( Equa t io n s 3 . 1 0 and 3 . I ) that t h e inherent range and ve l o c i t y precision s are r = 0 . 5 cd ( 5 .3 ) v
= 0 .6
c
f� d
( S A)
This pe rfo r m a nce can be ach ieved by using a n unweighted matched fi I t e r or Fourier t r ansform receive r.
5. 1 . 1
The Fourier transform receiver
The F o u ri e r t ransfo rm receive r of Fig. 4 . 7 is o ft e n used with this wave form in prac t i cal radar sy stems. So metimes the delay tv is se t at a con stant value and the system used t o me asu re the velocities o f t a rget s 68
Some examples of signal processing methods
69
in a fixed range bracke t . The Fourier transform calculator is often implemented as a bank of narrowband filters each tuned to a different centre frequency . To design such a system, given a specificat ion of the desired range and velocity precision, one should choose the pulse length from Equation 5 . 3 and the carrier frequency from Equation 5 .4 . This procedure will allow the fastest readout of velocity information- the required Fourier transform integration time being equ al to the pulse duration d. In the case o f a filter bank the filter bandwidth (in Hz) should be equal to abo ut l /d. Equation 3 . 5 gives _
I X( O , w) l - d
I
sin( wei12)
(weiI2)
I
( 5 . 5)
To avoid loss of p recisio n d ue to the inte rference of the two unce rtainty func t ions correspond ing to a given target (see F ig. 4.8) it is necessary to choose an offset frequency WL such that I x(O, w) 1 is ve ry small for w equal t o 2 wL ' The expression in Equation 5 . 5 will be more than 6 0 d B dow n for wei > 2000, thus a sui t able choice for WL (and hence the I F frequency) would be � 1 000 ( 5 .6 )t wL :::- dThe pulse repetition frequency period k woul d b e c ho sen high enough to eliminat e seco nd t ime round errors. Al tho ugh it is t heoret ically po ssible to carry out the above p rocedure, in m any cases the required carrier frequency will be too high fo r p ractical implementatio n . Comb ining Equat ions 5 .3 and 5 .4 give s the required carrier frequency as 0.3 fo = vr
c2
( 5 . 7)
Taking an uppe r prac tical limit of 1 00 GHz for fo (in t e rms of lase rs the upper limit could b e as high as 1 06 G Hz), gives vr
( 5 8)
� 0. 1
.
The units of v and r are mile/s and miles respectivel y . By w a y of example, a p recision specificat ion of 1 mile a nd 3 60 mph would lead to d = 1 0 . 8 J1s , t
fo = 1 00 GHz,
I t is usual to em ploy a lower value o f I F
h � 1 5 MHz (w V
than that indicated by
Equation 5 . 6 . This is permissible beca use the (sin x )/x expression of Eq u a t ion 5 . 5
applies to a pu lse with z ero rise and fall times.
Some examples of igllal pr ocessing metJlU ds
70
b d l f i l t e r i n t h e fi l t e r b a n k w u u l d r e q u i re a b a n d w i d t h o f a bo u t 1 00 k i l l . I f t he
r eq u i r e d c a r r i e r fr e q u e n c y c a n n o t b e a c h i e v e d a c u m b i n e d
r a n g e a n d v e l o c i t y p r e c i s i o n s p e c i fi c a t i o n c a n b e s a t i s f i e d b y u s i n g t h e
i m p r o ve m e n t
in
Doppl e r
p r ec i sio n
d u e t o r e pe t i t i o n . T o e x p l o i t t h e
p r e c i s i o n i m p ro v e m e n t i t i s o n l y n e c e ssa ry t o i n c r e a se t h e c a l c u l a t o r i n t eg r a t i o n t i m e t o 1 1 k . w h i c h g i ve s a n i n t e g r a t i o n o v e r
II
p u l se s . T h e
fi l t e r b a n k b a n d w i d t h s s h o u l d b e r ed u c e d t o a p p ro x i m a t e l y I
1( lIk ) H z .
T h e I l e w e x p r e s s i o l l fo r v e l o c i t y p r e c i s i o n ( r e p l a c i n g E q u a t i o n 5 .4 ) i s v =
0.6
j'
c
II � I
k
0 11 '
( 5 .9 )
e q u a t i o n 5 .9
i n d i c a t e s t h a t v e l ( ) c i t y p r e c I si o n c a n be b y l e ng t h e n i n g t h e i n t e g r a t i o n t i m e it s h o u l d b e n o t e d t h a t t h e r e ad o u t t i m e be co m e s p ro g ressive l y l o n g e r . h e n ce c h a n g e s i n v e l o c i t y w i l l t ake l o n g e r t o be c o me a p p a r e n t . A l t hough
i m p r o v e d i n d e fi n i t e l y
[O, w� /
E nv e l o p e of X
� ./ r
...
-
" , "
.,., /
_ _
-
./
, I ' I
" " , , , , , , , , I
:
, ,
,
, , , , , , '
.:�=:� .:: -=- r= -
E n velope of _ _ _
.
1' , . • "
" I' " .'
" " "
x [o. w - yJ
---:: ----....--.. - - - - / ,
� -=
"
;,
"
" " , , , , , ,
' ' ' ' ' '
-y
o
y
1
'2 PRF
w -
Fig. 5. 1 . Th e sli m of '« (0, w - y) a nd x((), the wciRh til/g fll nc t io n du e to r ep etit io n
w
, , , , , , , , , ,
\
+ y ) when bo th are multiplied by
I f o n e is rel y i ng u p o n t h e Doppl e r ' ba r' e ffe c t t o give
v eloc i t y
t h e syst em c a n be s i m p l i fied b y o m i t t i n g t he o ffse t a h o mo d y n e receive r- t h e p r i c e p a i d i s a l o s s o f k n ow l e d ge o f t h e sign o f v e l o c i t y c h a nges. R e fe re n c e t o F ig , 5 . 1 w i l l show I h a t . for t h i s spec ial case . o m i ss i o n o f t h e o ffse t o sc i l l a t o r d o e s n o t l e a d t o a l o s s o f p r e c i s i o n a s l o ng a s t he ma x i m u m Dopple r fre q ue n cy i s less t h a n h a l f t h e p u l se re p e t i t io n fr e q ue n c y . precision
osci l l a t o r - l e a d i ng t o
Some examples of signal processing methods
71
5.1.2 The matched filter receiver
It will be seen from the above discussion that the implementation of a Fourier transform receiver for a constant carrier rectangular pulse is relatively straightforward. The physically realisable form of the matched filter receiver would need a filter with an impulse response . given by a suitably delayed version off(-t). Reference to Equation 5.1 shows that, for this particular waveform, such an impulse response would be given by the expression for
f(t) itself. The corresponding filter
transfer function would be given by
HG*[j( -w -wo)] +G [j(w-wo)]}
(5.1 Oa)
where (5.1 Ob) The matched filter would actually be implemented at a suitably high IF frequency wI-rather than at wo-and used in the system of Fig. 4.1. It turns out that it is not particularly easy to construct such a filter (a method is given by Skolnik
[14] ) so most practical systems would use a 4.1 scheme.
non-matched filter in the Fig.
As an example of a non-matched system the performance using a single tuned circuit type of filter will now be considered. It is assumed that the complex analytic form of the filter transfer function is (S.11a) where
BUw)
=
a . a+Jw
(S.11b)
The corresponding baseband impulse response is
b(t)
=
a e -at u(t)
(5.12)
It is shown below that use of the above filter leads to a receiver
output envelope of the same shape as a cut (parallel to the r axis) through the following 3-dimensional shape
0,
11jI(r, w)i
r
=
r>d
72
Some examples of signal processing methods
To evaluate the above expression it is necessary to decide upon a value for the product ad. It is commo n practice to make the 3 dB bandwidth of the filter (in Hz) equal to the reciprocal of the pulse
width. The 3 dB bandwidth of the RF filter is equal to 2a rad/s, thus = 1T.
the above philosophy implies ad
An alternative approach would be to use a filter having the same noise bandwidth as the matched filter. It is shown below that this =
implies ad
2.
Figures 5.2 and 5.3 show the effects of the above choices of filter bandwidth upon the delay and Doppler performance; matched filter responses are also shown for comparison. The results are summarised in Table 5.1 in which the delay figures are values of figures are values of wd/21T.
T /d
and the Doppler
It can be seen that either choice of bandwidth leads to results which are not too different from those obtained with a matched filter. Neither
o
--------.--_
/
"
- 20
/
I
/
/'
-
"- "-
-"
Single tuned circuit
----- Matched filter
'\
I
dB
-40
-60 �------�--� o
o
-20
------ - --,-_ _
I , I
/
I
"" I
'"
"- "-
/
""
\
\ \
\
\
dBl -40
-60 �------�--� o 2 3 4
'tId
(b)
(h)
Fig 5.2. Delay performance (a) ad = rr (i.e. IF bandwidth lid Hz) =
ad
=
2
(i.e.
IF
bandwidth
=
21rrd Hz
Some examples of signal processing methods o
Single tuned circuit Matched filter
... , ""
\
-20 dB
-40
\ I \ I I I II I '' I
\
"
11 ,
II I I
-60
73
\
I \ \ ,' \ \II \I II 11 i I
I I
I I
....... \"---__-: , ,. , :=c .... ... ..___ . :o.. c-... \ , /-', I \ , \ '' \ \ \, I , \,
1/
\
W
II II \I II "
II \I
:
I I
, I
II II
II I I
I
\ , II 11
I
U
�---L----�--��o 2 4 6 e
wd/2n.
(a)
\ \ \
\ /" .... , \,' \ /'-, \: \\ '/ \\ '/._\\ \1 I , \ II I, II II \
-20 dB
-40
-60
I I I
:
I
II
II II \I
U •
I
H
: I
:
I I
/-\\
I
\ \
II II \I
\
"
I
/' ....,
I I II 1/
i
I
1/
I,,'
\
\
I
/
-
\
"
\
I
\/ II II
\ , I I
., II 1/
,
/-,
\ \
\
\ \
I
I
: I , : �I �I : I I I I I I �---L----�--L---�--� o 2 4 6 eWd/2n (b)
Fig. 5.3. Doppler performance (a) ad (b) ad rr (i.e. IF bandwidth l id Hz) =
=
2 (ie.
IF bandwidth
=
2/rrd Hz
=
Table
5.1
Matched filter Delay precision dB delay resolution Doppler precision Comparative SiN (dB)
60
1
2
1.2
o
ad
=
2
1.05 4.5
1.4 -1.26
ad
=
rr
1
3.2
1.6
-2.34
the matched filter nor the single tuned circuit will give good Doppler resolution, since both IX(T, w)1 and IW(T, w)1 fall off as l/wfor large values of w.
5.1.3 Derivation of results
The equations leading to Figs 5.2 and 5.3 will now be derived.
]4
Some examples of signal processing methods
Single pole filter The expression for 11/1(7, w)1 can be derived by using Equation 5. 12 with Section 4.5 (Equation 4. 15 ); this gives the output RF envelope as
Remembering that u(t - x assuming that (wo-
=
For 0 < (t
-
=
=
-
z) is equal to zero for z> (t - x) and
l
�
!!..- e-
1
x) < d
lh.,(t )1
For (t
-
;e
_a e 2
- ( , - x )a
lr
x
e(a+ iY)' dz
1
-( t - x ) a e
I
x) < 0 hJt)
a+.iY
=
l
0
Putting (t - x) 7 and y w and simplifying the above expressions leads to the expression for 11/1(7, w)1 quoted in Section 5.1.2. Since =
=
(5.13 )
the filter noise bandwidth will be given by
r b2(t) dt 00
J Noise \ Bandwidth
}
rr =
_00
IB(o) l�
=
rra 2
Equation 5.14 follows from Section 4.4, Equation 4.12.
(5.14 )
75
Some examples of signal processing methods Matched filter The baseband impulse response of the matched filter is given by
s-LJl
=
bCt)
o
Hence
J00
b
_
112
2
d t-
( t) dt =
1l2d
To equalise the impulse response energies of the two filters it is necessary to put d equal to a12, this gives (5.1 5) The matched filter d.c. gain is given by 8(0)
=
r
bet)
dt = J1d =
00
J( :) a
The expression for noise bandwidth is
{ No ise Bandwidth
}-_ r
-
1T
2)1 _
b\t)
-,--c..:oo ----,-
IB(o
dt
_
=
d rad / s 1T
( 5.1 6)
d -d, w)1a
The noise bandwidths of the two filters can be equalised by combining Equations 5.14 and 5.1 6 to give a = 2. The output e n velop e of the matched filter receiver can be found either by s ub st itutin g the filte r b aseb n d i mpulse response in Equation 4.15 or by eval uating ll / 2 I X(T from Equation 3.3. The
result is
{ output Matched filter } J JC�2d) envelope 1 0, =
sin
[�d ( -1(:) - I )] I
I
otherwise.
As the filter gains have been adjusted to give equal impulse response
energies, the peak signal-to-noise ratios can the peak output signals (see Section 4.4. I).
be compared by comparing
Both filters will give their peak outputs for
w
equal to zero and T
equal to d. The peak output from the matched filter will while the peak output from the single pole filter will be
be
y(ad/8),
Some examples of signal processing methods
76
Using these results with Equation 4.1 4 gives
(�) (�)
for single pole filter for matched filter
2 = - [1 + e -2ad - 2 e -W] ad
5.2 THE RECTANGULAR PULSE WITH SAWTOOTH
A rectangular pulse
FM
with sawtootht FM can be represented by
[-fl-J [
f(t) =
c os
t-
wo
t+
� J t2
(5.17 )
� is the total change in frequen cy
(ra d/s) during the pulse. The often mentioned, this is defined as
'dispersion factor' of such a pulse is
the dimensionless product d�. If W o is sufficiently high
[ n]
!aCt) = The
uncertainty
illustrated in Section
hi gh ) are
t-
function
3.2.
of
j(t2 iJ.j2d)
this
The inherent
c::: T� _1T ,
Alternatively
e
= =
v=
r =
e
jwo
t
(5.18 )
w a vefor m is discussed and precision relationships (for d�
1.2
(5.19 )
I '
(5.20 )
.
O.t)C
--
(5.21 )
0.6c -�/21T
(5.22 )
fad
5.2.1 The Fourier transform receiver The Fourier transform receiver of Fig.
4.7 will exhibit the above characteristics together with the poor resolution predicted by the uncertainty function. Resolution can be improved by time weighting the output of the main mixer before the Fourier transform is 1" i.e. successive pulses have
the same initial frequency.
77
Some examples of signal processing methods
calculated. This effect will now be discussed, together with a fuller treatment of the effects of delay-Doppler coupling which were discussed qualitatively in Section 4.2. It was shown in Section 4.7 that, provided the offset frequency WL was high enough, the output of the Fourier transform receiver was a cut through the modulus of a displaced uncertainty function, specifically (5.23) where h(w) is the receiver output function and the other variables are defined in Fig. 4.7. In practice the receiver output will be a function of time but the variable is left as wto avoid confusion later on. Expanding Equation 5.23 by means of Equation 2.1 5 gives
h( w)
=
where
! I ff{e(t) ej(r Cl/2d) e(t + x - tv) e -j(Cl/2d) (t+ x - tV)2 }Iw->W e(t)=
_
Y _ WL
rL o
d t-
Multiplying out the exponential terms gives
Hence
h( w) h( w)=
where
Ic [j( � w+
(x - tv) -y -
wL )Ji
CUw) =! ff{e(t) e(t+x-tv)}
(5.24)
(5.25)
As was shown in Section 4.7, the complete receiver output waveform is given by the above expression together with another symmetrically displaced uncertainty function. This provides the 'balancing' term necessary for the Fourier transform of a real function. It is usual to set (x - tv) equal to zero and make the offset frequency WL high enough to ensure separation of the two uncertainty functions. It can be seen that, for this special case, one of the effects of target delay x is to cause a shifting of C(jw) along the waxis proportional to the frequency sweep �. If the variable delay and offset oscillator were dispensed with Equation 5. 24 would become (5.26 ) As long as x � is large enough to ensure that h( w) has negligible amplitude at low frequencies this function will have enough inherent offset to overcome interference from its 'mate'.
Some examples of signal processing methods
78
It can be seen from Equation 5.26 that changes in target delay x and Doppler offset y have the same shifting effect upon the output waveform. The delay-Doppler coupling is not complete since changing delay has an effect on the shape of C(jw) ( Equation 5.25). It can easily be shown, however, that for large dfj, products the change in shape will be negligible. Since e(t) =
ILl d 0
t--
it follows that
e(t+x)= rL' -x
d-x t--
Hence e(t ) e(t +x) =
n'
(5.27)
t-
Thus the length of the e(t) e(t + x) pulse becomes d
[ ;�] I-
(5.28)
The question of change of pulse shape with change of target delay can now be resolved by comparing the Fourier transform of a pulse of length d with a pulse whose length is given by Equation 5.28. Clearly, for moderate values of x fj, and large values of d fj, the difference is insignificant. Since the Fourier transform of a rectangular pulse is of sin(x )/x form, Equations 5.25 and 5.27 confirm the poor resolution performance predicted by the uncertainty function. If the signal at the input of the Fourier transform calculator had been multiplied by a time weighting function m(t ), Equation 5.25 would have become
C(jw)
=
! .?"{m(t) e(t) e( r + x - t v)}
(5.29)
One is now at liberty to choose a suitable met) such that the above Fourier transform has low level sidelobes. Fortunately there are many well known functions which have this property-a good treatment is given by Blackman and Tukey [IS]. One suitable function is met) m(t)
=
=
[
I + (X I -(X 21T -- + -- cos d (t - d /2) , 2 2 0,
J
::� t>d
}
(5.30)
79
Some examples of signal processing methods
/\----1
that is
met)
A suitable value for
a
=
J" 0
'r
t---
d
is 0.08 which results in the Hamming function
[15]; the sidelobe performance is shown in Fig. 5.4. Strictly met) Hamming curves
Sin
(xlix
-20
dB
/
--- ... .... ,
"',
I I
I
'\
\ \
\
-40
, , I I I I I
-60 �------�--�L---����-o 3
Fig. 5.4. The inverse F. T. of the Hamming function of Equation 5.36
should be varied according to the target delay to obtain the optimum shape (i.e. the function met + x - tv) should be used) but for large values of
dt::. the reasoning which predicted no significant change for
the unweighted receiver also applies to the weighted receiver.
5.2.2 The matched filter receiver
It can be seen that the manufacture and weighting of a Fourier transform receiver is relatively straightforward. Assuming that broadband mixers are available, the main problem is the manufacture of a 'real time' spectrum analyser or having to accept the slow readout of a conventional - one. By contrast, the matched fIlter receiver is more difficult to implement. However, once the filtering problems have been overcome the output is inherently 'real time'. It is virtually impossible to manufacture an exact matched filter for the rectangular pulse with linear FM. The complex analytic form of the required matched filter transfer function is
G* [j(w-wo)]
80
Some examples of signal processing methods
where GOw) is given by Equation A3.6 of Appendix 3. The requirement is for a filter with a baseband response ej(w2d(2ll) cascaded with a complicated Fresnel integral function. Matched filter receivers for linear FM employ unmatched filters called pulse compression networks. The term 'pulse compression' arises because the final decision making envelope is short, like the uncertainty function cut, while the original transmitted envelope is long. It is not only the difficulty of implementing a matched filter which makes the use of an unmatched filter necessary-a big disadvantage of a true matched filter receiver would be the poor resolution due to the slow rate of fall off of the sidelobes of Ix(r, 0 )1. A philosophy for the design of pulse compression networks is outlined in Section 7.7. A linear FM waveform with an arbitrary envelope e(r) is considered, i.e.
[
f(r) = e(r) cos wor +
� t2]
(5.31 )
pulse compression filter would have a corresponding complex analytic transfer function of G[j(w - wo )] where A suitable
(5.32) Note that the exponential term in Equation 5.32 is the same as that required for the matched filter. Group delay [ 1 I] is defined as -(d¢/dw) hence the above exponential term corresponds to a dispersive line having a delay proportional to frequency. Section 7.7 (Equation 7.52) shows that if a signal of the form of f(r), but having a Doppler offset of wd, is applied to such a pulse compression network, the RF envelope of the output will be given by
(5.33)
a given transmitted envelope e(r), EUw) can easily be calculated, thus to choose a suitable B(jw) it is necessary to find a bet) which, when substituted in Equation 5.33 leads to a low sidelobe level. The above procedure is a daunting task since in general Equation 5.33 can only be evaluated numerically. However, for very large values of dispersion factor it turns out that the term For
Some examples of signal processing methods
81
can be considered equal to unity over the range of the other two functions in Equation 5 .33. Section 7.7.3 shows that if this is true a substantial simplification results; Equation 5.33 becomes
Ihjt) I = �J�� 1m (t d:d) I
(5.34 )
+
where
(5.35 ) Note that Equation 5.34 indicates complete delay-Doppler coupling, the effect of a Doppler offset being merely a shift in time without any change in shape. In a practical situation Equations 5.34 and 5.35 could be used to make preliminary investigations of possible choices of BUw), a numerical evaluation of Equation 5.33 could then be carried out for specific values of dfl.. As in the case of time weighting, the Hamming function would be a good choice for BUw), the form would be
BUw) =
1
+a + (I-a) cos [21TW] 6 ' 2 2
a
=
!}
( 5.36)
Iwl> "2
BUw) = 0, where
Iw l <
0.08. Hence
w---
t,.
2
For a rectangular transmitted envelope let
�1 -t,.
2"
t,. 2
w---
Hence Equation 5.35 reduces to
met) =
.�-I
{BUw)}=
2(i +a)] sin (fl.t) 2fl.2-21T-41T [at1Tt[r 2 2fl.2 2]
(5.37)
Equation 5.37 is illustrated by Fig. 5.4-which also shows a sin(x)/x curve for comparison. Equation 5.34 predicts that the compressed pulse will have the shape of for large values of dispersion factor; it can be seen that this waveform is a great improvement over the output from
met)
82
Some examples of signal processing methods
a matched filter since the sidelobes start at -42 dB rather than -13 dB. Figure 5.4 shows, however, that the main lobe is broader and that the 13 dB resolutions of the two waveforms are similar. Figure 5.5 is the result of a numerical evaluation of Equation 5.33 for the Hamming function and a dispersion factor of 100. It can be seen that dt:. must be much larger than 100 for Equation 5.34 to be valid. The author's numerical integration program is limited by computer rounding errors to a maximum value of dt:. of 100. o
-20
dB
-40
de. .......
.......
....
....
....
"
\
\ \ \ \
",,--- ...
\
I \ I \ , " "
,
=
\ \ \ \ \ \
100
I , ' I I \ ' ' I
,
tI "
, I I
"
,
�, I
".'
/
" \ \ \ \ \ \ 1 I I I , I I I I I
-60 �----�--�14 3 2 o
te.hn:
Fig. 5.5. The result of using the Hamming function with a low dispersion factor
An important practical consequence of Equation 5.35 is that, for large values of dt:., imperfections in the actual shape of the pulse compression network amplitude response can be corrected by modifications to the transmitted envelope. Drastic envelope modifications defeat the object of transmitting more energy in a given bandwidth, but small corrections to optimise the sidelobe level can probably be carried out more conveniently on the envelope than on the filter. It should be particularly noted that a rectangular envelope provides the sharp cut off reqUired in the Hamming function frequency response. 5.2.3 The effect of duty cycle
One further point to be discussed-applicable to both types of receiver-is the question of the pulse repetition frequency period or, what is more relevant, duty cycle.
Some examples of signal processing methods
83
As in the case of the constant carrier frequency pulse of Section 5.1, the pulse repetition frequency period k should be chosen to eliminate second time round ambiguities. The problem of duty cycle did not have to be faced in Section 5.1 since, for good precision, short pulses had to be used and the duty cycle was inherently low-this in fact is the big disadvantage of the constant carrier frequency pulse. If the linear FM waveform is used duty cycles can approach unity and still gi�e good precision and resolution if the dispersion factor dD. is large enough. The advantage of using duty cycles of much less than 0.5 is that the resultant uncertainty function has areas of no overlap, hence, no swamping can occur for targets separated by delays greater than 2d. Duty cycles of greater than 0.5 lead to overlapping uncertainty functions (see Section 2.3); the sidelobes of the uncertainty function fall to a minimum value for r equal to approximately k/2 and then begin to rise again, with no clear region. It is common, in the established literature, to refer to FM systems having a duty cycle much less than 0.5 as 'chirp radars'. The term 'FM CW' is used to refer to FM radars having duty cycles approaching unity. Whether the effect of overlap is important depends upon the dispersion factor used. It can be seen from Section 3.2 that X(O, 0) is equal to d, and that the envelope of Ix(r, 0)1 falls off according to the law 2d
(dD.)(r/d) Thus at the critical value of r /d = 0.5 the envelope of Ix(r, 0)1 is
20
10gIO el})
dB down with respect to the amplitude of the main lobe.
S.2.4 The effect of high dispersion factor
short discussion on the circumstances under which E q uation 5.33 reduces to Equation 5.34 will now be given. It follows from the definitions of J and � that, for large values of the dispersion factor, most of the transmitted energy will lie in a band roughly D. rad/s wide: thus the bandwidth of a use ful 8(jw) w o uld have 1.0 be in the region of D. to yield a good output signal to noise ratio. E(jw) and b(t) for the extreme examples of a rectangu13r envel()pe and a rec tan g ular BUw) are illustrated ill Figs 5.6. 5.7 and 5.8. A
84
Some examples of signal processing methods
r---t--.....,
- -1
o
o
t_ e
Wd/2Tt
( t)
E(jw)
Fig. 5.0. A tl'pica' transmitted en�'elope and its spectrum
r-----1r-----, --
-1
o w B(jw) Fig. 5. 7. A possible B(j w) and its impulse response
I_[�
�,/ ,�
dW d -- +1 6
I I I
, I
I I I I I
I
I I I I I I ,
\
2 IT
+
d Wd
2n
'
. \ \
J
\
\ \ \ \ \ \
\
\
/
xd 2IT-
Fig. 5.8. The functions to be multiplied in t:q uation 5.33.
It is only necessary to carry out the integration in Equation over those values of
x for
which the product
[
dX E(jx)b �
+
dWd
T
+t
5. 33
]
5.8 shows that, for moderate values of t and Wd the product will be small for large values of xdj2rr, hence it is possible to have a sufficiently large value of dt:J. for the exponential term in Equation 5.33 to remain sensibly constant over the effective
is Significantly large. Figure
range of integration.
Chapter 6
LAPLACE AND FOURIER TRANSFORMS
Most electronics engineers are familiar with the one-sided Laplace transform which leads to the modern pole-zero method of circuit design. Signal theory is usually written in terms of the Fourier transform which can be regarded as a special case of the two-sided Laplace transform. The two-sided Laplace transform is introduced in Section 6.1 and its relationship to the Fourier transform is explained in Section 6.4. It is suggested that the two-sided Laplace transform definition be accepted as a 'personal' definition even when one-sided Laplace transforms are being discussed. Two simple rules, given in Section 6.3, show the slight change of viewpoint necessary for the above to be successful. A thorough treatise on the two-sided Laplace transform is contained in the excellent book by van der Pol and Bremmer [8] . The advantages of the two-sided Laplace transform may be summarised as follows: (l) The class of functions suited to an operational treatment becomes much larger. (2) The transformation rules are considerably simplified. (3 ) The entire treatment becomes more rigorous than the usual presentation of the one-sided integral in technical books. 6.1 THE TWO-SIDED LAPLACE TRANSFORM
If
J e-pth(t ) dt 00
f(p)
=
p
converges for 0:: < Re(p) < (3, then h(t)
=
1
__ .
21TJ
f
c+
c-
joe
joe
ept f(p) dp P
where 0:: < c < (l The infinite strip of the p plane bounded by 0:: < Re(p) < {3 is termed the 'strip of convergence'. The precise meaning of convergence, 85
Laplace and Fourier transforms
86
and the necessary restrictions on h(t) for the above to be true are set out in [8J. The above notation is known as the 'p multiplied' form of the Laplace transform. It was used to allow the results obtained to be more easily identified with the work of Heaviside (1850-1925). It is more usual to call the function f(P)/p 'the Laplace transform'. It is also convenient to refer to h(t ) as a time function and H(P) as its Laplace transform. The Laplace transform of j(t) is defined by
J j(t) e-pr dt
f{Je/)} = F(p)=
(6. 1)
If the integral converges for Q< Re(p)< (3 the inversion integral is J{r)=
_I
c+
27Tj
j>o
(6.2)
. c _)00
where Q< C < {3. For all practical purposes the above formulae define unique functions. Thus, if two functions have equal Laplace transforms (regarding the strip of convergence as part of the transform definition) then they are equivalent. 6.2 THE SIGNIFICANCE OF THE STRIP OF CONVERGENCE A specification of a two-sided Laplace transform is not complete without reference to its strip of convergence. Although the limits of the required strip of convergence can be formally calculated they usually become apparent as the Laplace transform is evaluated. The above remarks, together with the effect of choosing the wrong strip of convergence can be illustrated by a simple example. Consider
u( t)= I.
I> 0
u(r)= 0.5,
r= 0
1I(t)= 0,
r
Applying Equation 6. 1 lead s to U(fJ)=
JOO
1I(/)e-ptd/=
Joo
l- -prj'
e-ptdr= C
o
_p
0
00
TIle upper limit gives an infinite value unless Re(p) > O. Hence £{II(t)} =
I
p -
,
Re(p) > 0
Lap/ace and Fourier transforms
87
Contrast the case for u( -t), i.e. u(-t) =0,
t>O
lI(-t) = 0.5,
t = 0
u(-t)
=
I,
t
This time the lower limit will give an infinite value unless Re(p)
£{u( -t)}
I
= - p'
Re(p) < 0
Thus the function whose Laplace transform is l/p could either mean u(t) or -u(-t). Specification of the strip of convergence removes the ambiguity. It can be shown that all one-sided functions (i.e. functions which are zero for t < 0) lead to a strip of convergence which lies to the right of all the poles of the resulting Laplace transform. 6.3 THE ONE-SIDED LAPLACE TRANSFORM
It is more usual to define the Laplace transform of f(t) as
J Jet) e-pt dt.
o
The expression, having a lower limit of zero, is known as the 'one-sided' Laplace transform since values of J(t) for t < 0 are ignored. It is perfectly feasible to always regard the Laplace transform in terms of the two-sided definition of Section 6.1 , and read work based upon the one-sided definition as if it had been written in terms of two-sided transforms. It is only necessary to remember the following two points: (I) References to f(t) in the one-sided treatment mean f( t )u(t ), where u(t) is the unit step function. (2) The strip of convergence of the Laplace transform of f(t)u(t) is that region of the p plane which lies to the right of all the Laplace transform poles.
88
Laplace and Fourier transforms
6.4 THE FOURIER TRANSFORM
The Fourier transform of f(t) is defined as 00
ff {j(t)} =
J 1(t) e-jwt dt
(6.3)
If the integral converges, the inversion integral is given by 00
If
t = j{ ) 2rr
.1}
'
eJw t dw
{j{t)}
(6. 4 )
To see the relationship between Equations 6.3 and 6.4 and Equations 6.1 and 6.2 (defining the two-sided Laplace transform) it is necessary to remember that pis a complex variable which may be written in the form p= c + jw where c and ware real variables. If the definition integral of the Laplace transform (Equation 6.1) converges for Re(p) = 0, it is permissible to put p =jwand c = 0 in the Laplace transform inversion integral. The (two-sided) Laplace transform then reduces to the Fourier transform and one may write .qF
U{t)}
= F(jw)
If the Laplace transform of J(t) does not converge for Re(p) = 0, the Fourier transform does not exist. Thus the Laplace transform definition of Section 6.1 embraces a larger class of functions than does that of the Fourier transform. Some examples are
(1) J(t)= e-a t u (t ). Therefore I F(p)= p+a'
Re(p)
Hence
.:F{j(t)} = F(w j )
(2)
J(I)= eat u(t).
_ 1._ ,
Therefore
1
F(P) = -. p-a Hence
(3)
for a> 0
a+Jw
.qF{J(I)} does not exist, for a
Re(p) > O.
J(t) = e-at,
t> 0
j(t)=ebt,
t
Lap/ace and Fourier transforms
89
Therefore
F(p) =
p+a
+
--
b-p'
-a
Hence, for the Laplace transform to exist it is necessary to have -a< b. For the Fourier transform to exist it is necessary to have
-a< 0< b. 6.5 THE
PHYSICAL
INTERPRETATION
FOURIER TRANSFORMS
OF
LAPLACE
AND
It is possible to visualise both the Laplace and Fourier transforms in terms of frequency spectra. Equation 6.4 can be regarded as expressing jet) as an infinite sum of time functions of the form ejwt. The term 'dw' ensures that the contribution at any one frequency is vanishingly small, and the multiplier FGw)/21T may be regarded as the spectra] density, measured in volts per rad/s. Hence the units of FGw) are volts/Hz. The function ejwt is a complex quantity, i.e. ejwt= cos(wt) which can be visualised as a rotating vector. For +ve frequencies w> 0 and the rotation is anticlockwise, while for -ve frequencies w < 0 and the rotation is clockwise. For real functions of time the Fourier transform symmetries (Section 6.6) are such that terms in dwt combine with terms in e -jwt to give real sinusoids. The interpretation carries directly over to the Laplace transform, the only difference being that the basic time function becomes
which can be regarded as an expanding rotating vector (c> 0 ) or a contracting rotating vector (c < 0 ). Thus the physical interpretation of Equation 6.2 is that f(t) can be regarded as consisting of any one of a number of spectra of the form eet ejwt; c being in the range Q'< c< {3. The spectral density is given by the expression F(P)/2rrj. It should be noted that it is not essential to have a physical explanation for a mathematical result; indeed some authors scorn a physical interpretation of the Laplace transform. The big advantage of physical interpretations is that they allow reasoning by analogy-there is nothing wrong with this as long as the results obtained are subsequently verified by rigorous methods. In many cases the most difficult part of a mathematical proof is deciding what has to be proved in the first place!
Lap/ace and Fourier transforms
90
6.6 FOURIER TRANSFORM SYMMETRIES
This section will be used to prove the following results:
g;-{f(t)} = FGw)
If
g;-{FUt) }
=
.� {flat)}
2rrf(-w)
(6.5)
1 w _ F j a lal
(6 .6)
( )
=
Also that the following necessary and sufficient conditions apply
F( -jw)
F( -jw)
=
=
F*( jW), -
F* (
jw),
f( - t ) = j*( t), j{-t) = -1*( t), Other results are stated in
(6.7)
for f(t) imaginary
(6 .8)
for FUw)
(6 .9)
for
real
F Uw) ima gina ry
(6 .1 0)
Section 6 .10.
6.6.1
Proof of Equation 6. 5
Using
Equation
6.3 and
for f(t) real
remembering that the
integration varia ble t is a
dummy variable, gives
r F(j.x) e-jwx d.x 00
.F {FUt)} = Similarly, from
I
00
( )= 1 j t 2rr Hence .
�{FG t) }
FUx)
ejwx d.x
=2rrj(-·w).
EXAMPLE Since
. .,-!-IL'! "1'
then
-a
a
t-
=
,.,
�
w
sin ( aw)
(6.1 1 )
(6.] 2)
91
Laplace and Fourier transforms
6.6.2 Proof of Equation 6.6
f j(at) 00
9'{t{at)}
=
e-jwt dt
Changing the dummy variable to tfa gives
J j(t) e-jwt/a(Ifa) dt, 00
9'{j(at)}
=
f j(t) e-jwt/a(I/a) dt,
a> 0
00
9'{j(at)} = -
which is expressed compactly by Equation
a <0
6.6.
A frequently used application of Equation 6.6 is the case resulting a = -1, i.e.
from
9'{j( --t)}
=
F(-jw)
(6 .13)
6.6.3 Proof of Equations 6.7 to 6.10
The proof of Equations
6.7 to 6.10 follows from the formulae for the
real and imaginary parts of a general function g(z ) Re {g( z )}
=
1m {g(z)}
=
1 2 [g( z ) + g*( z) ] 1 2j [g(z)-g*(z)]
Thus necessary and sufficient conditions are
g(z ) g*(z),
for g(z) real
=
g(z) It is shown in Section
=
6.7 that .q,;;-{f*(t)} .�-I
Hence Equations
for g(z) imaginary
-g*(z),
=
F*( -jw)
{F*Uw)} f*( -I)
6.7 to 6.10 follow.
=
( 6.1 4) (6. 15)
92
Laplace and Fourier transforms
6.7 THE FOURIER TRANSFORM OF A CONJUGATE FUNCTION The following results will be proved
ff{f*( t)} = F*( -jw)
( 6.16)
ff-I {[F*Uw)]} = f*( - t)
( 6.17)
The definition integral (Section 6.4) gives 00
F( -jw)
J f(t) ejwt dt
=
_ 00
Hence 00
F*( -jw)
=
J f*(t) e-jwt d t _ 00
which proves Equation 6.16. Since Equation 6.16 is true, Equation 6.13.
Equation
6.17 follows from
6.8 LIMITING CASES OF THE FOURIER TRANSFORM
Certain useful functions do not, strictly speaking, have Fourier transforms. They can however be regarded as limiting cases of functions which do. Consider, for example, u(t), in Section 6.2.
£{u(t)}
=
-1
p
,
Re(p)> 0
Since the strip of convergence does not include Re(p) = 0 the Fourier transform of the unit step does not exist. If, for the purpose of analysis, one is prepared to accept that a decaying function with a 'half-life' of millions of years will have the same effect as a unit step, it is permissible to write
Since
£{e-ft u(t)} it follows tha t
=
1 _' P+f
Re(p )>
-f
Laplace and Fourier transforms
93
The first expression in the right-hand limit will be recognised as 1T<5(W) (see Section w
=1= 0, and zero for
6.9) while the w = O. Hence §'
second expression becomes
{u(t)} = 1T<5(w)+i(w)
(6.19)
g-{u(-t)} = 1T<S(w)-i(w)
(6.20)
i(w) =
�,
W=l=O
JW
i(w) = 0,
Combining
(6.18)
}
where
Similarly,
-j/w for
w =o
for a reversed unit step
Equations 6.18 and 6.20 to give the Fourier transform of
unity
ff{l} = §,{u(t)+u(-t)} = 21T<S(w)
(6.21)
Alternatively
£{ l} = Lt £{e -fl11 } 1'
"{
oLe
-fltl
}=
.... +0 2€
€ -p 2'
Therefore � {1 }
-€< Re(p)< €
2
= Lt 1'
.... +0
{--f�} = 21T<S(w) € +w
(6.22)
The above results are consistent with the physical interpretation discussed in Section 6.5 where FUw)/21T had the significance of spectral density. Equation 6.18 credits the unit step with a spectral density of 0.5, concentrated at d.c., plus some high frequency components due to the sharp transition at t = O. Equation 6.22 shows the unit d.c. level to have a spectral density of 1, concentrated at d.c., and no high frequency components. The precise behaviour of 9" {u(t)} at W = 0, given by Equations 6.18 and 6.19, is used in Section 7.2; it also allows a quoted result for the signum function to be verified. Consider
{I, t>O sgn(t) =
0,
t = 0
-I,
t
94
Lap/ace and Fourier transforms
Hence
sgn(t) = 2u(t) - I Using Equations 6.18 and 6.22 .?{sgn(t)} = 2i(w) =
2/jw,
W*O
°,
W= °
}
( 6.23)
which, together with the remarks on notation in Section 6.11, verifies a result given by van der Pol and Bremmer [8] (page 114, Equation 54). The Fourier transform symmetry equation (Equation 6 .5) can be used with Equation 6.18 to prove another result which is required for Section 7.2. Since .?{ u{ t)} = m5( w)+i{ w) it follows that
.?{m5(t)+i(t) } = 21TU(-W) Hence .? - 1 { u{ w)} =
1 0 (t) -
_I
}
where 1
i(t) =
6.9 THE DELTA
jt'
t *0
0,
t= °
(6.24)
21T
(6 .25)
FUNCTION
This section will be used to give a short description of the delta function, oct), and some of its properties. A full treatment will be found in van der Pol and Bremmer [8] . The delta function is defined through the following properties
o(x) =
{a,
00
,
J o(x) coo
x*O x= ° dx =
1
}
(6.26)
( 6.2 7)
Although Equations 6.26 and 6.27 do not define a function in the ordinary mathematical sense, the delta function does have a rigorous mathematical basis [19].
Laplace and Fourier transforms
95
It is in order, and perhaps more satisfactory, to regard the delta function as a 'short-hand' notation for a limit, involving a normal function, which possesses the above properties if the limit is taken as the last operation. As an example o f such a function, consider the integral
f�a2+x2
=
!
tan-I
a
(�)a
where the principal value of tan -I is intended [10]. Hence
[rr rr ]
":' a dx I a2+x2
= "2 + "2 = rr
_;,
The above results lead to the following representation of the delta function
(6.28) A similar treatment allows trains of delta functions to be represented as limits. As an example 00
n=�oo
oct - 2rrn) =
Lt
y--->(-o
{
I _r2
}
2rr(l-2rcost+r2)
(6.29)
The most useful characteristic of the delta function is its sifting property,
f h(x)o(t -x) dx = h(t) 00
(6.30)
Equation 6.30 follows from Equation 6.2 7, remembering that oCt - x) is zero except at the point x = t. An alternative representation of the delta function can be obtained by considering its Fourier transform.
J oCt) 00
.7{0(t)} =
e-jwt
dt = 1
(6.31)
The result of Equation 6.31 follows from Equation 6.30 by noting that e-jwt =
1,
for t
=
0
Equation 6.31 can also be written in the form of an inverse Fourier transform giving
1 oCt) = _
2rr
f
00
ej wt
dw
(6.32)
Lap/ace and Fourier transforms
96
Equation 6 .32 may be used to obtain another useful result-the interpretation of o(a t + b).
o(at +b) = +-�rr f
00
ej(at+b)W dw
Changing the dummy variable to wla gives
o(ar+b) = �1rra f
ej(t+bja)w
o(ar+ b) = -=-� 2rra f
ei(t+b ja)w dw,
a>O
00
dw.
_ 00
Hence
a
5(ar+b)= �o lal (r+!2)a
(6.33)
Failure to appreciate the result given by Equation 6 .33 once led to the discussion of a 'paradox' in the correspondence section of Proc. I.R.E. (November, 1 96 1 ; February, 1 96 2). 6.9.1
Derivation of the Fourier series
The delta function is used by recognising the significance of limits such as Equations 6.28 and 6 .29 and using properties such as Equations 6 .30 to 6.33. A good example of this procedure is given by a derivation of the Fourier series from a consideration of the two-sided Laplace transform of a repetitive function. Consider the function n =:
(6.34) - 00
where n takes on integer values (including zero), c, k are real +ve constants and Lif(t)} converges for -(}: < Re(p)< 00, with (}: > c. Equation 6.34 may be written in the form
jR (t) where
II (t) f2(t)
= =
= r
(6 .35)
II ( ) +/ir) - jet)
f(t) + e -Ckf{r k) + e -2ckf(t _.
-
2k) + ...
f(t) + e -Ckj(t + k) + e -2Ckf(t + 2k) + .. .
Laplace and Fourier transforms
97
It will be noted that the factor e-Inlck ensures that the 'pulses' decrease in amplitude each side of the point t = O. This effect can be made negligible by allowing c to tend to zero. From Equation 6.35
+e-cke-pk +e-2ck e-2pk +... ]
F 1 (p) = F(p)[ 1
(6.36 ) (6.37)
Equations 6.36 and 6.37 are geometric progressions which summed to infinity give
(6.38)
F2(P) =
F(p)
I -e -ck epk'
le-ckepkl 1
(6.39)
<
The convergence conditions of Equations 6 .38 and 6.39 reduce to Re(p) > -c and Re(p)< c, respectively. Hence
FR(P) = F(P) where Re(p) =
[ 1 -e-;k e-pk + 1 �Ck pk -1] e
-e
(6.40)
-c
< Re(p) < c. Since the strip of convergence includes O,fR(t) has a Fourier transform given by
FRUw) = rC'U ' W)
[ 1 -e -ck1 e-J"
1
+
-e e
"
-
1]
(6.41)
Equation 6.4 1 simplifies to
FRUW) = FUw)
e-ck
[1_2e-lc�c:�2:: +e-2Ck] + 0, r 1 - O.
(6.42)
21TFUw) L 8(wk-21Tn)
(6.43)
Putting = r it is seen that for c -+ and 6.29 are compared, it follows that
c-+O {FRUW)} Lt
=
"
-+
If Equations 6.42
n =--
The Fourier transform inversion integral gives
(6 .44) Substituting Equation 6.43 in 6.44 and using Equation 6.33 gives
c ��
0
{fRet)}
=
-
t f FUw) ejwr f
-
-
)
8 (w - 21Tn dw k \ n=--
(6 .45)
iwo t}
e
.
The
'
{f(��)}
expression
-
formed
from
J .
I
J
'
Fourier
transform
FUw) eJwt dw
F(�j27Tt)
2n
1
W
()
T
J
F,(x)G,(w� x)dx
J
w F,(w) ei t dw F,(�27Tt)
27T
..'F -'{F, (w)G, (w)}
27
F, (w �wo)
e
� iWT F (w) ,
F
!at 'a
1
( J
F2(f) F2 (�t)
J
F 2 (x)G2 (f� x) dx
i e 27T!t df
§. -, {F 2 (f)G 2 (f)}
00
��)
�i27T!T
F2 (f)
() f F2 a
F2 f �
e
!at
7T t
in this way does not necessarily represent the Laplace transform
FUx)G[j(w�x)J dx
.'7 -'{FUw)GUw)}
27T
'"
F[j(w � wo)J
e�iWT FUw)
.
( )
1 [F W Ja j(i
the
throughout the strip of convergence.
t
.�-'{f(f)} = '7
f(t)
f(x)g(t -x)dx
00
J
.�{f(t)g(t)}
) .� (rU
.� {f(t� T)}
.� {f(at)}
F2* (�f)
F* ( �jw)
.'F {ru>} F,*( �w)
F2 �j
F, (�jp)t
F(p)t
£ {f(t)}
( f)
F2(f)
F,(w)
FUw)
Result
.�{f(t)}
Property
Table 6.1
� ""
....
� <::>
�;::s
�.
� l:::
� � I::l ;::s I::l...
E)
t--o
-§
\0 00
Laplace and Fourier transforms
99
Applying the result of Equation 6.30 to 6.45 leads to the Fourier series Lt {fRet)} = 1 k c-+o
I: [j 2rrn ]ej(21Tn1k)t
n=-oo
F
_
k
(6 .46 )
6.10 FOURIER TRANSFORM NOTATION
The expression defined as the Fourier transform in Section 6 .4 can be called FQw), F(w) or F(f) depending upon the author. Although the differences are mathematically trivial they can lead to confusion when theorems are quoted. For the convenience of the reader, various results are quoted in Table 6 .1 in the three 'languages'. DEFINITION g;- {lU)}
= FUw) = F I( w) = F2(f) =
where
6.11 USING
f fit) e -jwt dt 00
w = 2rrf.
THE
P-MULTlPLIED
LAPLACE
TRANSFORM
NOTATION
Van der Pol and Bremmer [8] have given extensive lists of Laplace transform theorems and results under the respective headings o f 'grammar' and 'dictionary'. A s the information is given i n the p-multiplied form it is useful to be able to translate it into the standard notation. A typical 'dictionary' entry is valid for 0 < Re(p) < 00
u ( t) --+ 1,
To convert this to the standard notation, simply divide the given Laplace transform by p. Thus
£{u(t)}
=
1
-
p
,
valid for 0 < Re(p) <
00
The change in notation does not affect the strip of convergence.
Laplace and Fourier transforms
100
The procedure in the case of theorems is slightly more complicated. Some results in the p-multiplied notation are If
h(t) -+ f(p), Then
valid for
h(t + A) -+ eAP f(p),
Also
e-At h(t) -+ p
�A f(p +A),
a
< Re(p) < (3
valid for a < Re(p) < (3 valid for
a-
Re(A) < Re(p) < (3- Re(A)
To convert to the standard notation
(I) Divide the right-hand expression by p. (2) Replace fez) by zf(z). The above examples then become, if
£{h(t)}
= f(p),
valid for
a
< Re(p) < (3
then
£{h(t+A)}
=
eAP jCp),
valid for a < Re(p) < {3
Also
£{ e -At her)}
= f(P +A),
valid for
a
-
Re(A) < Re(p) < (3 - Re(A)
.
6.] 2 THE DISCRETE FOURIER TRANSFORM (OFT)
The recent advances in integrated circuit techniques are such that it is becoming more and more economical to adopt methods of signal processing in which the quantities being handled are discontinuous (sampled). A processor of sampled signals operates on batches containing a finite number of samples; accordingly a Fourier transform calculator using these techniques would be required to evaluate the discrete Fourier transform (DFT) rather than the continuous Fourier transform. This section contains a description of the DFT and some of its properties, together with a consistent notation. The approximations involved in evaluating the Fourier transform by means of the DFT are discussed in Section 6. 1 3. The DFT is normally evaluated by means of an efficient procedure known as the fast Fourier transform (FFT) method; this is described in Section 6. 1 4 . Finally Section 6 . 1 5 contains the proofs of various formulae quoted in Sections 6 . 1 2-6. 1 4 . From a purely mathematical point of view, the N point DFT can be regarded as a relationship between two infinite sequences, {ai} and {A n },
Laplace and Fourier transforms
101
which have a period of N. The reference to 'a period of N' means that
ai = ai± N = ai 2N' . . . An = An±N = An±2N, ... ±
The relationship is stated as follows: If N-J
An = L
i=O
ai e-j (2rrn/N)i = 9}N {ai}
(6.47)
then ai
=
1 N
N-J
L An ej(2rri/N)n
n=0
=
9} r.J {An}
(6.48)
Equations 6.47 and 6.48 define an exact relationship which is proved in Section 6. 15. An and ai can take either real or complex values.
6.12.1 The relationship between the DFT and the Fourier transform
A tie-up between the OFT, as defined above, and the Fourier transform can be obtained by considering the Fourier transform of an infinite train of delta functions. The relationship is derived in Section 6. 15.3 and illustrated in Fig . 6. 1. It can be seen that if the time delta function strengths follow the sequence ai> the Fourier transform is another delta function train having strengths pr oportional to the sequence A n'
a N-1
I ao t -k
a,
1
Ao \'
I
-%
A,
1
1
aN
a N1
aN-l
-
I I I ao ! ! I I I t 1 ! I I 1 a2
a,
-kiN
�
0
Ao
A2
kiN
t-
(0 )
A2
I 1 1 Ar I t 1 -11k
0
A, !
h
1-
a a ,
1
a2
k
Ao
I
A A I 1- , .
A,
N/ k
i
I
A 1 ! N-1 t
(b)
Fig. 0.1. (a) the time function fm(t); (b) a representation of kFm(jw)
102
Lap/ace and Fourier trans/arms
The Fig. 6.1 relationship can be written in the form ff
where
N-l A {N-l ( 1 ik)} a 2: J>k � n °N/k (-n) .
1=0
t
-
=
N
o�t) = L m
=_00
k n-O
/
o(t - mk)
k
( 6.49)
(6.50)
If it is understood that delta functions are 'attached' to the sequence terms, and if the group repetition period (k) is unity, then the sequence {A n } can be regarded as the Fourier transform of the sequence {a;}.
6.12.2 Properties of the DFT
The physical significance of the OFT can be used to obtain many OFT properties straight from known Fourier transform properties. For example, it is clear that the form of kFm Ow), illustrated by Fig. 6.1 b, is purely symbolic. If the val ues ai are real then /m (t) must be a real function of time; this means that Fm (-jw) is equal to F! Ow) (Equation 6.7). Using this information in conjunction with Fig. 6.1b, it follows that (6.51) Other necessary and sufficient conditions, based on the results of Section 6.6, are
for ai i m agi n a r y
(6.52)
for An real
(6.53 )
for An imagi n a r y
(6.54)
A_n =AN-n = -A:, a_i =aN-i =a;, *
a_i =aN i = -ai,
There is also a convolution theorem applicable to OFT's. This theorem is proved in Section 6.15.2. The result is 9 N{aibi}
N-l
I = IV
qN -l{A n Bn}
L
111 =0
AmEll-Ill
N-l
-
'\' am i_ L b 11I m=O
(6.55)
(6.56)
The convolution theorem reduces to a particularly simple form if the multiplying function involves exponentials. For example, it is shown in Section 6.15.2 that if
(2Tri)
bI· = cos I V
( 6.57)
Laplace and Fourier transforms
then
Bn
2 / 2,
= N/
Bn = N
,
n
= 1,
n
= -1,
1
±N, .
103
.
.
- 1 ± N, .. .
} (6.58)
otherwise Hence, if
(21Ti)
c-I = a·cos I N
(6.59)
then (6.60) Thus, in the case of a time weighted function of the form discussed in Section 5.2. 1, (6.61 ) Hence (6.62) showing that the effects of time weighting can be obtained by spectral processing, if desired. 6.13 EVALUATION OF THE FOURIER TRANSFORM BY MEANS OF THE DFT
It is shown below that if the sequence {ai}is formed from samples of j(t) taken T seconds apart then, under certain conditions, FGw) is represented by the OFT sequence {An}. The necessary conditions are ( I ) liT must be greater than the total (i.e. +ve and -ve frequency) bandwidth (in Hz) of FGw). (2) The number of samples N must be such that the truncated time function, of length (N - I)T seconds, has essentially the same spt�trum as f(t).
For the usual case of a real time signal, bandlimited to B Hz, the total bandwidth is 2B Hz and ( I ) becomes 1 /T > 2B, i.e. the normal sampling condition. The implications of (2), from the radar point of view, can be obtained from Sections 4.2 and 4.3 by noting that the signal processing time is (N - I )T seconds. If the above conditions are satisfied then FGw) can be evaluated for N values of w by using Equations 6.67 and 6.68 in conjunction with Fig. 6.2.
104
Laplace and Fourier transforms
o f(Q)
o
f---(b)
Fig. fl. 2. (a) the form of FUw) in Equation 6.64; (b) the form ofrFs(jw) in Equation 6. fl4
In a similar man n e r , the inverse Fo u r i e r t r a n s form of FUw) is represented by the seque nce {ail if the sequence {An} is fo rmed from samples of FUw) t aken � Hz apart . The necessary c o n d i t ions a re
( I ) I / � must be grea t er than t he t o t al d u r a t i o n o f f(t). (2) The nu mber of samples (N) must be such that the truncated spect rum, of width (N I )� Hz, has esse n t ially the same inve rse -
Fourier transform as FUw).
The res ult is summarised by Equ a t io n s 6.73 a n d 6.74. 6.13.1
Deriva tion of Fourier transform results
In the t rea t ment which fol l ows sampled signal s are represe n t e d by t rains of delta funct ions where as re al sampl ed systems employ n a rrow pulses. It ca n e asily be shown t hat samp l i n g w i t h a pulse of w i d t h d seconds is equ ivale n t t o impul sive sampl in g, provided that i n t e rest i s concen t ra t ed a t frequencies m u c h lower than I/d Hz and that a st re ngth of d (ra t he r t han u n i t y ) is all oc a t e d to the sampling d e l t a funct i o n s. The la t t e r m od i fication also serves to restore the correct dimensions ( t he d i mensions o f the d e l t a fu nct ion are t i me-I).
Lap/ace and Fourier transforms
105
Denoting the impulsively sampied form of f( t) by h(t), it follows that
fit) = f(t) 2:: oCt -i7) ;=-00
=
f(t) 1 7
�
i=-oo
ei(21Ti/r)t
(6.63)
The second half of Equation 6.63 follows from Equation 6.46. Hence the Fourier transform is given by
(6.64) Equation 6.64 is illustrated by Fig. 6.2 for the general case of f( t) complex. It can be seen that provided 1/7 is greater than the total bandwidth (Hz) of FGw), no spectral overlap occurs in Fs Gw) . Hence FGw) can be obtained by evaluating 7 FsGw). If, for an appropriate value of time displacement ( to ) , and large enough N, h(t) can be represented by
fs(t) Then FsUw)
� f(t) L oCt - to N-J ;=0
00
� f j(t) ,2:: oCt-to N -J
_00
=
1=0
N-l
(6.65)
-iT) e-iwt dt
2:: f(to +i7) e-iwto
;=0
- i7)
e-iw;r
(6.66)
Hence
Thus, if ai =
then 7 Fs
j(to +i7)
(J' 21Tn) � N7
e-j(2rrnto/Nr) 7A n
(6.67)
(6.68)
Note that it is often possible to choose the time origin such that to is zero .
Laplace and Fourier transforms
10 6 6.13.2
Derivation of inverse Fourier transform results
A similar proce d u r e can b e carried o u t fo r the case o f the inverse F o u r i e r t ransfo rm. By re-a r ranging Equations 6.34 and 6.43 it follows tha t if
F/jw)
=
FU w)
L
n=-oo
o(w-27Tn�)
(6.69)
then
(6 . 70) The units o f � are Hz. The rela tion ship given by Equation 6.70 is of the same form as that illust rat ed by Fig. 6.2, showing that .I(t) can b e o b t ained by evaluating 27T� .I�(t), provided that I / � is gre a t er than t he t o t al durat i o n o f f(t). The same reasoning used for t he t ime fu n c t ion case shows that if, for a n a ppropriat e val ue o f Wo a n d large en ough N , FrUw) can be represen ted by
Fr(jw) � FUw)
N -I
L
1/=0
o(w-wo - 27Tn�)
(6 . 7 1 )
t hen
Thus, if
(6.73 ) then
(6.74)
6.14
THE FAST FOURIER TRANSFORM (FFT) PROCESS
The fast Fourier tran sform (F FT) process is an e fficie n t method o f evaluat ing t he discre t e Fourier t ransform; it i s based u p o n two t h e ore ms which are de rive d in Sectio n 6.15.4. Reference t o the OFT de finition re l a tionship (Equa t i o n 6.47) shows t hat N operat io ns are req uired t o calculate a si ngle value o f the sequence {An}. Th us i t would appear that N2 ope rations are required to cal culate the full N val ues o f the sequence. H oweve r , it is not always
Lap/ace and Fourier transforms
107
necessary to carry out N'l operations. It is shown in Section 6.15.4 ( Equations 6.95 and 6 .96) that, if N is an even number (6.75 ) (6.76) Equations 6.75 and 6.76 are illustrated by Fig. 6 . 3, for the case of N = 8 . The multiplier W is equal to exp (-j2rr/N). It can be seen that the original N2 = 64 operations has been reduced to 2 x (!N)2 + N = 40 operations; furthermore there is no reason why the two 4-point DFT's should be evaluated by the direct method. If the FFT process is repeated each 4-point DFT will need 12 operations rather than 16. Thus the number of operations could be reduced from 64 to a total o f 3 2. The saving in processing time becomes more dramatic as N is increased. If N = 2 m the FFT process needs a total of (m + 1)N operations, rather than N2 .
ao a, az a3
bO
(a, + as)
b,
(aZ+a6 )
bz
(a3 +a7)
b3
8 8
4 Point OFT
a6
Fig. 6.3.
The
exp(-j21T/8)
reduction
An alternative 6.99 6.101.
Z
C3
(a3 - a7)
a7
=
C
OFT
of
an
procedure
8-point
DFT
to
two
Az
83 0
AO
=
8
C,
4 Point
(az - a6)
,
=
=
C
(a, - as)
as
0
Z
(a O-a4)
a4
W =
(ao + a4 )
A4 A6 =
A, =
A3 =
AS =
A7
4-point DFT's
can be obtained using Equations
-
(6.7 7 ) Hence, if hi = a2i and Ci = a2i+l, then A m = Bm + e-j(21TmIN)Cm,
A m+Np = Bm - e -j(2mn IN) Cm,
o �m �N/2-1
(6.78)
o �m �N/2-1
(6.79)
108
Lap/ace and Fourier transforms
The above relationships (for the case N = 8) are illustrated by Fig. 6.4. It can be seen that this method is a 'reverse' form of the first method.
a
b
2
a
b
4
a
1
B 1
+
0
B
+
0
B 3
+
0 3
B
-0
C
5
OFT
o
C,
3
4 Poin t
3
C
,
a
B +0 = 0 0
2
b
a 5 a
0
b
a o
2
B
4 Point
Fig.
6.4.
o 1
2
0
-0
1
B -0 2 2
OFT
( 3
a 7
2
1
B
An altcrnativc reduction of an 8-point
DFT W
=
3
exp(
-0
3
=
=
=
=
=
=
=
A
o
A 1 A 2 A
3
A 4 A A A
5 5 7
-j2rr/8)
Eq u ati o n s 6.75-6.79 a p p l y t o t h e cal cul a t ion of the OFT; very
s i m i l a r r e s u l t s a re a p p l i c a b l e to t h e c a l c u l a t io n o f the inverse OFT. The r e l e v a nt equati o n s a r e 6.97, 6.98.
6.102,6.103 and 6.104. To convert
Figs 6.3 a nd 6.4 to th e i n ve rse OFT c a l c u l a t i o n it is o n l y necessary to (1) Re p l ace ai by Ai a n d Ai by 2 ai. (2) Re p l ace th e DFT boxes by inve rse OFT boxe s.
(3) Cha nge the W m u ltipl i e r to W = expU27TjN), i.e. W = expU27T/8),
forN=8 .
Fo r
fu rth e r
i n fo rm a t i o n
F FT
a b out the
p ro c e s s
Gol d and
see
Ra d e r 19] .
6.15 PROOFS OF THE PROPERTIES OF THE DFT AND FFT
The formal p ro o fs of the
}
OFT tra n s fo r m p rope r t y a n d t he OFT
conv o l uti on th e o r e m d e p e nd u pon th e fol l ow i n g r e l a t i o n s hi p .V
�
I
11/=0
ei(2 rrkjN)J/1
O. ±N, ±2N, etc .
,V,
k
0,
k = a n y other i ntege r
=
In th e first case t h e a rg u m e nt o f th e summation i s
I fo r a l l
(6.80) m,
hence the
109
Laplace and Foun'er transforms
sum is clearly N. In the second case the summation is a geometric progression and it follows that N-l j21Tk 1_e----,L ej(21Tk/N)m = ----,N) i m =0 1 e (21Tk/ _
which is equal. to zero, for
k *-
0, ±N, etc.
6.15.1 The basic DFT property
Using the defini tion of the DFT given in Section 6.12, one can write _I
fl} N
N -I N-I
1
e-i(21T n/N)r ei(21Ti /N)n { f'jJ N(aJ} - N L L ar n=O r=O _
_
-
1
N-\ i ar L ei!21T( - r)/N-)n r=O n=O N-\
NL
(6.8 1)
Equation 6.8 0 shows that the inner summation is zero unless (i - r) is equal to 0, ±N, etc . Hence Equation 6.81 becomes (6.82) It also follows that ai = ai+N= ai+2N , etc. A similar procedure can be used to show that fl}N{ fl},-\/(An)}
=
(6.83)
An
6.15.2 The DFT convolution theorem
To prove the convolution theorem
I N-I LAm Bn N
m
-
m=O
iV-I
=
=
I L
N m=O
J.-
N-I
N-I
L
i=O N-I
L L
N i=O
k=O
aie-i(21T111/N)i
N-\
L bke-il21T(n-m)/Nlk
k=O
aibk e -i(21TI1/N)k
N-I
L
111=0
e -iI21T(i-k)/Nlm
Hence, using Equation 6.8 0
1 N-I N-I . . -J(21T1//N)/ = f!j {a/·b/·} a· ·e = � b B - � Am n-m / / N L N 11L i=O =0
1
(6.84)
110
Laplace and Fourier transforms
A simil ar procedure can be used to show that
l �N {AnBn} = For the case bi
=
N-l
L
m=O
ambi-m
COS(21Ti/N)
N-l = Bn .L 1=0
=
!
�
N-l
! iL
Hence, using Equation
Bn =
=O
( 6.85 )
]
ej(2rri/N) + e-j(2rri/N) e-j(2rrn/N)i
d[2rr(l -n)/Nli.+ e-j(21T(1 +n)/Nli
6.80
{ 12
NI2,
n = 1, 1±N, etc.
N ,
n = -1, -1±N, etc .
0,
otherwise
Similarly, for b i = sin(21TilN)
Bn
=
n = I, }±N, etc .
{-)N)N121,2,
n =
-1, -1±N, etc.
otherwise
0,
6.15.3 The relationship between the D FT and a delta function train
The proof of the relationship between the OFT and an infinite train of delta functions will now be given. Consider a delta function train f m(t) derived from a sequence {ai} of N sample values . Let fm(t) =
N \
i�-
( - ik'J)
aJ>k t
N
( 6 .86)
where
n = -oo
(6.87)
It can readily be shown that (6.88)
111
Laplace and Fourier transforms Hence
FmGw)
=
Since the
N-I
=
j
�
2rr / aj e-j(wik N) 821T/k(W) k
� (
8 W-
n= -oo
)
2rrn
2rr
k
k
f
N
l
j=O
21Tn/ j N) aj e-j(
i summation is periodic in n, it follows that
written in the form
FmGw) can be (6.89)
where An
By putting
w
=
N-J " L., a.I j=O
=
e-j(21Tn/N)j
2rrt, and using Equation
written in the form
F,nGw) =
1
k
N-J n
�
(6.90)
6.33,
Equation
6.89
can be
� �)
(6.91)
8J/dt)
(6
An8N/k
-
Using the relationship
G;-1{82ndw)}
=
�
2 C
it is possible to work backwards from Equation
which is equivalent to
fm(t) = If
Equation
6.93
N
-J
L
j=O
� - _·k) -
8k t
I
1 N -J
L
N N n=O
is compared with
6.89
92)
and say
An eK2ni/N)n
Equation
.
6.86
(6.93) (the defining
equation) it can be seen that the sample values are given by a· I
Thus Equations and its inverse.
6.90
=
and
- N-J
1 " ej(2ni/N)n L., An N n=O 6.94
(6.94)
form a logical definition for the
OFT
Laplace and Fourier transforms
112
6.15.4 Proof of the FFT theorems
The theorems leading to the FFT process will now be proved assuming that N is even. Equation 6.90 can be written in the form A
n
Nj2-I
L
=
i=O
N ai e-j(2rrn/ )i
+
N-I
L
i=Nj2
ai e-j(2rrnjN)i
which, with a change of variable in the second term, becomes A = n
=
Nj2 -I 2 " a·I e-j( rrnjN)i L i=O Nj2 -I
�
i=O
2) a·I+Nj 2 e-j(2rrnjN)(i+N/
- 1 -j(2rrnjN)i (a·I +a·I+Nj2e j1TI )e
" L
i=O
Hence A2r
A2r+1
Nj2-1
+ )'
NJ2 -I
" L
=
i=O
Nj2 -I
2 ijN) e-j]2rrrj( N j2) ]i (a i-ai+Nj2)e-j( rr
L
=
(a· +a· )e-j[2rrr/(NJ2)]i I I+N/2
i=O
(6.95)
(6.9 6)
A similar process, starting with Equation 6.94 gives a2r =
I
a2r+1 = -
�
Nj2 -I
N
" L
11
=0
NJ2-1
L
N
n=O
(A
n
+ An+NJ2)ej[2rrr/(N/2)]n
(2rrnjN)ej[2rrrj(NJ2)]n (AI1-An+Nj2) ei
(6 .97)
(6.98)
An alternative way of writing Equation 6.90 is AI1 =
NJ2-1
L
i=O
2 N)2i a2i e-j( rrl1 /
+
L:
NJ2-1 i=O
a2i+1 e-j(21mjN)(2i+1)
(6.99 ) Putting form A
bi = a 2i
and
Ci = a 2i+I,
Equation 6.99 can be written in the
-j( 2m n /N) C A 111 = Bm +e m,
e-j(2rrm/N)C , m + Nj2 - Bm m
o o
m
/2 - 1
(6.100)
m
/2 - 1
(6.101)
<.<. <.<. NN
Laplace and Fourier transforms
113
Equation 6.101 follows since the period of Bm and Cm is N/2, rather than N. As before, a similar process can be carried out on Equation 6.94 giving (6 .102) Also, if Bn
=
A2 n
and Cn
2 am = bm
+
=
A2n+1,
then
ej(2rrm/N) cm,
2 am+N/2 -- bm-ej(2rrm/N) cm,
o �m �N/2-1
(6.103 )
o �m �N/2-1
(6.104)
Chapter 7
HILBERT TRANSFORMS AND COMPLEX ANALYTIC SIGNALS
When dealing with real RF signals it is mathematically convenient to work in terms of complex signals having one-sided spectra. This chapter deals with such signals and also considers the Hilbert transform which occurs in the study of their properties.
7.1 SUMMA R Y OF THE MAIN RESULTS OF C HAPTE R 7
(I) The spectrum of a real signal consists of positive and negative frequencies which are related by the symmetry of Equation 6 . 7 . Any spectrum not possessing this symmetry must necessarily belong to a signal which is either complex or purely imaginary. The term 'complex analytic signal' is used in the literature of signal theory to denote a particular class of complex signals having spectra which contain no negative frequencies. In particular the complex analytic signal faC t) corresponding to the real signal f(t) is defined through the following property
,�{fa( t)}
=
where
FaUw)
=
FUw)
2FUw),
w>O
FUw) ,
w=0
0,
w
= .,¥,
Ir
(7 .1)
{f(t) }
It is shown in Section 7.2.1 that this definition o f the Fourier transform of j�(t) leads to
faC t) = f( t) +jj( t)
(7.2)
where -
j(t)
f (t 00
1
= 1T
x
j( )
_
00
-x )
dx
(7.3)
The integral in Equation 7 . 3 has to be taken in the sense of a principal value . ./(t) is called the Hilbert transform of f(t). (2) Some properties of the Hilbert transform are listed and proved 1 14
Hilber t transforms and complex analytic signals
115
in Section 7.2. It is shown in Section 7.2.2 that !( t) describes the waveform which would be obtained if f(t) was passed through a perfect, broadband, 90° phase lag circuit. It is also shown that the real and imaginary parts of the transfer function of a physical network are related through the Hilbert transformation. (3) The complex analytic signal concept is particularly useful in the study of RF signals of the form /{t) = laC t)1 cos[wot+rJ>(t)]
(7.4 )
where la (t ) 1 and rJ>(t) are the real signals applied to the amplitude and phase modulation terminals respectively . Provided that the carrier frequency Wo is so high that there is negligible low frequency energy, Equation 7.2 does not have to be used to calculate faC t); rather faCt) can be written down by inspection as (7.5) It should be noted that one application where Equation 7.5 may not be valid is in the study of sonar [2] . (4) Once the form of the complex analytic signal has been calculated, results obtained from operations upon such signals can be related to the corresponding real signals through various properties of the complex analytic signal. The modulus of the complex analytic signal is a real function which corresponds to the function engineers would call the 'RF envelope'. If a real filter processes a real signal f(t) and gives a real output signal h(t), the same filter would process faC t) to give an output of haCt). The effect of multiplying two real signals f(t ), get) is to produce an output signal having high (sum) and low (difference) frequency components . In terms of complex analytic signals the high frequency component is given by 1 Re {!a ( t) ga (t) } while the low frequency component is given by
A key property of the complex analytic signal, from the point of view of the r.m.s. error criterion is that
J
00
J [fa( t)-gaC tW dt 00
[t( t ) g( t ) ] 2 d t = 1 -
(7.6)
116
Hilbert transforms and complex analytic signals
7.2 THE HILBERT TRANSFORM
The Hilbert transform of
f{t) is defined as
]({f{t)} =j(t)= 1 1T
Joo 11& (t-x)
_ 00
dx
(7.7)
The integral has to be taken in the sense of a principal value, i.e.
This section will be used to show the significance of the Hilbert transform to the study of complex analytic signals and physical transfer functions. The following properties will also be proved
w>O
-jFQw),
}
w=o
g; {j(t)} = 0,
(7.8)
w
jFQw),
let)= -f{t)
(7.9)
J f(t)g(t) dt= - J j(t)g(t) 00
00
dt
(7.10)
_00
J j(t)g(t) dt= f f{t)g(t) dt 00
00
(7.11)
(7.12)
7.2.1 The complex analytic signal The relationship
between the Hilbert transform and the complex
analytic signal can be proved rigorously by the theory of contour integrals ( Gouriet
[11]). Since rigour-like beauty-is sometimes in the
eye of the beholder, engineers may find the following treatment more convincing. The Fourier transform of
fa(t)
is defined by Equation
be re-written using the unit step function ( Section
g;{fa(t)} = 2FQw)u(w)
7.1.
This may
6.2) as (7.13)
Hilbert transforms and complex analytic signals From Equation
6.24
{u(w)} = H(t)
.�-1 where
i(t)= Hence
{
faCt) can be found from
theorem ( Section
- +- i(t) ... 1T
1/jt,
t* o
0,
t=O
Equation
117
7 .13
by use of the convolution
6.10).
faCt)= 2g-1{FGw)u(w)}
f f(x)o(t-x)dx- * f f(x);(t-x)dx 00
00
=
The sifting property of the delta function, Equation first integral in Equation definition of
;(t)
7.14
to be equal to
f(t),
(7 .14 )
6.30, shows the
while the precise
shows the second integral to be the principal value of
f
00
! 1T
fix) dx = -J(t) J (t - x) .
Hence
faCt)=f(t) + J(t)
(7 .15)
7.2.2 The phase shifting action of the Hilbert transform From Equation
7.15
g; {/(t)} = -j[g;{fa(t)} - ·�{f(t)}l Using Equation
{
7.16 with 7.13
g;{j(t)}=
leads to
-j[2FGw)-FGw)]' -j[FGw)-FGw)],
.
-J
7.8
w= 0
7.8.
Since
Equation
w> ° w< °
-j[O-FGw)], which reduces to Equation
(7.16)
- e -jrr/2
_
credits the Hilbert transform with retarding the phase
of all positive frequencies by 90
° negative frequencies by 90 .
° and advancing the phase of all
Hilbert transforms and complex
I 18
The above is the action of a perfect from Appendix
analytic signals
90°
phase lag network. In fact,
4
K{cos(wt)} = sin(wt ) ,
w>O
(7.17)
Incidentally the above operation, yielding a real output for a real input, should not be confused with merely multiplying by j which changes a real signal into an imaginary one. Multiplying by j advances the phase of both +ve and -ve frequencies by
90°
and thus destroys
the essential spectral symmetry of a real signal.
7.2.3 The inverse Hilbert transform
{
Applying Equation 7.8 to the double Hilbert transform gives
.qF
{le t)} =
-FUw), o,
w>O
-FUw),
w
w=
(7.18)
0
Thus Equation 7.18 leads to
!( t) = -f(t)
(7.19)
Equation 7.19 may be regarded as an inversion relationship, i.e.
00
f{t) = - � 1T
J'
•
f(x) ( t-x)
dx
(7.20)
The loss of d.c. component shown by Equation 7.18 is of no practical significance. If one insists on dealing with signals which possess a d.c.component and exist for all values of the
d.c.
component
is
lost.
However
all
t
from _00 to +00,
practical
signals
transients-even repetitive ones-thus the suppression of the
w=0
are line
will show up as a droop which will be infinitely small over any finite interval.
7.2.4 One-sided time function s Equation 7.15 can be regarded as a way of writing the inverse Fourier transform of a one-sided spectrum. A similar result can be obtained by considering the Fourier transform of a one-sided time function. Let f(t)
be a real one-sided, Fourier transformable time function, specifically
t>O t=0 t
(7.21)
Hilbert transforms and complex analytic signals
119
A two-sided function g et) can be formed by reflecting f(t) about the origin, i.e.
get) = f(t) + f(-t) in other words
g(t) =
{ f1(t),
(7.22)
t�O
f1(-t),.
t
(7.23)
Equations 7.22 or 7.23 allow f(t) to be written (7.24)
f(t) = g(t)u(t)
Since f(t) is Fourier transformable, it follows from Equation 7.22 that g(t) is also Fourier transformable. Also, from Equation 6.18
no(w) i(w) i(w) {I/0,jW, w 0 g;{u(t)} =
where
+
=
=
Using the convolution theorem with Equation 7.24 gives
2In [ J GUx)mS(w -x)dx J GUx)i(w-x)dx J (7.25) precise definition of i(w) shows the second integral in 00
00
ff{f(t)} =
+
_ 00
The Equation 7.25 to be the principal value of 00
GUx) dx j J ew-x) Hence Equation 7.25 can be written in the form
FUw) R(w) -jR(w) =
R
(7.26)
where (w) = -!GUw). Since f(t) is real it follows from Equation 7.22 and Section 6 . 6 (Equation 6. 9 ) that GGw) is real; thus R(w) :s the real part of FG ) while -R(w) is its imaginary part. It can also be seen that is the Fourier transform of f(t) + f( -t). It should be stressed that f(t) of Equation 7.24 must be Fourier transformable for Equation 7.26 to hold. Sections 6.3 and 6.4 show that this means that the poles of F(P) must lie in the left half of the p plane. A further requirement is that the Hilbert transform of R( be convergent; this means that F(P) should have more poles than zeros.
2R(w)
w)
w
120
Hilbert transforms and complex analytic signals
7.2.5 Proof of Equations 7.10 to 7.12
To prove Equation 7.10 we write, using Equation 7.7
f
f � J (;�;)
00
00
f(t)g(t) d t =
-�
00
f( t )
- �
dx
- �
dt
The order of integration is immaterial, hence
f
J g(x) ; J (�)x) dt
00
00
f( t) g(t) dt =
00
_00
_(X)
_00
The inner integral on the right-hand side of Equation all terms which are functions of the variable separately. Comparing Equation
7.27
t,
J f(t)g(t) dt= J g( ) j{ ) x
-
since
Equation 7.10.
t
and
contains
7.7 gives
00
_00
which,
7.27
and so may be evaluated
with Equation
00
(7.27)
dx
x
_00
x
are
dummy
dx
variables,
is
equivalent
to
Equation 7.11 follows directly from Equations 7.9 and 7.10, i.e.
J f(t)g(t) dt= - J .f(t)g(t) dt= J j(t)g(t) dt 00
00
_00
00
_00
A corollary of Equation 7.11 is
J i.e.
J fJtt)]2dt 00
00
r/(t)Pdt=
(7.28)
_00
Equation 7.12 can be proved using Equations 7.11, 7.15 and 7.28,
J !fa(t)-ga(t)12dt= J 1 [f(t)-g(t)] +jLttt)-g(t)] 12 dt 00
00
_00
_ 00
The right-hand side becomes
J {(flt)P + [g(t)]2 - 2f(t)g(t) + [/(t)]2 + [g(t)]2 -2j(t)g(t)}dt 00
_ 00
which, using Equations 7.11 and 7.28 reduces to
J {fJtt)]2 + [g(t)] 2- 2f(t)g(t)} dt 00
2
_00
proving Equation 7.12.
Hilbert transforms and complex analytic signals
121
7 . 3 E N V E LOPE A N D P H AS E FUN C T I O NS When engineers refer to the envelope of a real
RF signal
they mean the
waveform traced out by the tips of the +ve excursions of the alternating signal. This is the waveform which is recovered by a physical envelope (or peak) detector.
A
complex analytic signal has a one-sided spectrum, by definition ,
and can be visualised as a collection of vectors each rotating in the same direction
(anti-clockwise)
frequencies .
RF
A
but
with
different
amplitudes
and
complex analytic signal which is confined to a narrow
band , can be visualised as a single vector rotating at a nominal
carrier frequency with short term fluctuations in the speed of rotation caused by any phase modulation. The length of the carrier vector will fluctuate in accordance with any amplitude mod ulation. The above visual interpretation , which is compa tible with a real signal being the real part of the complex analytic signal (Le. the proj ection of the rotating vector on its baseline) allows a ma thematical definition of signal envelope and phase which can be applied also to the case of broadband signals where a physical definition would not be so easy. If
f� (t) = j( t) + j /( t) The envelope of f(t) The p h ase function o ffet)
V [{f(t)}2 + {/(tW ]
= lfa ( t ) I =
= Arg{J� (t) } = tan
-1[;���J
EXAMP LE
Let
j(t ) = cos(wo t) Then
le t) = si n ( wo t )
( from
Append ix 4)
Hence
Thus
fa C t) = c os (wo t) + j s i n (wo t) Envel o pe
The phase function
=
[fa ( t) I
e (t) is
=
V{ cos2 (wo t) + sin 2 (wo t ) } = I
given by
Tan [ 8 ( t ) ]
=
si n ( wo t )
c os(wo t)
= t an(wo t )
Wo t . a spe c i fic frequency
Hence , the principal val ue of the phase function is It is normally only necessary to identi fy
as the
122
Hilbert transforms and complex analytic signals
carrier freq u e n c y when one wants to consider t h e properties of a low pass m o d u l a t i n g fu n c t i on i n d e p e n d e n tly of t h e RF signal to which it i s appli e d . When considering a real , c o m p l e x , o r com plex a n aly tic low pass m o d u l a t i n g signal , one may define the carrie r frequency as that fre quency by which t h e l ow-pass spectrum has to b e s h i fted u pwards t o obtain the complex a n al y t ic signal co rrespo n d i n g to t h e real modulated R F signal . A n exam ple o f the use o f the above d e finition o f carrier frequency may b e given b y summarising the resul ts which will b e obtained i n Sec t i on 7 .4. I f t h e spe c t ru m o f a m o d u l a t i n g signal e x t e n d s t o -f t h e n t h e carrier
fr eque n cy used w i t h that m o d u l a ting signal must b e greater than f i f
the e x p o n e n t ia l and c o m p l e x anal y t i c signal s , c o r responding t o the r e s u l t a n t real RF signal , a re to b e equal .
7 .4
THE EX PONENTIAL APPROXIMATION TO THE COM PLEX ANAL Y TIC SIG NAL
Conside r the complex signal 0:( r )
wh e re
=
l aC r ) I e i I Wo 1 + 0( r ) I
la ( t ) l , ¢J( r ) a re r e a l a r b i t r a ry fu n c t i o n s o f t i me . By insp e c t i on Io:( t ) I Arg l o:( r ) 1 R e I O:( f ) 1
=
la ( r ) I
= Wo f + 6( r ) =
la( ! ) 1
C OS I Wo f + 6( t ) 1
Th u s , i f o: ( t ) w e re t h e c om p l e x a n a l y t i c s i g n a l
= j( r )
c o r re s p o n d i n g
to j(t), la( ! ) 1 w ou l d b e t h e e n v e l o p e o f j( l ) a n d [ w o t + ¢( r ) ] w o u l d be i t s ph ase
fu n c t i o n . S i n c e j ( l ) i s i n a fo rm w h i c h is n o r m a l l y e n c o u n t e r e d in pract ical s y s t e m s , t h e above s a t i s fy i n g r e s u l t s w o u l d a l l ow one to go d i rec t l y to t h e c o m p l e x a n a l y t i c signal w i t h o u t h a v i n g to eval u a t e t h e re levant H i l b e r t t r a n s fo r m .
H oweve r O:( l ) , al t h o u gh a l w a y s c o m p l e x , w i l l o n l y b e a complex
a n a l y t ic s i g n a l i f i t s s p e c t ru m is z e r o fo r n e g a t ive fr e q ue n cies . The latter
e ffe c t w i l l o n l y oc c u r i f t h e c a r r i e r freq u e n c y ,
wo , is so h i gh that the
spec t r u m o f j ( l ) i s ze r o a t l ow fr e q u e n ci e s . If this is not so, the Hil b e rt
t r a n s fo r m a t i o n m u s t b e u se d t o c a l c u l a t e t h e c o m p l e x a n al y t i c signal . To b e p e d a n t i c t h e spe c t r u m o f J ( r ) w i l l n e v e r be z e r o a t l o w
freq u e n c i e s s i n c e p r a c t i c a l sign a l s a re n o t
t ru l y b a n d l i m i t e d ; t h u s ,
s t r i -: t l y , t h e H i l b e r t t r a n s fo r m a t i o n sh o u l d a l w a y s b e use d . How ever the e r rors i n v o l ved i n t a k i n g s m al l .
0:(1 ) as the c o m p l e x a n a l y t i c s i g n a l are oft e n
Hilbert transfonns and complex analytic signals
123
By way of exampl e , consider the low frequency compone n t s of a signal consisting of a burst of carrier which has a duration of n radio frequency cycl es. A study of the (sin x)/x fu nction shows that the l ow frequ ency components will b e 20 1 0g l O (m r ) d B down on the radio fre quency components. A 30 nS X-Band pulse would have low frequency components at least 60 d B down. I n practice the attenuation would b e m uch greater, as the (sin x)/x function applies t o the spe c t rum of a pulse having zero rise time . The rel ationship b e tween the exponential and the complex analytic signals will now be show n . Let
f(t)
=
laCt) 1 cos [wot+d>(t)]
(7.29)
The corresponding exponential signal is defined as
F rom 7.30
=
la(t)1 exp (j(wot+d>Ct))]
(7.30)
f(t) = R e{o{ t) } = -! [ aCt)+a *(t) ]
(7.3 1 )
aCt)
Al so, from Equation 7. 1 5
f(t) = Re{fa(t) } = -! Ifa (t) +fa(t) ]
(7. 3 2)
From Section 6. 7
ff{g* Ct)} = G *(-jw) Th u s , Equations 7.3 1 and 7.3 2 give
FU w) = H dU w)+ d*(-jw) }
(7.3 3 )
FUw) = -! {Fa U w)+ [Fa * (-jw)] }
(7.34)
where
dUw)
=
,�{a(t) } ,
Fa G w)
= .?l" {fa(t) }
That Equations 7 . 3 3 and 7 . 34 d o not imply dOw) = FaOw), and hence aCt ) = fa Ct) can be seen by reference to Figs 7.1 and 7 . 2. Re-arrangi ng Equation 7.33 gives
slOw) = 2 FOw) - d *( -jw) For the special case of
Wo
high enough for
d Ow)
= 0,
w�o
it fol lows that
�/ (-jw )
=0 =
S'1 * ( -jw),
w�o
(7 .35 )
124
Hilbert transforms and complex analytic sig nals
and , from Equa t ion 7 . 3 3
FU w) = 0 ,
{
w=O
H e n c e , Equation 7 . 3 5 can b e w r i t t e n i n the fo rm
dG w ) =
2FG w ) ,
w>O
FG w ) .
w=0
0,
w
which i s the d e fi ni t i on o f FaG w ) given b y E qu ation 7 .1.
2
---- -
o
o
w --
w --
w --
Fig. 7. 1 . Sp ec/ra oj n t ) alld its related complex analytic alld exponential sigllals
!F(jWI !
o
w -
W --
Fig. 7. 2. The jormat iO I1 oj the spectru m of
f( t )
Th us fo r t h e special case o f a sufficien t l y high carrier fre quency a(t) = fa U ) , i.e. t h e expone n t ial and complex anal y t i c signals c o r responding t o f( t ) are equal. As a m a t t e r o f i n t e rest i t sho u l d b e noted that i f W o is fixe d and the ab ove req u i rem e n t s are t ransfe r red t o a restriction o n the spe c t rum of the complex m od u l a t i ng signal , the res t r iction need only b e applied to the -ve modula t i ng freq uen cies.
Hilbert transforms and complex analytic signals 7. 5
THE
R E S U LT S OF
125
M U LTIPLYING R E A L O R
C O M P L E X A N A L YTIC S I G N A L S
Consider the effect of mUltiplying two real R F bandlimited signals andg et) having spectra as indicated by Fig. 7 . 3 . Note that WI
= bandwidth of f(t)
W2
=
bandwidth ofg et)
W3
=
lowest frequency in spectrum of f(t )
W4
=
lowest frequency in spectrum of g et )
f(t)
With the above notation W I to W4 are all +ve. Fl jw)
W3
w3
w
I I w
0
I
I
�
--
G l jw) l2 Ga* I -j w )
� Ga l j w)
' wFig.
7.3. Spectra of real signals at the mixer input
Experience with physical multipliers (Le. mixers) shows that the result of such a process is the sum of a high frequency signal low frequency signal
let ).
Both signals have a bandwidth of
and are separated in frequency by
(2W4 - WI)'
h (t ) and a
(WI + W2 )
The above statement will be proved in this section together with the facts that,
(1)
(7.3 6 a)
Hilbert transforms and complex analytic signals
126 and hence
Re{!a(t)gaCt)}
=
2h (t)
(7.36b)
and (2)
Re{!a (t )g;(t) } !a ( t)g;( t )
=
=
21 (t),
(7.37a)
always
21a(l),
if W3> (W2 + W4)
(7 .37b)
Note that the validity condition for Equation 7.3 7(b) is that the lowest frequency in f(t) must be greater than the highest frequency in g et). Since the spectral separation between let) and h(l) is equal to (2W4 - WI), the physical significance of the two signals is lost if WI> 2W4. However, the mathematical identity of let) and h(t) is still preserved by Equations 7.36 and 7.37.
EXAMPLE
A simple demonstration of the truth of Equations 7.36 and 7.37 can be obtained by considering single sinusoids. Let f(t) COS(W3t) and get)
=
co
=
s(w4r ). Then
j�(t)
&(r)
cos(wJr)+j
=
=
w ) r)
s in (
COS(W4r)+j sin(w4r)
Now
giving
1I(r)
=
l(r)
=
1 COS(WJ+W4)t 1 COS(WJ-W4)r
Thus
This can be seen
to be e q ual to �j�(r)gAr). Also
j�(r)g(�-(r) =
(OS(W3 -W4)r+j
sinew) - (4 )r
giving
Re{J�(r)g:(r)} j�(r)g�:(r) is only equal K{cos(wr)}:j: sin(wr) if w < o. Note that
to
=
21(r)
2Ia(r) if W3 > W4, this is because
127
Hilbert transforms and complex analytic signals 7.5.1 Proof of Equations 7.36 and 7.37
The general proof of the above results follows from the convolution theorem
�{ftt)g(t)}
=
f
I 2 1T
00
FOx)G[j(w-x)] dx
Expressing FOw), COw) in terms of complex analytic spectra, this becomes
ff{ftt)g(t)}
=
J
00
1
8 1T
{FaO x)+F;(-jx)}{Ca[j(w-x)]+C:[j(x-w)]}dx
Performing the multiplication yields
J
00
,9'{ftt)g(t)}
=
8I1T
+
1
8 1T
{FaO x)Ca[j(w-x)]+F;(-jx)C;[j(x-w)]} dx
J
00
{FaOx)Ca*[j(x-w)]+F;(-jx)Ca[j(w-x)]}
dx
_<X>
(7.38) For convenience of reference the first integral of Equation 7.38 will be defined as HOw) and the second integral as LOw). Thus Equation 7.38 becomes
.�{f(t)g(t)}
=
HOw)+ LOw)
(7.39)
If, in Equation 7.38, -w is substituted for w and the dummy variable is changed to -x it is clear that
H(-jw)
=
H * Ow)
L(-jw)
=
L * Ow)
Thus from Section 6.6, both HOw) and LOw) are Fourier transforms of real signals. The physical significance of HUw) and LOw) will now be shown. Using the convolution theorem (7 AO)
_i
<X>
y{fa(t)g;(t)}
=
211T
Fa U x)Ca[* j(x-w)] dx
(7 Al )
128
I 1 I
(.
Hilbert transforms and complex analytic signals
W2
I
I 1
'II
I
:W4
'jI 1
W3
'I·
(W, +W21
w-
Spectra involved in the multiplication > 0
7.4.
all
,_
X ------..
Fig.
,,
;
(W3+W41
w,
fa(t)ga(t). Note that
GaU(w - x) J is Ga( -jx ) moved to the right. for w
It can be seen from Fig. 7.4 that Equation 7.40 defines the spectrum of a complex analytic signal (since W3 + W4 > 0). Also
Re{!a(t)ga(t)}
1 ffa(t)ga(t) +fa'tt)g:(t)]
=
Hence
J F;\-ix)C;li(x - w)] dx} 00
+
Comparing the above with the first integral in Equation 7.38 shows .�
{Reffa(t)ga(t)]}
=
2HUw)
The above results, together with th e complex analytic nature of
!a(t)git), establishes the truth of Equations 7.36(a) and (b). The proof of Eq u ati o n 7.37(a) follows the same lines.
.7{fa (t)9 a"It)}
1 w2
(.
I I I.
(W3 - W4)
x_
Fig. 7. 5.
G�[i(x - w)J
Sp ectra
is
G�(jx)
I I'
.
I
1
I I
W,
I ( ' ( ·1, - W - W2 )
(
(W3,
'1
I
(w, tw21
( I
·1
4
w_ involved
in
the
multiplication
moved to the right for
w
> 0
fa( t)g; (t).
Note
that
Hilbert transforms and complex analytic signals
129
Figure 7.S shows that while g;{fa(t)g:(t)} defines an asymmetric spectrum which is lower in frequency than g;-{fa(t)ga(t)}, it is only one-sided if (W3 - W4 - W2 ) > O. This establishes Equation 7.3 7(b). 7.6 FILTERING EITHER REAL OR COMPLEX ANALYTIC SIGNALS
When working in terms of complex analytic signals it is essential to know how the results obtained relate to the corresponding operations using real signals. For the purposes of illustrating the effect of filtering it is convenient to define two hypothetical signal processors, shown in Fig. 7.6. The ASF (analytic signal former) converts a real signal into its complex analytic counterpart, while the RSF (real signal former) performs the inverse operation. f it)
----11
ASF
--o..---�_R_SF___' f-----. fit)
1-1
f It) a
=
fit)
+
jflt)
Fig. 7. 6. Hypothetical signal processors
It is also convenient to define the complex analytic transfer function
{
Ciiw) corresponding to a physical transfer function CUw). By definition
CaUw)
=:
2CUw),
w>O
CUw),
w=0
0,
w
(7.42)
It therefore follows from Equation 7.15 that the relevant impulse responses are related by
ga(t)
=
get) +ji(t)
(7.43)
The above definitions have been used in Fig. 7.7 to present some equivalent systems. In the case of Figs 7.7(b) and (c) it can be seen that the spectrum applied to the RSF'input is
HaUw) =
{
2FUw)CUw),
w>O
FUw)CUw),
w
0,
w
=
0
Hence equivalence is proved. The RSF input spectrum in the case of Fig. 7.7(d) is
2FUw)CUw),
w>O
�FUw)CUw),
w 0 w
0,
=
13 0
Hilbert transforms and complex analytic signals fit)
------�I
G Ij
W)Ir----------. hit)
la ) Rea l signa l into rea l filter
fit)
-----1
ASF
t-------i\ G Ij w)
RSF
� hlt)
RSF
� hlt)
"SF
� hl<'
Ib) Ana lytic signal into rea l filter
fit)
� G l j W ) �---�
--------
\
a
1
h a It) Ic) Rea l signa l into a na lytic filter
'I<'
-----i
ASF
HG"'jW'� ha lt)
fa it)
Id) Analytic Signal into a na lytic filter
Fig. 7. 7. SOllie eq1l i v alen t SI'stellls (a) real signal into real filter (b) analytic si!mal iI/to real filfer (c) real sigl/al iI/to analytic filter (d) analytic signal into analytic ji'lrer
from HaUw) in the infinitely narrow band
Although this differs surrounding
w
=
0,
the
discussion
in
Section 7.2.3
(following
Equation 7.20), coupled with the fact that th e c o m plex analytic signal method is normally lIsed only for RF signals, means that Fig. 7.7(d) is
equivalent to Figs 7.7(a), (b) and (c) for air practical purposes.
EXAMPLE
As a simple e x a m p le of the lise of Fig. 7.7, consider a sinusoid applied to
an arbitrary filter. Let lU)
=
COS(Wol),
ft.t) sin(wol), =
fa(t)
=
GUw)
=
IGUw)1 ejC/l(jw)
frolll Appendix 4
COS(Wol) +j sin(wo t )
=
e jwo(
Hilbert transforms and complex analytic signals
13 1
From Fig. 7.7(b) and the convolution theorem
J g(x) ejwo (t-x) dx 00
ha(t )
=
=
ejwot
00
J g(x) e-jwox dx
Hence
h( t)
=
Re{ha(t)}
=
ICUwo)1 cos[wo t +rt>Uwo)]
as expected.
7.7
THE COMPRESSION OF SHIFTED PULSE
A
DOPPLER
It is a common practice in sophisticated radar systems to transmit a pulse having a long envelope-allowing high mean power-with some form of phase modulation. The uncertainty function of such a waveform often shows a potential for high precision, if the radiated bandwidth is wide enough. One method of processing such a pulse, so as to exploit the inherent precision, is to fil ter it in such a fashion that the wide envelope is converted into a short envelope. The physical implications of this method are discussed in Section 5.2.2; it is the purpose of this section to develop the necessary mathematics. Consider a real, Doppler shifted, RF signal
f( t)
=
la(t)1 cos [Cwo + Wd) t + rt>( t)]
(7 .44)
which is filtered by a real filter, centred on the carrier frequency woo The fil ter transfer function will be defined as CUw)
=
-HB[j(w-wo)] ejli[j(w-wo)] + [B[j( -w - wo)] ejli I j(-w -wo) ]] * }
It is assumed that Wo is high enough for the complex analytic version ofCUw) to be
(7.45) Equation 7.45 implies that the response of the filter about Wo is the same as is a baseband filter
(7.46 )
132
Hilbert transforms and complex analytic signals
about w = O. In the normal practical case the exponential term in expression 7.46 represents some ideal dispersive characteristic which is required for pulse compression, while BGw) represents the deficiencies of a practical dispersive network together with an additional bandwidth weighting term which is required for noise and sidelobe reduction. Equation 7.44 is written in the form mainly used in this book. It is convenient, for the purposes of this section, to call the real envelope function e(t) rather than la(t)l. Thus f(t) becomes f( t)
=
(7.47)
e( t) cos [(W o + wd)t + ¢( t)]
Also Wo is assumed high enough for
fa U )= e(t)
ej(t) ejwot ejwdt
(7.48)
A
final piece of special notation is to call the phase modulation term pet) instead of e x p[jC/>(t)] , giving (7.49) With the above notation and using Section 7.6 (Fig. 7.7(d)), the output complex analytic spectrum can be written in the form .0/' {/ta< t)}
= t .?"le(t)p(t) ejwo( dWd(}// [j( W - wo)]
ejO Ij(w Wo) I (7.50) _.
The envelope of the real output signal is then given by Iha(t)1. The rest of this section will be used to show ( I) In the general case
f
""
/taU)
8-�2
=
x
f
00
F(jx)
//U(w-wo)]
ejo)j(w-wo)l
PU(W-WO-Wd-X)] ejwt dwdx
C) In the special case of linear
(7.51 )
FM,
i.e. 2 p(t)= ejb(
it is desirable
to
choose 0Gw) such that OGw) =
2
�b
whereupon Equation 7.51 reduces to /t a (t)
=
_I
4rr
b foo00 FG' x)b [-�2b 2b t] ,\/\/(�)
ejwot
+
W
d+
_
(7.52)
Hilbert transforms and complex analytic signals
133
(3) For the common case of wideband linear FM, Equation 7.52 becomes (7.53) where
The physical implications of Equations 7.52 and 7.53 are discussed in Sections 5.2.2 and 5.2.4. 7.7.1
Proof of Equation 7.51
To show that Equation 7.50 leads to Equation 7.51
f EGx)P[j(w-x)] 00
�{e(t)p(t)} =
21n
dx
_ 00
Hence Equation 7.50 becomes
f EGx)P[j(W-WO-Wd-X)] 00
x
dx
(7.54)
hit) can be found from Equation 7.54 by using the Fourier transform inversion integral, i.e.
h a (t) =
f B[j(w-wo)]
8�2
�e[j( w-wo))
_ 00
f EGx)P[j(w-wO-wd -x)] 00
x
dx
ejwt dw
Re-arranging Equation 7.55 into a form where all the contained in one integral gives Equation 7.51. 7.7.2
Linear
(7.55) W
variable is
FM
For the special case of linear FM
(7.56)
134
Hilbert transforms and complex analytic signals
'Campbell and Foster [ 12] (pair 760) give
{
}= e±j(W2/47T) e+j7T/8
(7.57)
�
(7.58)
� e +j7Tt2 e±j7T/8
Hence using Equation 6. 6
{
ff e+j7Ta 2t 2
giving
}=
I I
J(*)
PUw)=
e±j(W2/4 7Ta2 ) e+ j7T/4
2 ej7T/4 e-j(W /4b)
(7.59)
Substituting Equation 7.59 in Equation 7.51 the inner integral becomes
(7.60) The above integral may be evaluated se par ately as it contains all the w variable. The method to use in the particular case of linear FM-and possibly also in the general case-is to choose e [jew - wo) ] such that the non-linear (in w) terms in the other e x ponentia l are cancelled. This le a ve s an integral which is of the form of a Fo urier transform of a wide band function (i.e. BOw) and which will lead to a sho rt time function. Accordingl y, we c hoose
(7 .61) Substituting Equation 7.61 in 7.60 gives eiTT/4
J( �) S CoO
CoO
H[j(w-wo)]
eil(w-wo)(Wd+X)/2b)
_
With a change
of
eio!4 e-ji(wJ'
(7.62)
dummy variable to (w x}'/4b I
J(�) rBU w)
+
wo), Equation 7.62 becomes
eil w(wd > xll2b I
e i (w >w,}, de;
(7.63) From the definition of the Fourier transform 00
bet)
=
_I 21T
135
Hilbert transforms and complex analytic signals
00
Substituting Equation 7.64 for the inner integral in Equation 7.5 1 gives ha(t)=
1
ej1T/4 ejWo 4 rr
J(�) J EUx)b [ �: �] _
t+
00
+
which is Equation 7.52.
7.7.3
The case of high dispersion factor
Equation 7.52 can be evaluated numerically for specific functions. An example is given in Section 5.2.2. It is shown in Section 5.2 .4 that physical considerations are often such that the exponential term in the integral of Equation 7.52 can be considered equal to unity over the range of the other two functions. This leads to a substantial simplification; Equation 7.52 becomes
�rr ej·/4 e W ''J(�) {£(jx)b [;b �; t1 00
h,(t)"
j
+
+
dx
_
(7.65)
bet)
If the Fourier transform and its inverse are used to express in terms of and in terms of e(t), the integral in Equation 7.65 becomes
BUw),
00
II
j
1 e(y ) e- xy dy rr 2
EUw) 00
I BUw)
dw
ei[ (x/2b)+(W d/2b)+ tlw
dx
(7. 66)
Re-arranging expression 7.66 to put all the x variable under one integral gives 00
00
I
_00
e(y)
21rr I
ej(w/2b)-y)x dx
I BUw) 00
The integral in x will be recognised as Hence 7.67 can be written
I e(y)8 [2� -Y] I BUw) 00
_
00
<5
ej( Wd/2b)+t )w
[(
dwdy
(7.67)
w/2b) - y] , Equation 6.32.
00
ei[(Wd/2b)+t)w
dwdy
(7.68)
136
Hilbert transforms and complex analytic signals
Carrying out the y integration in 7.68 by the use of Equation 6.30 means that 7.68 is equal to
f B(jw)e b�] 00
ej[( Wd/2b)+t)w d w
If now M(jw) is defined as
MUw) = BUw)e
(7.69)
[21]
00 met) = _1 f BUw)e [W]
The Fourier transform inversion integral gives 2b
2,"
e jwt dw
(7.70)
Substituting expression 7.69 for the integral in Equation 7.65 and using Equation 7.70 gives
J( ) m [t ��]
ha(t) � ej1T/4 ejWot which is Equation 7.53.
'" 4b
+
Appendices
APPENDIX 1
COMPLEX CONJUGATE TERMINOLOGY •
In this book the complex conjugate of a function or a number is denoted by a single asterisk. Thus, if z= x+ jy, x and y real z*=x- iY,
by definition
Similarly, if f(z) =z + jz,
z complex
[*(z) =(z + jz ) *=z*-jz*,
Choosing a real variable,
by definition
t, f(t)= t + jt
then for t real
[*(t) = t* -jt*=t- jt,
Typical examples used in the present text are f(t)= ejbt,
[*(t)=e-jbt,
for band t real
a FUw) = _._, a +Jw
a F*U w)= _. _ , a-Jw
for a and w real
The above form of notation is frequently employed by authors of engineering books (see for example James, Nichols and Phillips [16]). Authors of more theoretical books often use a different form of notation, as typified by Milne-Thomson [17] . Following the notation of Reference 17, if f(t) = t + jt,
t real
then J(t) = t- jt,
by definition
Since the definition has been applied to the case of a real variable, rather than to the more general case of the complex variable, the normal rules of functional notation lead to f{z)= z + jz
,
Rz)=z-jz,
z complex
which is not the complex conjugate 1 37
Parseval's theorem
138
The complex conjugate of the given fez) is z - jz, i.e. /(z). It therefore follows that the relationship between the bar notation of [ 17] and the asterisk notation used here is f(i)={*(z)
APPENDIX
2
PARSEVAL'S THEOREM
Parseval's theorem as normally quoted relates the area under the squared modulus of a time function to the area under the squared modulus of its Fourier transform.
I
00
la{t)12 dt
=
f
00
1 2 1T _
00
(A2.I)
lA Uw)12 dw
A more general form is
f
00
a{t)h*(t)dt=
f
00
1 2 1T
(A2.2)
AUw)B*Uw)dw
special case of bet) a (t ). is used with the §{b*(t)}= B*(-jw). Hence
Equation A2.1 follows from A�.2 as the
=
To prove Equation A2.2 the convolution t heorem
Section 6.7 result that
J
00
.F{a(r)h*(r)} =
211T
A(jw)B*I-j(x- w)1 dw
The Fourier transform definition in t e gral g ives .00
J
{a(r)b*(r)} =
Equation A2.2 follows as the
f
a(r)b*(r) e-ixr dr O.
c a se x=
00
APPENDIX 3
.I
00
A sim il ar procedure shows that
00
a(r)h(r) dr=
21T
I
A
(jw)B( _·jw) dw
(Al.3)
FRESNEL INTEGRALS AND THEIR
RELATIONSHIP TO LINEAR FM
The Fresnel integrals
C(x), Sex), are defined
as
(A3.I)
Fresnel integrals and their relationship to linear FM
139
x
Sex)
=
f (1T�2) sin
(A3.2)
dt
o
The above integrals cannot be expressed in closed form, but tabulated results are available [13]. C(x) and S(x) are sketched in Fig. A3.1, for x> o.
1.0
C(x)
0.8
06 I
/
/ I
---I-I 0.4 I I I
/ I
I
i
!
.,
/
I
!
4
3
2
0
5
Fig. A3.1. Fresnel integrals
It can be seen from Equations A3.1 and A3.2 that
C(-x)
S( -x)
-C(x)
(A3.3)
-Sex)
(A3.4)
=
=
An integra] which occurs in the study of linear FM can be expressed in terms of C(x) and S(x) x
r
e±j(rrf/2) dt
=
C(x)
6
The above results are used to show that
±
jS(x)
(A3.S)
140
Fresnel integrals and their relationship to linear FM
wh ere UI =
U2 =
J(b�2) II �J J(��2) [ 1 �]
( A3 . 7)
+
(A3.8)
±
Putting b = 6./2d (see Section 2.4)
J( ��) [ :J J(!�) [ 2:J
UI=
I+
U2=
I±
2
(A3.9)
(A3.IO)
For ji(t) = � e ±jbt 2 , -.I. d .!. d 1
t--
1
b> 0
l/,d
FUw) =
r
-1;,d
e±bjl t 2+( w/b)t 1d t
Compl eting the square of the exponent gives FUw)=e+j(W2/4 b)
'I,d
J
e±bilt +(w/2b)12dt
-l/,d
Changing the dummy variabl e such that (A3.II)
gives
J(2b--:;;-) v(2rrb)
y =t
Hence
-
w
+
(A3.I2)
It fol l ows, from Equations A3.5 and A3.II, that the integral in Equation A3.I2 can be written in the form (A3.I3)
A
short table of Hilbert transforms
141
where (A3.14)
- J(2rr�)+-y(2rrb) w
U2-d
(A3.IS)
Equations A3.3 and A3.4 allow A3.13 to be written in the form (A3.16) Substituting Equation A3.16 for the integral in Equation A3.12 gives Equation A3.6.
APPENDIX 4
A SHORT TABLE OF HILBERT TRANSFORMS
The following examples are quoted in the extensive table given by Erdelyi [18]. /(1)
f(t)
0
�I a
t ---+
b
1 -log rr
b> a
5
l a- t\ b-t
-
expUat)
a>O
-j exp(jat)
sin(at) t
a>O
1 - cos(at) t
cos(at),
a>O
sin (at)
-get)
APPENDIX
e
g(t)
A SUMMARY OF THE MAIN NOTATION
The main notation is listed below. Any deviations are indicated in the appropriate sections of the text. Laplace transfonns
[{f(t)} = FCp) =
J f(t) e-pt dt, �
[-1 {F(o)} = f(t) = 2 j
f
c +
c
-
0' <
ReCp) <�
joo
JOO
FCp) ePtdp,
O'
142
A
summary
of the main notation
Fourier transforms 00
.qF{f(I)}
=
FUw)
r f(t) e-iWTdt
=
_00
217T f
00
.�-I{FUw)}
=
J(I)
=
FU w) eiwt dw
Hilbert transforms
7T I
00
K{.t(l)}
=
/(t)
=
x) dx -.l( (t-x)
-
f
00
K-1 utt)}
=
.I(t)
=
-
� 7T
Jx t ) dx (t-x)
TIle above integrals are to be taken in the sense of principal values.
Fresnel integrals
C(x)
f [ 7T�2]
dt
J [7T�2]
dt
x
=
cos
o
Sex)
x
=
sin
o
Real functions
J(I )
,
get),
h(t)
Complex functions
a(l),
bet),
e(t)
Complex analytic functions
J�(t)
Fq Uw) faCt)
=
=
=
the complex analytic function corresponding to the real f(t)
,qF{fa (I)} f(t) + .if(l)
A
{
summary of the main notation
FaQw) =
2FQw),
w>O
FQw),
w
0,
w
=
143
0
General fonn of transmitted signal
f(t) = la(t)1 cos [wot+1>(t )] where la(t)1 and
e i¢(t) eiwot=
te aCt)
=
a (t) eiwot
ei ¢(t)
la(t)1
An alternative form of aCt) is used in Sections 5.2 and 7.7
aCt)=e(t)p(t) where e(t) = la(t)1
pet) = ei¢(t) Coded wavefonn notation
Ci=ith (complex) code element Xb(T, w)= X(T, w) for a single bit pulse L = number
d(m, w) = A(m, w)
L-\-m
L
i=O L -\
=
L
i=O
of bits
in
ci c7+ m e-jiwd
cic7+m e-iiwd
DFT notation
N-\
gN{ai} =An= I
i=O
�Nl{An}
one code word
=
a. I
=
I
N
aie-j(2rrn/
N-\ " L.
n =0
A11 ei(2rri/
144
Other notation
Other notation
b c d
quadratic phase multiplier, b = (AI2d) velocity of propagation, also real part of p pulse duration
E
signal energy i.e.
00
_
f j
k
f
00
[f( t) F dt
real frequency variable (Hz), = (wI2rr)
v=r
PRF
period L number of bits in one codeword n number of pulses in a repetitive train, also DFT variable N number of DFT samples complex frequency variable, p = (c + jw) p r range (measured with respect to target) t real time variable tv value of Fourier transform receiver variable delay 7 time co-ordinate (measured with respect to target) v velocity (measured with respect to target) w real frequency variable (rad/s), = 2rrf Also Doppler co-ordinate (measured with respect to target) frequency of Fourier transform receiver offset oscillator WL carrier frequency (see Section 7.3) Wo frequency of matched filter receiver variable oscillator Wv x time co-ordinate (measured with respect to receiver), also a dummy integration variable Doppler co-ordinate (measured with respect to receiver), also y a dummy integration variable z a dummy integration variable A total frequency sweep (rad/s) of an FM pulse. A = 2bd. In Section 6.13, A = distance between spectral samples (Hz) oCt) the delta function (see Section 6.9) u(t) the unit step function (see Sections 6.2 and 6.8) IX(7, w)1 the uncertainty function (see Section 2) * Denotes complex conjugate (see Appendix 1) e !wk 1/1(7, w) the function, similar to X(7, w), obtained from a practical receiver 0k(t) a train of delta functions, with period k (see Section 6.12.1) matched filter gain constant (see Section 4. 4) J1
References
145
REFERENCES
1 . WOODWARD, Probability and Information Theory. with Application to Radar. Pergamon Press ( 1 955). 2. KRAMER, 'Tolerances of FM Correlation Sonars', Proc. IEEE. 627-36 (May 1 967). 3. STUTT, 'A Note on Invariant Relations for Ambiguity and Distance Functions',lRE Trans. Information Theory. 164-7 (Dec. 1 959). 4. STUTT, 'Some Results on Real-Part/Imaginary-Part and Magnitude-Phase Relations in Ambiguity Functions', IEEE Trans. Information Theory. 321-7 (Oct. 1 964). 5. SIEBERT, 'Studies of Woodward's Uncertainty Function', MI. T. Res. Lab. of Electronics Quarterly Progress Report. 90-4 (April 1958). 6. REKTORYS, Survey of Applicable Mathematics. Iliffe, 695 ( 1 969). 7. GUILLEMIN, The Mathematics of Circuit Analysis. Wiley, 485-495 (1949). 8. VAN DER POL and BREMMER, Operational Calculus. Cambridge University Press ( 1 964). 9. GOLD and RADER, Digital Processing of Signals. McGraw-Hili ( 1 969). 1 0. DWIGHT, Tables of Integrals and Other Mathematical Data. Macmillan ( 1 965). 1 1 . GOURIET, 'Two Theorems Concerning Group Delay', lEE Monograph No. 275R. (Dec. 1957). 12. CAMPBELL and FOSTER, Fourier Integrals for Practical Applications. Van Nostrand ( 1 948). 13. ABRAMOWITZ and STEGUN, Handbook of Mathematical Functions. Dover (1965). 1 4. SKOLNIK, Introduction to Radar Systems, McGraw-Hill, 416 (1962). 1 5 . BLACKMAN and TUKEY, The Measurement of Power Spectra, Dover (1959). 1 6 . JAMES, NICHOLS and PHILLIPS, Theory of Servomechanisms. McGraw-Hill, 41 (1947). 17. MILNE-THOMSON, Theoretical Hydrodynamics, Macmillan, 119 (949). 1 8. ERDELYI, Table of Integral Transforms, Vol. 2,McGraw-Hili ( 1 95 4). 19. LIGHTHILL, Fourier Analysis and Generalised Functions. Cambridge Univer sity Press (1958).
INDEX
Achievable precision 5, 50, 70 Advance network 45 Aerial polar diagram 2 Ambiguity 2,5,17,44,83,87 function 4 wavelength 4 Amplitude 1 3,82, 1 21 modulation 2, 9, 1 15 Analogy, reasoning by 89 Analytic signal. See Complex analytic signal Arbitrary functions 9, 1 2 2 Area of uncertainty I, 2 , 5-6, 1 5 - 1 7, 33,36 ASF 1 29-130 Asterisk notation 137- 1 38 Asymmetric spectrum 129 Attenuation, range 2,17 b,
value of 19 Bandlimited signals 103, 122 Bandwidth 2, 65, 69, 70, 72, 83, 103, 105,125,131 noise 61,72, 74-75 Bank of filters 43,65,69, 70 Bar effect. See Doppler bar effect Bar notation 13 8 Baseband filter 71,7 5 , 80,13 1 Baseband function 4 , 30, 34, 37, 122, 124 Bit 18, 19,54,55 Blip 43 Broadband signals 20,1 21, 133 Broadening, of main lobe 3,82
Calculator, Fourier transform 47, 49, 50 , 64-65, 67, 69, 70, 78,100 Carrier frequency 4, 7, 10, 13, 30,33, 44, 46, 68, 69, 115 , 12 1- 122, 123, 131 Chirp radar 83 Clear region 83 Clutter 3,6, 4 6 , 4 8 Coded waveforms 18, 19, 53-55 notation 143 Complex function 4 , 50, 63,1 22 �equence terms 13,101 variable 88, 137
Complex analytic fllter 5 1 , 62,63,67, 71,79,80,129-131 matched 62 Complex analytic signal 9, 10, 11, 1 1 6- 1 17, 1 21, 1 2 2, 1 25-129, 142 definition of 114 exponential approximation to 10, 27-28, 62, 68, 76, 1 1 5, 1 2 2- 1 24, 1 3 2,143 Complex conjugate Fourier transform of 9 2 notation 137- 1 38 Composite function 13, 28-29 Compression, pulse 80-8 2,131- 1 36 Computer plots constant carrier pulse 3 2,33 Hamming, low dispersion factor 8 2 linear F M pulse 36 Contour integration 85 ,86,1 1 6 Contour of uncertainty function 5, 6, 33,36 Control loop 43 Control variable delay 47, 48,50, 5 1 , 68,77,79 Convergence 85-88, 97, 98,99, 1 1 9 Convolution theorem 21, 59, 63, 98, 117, 119, 1 27, 1 31, 1 38 DFT 102,108, 109-110 Corresponding variables 51, 67 Coupling, delay-Doppler 37,43-44,48, 77-78, 81 Cross uncertainty functions 37, 38,40 DC component 118 Fourier transform of 93 gain 61, 75 Hilbert transform of 14 1 Delay 1-3, 7-8, 43, 44, 48-49, 68, 72-73,78,79,105 control 47, 48, 50, 51, 68, 77, 79 for matched filter realisation 43, 45, 60,71 group 80 line 45, 4 8, 49,80 measuring system 2,4 1 precision 4,7, 32,37 Delay-Doppler coupling 37, 43-4 4, 48, 77-78,8 1
14 6
Index Delta function 94-99 dimensions of 104 Fourier transform of 95 sifting property of 95,117 train of 95, 101-102, 104,110-111 Detectability 44 Detector, envelope 121 DFT convolution theorem 102, 108, 109-110 definition of 101,111 evaluation by FFT 106-108 of sin and cos 110 properties of 102-103 relationship to Fourier transform 101-102,103-106 Difference, Lm.s. See R.M S error D fference frequencies 115 ,1 25-128 Discrete Fourier transform. See DFT Dispersion factor 34, 39, 48, 53, 80-83,135 definition of 20,76 Dispersive network 80' 13 2 Displaced uncertainty functions 46-47 ' 67,69 due to repetition 14, 17,18,5 2-5 4 ' 70, 83 Dominant part of plane 2, 17, 35, 39, 52 Doppler delay coupling 37, 43-44, 48, 77-78,81 effect 1 measuring system 2, 41, 46, 68-70 performance 72,73 precision 7, 16, 32,35, 44-46, 49, 50 resolution 33,35,73 shift (or offset) 1-3,4 2,43,47,62, 70,78,80,81,131 sign, I, 70 Doppler bar effect 16-18,53,70 precision due to 16,46,49,70 Droop due to Hilbert transform 118 ' 130 Duty cycle 5 2,53,82-83 .
�
.
Effective number of pulses 49 " 51-53 64 Energy of impulse response 41, 59, 61 ' 75 of repetitive signal 17 low frequency 69, 77, 115, 1 2 2-1 23,130 transmitted 4-7 1 2 Envelope 70, 83. See also R F envelope Error, r.m.s. See R M.S. error .
147
Exponential approximation to com plex analytic signal 10, 27-28, 62, 68, 76, 115,12 2-124,132 ' 143 Fall time 69 Fast Fourier transform (FFT) 65,100, 106-108,112-113 Filter 51, 59, 60-6 2, 6 2-63, 67,71-74 ' 115,129-131 bank 43, 65,69,70 complex analytic. See Complex analytic filter matched. See Matched filter non-matched 46, 50,71,80 Finite logic speed 50,65 Finite processing time 49-50, 51-55, 64,69-70,103,107 FM linear. See Linear FM sawtooth 76-84 triangular 37-40 FM CW 83 Fourier series 96-99 Fourier transform 11, 85, 88-89, 134, 1 36,138 calculator 47, 49-51, 64-65, 67, 69-70,78,100 definition of 88 discrete. See OFT evaluation by OFT 103-106 fast. See FFT inverse 88,104,106 limiting cases 92-94 nota lion 98-99 of conjugate function 92 of delta functions 95,1 1 0-111 of signum function 94 of step function 93 of unity 93 other properties 98 physical interpretation 89 receiver 2, 40, 46-51, 5 2, 65-67, 68-70,76-79 relationship to OFT 101-102 symmetries 89,90 Frequency, carrier 4,7,10,13,30,33, 44, 46, 68, 69, 115, 1 21-122 ' 123, 131 IF 41,69,71 instantaneous 19,37 local oscillator 4 2, 43,51,65 modulation. See FM of �et 46,48,49,69,77 positive and negative 47, 89, 103, 1 14,1l7-118, 1 2 2,124 sum and difference 115, 1 25-128 sweep 19, 20,37,65
Index
148
Frequency weighting 46, 50, 67, 81-8 2,103,13 2 F r es n e l cross functions 39 integrals 38-40,80,138-141 pedestal 39 Function ambig ui t y 4 arbitrary 9, 1 2 2 baseband 4 ,30,34,37,1 2 2,124 complex 4,50 , 63,1 2 2 composite 13, 28-29 delta. See Delta f un c t i o n H a m ming 79, 81,8 2 m ul t i pl y i n g 16, 51, 53, 67, 78-79, 115,1 25-1 29 sig n u m 93-94 step 86-87,92-93, 116 uncertainty. See Un c e r t a i n t y fun c t io n un i que 86 w e i g h t i n g . Sec Weig h t i n g
1 3 ,17,27 p h en o m en o n 16
Ga t ed o s c i l l a t o r Gibb's
Gr o up d e l a y RO
79,81, 82
Ham m i n g f u nc t i o n He a v i si d e
86 114-120, 122 116
H i l b e r t t r a n s fo r m d e fi n i t i o n of
n o t a t i o n 142 p h as e sh ifti ng pro perties I16 t a b l e 141
a c t ion
Ho m o d y n e rec e i v e r IF f r e q ue n c y
115, 117-118
70
41,69,71
of t r a n s fe r fun ct i o n 115, 119 I magi nary signal or transform 90, 102, 114 I m pul se r e sp o n se 59,71, 84, 129 co m p l e x a n a l y t i c 51, 62, 63, 67 , 1 29 en e r gy 41,59,61,75 of m a t c he d fi l t e r 41,60 , 7I,75 I m pul si ve sa mpling 104 I n e qua l i t y , S c h wa r z 2 2,59 In sta n t a n e o us freque n c y 19,37
I m a g i na r y
pa r t
I n t eg r a t i o n
c o n t o ur 85,86,116 time 49-50, 51-55, 69 In v e r s i o n of D FT 101,108,109,III of Fo uri e r t r a n sfo r m 88, 104 of Hilbe r t t r a n sfo r m 118 of L a pl a c e t r a n sfo r m 86
Laplace transform 85-87, 88, 98, 119 physical interpretation 89 Lasers 69 Limitations of. uncertainty function 1-3 Limiting cases of Fourier transform 9 2-94 Line spectrum 49,97,101-102 Linear FM 43-44,48,53,83,13 2-136, 138,139 effect on uncertainty function 19-20, 23-24 rectangular pulse with 34-37, 76-84 Local oscill a t or 4 2, 43,65 Location of received signalS Matched filter complex analytic form of 6 2 -64 c o n c e p t 59-6 2 fo r repetitive pulses 44-4 6 r e c e i ve r 2 , 40, 41-46, 48-51, 52, 62-64,67,71-76,79-8 2 t r a n sfe r fun c t i o n 60 Ma thematics o f F o uri er transform receiver 65-67 of rnatched filter receiver 62-64 of un c e r t a i n t y function 9-29 M a x i m um l e n gt h code 54-55 M ea n s qua r e error. Sec R. M . S . error M ea sur e , of d i ffe r ence 3 M i x e r 47,50,79,1 25 M o d i fi ca tio n s to envelope 82 to optimum receivers 3, 6, 41, 46. See also Weighting to signal I, 2 M od ul a t i ng si gna l , restriction on 1 2 2 Mod ul a t i o n 2, 4, 9,19,30,34,37,8 2, liS, 1 21-1 2 2, 124, 131, 132. See also F M ; and Coded wave fo r m s M ul t i pl e a r e a s of un cer t a in t y 2 , 15-17 M u l tiple t,Hgets 3,6,46,48 M ul t i pl i c a t i o n by j 118 M ul t i pl i er 13, 39, 46,50,79,107-108, 1 25 M ul t i pl y i n g fun c t i o n s 16, 51, 53, 67, 78-79, lIS, 1 25-129
N ega t i ve
fre que n c y
89, 103, 114,
I 17-118, 1 2 2,124 N o i se
bandwith 61,7 2,74-75 system 5,46,83,132 white 41, 60-62 Non coherent oscillator 14,17,28 Non matched filtt!r 46,50,71,80
Index Notation complex conjugate 1 37- 1 38 Fourier transform 98-99 p multiplied 85-86,99- 1 00 summary of 1 4 1 - 1 44 Offset Doppler 1-3,4 2,43,47,62,70,78, 80,8 1 , 1 3 1 frequency 46,48,49,69,77 inherent 48,77 oscillator 4647,48,70 One-sided Laplace transform 85,87 One-sided spectrum 1 14, 116- 1 17, 1 18,121,12 2,129 One-sided time functions 60, 87, 118-119 Operations 106-107,115, 129 Optimum receivers 3,6,41,46,5 0 Origin of T, W plane 12 time, in D FT 105 Oscillator gated 13,17,27 local 4 2, 43,65 offset 4647, 48, 70 pulsed 14,17, 27 non coherent 14,17, 28 Output envelope 4 2,6 2-63,64,71,74, 75,80,132 Output waveform 2-3,6-7, 40, 41,43, 44, 4647, 48, 49, 5 0, 5 2, 59, 67,72-73,77,78,81 Overlap 6,17,39,46,53,83,105 P multiplied notation 85-86,99- 1 00 P plane 87,119 Paradox 43,96 Parallel processing 50,64-65 Parameters, target l, 2,4,5,6,7-8,50, 69 Parseval's theorem 1 1, 2 2, 23,61,138 Peak signal-to-noise ratio 41, 61-62, 73,75 Pedestal, Fresnel 39 Penalty in precision improvement 49,70 in resolution improvement 3, 41, 46,50,82 Period, of sequence 57,101 Phase 1 3, 1 2 1 - 1 2 2 code 54-55 modulation 9, 1 1 5, 1 21, 1 3 1 , 1 32 quadratic 19 shift, 90° 1 1 5, 1 17-1 1 8 Physical significance of Fourier and Laplace transforms 89
149
Physical significance-cont. of Hilbert transform 1 1 5 , 1 17-118 of uncertainty function 4-5 Point target 1-3,50 Polar diagram, aerial 2 Pole 74, 75, 85, 87,119 Positive frequency 47, 89, 103, 114, 1 1 7- 1 18 Precision 6-7, 1 7,39,43,47,13 1 achievable,S, 5 0,70 definition of 7 delay 4,7,3 2,37 Doppler 7, 1 6, 32, 35, 4446, 49, 50 bar, effect on 1 6,70 full 46 improvement, penalty 49,70 inherent 2-3, 7, 32-33,35, 37, 45, 46,49,68,76 range and velocity 7,33,68,70,76 PRF 17, 44, 49, 64, 69, 70, 82, 83, 102 Principal value 95,114, 1 16,1 1 7,119, 121,14 2 Processing time 49-50, 51-55, 64, 69-70,103, 1 07 Propagation in air 1 velocity of 1,7 Properties of DFT 102-103 of Fourier transform 98 . See also Fourier transform of Hilbert transform 1 1 6 of uncertainty function 4, 1 1 - 1 3 Pulse compression 80-8 2, 1 3 1 - 1 36 Pulsed oscillator 14,17, 27 Quadratic phase 19 Radar chirp 83 FM CW 83 target 1-8, 4 243, 46, 47, 48, 50, 68,69,77,78,79 uncertainty function. See Uncer tainty function Range 1 -3,5,7-8,43,46,47,48 attenuation 2,17 measuring system 2,4 1 of integration 50,8 1 , 84, 1 35 precision 7,33,68,69-70,76 resolution 7 Readout time 46,69,70 Real part of transfer function 1 1 5,1 1 9 Real signals and transforms 1 1 , 90, 1 0 2,1 1 4
Index
ISO
Receiver Fourier transform 2, 40,46-51,52, 65-67,68-70,76-79
homodyne 70 matched filter 2, 40, 41-46, 48-51, 52, 62-64, 67, 71-76, 79-82
optimum 3,6,41, 46,50 tuning 40,43, 44,47,48 Rectangular pulse constant carrier 30-33,68-76 linear FM 34-37 sawtooth FM 76-84 triangular FM 37-40 Repetition 12-17, 44. 45, 49, 51-55, 64,69,70,82-83,96-99
matched filter for 44-46 of delta functions 95, 101-102, 110-111
Resolution 3, 6-7, 41,46,48, 50, 76. 80
definition of 7 delay 33 Doppler 33. 35 improvement by weighting 3, 41. 46.
50.
51, 67, 76, 78,
81-82.103.132
inherent 3 range and velocity 7 RF envelope 42, 62, 63, 64, 71, 74. 75, 80-82,115, 13 I, 132
definition of 121 Ridge, of uncertainty function 36, 40. 48
Rise time 69, 123 R.M.S. error 3-5,9,10.115 definition of 4 Rota ting vector 89,121 RSI: 129-130 Rx. See Receiver Sampling 100.103. 104 effect of pulse width 104 Sawtooth FM 76-84. See also Linear FM Schwarz inequality 22,59 Second time round 17,44,69,83 Separation of functions 46, 77, 125, 126
Serial processing 50,64 Series, Fourier 96-99 Shape of pulse 6, 19, 40. 41, 43, 44, 46-47, 48, 50-51, 54, 71, 78, 81-82
Sheared uncertainty function 19-20,
Sifting property of [) (x) 95,117 Signal to noise ratio 41,46,61-62,73, 75,76,83
Signum function 93-94 Single tuned circuit 71-75 Sonar 1,10,115 Spectral density 89,93 Spectrum 11, 45, 46, 49, 84, 89, 103-104, 105, 122, 125-129, 132
analyser 64. 67,79 one-sided 114, 116-117, 118, 121, 122,129
Spike 2,12,50 Step function 86-87,92-93,116 Strip of convergence 85-89, 97,98,99 Sum frequencies 115, 125-128 Superhet 41 Swamping 6,48,83 Sweep, frequency 19,20,37,65 Symmetry 12, 89, 90, 102, 114, 118
Target parameters 1, 2, 4,5,6,7-8,50,69 point 1-3,50 radar 1-8,42-43,46,47,48,50,68, 69,77, 78, 79
Theorems See convolution. Convolution theorem leading to FFT 106-108, 112-113 p multiplied 100 Parseval's 11, 22. 23,6 J , 138 Sce also Properties Threshold 3-5,6,7,17,43 Time limited 4 J ,60,103 processing 49-50, 51-55, 64,69-70, 103,107
weighting 50, 51, 67, 76, 78, 82, 103
Transfer function 60,115,119,131 complex analytic 71, 129-130, 131 matched filter 45, 60,71 Transform. See Fourier; Hilbert; or Laplace Transients 118 Triangular FM 37-40 Truncation, DFT 103,104 Tuning 40, 43,44,47,48 Two-level effect 54 Two-sided Laplace transform 85-87, 88,96
39
Shift register code 54-55 Sidelobes 3, 6, 16, 46,48, 78, 79, 80, 82,83,132
Uncertainty, area of 1, 2, 5-6, 15-17, 33,36
Index Uncertainty function contours 5, 6, 33, 36 cross 37, 38, 40 definition of 4 displaced. See Displaced u.f. effect of linear FM 1 9- 20, 23- 24 effect of repetition 1 3-19, 51-5 5 graphical interpretation 4-5 limitations 1 -3 mathematics 9-29 of constant carrier pulse 30-33 of linear FM pulse 34-37 of triangular FM pulse 37-40 overlap 6, 1 7, 39, 46, 53, 69, 83 properties 4, 1 1 - 1 3 shearing of 1 9-20, 39 symmetry 1 2 Unique functions 86 Unit step function 86-87, 9 2-93, 1 1 6 Unmatched filter 46, 50, 7 1 , 80 Vector, rotating 89, 1 2 1
151
Velocity 1 , 2, 7, 48, 68 measuring system 2, 4 1, 46, 68-70 of propagation 1, 7 precision 7, 33, 68, 70, 76 readout time 69, 70 sign of 70 Volume, of uncertainty function 1 2 Waveform. See Output waveform; or Shape of pulse Wavelength, ambiguity 4 Weighting due to repetition 15-17, 53, 70 frequency 46, 50, 67, 8 1 -82, 1 03, 13 2 Hamming 79, 81, 8 2 time 50, 5 1 , 67, 76, 78, 82, 1 03 x axis 2 White noise 41, 60-62 X
band 1 23
