Signal Processing for Intelligent Sensor Systems DAVID SWANSON The Pennsylvania State University University Park, Pennsylvania
a% M A R C E L
D E K K E R
MARCEL DEKKER,
NEWYORK BASEL
ISBN: 0-8247-9942-9 book
on
Headquarters 10016
270
2
2
Eastern Hemisphere Distribution AG
8 12,
4, 1-8482;
4
filx:
4
World Wide Web
on
book
;i
Copyright
(.
2000 by Marcel Dekker, Inc.
All Rights Reserved.
book
10 9 8 7 6 5 4 3 2
by
1
PRINTED IN T H E LiNITED STATES OF AMERICA
Series Introd uction
50
As
on,
DNA
on
0
0 0 0 0 0
I by
iii
Preface
Sigriul Procvssing.Jiir Iritelligent Sensor Systems
book
by
book
on
on book
book v
vi
Preface
book
on
no
on
by
by
on
do do as:
by
on
Preface
vii
on
on book good
by
book
works
This page intentionally left blank
Acknowledgments
1993
by
on 6,
I
3
3, on
book
PC on
on
you
book.
ix
This page intentionally left blank
Contents
K. J. Rajq Liu
iii v ix
Fundamentals of Digital Signal Processing
1
Series Introduction Preface Acknowledgments Part I
Chapter 1 1.1 1.2 1.3 1.4
Sampled Data Systems
3 5 7 11 14 19
Chapter 2 The Z-Transform 2.1 2.2 2.3 2.4
20 28 32 39
Chapter 3 Digital Filtering 3.1 3.2 3.3 3.4
43 44 47 50 53
Chapter 4 Linear Filter Applications 4.1 4.2 4.3 4.4 4.5
55 56 62 69 80 85 xi
Contents
xii
Part 11
Frequency Domain Processing
Chapter 5 The Fourier Transform 5.1 5.2 5.3 5.4 5.5 5.6
87 93
96 101
106 111 115 123
Chapter 6 Spectral Density 6.1 6.2 6.3 6.4 6.5
127
Chapter 7 Wavenumber Transforms 7.1 7.2 7.3 7.4
183
Part I11
Adaptive System Identification and Filtering
130 144 158 169 179 186 196 203 21 1 215
Chapter 8 Linear Least-Squared Error Modeling 8.1 8.2 8.3 8.4
217
Chapter 9 Recursive Least-Squares Techniques 9.1 9.2 9.3 9.4 9.5
237
Chapter 10 Recursive Adaptive Filtering 10.1 10.2 10.3 10.4
275
Part IV
323
Wavenumber Sensor Systems
Chapter 11 Narrowband Probability of Detection (PD) and False Alarm Rates (FAR) 1 1.1
217 22 1 324 233
238 24 250 257 27 1 27 7 393 31 1 318
327
328
xiii
Contents
11.2 11.3 1 1.4
339 347 356
Chapter 12 Wavenumber and Bearing Estimation 12.1 12.2 12.3 12.4 12.5
361
Chapter 13 Adaptive Beamforming 13.1 MUSIC 13.2 13.3 13.4
407
Part V
Signal Processing Applications
449
Chapter 14 Intelligent Sensor Systems 14.1 14.2 14.3 14.4
451
Chapter 15 Sensors, Electronics, and Noise Reduction Techniques 15.1 15.2 15.3 15.4
521 522 532 563 583
Appendix Answers to Problems Index
589 61 1
362 368 385 396 404
409 414 430 444
454 478 50 1 516
This page intentionally left blank
Part I ~
~
Fundamentals of Digital Signal Processing
1
This page intentionally left blank
Sampled Data Systems
1
control
1
on
O K .
by A/ A /D
A/D 3
4
Chapter 1
Input Sensor System
Input
Convertor
' control i8e
In ell ut ain
Information, Patterns, Etc.
Adaptive Signal Processing System
> Commands, Digital Data Input
I
Intelligent Output Gain Control D/A Convertor
Control Actuator Figure I
A
by
If (D/A)
5
Sampled Data Systems
1.1
A/D CONVERSION
A/D
D/A
2
A/D
A/D
D/A A D/A 1.
A/D up
D/A (LSB) 0
1
D/A
0
Digital Output
1
E Convertor
-
0 bit
Counter
if a>b: count down
if a
Analog Input Comparitor
Figure 2 D/A
A
A/D
Chapter 1
6
A/D
coded in either offset binary or in 0, 11
by conzplenir~ntforrzicits.
2 VmaX
D/A b’,,,,,, 255 ( 1 1 1 A/
tttlo ’s
by
AID. 2M-1.
50% 8, 12, 255,4095, 65535, 1000, on
A/D A/D, 60
1,
0 by
1
0
1
1.
do
N2 by 1
f3.5 V A/D
+2
V) (101)
1 V
(1
1,
As
Table 1
2 AID
+3.5 +2.5 +IS +OS -0.5 -1.5
-2.5 -3.5
01 1 010 00 I 000 111 110 101 100
00 1 000
5
111 110 101
100 01 1 * 010*
+1.5*
+0.5*
7
Sampled Data Systems
no 1
good 0.5
by 1
0
-2M-*.
A/D
good by
1.2
SAMPLING THEORY
< rn <
+
Af,,d“’f”‘;-00 OQ.
x(t)
T
n
x[n] x[n] = x ( n T ) = A
= C2 = 1.1~
3 3
1. 4 100 Hz
8
Chapter 1
It
0.8 0.6 0.4
0.2 0
-0.2 -0.4
-0.6 -0.8 -1
0.015
0.01
0.005
0
Seconds Figure 3
A 75 Hz
1000 Hz
(-)
1000
4
(*)
950
1000 Hz
f’5 j,’,/2, uncrliusrd, fs/2 by
-3
A/
.r[n] = ‘4 cos(Sln)
0.4.f;
9
Sampled Data Systems
1;
'I'
"A
0.8 0.6 0.4 0.2
0 -0.2 -0.4 -0.6
-0.8 -1
0.015
0.01
0.005
Seconds Figure 4
A 950
1000
as
50 Hz
no
=
T
2qf/.f,
A
= A cos(Rr2 f 271171);
f
111
112
= 0, 1 , 2 . . .
= 0, 1,2, . . . 0 -f , / 2
.f
no
-3
1/ 3 II3.f;
1 /2,f;
Chapter 1
10
fs/ 2. 5 1000 Hz
100 Hz
950 Hz
Figure 5 A 600
3
4, ( n on
Q
100, 300,495,600 950 Hz.
950
1000 Hz
11
Sampled Data Systems
1.3
COMPLEX BANDPASS SAMPLING
by
1500
300
Also
345
1
f0.15
1500
f 100 999.9 kHz 1.0001 MHz, 1 kHz,
on
2 MHz 1 MHz rpal
f 100 Hz.
MHz 999.9 kHz
1.0001 0
As
5,
6 1 MHz, f 100 Hz
MHz,
on 999.9
1000.75 Hz
1.0001
Chapter 1
12
-mfs
-2fs
-fs
0
fs
2fs
mfs
Figure 6
0
j ; / 2 Hz
250.1875
1.0001
350.1875
150.1875
1.0000 999.9
150 6
A/D
a
do do by
AID
iis
iis
a
Sampled Data Systems
13
(6 ,f = f s ,
)
+ (6)
~ ~ [=nA ]
3.2)
+4)
=A
As
0, n/2. it
LR
+ 4)
sR[t2] = A
( 13 . 3 )
( 1 3.4)
by
7 Hz, .f2
7
fy
= .fi - f I
0 Hz do
,f, Hz,
f2
by
6’
QI
27cfl/.f;.
(13 . 5 )
14
Chapter 1
Complex Bandwidth of Interest
Complex Baseband
2n
0
4n
6n
Figure 7 nd .
by by
1;
cos(27rf;r)
D/A
by It -1
1.4
t
d
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
Sampled Data Systems
15
on
(LSB)
on
PROBLEMS
1.
10
9.801
by 1 1.5 0.305
1.5 2.
no
100,000
you 3. 1
1500
body. 1
f 5
16
Chapter 1
+
v/c.),
13
<
\’
<
+
#a 12;
1’
I-
4.
12
=1
1
94
10 525
5. odd 1 .0
30
6. 12
30
0.7 7. by
114
on.
on 10,000 Hz,
0
1
44,100 8.
0 0
0
65535
A/D. 32767 A
iis
BIBLIOGRAPHY Mag,
1996.
pp. 6183. A.
1.
1978. 1979, p. 71.
C’liffs:
Sampled Data Systems
A.
1973.
REFERENCE 1.
S. 1984.
17
This page intentionally left blank
The Z-Transform
by
by
1
e", s = 0 1.
as 0. s = 0 +jco, (0= - 10.0
+ jru,
stuhlc.
0
c o = 50n
25
1.
by 1,
0.
c = 0,
0
CT
< 0
in
> 0,
sigrwl
r.e,sporzses
19
Chapter 2
20
.w
1
0.80 0.60 0.40
aJ
0.20
K
0.00
(I)
01
-0.20 -0.40
-0.60 -0.80 -1.w
0.00
0.05
0.15
0.10
0.20
0.25
Time (sec)
Figure 1
A
(7
50
.
its
on 0 Hz
all a
.s~~.steni irupirl.sc. r.c'.spom~.
has
2.1
COMPARISON OF LAPLACE AND 2-TRANSFORMS
1) K ( s , I). =
y
K ( . s . r)1r(r)cir
--2
K(s, 1
= (I",
21
The Z-Transform
~ ( s= ) ~
1
{ y ( t ) l = y(t>e-.\‘dl 0 n+/x
y ( t ) = 9-1 ( Y ( s ) ]= -
Y (s)e’“’cls
t
rzT
z = c’‘.
[n],
t1=0
Eq. y[s]
I.
Y(s)
vz]
Eq.
t = 0,
Eq. 11
Eq.
= 0.
no
arid . ~ p c c i f i ~r ~ io~ t .future ~ l l ~ ~iiipirts. on
= a+jw),
by As
(jw
on
irriit
circle. :=
e\ 7’,
analog
on
22
Chapter 2
( f ;= 1 / T
T
1
T. a
Table 1 z
s
1 .v
- .S() 1
(.s - .so)?
The Z-Transform
23
on
on
on no
z-
Linearity
+ h g ( t ) }= ctF(s) + hG(s) Z { u f [ k ]+ hg[k]}= nF[z] + hG[z]
-Ip{c!f(t)
(2.1 A . )
Delay Shift Invariance
, f i r ) =J [ k ] = 0
t,
k < 0,
Chapter 2
24
Convolution
A
Eq.
If‘f’[k]
g[k]
g[k]
by
.f’[k].
g[k]
k
k
Eq.
Initial Value
by ;is ,s
I
/‘(I) =
sF(,s) \”rL
n =
.
F[:]
Final Value a
as
.sF(s)
f ( i )= I - 2
on
(T
j c l ,
i =
(1
-
2
x.
).
on
:-I)F‘[z] 1 1 1
ci
sF(s) F(s)
3 0, on
2 1 . on j(o
on on
F(s)
The Z-Transform
25
s =0 z
flz]
z=1
=e.'T. s
r
on Frequency Translation/Scaling
by Y{e-''Y(t)) = F(s
+N) by
Ix
2 { r k f [ k ] }= E f [ k ]x ( : p F[z/rx] k
Differentiation fit)
by
9
-
= sF(s) - f ( O )
by
..($J
,Y-
= s N F ( s )-
SN-'-kf(k)(0)
k=O
t = 0.
Z { x [ n+ NI) = e Y N ; x [ z ]-
N-l
z.Y-k.Y[k]
on i [ n+
1 T
= -(s[n
+ 11 - x [ n ] )
xk
26
Chapter 2
1
Z ( i [ n+ 11) = -((I T
- zs[O]}
by 1 .\-[II]
11.
1 Z{.i.[IZ]}= - {( 1 - z-l)x[I] - .u[O]} T
1 Z{.;;.[t?]) = -( ( 1 - z - ' ) ' X [ z ] T' -
+ z-'x[I]]]
-
s[iz],
by
z = 1.
z(s[iz]} = __ (( 1 - z - ' ) 3 x [ z ] T-7
-
+ 3Z--').Y[O]
(1 -
- (z-'
-
3z-').u[l] - z - ' s [ 2 ] )
on X[IZ]
z=1
.V
.Y = 0
IV
Mapping Between the s and z Planes
As
1.1 by
t
T = 1 /.f;
11
As f , Hz by
z" =
c"'
nT,
27
The Z-Transform
0 Hz
Hz Hz
c’”
j
0. by
0 CJ
f0,
Hz.
5 tuS/2
on
Hz on
on w,
As
0
1,
0
71
“A”
- 71.
“I”)
on /A
“A”
on
S-Plane
Z-Plane
I 1.
I
Fm E.
Stable Signal Regions Figure 2
on on
“I” 2.
2
Chapter 2
28 s
z
H(s)
h(t),
2,
H(s)
h[rz].
H[=] h[n]
h(t)
11 T
2
jw
on
do
by (a< s
z
As
by s
by
z = 0‘7
2.2
SYSTEM THEORY
up up
M---
t) at’
+R! !! +I Ky( !! t ) = As( t ) at
)
The 2-Transform
29
t >0
As([)
)
A.v(t)
M
y(t)
K on
nT,
1
A.u(t) 1)
on ) ' ( I ) Eq.
+ R { s Y ( s )-
M{s' Y ( s )- s ~ ( O )-
=fo8(t) Y ( s )=
F(s)
+ ( M s+
{Ms' = H(s)G(s)
+ K Y ( S )=,/i) F(s) =.f&
+
+ Rs + K }
H(s) H(s) =
1
(Ms'
+ Rs + K }
G(s)
G ( S ) = F(s)
+ ( M s + R)lfO)+ M j ( 0 )
+ R)j'(O)+ + (MLy {Ms' + Rs + K )
Y ( s )= F ( s ) H ( s )
on F(s).
3
Eq. y(0) F(s)H(s)
y(/)
convolution integral.
F(s)H(s)
1 riT =
Chapter 2
30
H(s) =
Figure 3
1
Ms2+Rs + K
A
=t
-t
Eq. /I( t).
Also
Ms' If(/)
+ Rs + K H(s).
The 2-Transform
31
Eq.
=fo6(t);fo = 1,
jft)
h(r)
h(t).
Eq.
H(s) H(s)=
1
Ms2 i- Rs+ K
H ( s ) = ___ 1 $1
- s,
[-
1
s - SI
1 s - s,
sI
s2
2 =z*J/;-R (5)
2M
Eq.
h(t).
information
A/D
1)
Chapter 2
32
2.3 MAPPING OF S-PLANE SYSTEMS TO THE DIGITAL DOMAIN
Eq. z = e”.
N”’[I]
as
2) z
rz = k
+
1
Eqs
on
7‘. As
0, h‘”[n]
(‘J(/
1
h(nT).
4
on
Hz 57 Hz, 125 Hz,
;= 10
n
Eq.
125 co,,T
57
300
n/2, good
33
The Z-Transform
2.00 1.oo
57 samples/sec 0
.
0.00 -1.oo 0
.
-2.00 0
0.2
0.1
0.3
0.5
0.4
2.00 1.oo
L
125 samples/sec
0.00
-1.oo
-
-2.00
1
1
I
I
I
0 2.00
I
I
I
I
1
1
0.1
I
1
1
1
I
1 I
I 1 1
I
I
0.2
l
l
1
I
I 1
I
0.3
1
1
I
I
I
I 1
I
1 1
0.5
0.4
w
9.
1.oo
300 samples/sec
0.00 -1.oo
-2.00
I l l 1 I
0
I I I I I I
0.1
I
I
I
I
I
I I
I
l
I
1
0.3 Time (seconds)
0.2
Figure 4
CL),,T
I
l
I
I
1
I
1
1
I
I
I
I
l
I
1
1
0.4
1
0.5 on
0
on Eq. (2.3.6).
34
Chapter 2
5
on
2.00
1.oo
0.00 -1 .oo
-2.00
2.00 1.oo
0.00 -1 .oo
b
125 samples/sec
b
-2.00
0.1
0
0.2
0.4
0.3
0.5
2.00 1.oo
300 samples/sec
0.00 -1.oo
t
-2.00
I
0
1
1
I
I
I
1
1
I
I
0.1
1
I
I
1
I
I
l
I
I
I
0.2
I
I
I
I
I
I
I
1
,
1
0.3
Time (seconds) Figure 5
1
1
I
I
l
l
I 1
0.4
0.5
The 2-Transform
35
Eq.
w,I
by T = 1 /fs
3.
2?f;, good
z
H(s)
HI:]
5s5
on H[z]
T, H[z]
H(s)
H(s)
6
< = 10
Hz
H[2]
25
6
300 Wz],
by T2. As
6, At
7 H[z]
Eq.
by
p,
H(s)
57
Chapter 2
36
200 150 100 50 deg 0 -50 -100 -150 -200
t
0
i
k25
50
75
100
125
150
100
125
150
Frequency ( H t ) -60 -70
-80
dB -90 -100
-110
-120 0
25
50
75
Frequency (Hr) Figure 6
(A)
H(.s)(-)
H[:]
properly
300 Hz.
As
(T=
z;
+ 5, j c o / i z = f 130
pi
pi non-r?iinimuni phiist.
0
0
=
= - 10, j w / 2 z = f240
j m / 2 n = f 160 on
The 2-Transform
37
200 150 100 50 deg 0 -50 -100 -150 -200
0
5
10
15
20
25
30
20
25
30
Frequency (Hz) -60 -70 -80
dB -90 -100
-110 -120
0
5
10
15 Frequency (Hz)
Figure 7
H(s)
6,
57
H(s) =
- Zi*) (s - p ; ) ( s- pi*)@ - &)(s (s - Zi)(S
- py)
Eq. (2.3.11). h ( t ) = A,,&';'
+ B,\epi;"+ C,,epi' + D,ePy'
1)
Chapter 2
38
p;.
pi
causes
do
by
H[z]
by
(2 -: ,)(I = T--------
T
A,. ___ [r - p l
c‘, +--] D, +-: - p4,i * +--z-p; -py A ( . ,B,,
.4, =
B,. =
c,.= D, = Eqs.
A - , B,, C-,
D=
D,
Eq.
The Z-Transform
39
8
600
As
8, 9 As
8
9,
up
by As
10 5:1
2.4
11,
3 kHz,
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
0.005 0.004
0.003 0.002
0.001
I*+++ ++
Linear Scaled
+
++
+
+
O.OO0 -0.001
Modal Scaled
++
-0.002
-0.003 -0.004 -0.005 0.00
0.05
0.10
0.15
0.20
Seconds Figure 8
(-),
600 Hz.
(O),
(+)
40
Chapter 2
-100
-110 -120
dB -130 -140 -150
-160
0
50
100
150
250
200
Figure 9
300
( 0),
( ),
600 Hz.
(+)
0.0015 0.0010
1
,
Linear Scaled
0.0005 0.0000 -0.0005
4.0010 I
0.00
0.01
0.02
0.03
0.04
,
,
,
0.05
Seconds Figure 10
( -),
3000 Hz.
(o),
(+)
41
The Z-Transform
-90 -100
-110 -120
dB -130 -140 -150 -160
0
200
400
600
800
1,000
1,200
Figure 11
1,400
1,600
(-),
3000 Hz.
(+)
on
A/D by
0 <.f < , f i ! 4
ji. is (.fs
(0).
42
Chapter 2
PROBLEMS
1. 3 -.
1x1 = 1.
3. Eq. (2.1.12) 4.
by (2.1.15)
3
5. 175 1000
40
BIBLIOGRAPHY
York:
1977.
Engleuood 1 984. 1985.
Digital Filtering
on
on on
on
43
Chapter 3
44
book,
FIR DIGITAL FILTER DESIGN
3.1
go book.
on
a
on
:
(2.l.3), (:
= ( , l W / \ ) ;is ii
j-[ri].
(3.1.1)
)
5.
it
~“121
s[n]
by
by
K
jfn]
.u[rz]
h[ri]
ii
[ H [ R ] !=
I N[- a]). = -
-Q]).
e’” /1[11]
be
Any /2[11] it
1iri.1‘
by ;i
bj’
Digital Filtering
45
8 1024 Hz,
128 on (3.1.4) z
.~[n-l].
'X[z]
k
.~[r1]
~"111
/z[rz]
N
1
(3.1.5). 375
10 1024
3 1024, 128, 16
1
8 32
32
128 1024
1 A11
1024
N I I j~ k~y
do
r t c r i c J ~I-tisporzsc~C'NII he rcw1ist.d.
2.
by
2, 0 AI = 2ticosO
odd
46
Chapter 3
10
m
'0
0 '.
'\
-10 -20 -30 0
128
64
192
256
320
384
448
512
Hz Figure 1
1024, 32,
8
;I
ii
374
Figure 2 c :I 11 \e s c ii
HA€
a
;i
at
s
nd nhan
n
Digital Filtering
47
“d”
2
8,
O)( d
0.3
-
2
0),
0.
0
C
343 0.5196 1.5 10
up
5
15
H [ z ] = 1 + z-I5
s[n]
~“121
jf113 = S [ I I ]
+ .U[/?- 151 no
2. 666 Hz
odd
3 333 Hz.
do
3.2 IIR FILTER DESIGN AND STABILITY by
z
to
48
Chapter 3
f \
A
-36
0
500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000
HZ Figure 3
2
0 = 30 . d = 0.3 m.
= 343
.v[n]
iin
;in
13[11]
jl[n],
.Y[u].
any
\+ay
49
Digital Filtering
If[:]
M (3
h[n]
s[n], lz,/z[
jfn] z , / z ; = 0,1,2, ...,M
H[z]. Wz],
on 1.
coci =
+ z.)
750n,
N.
=e
=-
2cT.
0.99 3.19 x 10
ho,
J“121
i = 10
= 1 / 1024
= 2e
- 1.32
= 0.98.
= h()s[rz- 21 - a1y[12 - 13 - a2y[n - 21
1024
1. by
P
Eq.
50
Chapter 3
2.
as
4
A R M A IIR 0
r=o
3.3
c P
h,s[ri - i] -
=
qy[i1
-j]
WHITENING FILTERS, INVERTIBILITY, AND MINIMUM PHASE
A R M A IIR
=
H[z]
H”[3] = A [ z ] i ’ B [ : ] .
H”[z]
knoun
H”[r]
i~iiii-
IIR
u i i u ~p liusc.
FIR IIR as
a
j*[ii]
(WUW/,
IIR
.4[z]
on
A
a
iiiiiiimiitu p/i(iso
H”[z]
H[z] H[z]
W[:] a
B[z]
Digital Filtering
B[z]
51
s[n]
+ d -j]
H [ z ] = z '*B [ z ] / A [ z ] H"'[z] = r'"A[z]/B[z]
H[z] ARMA
A[z]
(3.2.10)
y[n],
H"[z] ~[II].
(33.1)
300
60
1000 e+Ooh,
1.062,
f
on
IIR by
all-pass
H""[z]
5, by by no H"'"'[z],
by
6
FIR Iii the i'.utrrnic C'USI' with pair of' cmjugute zeros ut irifiiiitj~,the all-pass Jilter itqill lraw u puir of' zrros ut tlic' origin \i*hich corwspond to a delay of 2 samples in the r?iiriir?iiri?ipIiuscj sjTstcni.
U
Figure 5
H'"'[z]
52
Chapter 3
180
All-Pass Phase
90
m
0
8
-90
Minimum Phase
-180
0
100
Hz
*O0
300
400
300
400
500
10
5
0
m '0
-5 -10 -15
-20
0
200
100
HZ
500
Figure 6
sysa
also
iis
As
5
6, 1
of
;I
a
Digital Filtering
53
by
3.4
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
on by
by by Af 1/Af
PROBLEMS
44,100 8000
2. 10
10
50 3.
by 4.
3
Chapter 3
54
ARMA
5.
BIBLIOGRAPHY 1980. 1974. 1983,
J. 1983.
Linear Filter Applications
book slw
by M.hen
an A) 55
56
Chapter 4
a
A
3 as A A
no
t+
on
+I
4.1 STATE VARIABLE THEORY
A state 1
a
ARMA A
as iis
by A ' + 1
N by ,I'
a
;i
a M
\tates
book
COL
a U
"r"
k
+1
f i ( k ) ." ~ i "
.i(k + 1 ) =,f".i-(k),fi(k)] = ,4.i(k) + Blc(k)
at
a
*?(k)
k
"H"
Eq. (4.1.1 )
Linear Filter Applications
57
J ( k ) = g[.v(k),i(k)] = C.t(k)+ Dii(k) A
n x n, B n x B nx I
C
Y,
yt.2
x
C
n,
D 1 x rz
nz x
I’.
D
(4.1.3) A P x P, B P x 1,
P C 1 x P,
D 1x 1
+
0 0
~ l [ k 13
Sp[k
+ 11
1 0
0 1
-aP
0 ...
-ap-
1
P,
-
44
...
J”k] = [bphp-l . . . ho]
P= ap
Q
bo Eq.
>
P>
P-
P
1
Eqs
1 on Continuous State Variable Formu lation
3 fit)
2. on
u(t)
+
+
u(t) = Mj;(t) Rj,(t) Ky(t) ,j = a2y/i3t2
j= ay/t
(4.1.6),
58
Chapter 4
Y
1 XP
/
Xn
A
Figure 1
ARMA .u,[k].
by ii
) ? ( I ) .
by
j-(l), jf(l).
(4.1.X) = A'.i-(t)
;I
t SX(.S)
+ B'zt(1)
n - S(O+) = A'X(s)
X ( s ) = [s/ - A']-'.i-(O+)
+ BCIy(.y)
+ [sl - A']-'
B'U(s)
(4.1.10) -<-(Of ),
59
Linear Filter Applications
u(t)
S( t ) =
6"'(f).(O+) + 6"'( t - T)BCu(t)nt 0
$"'(t) = Y '{[slA"] '}. u(t)
2x 1 j(t)
C by
Eq.
jPs
k ( t ) = 9-* -
-M2/K
Eq. A4
xZ(t)
<
*?(t)= C"-i-(t).
Discrete State Variable Formulation
T
sl(t) Eq.
60
Chapter 4
Eq.
b
2
IIA k- I
A
M =1 f;
K = 314
R =4
500
2.
Force Unit Impulse Response 0.05
0.04 0.03
0.02 0.01
0 -0.01
-0.02 -0.03 -u.w
0
0.2
0.4
0.6 sec
Figure 2 500 Hz.
0.8
1
Linear Filter Applications
500
61
T =2
A As
(T 3
75 Hz.
3 signijicant oversaniplirzg is required-for nn accurate andstable irwpulse response.
0 < T
T < 2C/o;,
T>0 Hz.
f,>
fo =
3, fo = 27.65 Hz = 75
78.75 Hz.
Force Unit Impulse Response 0.15
1
1
I
I
1
I
0.1
0.05
0 -0.05
-0.1 -0.15
0
1
1
I
I
2
I
4
3 sec
Figure 3
75 Hz
I
I
5
6
7
Chapter 4
62
/ mcurcrtc’ f;
Eq.
4.2
FIXED-GAIN TRACKING FILTERS
/
on Eq. (4.1.16),
on
no ilk
A
ilk,
on
r
/)
a-/) cx-/)
o,,.
Linear Filter Applications
63
by
a,
x = 0.10, CI
>1 zk
)
xk
H V ~
k.
xklk
k+1
k
k
x;:Ik
A$,,,,
T
r-P-1~
on I
xk+IIk,
x r,
process noise.
0,
y
Chapter 4
64
do
by 2-B
0,.
vA
n
0,.
1 x-p
x-p-;?
x-/I
on
Ix-[I-j*
;.,\[.
o,T’i2.
B, ii
10 on :
s
u
n
ng
; nd non
on ns nt
u
on on
a
ro;. x. /j =
- IX) - 4
6 [j‘ir
7 x
/j (
Linear Filter Applications
65
x,
o,,
on
c,, o,,. a, /?,
7
p.
AM,
a
7 = p2/a cc,
p,
2-B
ZMG x-p-;~
5 15 0.25 up on up 10
3 c,,= 13, o,,= 3,
13 T=0.1
i M= 0.0433,
fl = 0.0374,
-y = 0.0055.
a = 0.2548,
4
U-/?-?
4
good 4 by
on a-p-,,
o,,= 3
13,
(AM = 5. by
66
UJ
Chapter 4
50 40
8 30 *
p 20
10 0
0
1
2
4
3
7
6
5
Seconds
0 0
8
9
10
9
10
0 0
0
0
0 0 0
0
N U
6 4
t%
:
c 8
\
3
1
2
4
3
6
5
6
7
8
2 0
-6 -8 -10 -12 -14
0
Figure 4
1
2
3
4
Seconds
7
8
9
10
3
x p-7
13
on
;i
100
5
Seconds
0
( - - -),
6 (U,, = 100,
(O), =
on
A,,%{, 2, p,
i'
o,,
7,
67
Linear Filter Applications
70 t 60 50 40 0) 3 30 z 20 10 0 -10 0
E
150
1
2
3
4
5
6
Seconds
7
8
9
10
t
I
N i z 0
8
2 0 -2 -4
- 4 f - 8 -10 -12 -14 0
1
2
3
4
5
6
Seconds
7
8
Figure 5
9 CJ,,
10
=3
13
4.
3
7 3
40
8 3
8
o,,,
68
Chapter 4
60 50
40 $ 30 * v)
s ;; 0 -10 0
1
2
3
4
5
8
7
6
Seconds
9
10
c
00
0
0
0
1
2
3
4
6
5
Seconds
Figure 6
7
0
8
9 0,=
4.
on
10 100
13
69
Linear Filter Applications
140 120 100 80 a 60 % 40 20 0 -20
E
=
P
-40
0
1
2
3
0
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
Seconds
100
2 Q
s
-50
-100
Seconds
Figure 7
4 n,,. =
7
8
U,,, =
on
4.3
2D FIR FILTERS
70
Chapter 4
160 140
0
120
2100 0)
%
0
80 60
0
I40 20 0 0
1
2
3
0
1
2
3
1
2
3
6
7
8
9
10
4 5 6 Seconds
7
8
9
10
4
7
8
9 10
4
5
Seconds
tgn:
- 0 2 -20 0-40 5-60 -80 -100
6 ' v 4 U
E
2
- 4 Q) e-8 5 -10 -12 -14 0
5
6
Seconds
Figure 8
3 = 40m r , [j. 7
7
(T,,
c,,
8
i.2,.
A
Linear Filter Applications
71
by
by
on 9.
9 on
good by on
Figure 9 640 x
72
Chapter 4
B(.Y.J,),
-v
A
j*
B’(.Y-,J.)
w,, Eq.
on
B(.Y+;.J*)
+s B’(x,j-).
10
c/essinrcrtrd.
10
8x 8
9.
by ‘ 2
Eq.
11
Linear Filter Applications
73
Figure 10
8x8
12 on
by
A A =
N =M
1 Eq.
+
WO. 1
;”’;.‘I
=
[
1 0 2 0 1 0
Eq. by
74
Figure 11
Figure 12
Chapter 4
75
Linear Filter Applications
(4.3.3)
(4.3.4)
45' on estiniute
A x
y
A
by
9
13
(4.3.5). 7
V-B
a'B as-
7
a'B +-
(4.3.5)
(4.3.6).
s
(4.3.6)
by
Chapter 4
76
Figure 13
1.
on
Eq. -V2B=
[
-1
do
8
by Eq.
[:::::I
w3= 14
by 10, 8x 8
15.
As 16
A
77
Linear Filter Applications
Figure 14 A
Figure 15
Wy
Chapter 4
78
16 16
by
on 16
by
on
17
by
Figure 16 A
Figure 17
4
79
Linear Filter Applications
18
on on
7.3. by
on on
8, on
Figure 18
body
body
Chapter 4
80
7.3. body
a
D/A RECONSTRUCTION FILTERS
4.4
to
(A/
(D/A)
upon no
zc~ro-ortlc~r.
A
hold a
a a
A
A
DIA
as
1s a c
Js
a it ii
1000
400 Hz.
f IOOO), f 1400
1980s
by
by
a
a s o \ . c . r . . ~ r i i ? i p l i ~ Io gn
!'
5.
Linear Filter Applications
c 1
0.8 0.6 0.4
0.2
0 -0.2 9.4
-0.6 m0.8 -1 -1.2
~
"
0
"
~
"
~
~
20
40
"
"
'
60
~
"
"
~
80
~
"
"
100
11
1
"~
"
"
"
120
"
" '
' " "
1
140
160
180
200
Sample Number Figure 19
x
x
100
32 x 44,100 Hz do
2 2,822,400
32 x 2,778,300
D/A 2 x, 4 x, 8x 20
3
2x 8x by
4x
7
F I R interpolation filter. by A
by 2x
A
4x 1 x, 2 x , 4 x ,
8x
01,.
21
82
Chapter 4
t
i
0
20
60
40
80
100
140
120
160
180
200
Sample Number Figure 20 D A
21 8x
;I
by
1
ii
Di'A
a
by
22 \+';I\
;I
1
ii
2x
4 x,
23
8x
to
22 atid 23
3x
1x
2x
,
Xx
go
8x
ii
look
4x 24.
;IS
At M
seen
24
IK o\
;i
case. Also
of
leak
03
Linear Filter Applications
- -0 . . 0th Order (1X) - -*- 1st Order (2X) .
(F- . 3rd Order (4X) -+- 7th Order (8X)
-
0.8 0.6
0.4
0.2
0 -0.2
0
8
16
24
32
40
48
56
64
Sample Number Figure 21
0
D/A
1
2
3
Sample Number Figure 22
2x
4
84
Chapter 4
1
0.8 0.6 0.4 0.2
0 -0.2 -0.4
-0.6 -0.8 -1
-1.2
2
1
0
3
4
Sample Number Figure 23
3x Xx
1
1st Order (2X)
-+----
0.6
0.4
.
-
0.2 0 -
-0.2 ;
0
0.5
1
1.5
2
f/fS Figure 24 in
2.5
Linear Filter Applications
85
14 8x
1 8 8x A on
4.5
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
A
on on on
PROBLEMS
1. (h()+hlZ')l(+ l cI,zl
2.
?
R=
C=
L =2
Chapter 4
86
3. A
by
10%
ct-D-7
cc-a-y?
4.
(2x)
5.
2nd on.
2nd
on.
BIBLIOGRAPHY
X. 1993. A.
J. C.
W. 1989. 1992.
Part II Frequency Domain Processing
"4-E-A") by
to
do
.U( t ) ,
x(t)~~""~~lt X(co).
.I-[/?].
a7
Part II
88
on on
by by by on
2 on by e'"'
(s)
(t)
c)'"
(to)
(k=
t
(o
k
on by
L+
Frequency Domain Processing
89
by do
on by on by
on
12
1
12
121/2
up on
1.059
by
on on
15
do
by
on.
by book.
Part II
90
up (2,
1/3
on by
on
good body,
on
Frequency Domain Processing
by
91
This page intentionally left blank
5 The Fourier Transform
1807. 12. 1817. T/zL;oric’ Arici/j~riqirc~ ck
/U
1822.
Clzaleur
+cc
--oc
(5.0.1)
+oo
Y(w)
~ ( t )
‘y” by
“s”
/
“cu”
+%
y(f) =
-oo
+Y
y(t)e-i2rffdt
(5
-oo
93
94
Chapter 5
(5.0.2)
Hz y( t ) =
(5.0.3)
S
+Tl’
= T+OC -
e+/ W
7-12
2j
/
+ 7-12 p-/2;;+tlt
o -/
dt -
dl
7.-CC
- 7-12
by
Eq. (5.0.5).
(5.0.5)
&fo,Y(f) orthogonulitql
Y(f)
jft).
+fo
-fo,
(5.06)
(5.0.5)
The Fourier Transform
95
k*fo
(5.07)
+
f=fo f=
-.fil
-
-
+
k.fo f=fo
A
+
A
( J ~ I= ) c’2n/of)
j(t),
1 Y(f)
T
by by -
= 1/
T ,Y # 0, Y ( f )= -jS(f;,
6(s), .I‘ = 0.
jft)
=
u)+jn6((o0+ C O ) , Y ( f )= 6(fo + S(fo Y(co)= n6((00 - (U) n6(oo+ 0). Y ( . f )= S ( f o - f), Y(cu)= - CO) Y(co)=
+j6(.fo
+f ) / 2 ,
by 2n
-
+
It([) =
+oc’
=
As 7‘
As
c>’2nfof,
by
96
Chapter 5
5.1 SPECTRAL RESOLUTION ~“121,
T .f, = 1 / T
(5.1.1) y ( i z q , rzT 1/ N
?(n]
1/ N
t,
1/ N
N. hciw t/w
wr prqfrr lzm’ to m?iplititck qf’ tlw .Ji.c.yut.rzc.?’-tiomeiin siizusoichl signtil inck>pencknt qf N ,
6. J*[H]
6 on
0 5 f’ 5 0,5/ T = 0.5/ T,
. ,v-
~~[ii],
I
fLf- I
M j , / M Hz.
Eq.
1.
N
Af=I / ( N O
1/ ( N T ) 0
N
N =M
on 1
Af)
NT, N
97
The Fourier Transform
0.8 0.6 -
wldth 1K
0.4 -
0.2
-
-0.2
-
U
-0.4 -
f)
I
wldth 1K
4.6 -0.8 -
Figure 1
fT / 2
100
Hz 10
on. j*[n], N
Af=fs/ N,
N
Af n T = n l N ,
Y(kAf)= Vk].
(5.1.4)
N
Eq.
p =n.
p # n.
#n
p -n
- (N
-
1)
+(N
1
Eq. p = n,
n do
Eq. p =n
- 1 ), “a” aN e’2n@-t’), ah’=
N j*[n]=j”p]; p = n .
98
Chapter 5
T )=
=
Ajl Aj’=fs
/fs)
.fo
= moAj:
Hz fo=mo = 2 7 c r ~ z ~/r N z ).
by
mo,n, by
nz
fnzO, +uzo
- nzo
-
by N = kn; k =
n(mo -
0, f 1 ,
...,
+moAf Hz
fS
-mo.
Af = j ; /N ,
rziAj;
IT?
Hz up 5,
10,
,r!(f)
ussumes
on
Hz Aj’=
100 16, 6.25 160 2
25
16
10 6.25 Hz
The Fourier Transform
99
“*”
f50
16 m
Y[m]
“m”
160 8,
3. 4. 2, 3,
4
by
‘b*”
by N)
25 Hz
4
160
O3
I
-0.6 I
0 ms
-0.5
Figure 2
1
0.5
100
25 Hz
a (*)
0.4
,
1 -0.5
-0.4
Figure 3
I
0 f/fs
0.5
16
100
Chapter 5
28.125
4.5 100
28.125
5 As
5 16
up spectral Ieukuge.
do
0.6 0.5
-> 0.4 Y
9
0.3 0.2
0.1
flfs
Figure 4
25 (*)
0.6 0.5
-
0.4
9
0.3
t 0.2 0.1
8 Figure 5
28.125 (*)
100
by
The Fourier Transform
101
28.125 112.5
140.612
N.
N
on 4, 32 28.125
3.125 on
9
28.125
on 16 on
2,3,
4.
160
5.3. by
on
on
N2.
5.2
THE FAST FOURIER TRANSFORM
on
do
Chapter 5
102
2
512, 1024,
2
1,048,576 105,240
by by
by by
Hz,
j;
1.3
0
1.7
/
Eq.
W N= e j Z n ' ' ,
by on 2
by
on
odd
+
The Fourier Transform
103
odd
n=O
n=0
N=8,
N
go q
2,
N=
on 3, 4, 5, by
6
Input
Figure 6
8-Point Radix-2 FFT
output
Bit-Reversed Binary Addmss Address
OOO
OOO
100
001
010
010
110
011
001
100
101
101
011
110
111
111
Chapter 5
104
WE-DSP32C, 96002, 6 by
y[3]
y[O]
by
on. h e c m w lit
one node, the two twiddle -fuctors dujc)r on113 in sign! W! = -W$, W,?.= - W { , W i = -Wi. 7
Wi = -W:,
an?l
do by b@
by j*[O]
by
+
0-1 y[4] in-place
W:
-
2 no
-
1
( W:)
2,
unit circle
Figure 7
-
z plane
6,
The Fourier Transform
105
by
Wi,
by
Wd
W: non
N /2 on 2
=
WiT, 4
2Mog2N on
by -
1
0 N,
0.
1
N - 1
on
1.2,
odd
on.
R e [ Y[m]= Re( V
N - t ? ? ] ) , t?? =
0, 1, ... , N - 1. = - lrir [ Y [ N - n ~ ] ; ,
It71 [ in
= 0, 1,
... ,
- 1.
Re[ 1171[ y [ t ? ~ ] = It??[
- 1711
= - Ref y [ N - m])
. by
Eq.
106
Chapter 5
YI[rn]
Y2[nz] Re( Y,[O]}= Re{
Re( Y,[O])= I m [
Inz( YJO])= Inz{ Y2[0])
+
- nz]}}
I!)?(Y,[rzz])= - { I m ( Y[nz]}- I m ( Y" 2
- 1121))
Re( Y?[m]} = - (In2( Y" 2
Y
Im( Y2[tH])= 3 (Rcq Y"
5.3
I I
RP( Y,[r72]}= - {Re(Y [ m ] } Re{ Y" 2
+
1
- 4 ) Inz( Y[nz]}}
1
- nz]) - Re( Y[nz]}}
DATA WINDOWING
by by
also
As
by N A
as
The Fourier Transform
107
2,
8
8
by N 2.0317
2.00
N=64
N)
by N , nurrowhand correction factor is the ratio a rectangular window of the same length. by
the the windoit, integral to the integral of
1
0.8
0.6
0.4
0.2
0
-4
-3
-2
-1
0
1
2
3
4
Relative Bin
Figure 8 N = 64
to
Chapter 5
108
by
on
the brouciband correction .fuctor is determined bqi the squure-root of the integrul of the wincio\t*function squared, then divided by N (the intc>grulof the squured rectangulnr tipinciow jirnction). N = 64 1.6459, 1.6338 N = 1024. N,
Nurrowbund and broadband correction fuctors ure criticullj*importunt to power spectrum unzplitucle culibrution in the frequencqi doniuin. by N N = 64
1.5242
up
-
(
n-i(N{(N-
(5.3.3)
2.0323.
Eq
a by
(5.3.5)
The Fourier Transform
109
1.8768
W E ' ( n )=
8
-c>-;(tl-+-l)~(3.43
10
(5.3.5)
A)
k >_ 4, k = 3,
1.0347.
1.5562.
k
on N , N=64,
2.3229, 9
6 N = 64
1
by on
N = 64,
by by
on do
1
0.8 0.6
0.4 0.2
0 -0.2 -0.2
-+- Hannin
0
0.2
0.6
0.4
n/N Figure 9
0.8
1
Chapter 5
110
Table 1
N=1024
k = 2.6 ~
2.03 17 1.6459
1.5242 1.3801
2.0323 1.7460
1.8768 1.5986
2.3229 1.7890
1 .OOOO 1 .OOOO
2.0020 1.6338
1.5015 1.3700
2.0020 1.7329
1.8534 1.5871
k = 5.2 2.8897 1.6026
N = 1024
1
0.8
0.6
0.4
0.2
0
-4
-3
~~~
1 .OOOO 1 .OOOO
-2
0
-1
1
2
4
3
Relative Bin Figure 10
10
1
N = 64
All
N, 9
The Fourier Transform
111
10. rz = 0
II
=N -
10 on
on
up
7.1, 12.2,
5.4
13.
CIRCULAR CONVOLUTION
by
circwlcrr
(’012\70htk~1?1
Q
by 1
112
Chapter 5
Figure 11 U(.[),
11
N(,/).
12 U(f)
S(J‘)
R ( f ) ,S ( , f ) ,
R(f).
N(f‘)
8. C(j’) no
C ( f ) = S ( f ‘ )= U ( f ) R ( f ) . U(f) C(f) A ( f )= 1 / R ( f ’ ) ,
11
A(f)
by U ( . f )= C ( f ) A ( . f ’ ) .
N(f),
no by
U’(.f)
U’(.f)= C ( f ) A ’ ( f ) / R ( * f ’ ) .
U(J‘) A‘(f)
(5.4.1)
0 2 A’(f’) 2 1
E = [IS[’
(.f)
+ 2 R o { N S )+ ( N / z ] A ’ +z IS]?- 2 R e { S N } * A ’ * -} 2A’(SI‘ + ,4’(N(’ S
N
SN
A’(f)
E(.f‘)
by
E
The Fourier Transform
113
A’ aA’
= ISI’(2A’ - 2 )
aE’ aA‘’
+ 21Nl‘A’
-- 2(ISI’
+
up. on
A’(f)
R(S) B ( . f )= 1 / R ( . f ) . A’(f’)
U(f) H ( f )= A ’ ( f ) / R ( f ) ,
C(f) ’4’(.f’)
no h(t) A’(f) B(f). R”’(z) = E(cr’(t)h(t,- r ) ) A ’ ( f ) B( f ) . B(f)
a’(()
on It is o r i l j i ,fhr tlic~CNSC’ I ~ ~ > I u11 Y J narroii~haizdsignal freyuenclQcomponents are bin-aligncd tlicit the spctral prochrct of tiiv ,functions results in the equivalent linear correlcitio~ior c o ~ i ~ w l i ~ t iino nt/ic tinw-dor~aiii. circular convolution
circular correlation
4),
12, by odd
by
5 12 12
no
(
-
0
-
).
114
Chapter 5
0.4 0.2
0 -0.2
-0.4 -0.6
0
128
64
192
256
Figure 12
(
0 -)
by by
128 256
zc~ro-pudtlc~cl,
12.
128
a
128 a
a
13 (
~
0- )
t
and/ As
13, 128
128
115
The Fourier Transform
-0.6 1
1
1
1
1
1
,
1
1
Figure 13
~
(-
1
1
0-) 128
do
5.5
UNEVEN-SAMPLED FOURIER TRANSFORMS
good
As
Chapter 5
116
by
A good
do
I1
good
by
on
T no
by
t[tz] I I = 0.12, ...,
N N-
, 1 1 [ n ]
11
= 0,1,2,..., N
-
1, j3[11]
117
The Fourier Transform
by (5.5.
N-l
y Lonlh ( ( 0 )= 2a'
Cy[n]- y )
o ( t [ n ]-
tl -I'\
I
1
to(t[n]-
r1=0
(5.5.2)
J
n=o
by
t
(5.5.3)
7
t[n] ulgorithm
verj'
tlio Loriih t r w s f i ~ r uisi N T
by N . A
(5.5.4),
1 YL'"'(tu)I'/N = P"lh((u) (.f/,fi = 0. )
0.1
0.5 14
T 14 15
0.1
on
15.
118
Chapter 5
1.5 1.25 1 0.75 0.5 0.25
YWI
0 -0.25
-0.5
-0.75 -1
o = .5T f / f S = .1
-1.25 L - 1 . 5 - ' 0
~
'
"
32
"
"
'
"
128
96
64
Sample n Figure 14
0
0.5T
0.1.f:~
0.1
0.2
0.4
0.3
0.5
0.6
0.7
0.8
0.9
1
f Jfs Figure 15
14
good
0.9f.S.
-
16
do not SIZO~I, this "nzi,.ror.-ir?ztrKr."
119
The Fourier Transform
0
0.1
0.2
0.3
0.5
0.4
0.6
0.7
0.8
0.9
1
f I fs Figure 16
no
0.9fS
0.5 T
0.1,f.s
As
on
as
19.
17
18 0.45fs
T.
on
0.5T. 19.
14
19
SOMC-
on 20
10T
by
O.7fi
120
Chapter 5
1.25
o=ST f/fs=.45
-1.25
0
32
64
96
128
Sample n Figure 17
0.45fs
0.5T
0.8
Figure 18
14
0.57.
21 do
1
to
0.451s ;is
wave
0.9
The Fourier Transform
121
20 15 10
5
0
dB
-5 -10 -15
-20 -25
-30 0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f Ifs
Figure 19
1.25
on
1
0.5 I 0.25 I
Yml
0 ! -0.25 L -0.5 L
-0.75 I
-1 2
-20
0
20
40
60
80
100
120
140
160
Sample n Figure 20 10
0.7fs
on
122
Chapter 5
20
15
10 5
0
dB
-5 -10
-15 -20 -25
-30
Figure 21 20.
OST? 14 0. I.fs,
16
20
by on up. AI1 thut reullj* niuttc>rsis tliut the correct sample times ure used in the Fourier integrul. As
on by N , 10
N,
21 +I8
The Fourier Transform
123
up
5.6
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
by by by 10 Hz 100
0.1
10 on
by
no
1024
10,240
1,048,576
by
do
on by N
by
N
Chapter 5
124
by by 1
by /
no
PROBLEMS
1. A
100,000
20,000 20,072 Hz.
2.
25
N N = 128, 1024, by
3.
by
The Fourier Transform
125
1
4. 0.00
11
=1
n=N, 2.00
1.732.
3” BIBLIOGRAPHY
1989. 1977.
H.
B.
S. A. 1986.
J. 1, 1992, pp. 1437. 1974. S.
1975.
REFERENCES 1. J . J. Mclgcizine, 1 , 1992, pp. 1437.
ZEEE Sigttcii Proc*c.s.sitig
This page intentionally left blank
6 Spectral Density
Hz. As
-
T
+T
+
+
t= -
x(t)
on
on -
f
t.
+T/2
(6.0.1) -
-T
X(u)
+ T,
7‘) Sx(w), A good
g(t)
f(t) 127
128
Chapter 6
F(ru),
F(w)G(-co)dtu =
(6.0.2)
(o
Hz (r)
s(t)
- T/2 < t <
T/2, (6.0.3)
by T
i
TJ2
T
+lx:
(6.0.4) -l-j2
--3o
x(t)
X( +f’).
X( - f )
(6.0.5)
(
(6.0.6) Sx(,f)
two-sided pobtvr spectrcrl densit)? ( P S D ) x(t)
129
Spectral Density
/
+m -3
x’ = E{x2(2)}= 2
SX(f)df
(6.0.7)
0
one-sided P S D . ( fT fT l 2 )
Eq. (6.0.7) power spectrum
Gxx(f),
x(t),
x(t)
~ ( 2 = ) A
+ by
A’
+ B’/2.
x(t)
X(c0) = A T
wT/2
+-BT 2j
SAW)
[w1 - w ] T / 2
BT 2j
-__
-o]T/2)
[col - w ] T / 2
(6.0.9)
by Eq.
T
(6.0.10)
I 1) -cc
7’= PSD
by
df = d o / 2 n ,
(6.0.12)
Eq. (6.0.12).
Chapter 6
130
.x[n] = s ( n T s ) , s[n],
Ts N- I
1
N-l
n=O
1/
Eq.
N mrzz],
1
N-1
N
n=O
1
1/
N-1
m=o
~[n]
do on
by N = 6.1
STATISTICAL MEASURES OF SPECTRAL BINS
1.
131
Spectral Density
x,,, Y Y=
1
+ xz + x3 + . . . + Xn]
fi
Probability Distributions and Probability Density Functions
A
Phi)
Y
)!.
1 6 3 (a)
1
0.5 (50‘%),
0.1667
6,
1.0 1
6
A
P y b ’ ) = Pr( Y 5 y)
(b) Py(-OO) = 0
(4
-oo<~<+oo Py(+oo) = 1
< y 5 y 2 ) = P Y c Y 2 ) - PYCYl)
probability density function (PDF). A
--Dc
J’
p &-)
132
Chapter 6
1
1
1
6. 1 1
6
unifornzprobabilit?.i distribution. by (6.1.4)
- 3
Y - = E ( Y 2 }=
/
y2py(y)dy
-Dc
G)’,
1
0
6
j*2
0
6
y2,
6
~
1
,
6 3
12 - 9
12.
9
1 1 12
116
3. y3
0,
100 100
100
up
300. n 1
< 1000
ni kr = 0
kr
= 1.
Spectral Density
1
0.50
0.40 0-45
0.30 0.35
I I
0.20
L
0.15
L L
0.10
133
'
-5
-4
-3
"
-2
1
"
'
1
1
-1
1
1
1
t
0 X
Figure 1 A
Statistics of the NPSD Bin o,,
oR=oI=of.
1
'
1
1
'
4
1
'
2
3
4
5
Chapter 6
134
a;
1
N-1
s2[n]=
of = -
+ X,[m]'
N-'
N'
tn=o
ll=O
a: =
by
5.1 by
a? 1/ N
10
2/N
(N2) A
Hz)
N
(ZV?),
10 A
27 18
no on by a ' .
by by
y
fY'/'.
s
Y
]'=.U'
Y
135
Spectral Density
4’ = x2.
d ( + m p( Y ) = p ( x = +Jy) dY - p ( x = -Jy) dY
1) 1’ =
p(x21v= E{)>= ) E{.x2)= 0 2 .
1
s
1 1 .
on
x
odd E ( ( s- .?)‘I}
= 1 - 3 - 5 . . . (n - 1 ) ~ ”
v = 2.
1’
2 up v= 1
n”’
1
(17/2)
M = 2v.
(A4-
2 v= 1
M = 2, 4,
(A4 >
v=2 ( M =
32
o2=
Chapter 6
136 1.75
n
I:
1.50 -
~~~
~~~
+-vvl ~=2,M=l v = 4, M = 2 - -v=8,M=4 __-v = 64, M = 32
-+-1.00 > n
0.75
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
3.5
4.5
5.0
2=x2
Figure 2 (M > 1 )
M
M.
Eq.
on v= 1
1,
= 2.
by
1/2N
( M = 1, 2, 4,
skewness
2,
A4 .Y
204 4a4/M
1
2.
( M = 1, 2, 4, 3 2.
s'.
on
.Y'.
Confidence Intervals For Averaged NPSD Bins
M = 32 2.5
3, 0.99.
137
Spectral Density
1.10 1.oo
0.90 0.80 0.70 n
> 0.60 OJ
5 0.50 n
0.40
0.30 0.20 0.10
0.00 0.0
1.o
2.0
4.0
3.0
5.O
Z=X2 Figure 3 (T=
2
1.
1.5 1.5
0.02. 2.5 50%,
4
1
( M=
cmzfidmce intcwcrl
A
M =2 0.08 3.7,
0.04
M =4
0.49, 4
on
= 2.0.
0
4 A4 = 4
f3 M= 1
50%).
Synchronous Time Averaging
124
T,,, .J;, = 1 / 7‘’
.h,, 2j;,, 3J;,,...)
All
138
Chapter 6
-9.0 1.10
-6.0
0.0
-3.0
6.0 1.10
3.0
1.00 I
1.oo
L 0.80 L
0.90
0.90
-~=8,M=4
-
0.80
0.70 L
0.70
> 0.60 1 5 0.50 L n 0.40 L
0.60
0.30
0.30
0.20
0.20
0.10
0.10
n
01
0.50 0.40
0.00 -9.0
-6.0
0.0
-3.0
0.00 6.0
3.0
dB relative to 2(0)2
Figure 4
.v2 on
2.0.
10
by
ii
1/N
by 40
N 100
Higher-Order Moments
2nd
h gh -o
*’
*‘
s
s
s
ng
n s gh by /)(.\-).
--z
as
Spectral Density
139
/
+m
( X - X)’] = E ( ( X - X)”
=
(x - x)”p(.U)d.u
--cx,
As
d.
2nd sketvness by
A
kurtosis
Kurt = E ( (-)4}X - X
-3
A
A
- 1.
on
The Characteristic Function
Eq.
X
chrrructeristic .function 4(u). $ ( U ) = E{ eit‘Xf.
a
1)
a
Chapter 6
140
Eq.
zi=O.
U
As
by (6. by
Cumulants and Polyspectra
by
171;
=Els),
.u(k).
.u(k).
2nd order cwiulcint
uz; mi(7)
c o w r i m c o .seclircnc*o
As
Spectral Density
141
r71;
2nd
power spectrum trispectrurn
= 0.
bispectruiu
on.
po/jqspecatru
2nd
2
),
by cu1
col, LU?
+ 0 2 5 71
+m
+x,
> 0,
no
+oo
96 on
s ( k ) , X(co).
PSD) Ci((o) =
N,
).
1/ N
s(k)
at./; = 256
Chapter 6
142
A I = 12,J = 10, O1 = 4 5 ” , A 2 = 0.01.
z ( k ) = .y(k)+
w(k)
= 24, O1 =
(6.1.34)
s(k)’
= 0.10,
~(k). (20 48 34
14
5
z(k)
10%
6
s(k)
PSD PSD
10
Hurnionic Distortion, Intermodulution Distortion. 10 20
.f:
10
no
70
60
50 40
30
m 0
20
10 0 -10 -20
-30 0
0
16
24
32
40
40
56
64
HZ
Figure 5
10‘%,
Spectral Density
143
6
5, 6
on 6
=f2
20
48 Hz
6 24 Hz
7.
6 14
Figure 6
34
7
Chapter 6
144
Figure 7
6.2
6
24
5
TRANSFER FUNCTIONS AND SPECTRAL COHERENCE
on
by ‘j.,’’
Spectral Density
145
0
71
on on A
2,
good
Y(f) H ( f )= Y ( f ) / X ( f ) ,
H(f) X,,(f;r )
Y,,v,r )
X(f), 8.
H(.f),
T
“n“
)
up
x,,(t)
on
ergoclic rcinclorii prot~c~.sse.s,
146
Chapter 6
32,
PSD
11
= 32 A4 = 1
9
=
A4 = 32
0.002. M =1
N, 1
N,,( % f ‘ ) ,
Figure 8
M = 32 9,
by
A
H(f’)
X(,f)
Y(./’). 0.016
c
0.014 0.012 0.010 n rc Y
0.008
(3
0.006 0.004 0.002
0.000 0
64
128
192
256
320
304
HZ
Figure 9
1
32
1 Hz
440
512
147
Spectral Density
(6.2.2)
I
(6.2.2)
by 2 X ; ( f , T )
G,.(f) G,.\(.f).
Sdf)
x(t)
(6.2.4)
G,,(f),
s(t).
(6.2.5) f
G,,(f), ~ ( t )
(6.2.6)
G,,.(f) s
4'
(6.2.7)
(6.2.3), (6.2.6),
(6.2.7)
Chapter 6
148
H\,(f). G,,.(f)
G,\(-f).
by
H\,(f). by orc/iricirj*c~otit~rc~nc‘c~
.firnc>tiori
no 1
A/D
A
374.5
60.5 1024
240.5 10
(M=
11
4 4 240.5
149
n
% 2
0
64
128
192
256
320
384
448
512
0
64
128
192
256
320
384
448
512
50 40 30 20 10
0
op -10 -20 -30 -40 -50
1.00
cv
*
0.75 0.50 5 0.25 L 0.00
"
' '
"
'''
"
'
'
I '
'
' ' I
' ""' '
I '
I
'
'''' ' '' ''
I
'
''
"
' '
"
'
''
Figure 10 &.(f).
by
I ,*-I
I I
,r-1
12 240
"
''
' I 0
150
Chapter 6
180 135
-135 -180
0
64
128
192
256
320
384
448
512
0
64
128
192
256
320
384
448
512
40 30 20 10 0 -10 rn -20 0 -30 -40 -50
HZ
Figure 11
4
A/D A
151
Spectral Density
180 135
0
64
128
192
256
320
384
448
512
40 30 20
F 'x =m
0
-10 -20 -30 -40 -50
l I , , I / I I
0
64
.
I
,
1
,
1
1
128
1
.
1
1
1
1
1
1
192
,
1
,
1
,
,
,
,
/
,
256
,
,
,
1
,
,
,
.
,
,
320
,
1
,
384
,
,
/
,
,
1
,
,
,
448
,
,
512
Figure 12 by
13.
U(.() + N x ( . f )
X(f)
on V(.f')+
Nj,(.f')
Y(.f')
1) N.I-(,/') Nj*(.f')
C'( I')
152
Chapter 6
Figure 13
X(,f)
H(f’)
Y(,f)
V(.f’).
<
14
U(f)
on
N.Y(.~’)
0
3 0.5.
14 good on
H,,,(.f’) 15
on
4
H,,(.f)
UU). 16
250
SNR by 32 500
Spectral Density
-f
153
180 135 90
45 0
p -45 0
-90
-135 -1 ao 40 30 20 10
%
O
3 -10
m -20
-30 -40 -50 -60 0
64
128
192
256
320
384
448
1.25
512
1
Figure 14
4 0
SNR
16
SNR
by on
154
Chapter 6
180
-135 -180 0
64
128
192
256
320
384
448
512
0
64
128
192
256
320
384
448
512
40 30 20 10 0 -10 m D -20 -30 -40 -50 1.25
1
1
1.oo 0.75 0.50 0.25 0.00
% *
0
64
120
192
256
Hz
320
304
440
512
Figure 15 4
17
17 is by
“H”
nd,
Spectral Density
-
155
180 135 90
-45 -90 -135 -180
0)
8
0
30 20 10 0 -10 -20 -30 -40 -50
64
128
/ ! , , , , , I , , , , , , ,
, I I
192
, ,
I
t
,
256
, I # ,I
I , , , ,
320
, ,
I
,
384
448
, , , , , , , / , 1 , , 1 , ,
I
.
,
512
,
,
,
,
0
64
128
192
256
320
384
448
512
0
64
128
192
256
320
384
448
512
HZ
Figure 16 32
E{Nj?(f)V I (f))
ZMG
U(f)
Njff),
Ny(f)
H(f)
V(f) 17
(6.2.15)
Chapter 6
I
Figure 17 SNR.
H.yJf’,)
1/(2n,,) nLi
GNJ.s,.(/)/GI , { J )
by
8)
SNR
(
Spectral Density
157
no
by
0.1, 0.375 - 20
- 40
20
16. 90 16
on
158
Chapter 6
AJ’= 1 / 0 . If
6.3
INTENSITY FIELD THEORY
intensity by
by by
Pojwtirzg Vector,
E A
S = E x H. H acoustic
Point Sources and Plane Waves
poirzt source. by
Spectral Density
159
on on
18.
R R - RcosO
A4
%
19
A 1 100
Figure 18
R
1u
8
+ +
6
+ f
4
+
+ t
U
10'
Figure 19
102 Distance in Wavelengths
103
160
Chapter 6
1
0.34
30 40 19
no
12.3
12.4
Acoustic Field Theory
Equution
100,000 94
Stufc,
1
1
1
100
105 155 by
1
1’
R
8310
T udicihutic,
no /
161
Spectral Density
by
M
1’
by
p, d v l d p = - M / p 2
Linearized Acoustic Equution of State
c2
c‘
p =RT/
17,
( M=
349
326
( M=
345 on
155 by (f=mu) d.~.
Euler’s Equation. S on
Sp(s)
on
Sp(s+d-~).
on
duldt
162
Chapter 6
S[p(s)- p ( x +dx)]
Sd-x p SdY
p ( s )-p ( x
+
d.Y)
ci-Y
(Ip du = -SdY - = Sdxp (1.Y
3)
nt
duldt
U
du/d,r
du/cl.~ Euler’s Equution
The Continuitjq Equation dt
Sp(x + d x ) u ( s + c1-y)
S ~ ( . UK( X)) J t dt.
by apSd-y.
po
The Acoustic. Wave Equution (d%/dx)dt.
Acoustic Intensity
by
Spectral Density
163
Eq. (6.3.15)
t
k
U(X,
m/c,
s
-1 t) =p
ap(X, 1 ) ____
ax
1 dt = -p(x, PC
(6.3.16)
t)
by fur-Jield
pc.
(6.3.18)
ro uoP"
r
A
(6.3.19)
r = ro,
uod""
A (6.3.20)
by 112
A
1
(
9
( I @ ) ) ,= -- 1 +'A r2' 22pc (6.3.21)
(kr>> l), by
(kr<< I),
Chapter 6
164
by
20
neur-ekd reuctive intensity uctive interzsitji
21 20
on
p2/2pc
0
100
105 1 L
4
$! 100
L
10-10‘ 100
101
102
I
#
I
,
101 102 Distance in Wavelengths
Figure 20
103
I
103
1-0
10
= j.
165
Spectral Density
6o
500
!
‘
1
200 100
4
101 102 Distance in Wavelengths
103
Figure 21
A
10
19, 20,
21
100
(
+ Sources
(
Power) Out Of Surface
+
(
Rate Of Energy Storage In Field
)
=0
(6.3.22)
Structural Intensity
A
Chapter 6
166
0 p,
Eq.
on
s,
Spectral Density
167
on
a,,
a).,
a,,
1
6
on
(6.3.24)
E
I
p
y
S
x Fs = EI(a341/as3), us = (ay/&),
M S = EZ($y/a.u2),
R‘ =
($~~/8xily).
(6.3.25)
* EI by by -
Chapter 6
168
Electromagnetic Intensity S = E x M.
E U,, U , ,
U:
by djwunzic.
on
H
CD v= -N
N, @=pHS, S
1.1
4n x 10-7
(p
on
3 x
100 MHz
3
Spectral Density
6.4
169
INTENSITY DISPLAY AND MEASUREMENT TECHNIQUES
good by
on
book. by
Graphical Display of the Acoustic Dipole
A
( r o << 1..
kro <<
A
A
=
21
171.5
by 1
0.1 1
J’ = 0
.Y
=
f 0.5 22 by
x =0
343
by =0
23
112 ;I
no 343 Eq. .Y
- 1’
170
Figure 22 by 1 on
Figure 23
Chapter 6
171.5 Hz y =0
343Hz, y =0
x
y
24
10
1
171.5
1.078
0.0338
24 23,
171.5
Spectral Density
171
InstantaneousVelocity
5
1
I
4
3
2 1
0 -1
-2 -3 -4
-5 -5
5
0
Figure 24
171.5 Hz
mntinn 2lnncT
the
nil11
g x i c nf Y
=0
I(.YJ!,I)= p ( s , ~ ~ , t )
by U(.Y,J!,I).
25 26
(+0.5,0)
25
26
25 ( I ( - Y ,= JV 'A ) ) ~ ~ ( - Y , J ~ , Iu)* ( s , y , f ) ,
26,
25,
26,
.Y
to
don
=(
+
172
Chapter 6
hstantaneous Intensity 4-
3-
21-
0-1 -
-2-3 -4 -
-5'
I
I
Figure 25
171.5 Hz
on
Spatially Coherent Time-Averaged Intensity Snapshot cl
4-
3-
21-
0-1 -
-2-3 -4 -
-51 -5
I
0
Figure 26 17 1.5 H z
I
5
173
Spectral Density
= (s2+ y2)'
0
(l(r, 0, t ) u*(r, 0, t ) 0, t )
= 54
Eq. U(Y, 0, t )
A
171.5 Hz
27.
do on
27 22
s=O
24.
Figure 27
17 1.5 Hz
Chapter 6
174
24
27 by
box
S-J-:
25
Calculation of Acoustic Intensity From Normalized Spectral Density
by 2
27.
by
Ar,
p2(w)
by
by 1 / 16 by
1/4
Spectral Density
pl(w) p2(co)= p f ( w ) + j p i ( o ) ,
175
pz(co)
uctive irztensitj’,
’ (P 1 *
‘((!I)
( 4 P 2 ( 4t
reactive intensit??
Eq.
by
by
Calculation of Structural Intensity for Compressional and Bending Waves
I. At
by
=
Chapter 6
176
on
on
20 jo
--(I?,
Hz.
Hz,
on
(PZT)
6
by by
6
E
Spectral Density
177
-
by A A?(to) - A
2
,(m), by - 1 / ( A w ') S
1,
(6.4.9)
(6.4.10)
(6.4.1
1 by
Eq. (6.3.23) (6.4.9).
by E = 50 x
by
p=500 10
31,623
good (6.3.25) ii
1 jf.\-) -
1 /w'
A A3(co).
178
Chapter 6
on
,45((0)
Al((o),
wakve)
A,(w)
N
.4,, i = by (A
E.
by
A
21
I,
E.
S,
p,
(11,
(6.4.13 ) Uace.
;I ;i
Calculation of The Poynting Vector it iis ii
ii
ii
ii 21
\
ii
\
be
a
ii
along
ii ii
t
ii 17
d n ;I
s q's t 111
ii
by h
pa
t
on d
p ov d has
ii
;IS ii
of' he
\b;i\~
as
(6.3.27),
of ;I
may
179
Spectral Density
A.
by
~(CO)
SNR. by
6.5
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
by
ZMG by
ZMG
o;! o: / N
a:/N2.
M
M
by
180
Chapter 6
by
do
by
all S N R on
PROBLEMS
1.
2.37 15 on 44.1 p?/
2.
by 1.
1024
Spectral Density
181
3. A you
4. A
0.75.
5% on
5.
BIBLIOGRAPHY 2nd
S.
9
1984.
1984. 1989.
J. S.
1986. 1977.
REFERENCE 1.
S. 1993.
13,
This page intentionally left blank
Wavenum ber Transforms
by n7a\'e
k
((1)
(a).
as a
'i,
k
2n. k,, k,.
kz k
spciticil .sig11(11
11
by knokvn a
by
ii
"'.
c""'
r
c'
t, As t
on
k,,
/i\,
k,
s,
17,
=
I-
+.I*:+ ? ) I
'
1
knonm
a
on one 2n i ,
o\ 11 a\
183
Chapter 7
184
1.
on
box, on box
box.
on
on
I(kx, k y ) .
-cc --oo
by k,
on
by by
Figure 1 A
ky
x
y.
Wavenumber Transforms
185
2,
As
on
on
A
by
on
by
Figure 2 A
186
Chapter 7
point spread function by
1
7.1
SPATIAL FILTERING AND BEAMFORMING
+
j ’ = ~ r ~h ,
3). r = /(2a)
on
12.2.
by F
3.
+
1’= l ( 4 a ) h.
d 1 +2d/i,
1
r
187
Wavenumber Transforms
I I I I
I
I I I
I
I I I
I I
I
I
Y
b +X
-X
1
Figure 3
As
4.
3
“A”,
188
Chapter 7
Source A
0 Source B
0
Source C Yl,,
y2.t
Y3.t
0
4
0
I Figure 4
Array Processor Output
A
ds n
LI
.
100
If A
A B
on
A
5 0
90 ,
R
J ’ ~ . ~
k
kR,
k 27rf’lc =
j.
2xf =
2xlEb, c’
A , = 1;
i = Z,2,..., A!,
s,
(7.1.1)
189
Wavenumber Transforms
Figure 5
A on
d
R
As
kR on
R M
.Y,
dlE.
by d/A
j274171 -
by 2
1
by 0=
(=) 4
21
Chapter 7
190
M,
on
((//A) 0.05,O. 10,
A,,
0.50. As
2ci.
90
(//A
7 1
0.85,
1.50.
Y
20
0
40
80
100
80 theta
Figure 6 (d/;.)
16 0.05, 0.1,
0.5.
120
140
160
180
191
Wavenumber Transforms
U
0
20
40
60
100
80
120
140
160
180
theta
Figure 7 0.85, 1.0,
d/i, 1.5.
no
on (M>>
Md.
4
A,,,
Eq.
5 8 A,,,.
-
A,,, = ‘/z nz= 1, 2, ..., M .
M,
2, M, M = 16
8.5
1.8823. As M
2.0 A,,,
192
Chapter 7 16
n
“0
20
40
60
100
80
120
140
160
180
theta
Figure 8
A
Ocl,
A.
Eq. (7.1.1),
Eq. O,/
M
9 90 , 6 0 ,
30
0.5. 9. As )
180 ), Md
O,,
90 .
193
Wavenumber Transforms 16
c.
I
I
14
I
I I I I
12 10
-
-
I
8
i
i
i i
'.!
I
I I I I
! ! !
I
I
I
dA = 0.5
I I
!
;
!
I
! I ! I ! I
/*
6
I r
- 1 ! I !I \I
4
I\. I\
2
U
I
i i.
0
20
40
80
60
100
120
140
160
180
theta
Figure 9
look
on 10 9.
180 go 11
12 4
10,20,25,
15,
0.4 12
4 by
Md by
194
Chapter 7
Figure 10
360' -), (-.-)
180
Figure 11
d/;. = 0.05 0.1
0.5
195
Wavenumber Transforms
25
1
1
UI
1
40
20
0
I
1
I
I
,v
1
80 100 scanned theta
60
120
I
1
I
I
140
160
Figure 12 A
180
4
$ $ =0 up,
=
on
by 0,
by
13 4 by 4
16 x-y
c,, =
c
k,,.
x-y
2qf
27tflcxJ,,
d
13
A,,,.,,,
by
0
+.
196
Chapter 7
+Z
t
Figure 13 A
0
Y
+[
XI*=,( 1 -
[
1-
(7.1.5)
A.
Ai;i,,l
by 0‘1
A
(f)
by
by 7.2
IMAGE ENHANCEMENT TECHNIQUES
$(I,
Wavenumber Transforms
197
4.3,
As
on on
on.
on
upon
As by by by
PSF
PSF,
PSF
PSF
on book, 14.
256 by 256 15 10
Chapter 7
198
Figure 14
Figure 15 A
256x256
14
on
Wavenumber Transforms
199
i ( s , 1‘)
--x
-w’
k,,=O 14
k , =0 on
on do
15. by
M by
A4
[
d P ( m , , n?J = 1 -
~
1
-
N by N N
N
Eq.
by
16 on
on (in,,
A4 by A4 in,) = 1 (IN,.,m,,) d’(m,, mJ3),
(M N by N
N) I’p ilp(ti,., / I ) , )
17. by
Chapter 7
200
Figure 16
Figure 17 by
Wavenumber Transforms
201
llk.
18.
A
on by on on on
by
on 20
19 on
Figure 18
by
202
Chapter 7
Figure 19
Figure 20
19 by
203
Wavenumber Transforms
on book. on
7.3
up
2
COMPUTER-AIDED TOMOGRAPHY
body,
up 1895.
10 1
1
by on
by on rudi-
ologist
1917 50
1972.
by J.C.
on
1979,
book by
Chapter 7
204
CAT 21.
+ 40
22
k , = 0, k , . k,.
k , =0
40
22,
40 ), N x
23. N
+
t d , =I I I , C O S ~ -
111,
=
n‘, =
tn’, = 0.
Figure 21
+
ti:
=
205
Wavenumber Transforms
Figure 22
Figure 23
by
206
Chapter 7
23
(mi 24. 8 = 0, k, = 0.
k , = 0,
(k = 25 26.
65k by
256
8,
2048
28
29 128
27 32
30 A
by
Figure 24
23.
207
Wavenumber Transforms
Figure 25
Figure 26
8
8
Chapter 7
208
Figure 27
32
Figure 28
32
Wavenumber Transforms
Figure 29
Figure 30
209
128
128
29.
21 0
Chapter 7
by 256 x 256
2048 256 x 256
4 1024 x
up
on up
by
31
128
A 32 on
Figure 31
128
211
Wavenumber Transforms
Figure 32
7.4
128
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
on
on
book,
13 by
1x
1
1 x 1014 Hz, on
12
212
Chapter 7
by by up by
on by on by on
by
PROBLEMS
1.
1
2. A
50 Hz
16
1530
16
3. k
4.
0 w‘
by
Wavenumber Transforms
213
BIBLIOGRAPHY
199
S. 2nd
S.
1988.
REFERENCES 1. 2.
1985.
E. von 29, pp.
3. J .
1917. 1994.
This page intentionally left blank
Part 111 Adaptive System Identification and Fi ter ing
on
8, A 9.
9
10
All
21 5
Part 111
21 6
LMS 10
2
Linear Least-Squared Error Modeling
24, by
8.1
BLOCK LEAST-SQUARES
by
Xn b
Yn
Unknown System
En b
f
+
H[ZI
y'n
Figure 1 21 7
218
Chapter 8
1 s [ n ]= xI1,
H[z],
~’1~.
JT,)
it;,
-Y,,.
.Y,~
by least-squured error
6.2 10.3 on H[z]
M+ 1
Eq.
1 by A4
hcr.si.~Jictwtiorz
+1
M + 1 by 1
y; J,:,
(8.1.4)
= (b, H
n -M.
n
by
8.3. (8.1.5)
n
n - N+l,
N by 1
219
Linear Least-Squared Error Modeling
J’n
yt1- I J’t1-2
... ?‘ri-N+
I
Eq.
X by H
(8.1.8)
(M=
H
M = 1, h0 hl.
/zo.
M > 1,
H. H H up.
H H
H
HH H
HH
H
HH
220
Chapter 8
HH
1s
H
H
up (8.1.11)
H
(8.1.12)
R; = E
{-~-fi.~-,,
.I-,,. s,, .I-,,
-
by up, 1)
As
Ri’ = E
[-Y:?*,,
N H[z]
by 6.2.
no no
Linear Least-Squared Error Modeling
9
221
10,
( M = 1)
2 hl = - 2.5
ho = - 1.5
H on
by
N H,
2
book.
A4
8.2
on
PROJECTION-BASED LEAST-SQUARES
Figure 2
ho = - 1.5
hl =
222
Chapter 8
book 2 by by by
k A by
C.
do by
M
+1
on. m =1
111
on up
171 = O S
nz =
by
to
9. G
ff.
223
Linear Least-Squared Error Modeling
by G ,
y,]
G,
g G
g
7. by
j= g
-1.
g
G,
by E =f = -7 -g.
J. H,
G,
g
Px
by
G
Eqs hounded, having unit), norm,
k.
self-adjoint.
G
(I - Px)Px = 0, 3
I
Eq.
E = (I - Px)j
X
x +s = x +S(I
X
by
+S.
by - P*)
by
Chapter 8
224
0bsewation
Figure 3
S
+X
I - P { X + S ) = I - P/y - ( I - Px)S[SN(I - Px)q-'S"(I - Px) Eq.
9. 9. by
8.3
GENERAL BASIS SYSTEM IDENTIFICATION
s,,
Linear Least-Squared Error Modeling
225
s=
6.0 8.0 26.51
=
x
by
x="
2 3 41. X
4n= [s,,
1
by
4 1
0.2
1
2 3 411 1 1 1
1
1.5
j*=
-
9.15,
p* =
7.82
[
0.7 0.4 0.1
0.4 0.3 0.2 0.1
0.1 0.2 0.3 0.4
0.1 0.4 0.7
]
E = ( I - Px)?;=
[z] 16.26.
= [.Y,,~ x,
[
.:ij] 1
0.25
1.25
0.95 0.15 0.05
0.15 0.55 0.45
7.75
0.45 0.55 0.15
4 9 1 2 3 1 1 1
4
[ 26.5
E = ( I - PX),. =
0.15 0.95
4.75,
]
[ ] ;:LI;
226
Chapter 8
by H=
- 20.85 44.82 4
- 26.10
3, Px
1x
-
4
by
(3),
55
30 I
Figure 4
85
55
20
I
I
1
1
I
I
1
1
4
1
4
227
Linear Least-Squared Error Modeling
85
A A A
on
A
Mechanics of the Human Ear
by 0
20
130 106 20
20 5
by
tendon of Stapedious muscle As
300
6000 Hz
5
A
Chapter 8
228
Figure 5 by
on 6
20
0
f 1
50
4000 Hz
Least-Squares Curve Fitting
7
2
Linear Least-Squared Error Modeling
229
m
U
-50
‘
I
-60 10-2
’
’
‘
’””’
10-1
.
. . . . . . .1
100 kHz
4
,
,....
102
101
Figure 6 A, B,
Table 1
A,
HZ
19.95
50.12
A B
-24.2
- 1 1.6
100
-1.3
199.5
1000
1995
3981
1.2
+1.0 -0.7
0
0 0 0
7
A,
6310
10000 20000
C 6
l. on
As 85
55
kHz
171 = 1
10
kHz, “H”
A,
H,,,.
-11.1
-4.3
10
bbs”
kHz
Chapter 8
230
9
20
20 kHz.
"7"' by
[1
H,,,
We
ji?. .
N
E=D-Fjlr
Eq. A,
6
2.
Pole-Zero Filter Models
A, B,
C
SLM. no
C
Table 2
5th Order Least-Squares Fit Coefficients for A, B, and C Curves
m
HA
1
-0.1940747 +8.387643 -9.6 16735 -0.2017488 - 1 . 1 11944
2 3 4 5
HR 1807204
+1.257416
-3.32772 -0.7022932 - 1.945072
c -0.07478983 3047574 -0.25 13878 -2.4 I6345 -2.006099
Linear Least-Squared Error Modeling
231
f i = 20.598997, f4 = 12194.22,
K,
2.24288 1 x
K.
1.0251 19
fs
K3
1.562339, f2
107.65265,
158.48932.
f3
737.86223.
by
7. 0
SLM
7 0
0.6
0.4
0.2 0
-0.2 -0.4
-0.6 101
I
102
I
103 Hz
.
1
. . ....
I
.
.
104
rigure
ANSI S 1.4-1983.
I
.
..A.
105
Chapter 8
232
B, 1. ut the husisfunction sumplc
by points.
N N
>> M
M.N
M
As
N
=
LrtIl
y n - ... J ' , ~- ,v+
M)
(us
t
n'=n-N+l
by
5, N
(XHX)-' by I.
N-'
by
X(co), Y(to),
H(co). 6.2. X(co)
233
Linear Least-Squared Error Modeling
(8.1.14)
.V
X(to)
6.2.
N.
N
X(w)
Eqs N H(m)
N
8.4
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
9 on
by
234
Chapter 8
on
on by
PROBLEMS
/?
4 T ( t ) = Toe-' 'I. 1.7. t
T
To
19.2, 'C.
=
2. 3. 4.
[ 1257, 1189, 1205, do you by do you
5. CHC =
- PX.).?.
6. 7.
P,y go
Px-F = XHU,
8. 9.
buy
Ho
$105 $100. $105.
you
on
$1 10. up on
$108.
235
Linear Least-Squared Error Modeling
BIBLIOGRAPHY
1988.
E. 1970. 1980. 2nd
S.
1988. 1985.
S. REFERENCES 1. 2.
2nd
B. I.
1991, pp.
I. 1993. 1979.
3. 4. 335
10017,
This page intentionally left blank
9 Recursive Least-Squares Techniques
by 8.1 on
N
N
-
1
on. on
kt’,
by (9.0.1) 237
Chapter 9
238
N
/?=1,
(9.0.1) /J= %,
N = 2 x =%
on. unbiased x = 0.99
N = 1, (x = 0 /I= N = 4 x = ‘/4 p = $,
i,
N = 3, x =
N,
x
/?
100
fl= 0.01,
fix
100 by 1 / e 0.99 on
100
by
1 2 I)
9.1
THE RLS ALGORITHM AND MATRIX INVERSION LEMMA
(RLS)
(9.0.1) 8.1 (8.1.1)-(8.1.7),
do
/?
on
(9.1.2)
N xf
yr.
(9.1.3) A
(9.1.4) H
(9.1.1)
(9.1.4).
The matrix inversion lemma ‘‘Aneh =
+
239
Recursive Least-Squares Techniques
BCD". (A
+ BcD)-'
A, C
= A-' - A-IB(C-1
DA - 'B, by ( A
+ DA-lB)-'DA-'
BCD by
+ BCD).
A
+ BCD
K,,+1
K,,+I
A-,,
N
H,,
H,,+I,
Eq. 1)
Chapter 9
240
S,l+lH,l,
J*;~+~)
j ~ , ~ + ~ H.”
1 1).
on by
on
K,,+,,
on
1 .Y,,+~
“rz+l”
J*,~+I,
P,,= (Xf’X),;’
H,, Approximations to RLS
by
PI, (9.1.13)
on
2
-,’
2
K,,tl Table 1
(RLS)
I
a 11
r
;I
on
:
-
1.
- / I -
pfl = (X
RLS x = (,Y
. . . .V,I
= [.~,l+l.~,l-\-,l
.Y,~+I
=
l)/,Y Pn+1
Hn+ 1
.)*:It
3
=
+
pn4,/,:
+
(b,l+l
Pdb!!
= r p l [ lH n
+
k;i+
Li,,i+
- j v i l+
1Hn
Recursive Least-Squares Techniques
241
44H
ct
M. 1’ < 2
LMS
p
by
bound
LMS no p.
DSP
prc,l
<1 p,.,,/
N = 1 /p,.(,/,
on
p,,;,, = 1 /
[cT:,)
q!Ill+l
0;
prel
N = 1 /p,.,,,.
M+ 1
9.2
prC,/=1 / ( M
+
pr,,i,\plf8/<
LMS CONVERGENCE PROPERTIES
A
by
by by
Chapter 9
242
innovation. by
LMS
1. System modeling using adaptive system identification H[z]
1 E,
= J * , ~- j*i1
by
box
H[z]
H[z]
LMS
LMS
Unknown System
Xn b
Yn
I
Figure 1
Block
.\-,,.
j*,,,
cn.
Recursive Least-Squares Techniques
243
LMS
prc,/= 1
Eq. yir+l
$,l+lHn.
Eq.
R',
Eqs
D=
&, k = 0, 1,2, ..., M on QHQ= I Eq.
D.
p,
11 - 2 ,uR,
< 1.
LMS
0
/L
H
Chapter 9
244
LMS i,,,;,,
,H
LMS LMS p
up
LMS U
priori
p
on
Eb,,,x
LMS
,U
Norrnulized L M S Algorithm
p
25
ZMG 1024 RLS
LMS
5
on
f 100
166
333 hl
on
2. 2
LMS RLS - 1 1.2
38.2.
1+12.5)
27,
(2512).
prC,/ Eq.
0.05 20
(r = 0.95
RLS
RLS
LMS 20
2
3.
LMS p,',/ 1 /p,.',/)
p
245
Recursive Least-Squares Techniques
Figure 2
hl
s,,
5
25
*A
1024
1
1
I
1
I
100
150
200
250 300 Iteration
I
I
I
350
400
450
8 6 4
-
2
.g
0
E
-2 4
6
I
-8 -1oo
50
Figure 3 2.
500
246
Chapter 9
LMS RLS 25 RLS
256 4
no 13.5
LMS
ZMG 0.05
zt
27 0.95
25 Hz
p,.(,/
LMS
RLS
4.
LMS
RLS no LMS LMS
RLS
5. Signal modeling using adaptive signal-whitening filters
innovution by by
2.5
I
I
I
1
1
1
I
1
I
1
I
2 1.5 1
-j r
E
0.5
0 -0.5 -1
-1.5 -2 -2.5 0
Figure 4
I
50
LMS
I
100
1
150
1
200
250 300 Iteration
I
350
I
400
I
450
500
HI
256 Hz
1024 Hz.
247
Recursive Least-Squares Techniques 10
I
I
1
I
I
I
I
I
I
8 6 4
-
.E E
2
0
-2 4
El:
-6 -8 -10 0
I
50
I
100
I
150
I
I
I
250
200
300
I
350
I
400
1
450
500
Iteration
Figure 5
LMS
RLS
4.
6 6 ARMA
AR
MA hon~,, A4
10
ARMA jqf,
?'if.
AR LMS
248
Chapter 9
Innovation 1 + a$+
w"
I
Figure 6
Signal to be Modelled
b0
b
... + $
b
yn
I
Adaptive Whitening Filter
Block
(bn = [ - j ; r - l
-J-,~-?.
. . -I*,,-,bf] on
u k , , , = a k , , , - - 2pjirC,,I!,,-
by
Eq. E, = J * , ~- j1i1,
step-\tiisc>d u p t tlic. L M S cocjficaimt \tx>iglitsin the opposite dirwtiori q f t h e error g r m ' i w t .
LMS
by
AR A
7 0.01
2
up
= 0.8, - 0.4532
as
jiIC,/= 0.2
5 1.4534
8 up
0.1632
50 H z
1 .0
Hz
200 Hz
0.8370. A
9
LMS
249
Recursive Least-Squares Techniques
0.5
D L S --.IMs
0
I
-0.5
j
-1
-1.5
-2
-2.5 0
Figure 7
1
50
1
100
1
150
I
200
1
I
350
50 Hz
NI
LMS
Iteration
Figure 8
I
250 300 Iteration
50 Hz
1
400
I
450
500
250
Chapter 9 0.2
I
1
1
I
I
1
1
I
I
I
100
150
200
250
300
350
400
I
I
I
I
1
0
4.2 !
'r ! \
i
'0
'.
4.8 -1
-1.2 0
I
50
450
500
Iteration
Figure 9
200 H z
(11
LMS
RLS
256
LMS ji
by p by
9.3
LATTICE AND SCHUR TECHNIQUES
Recursive Least-Squares Techniques
251
no on
by
jlfl
A4
M
n-M+1.
on
M+ 1
J$-~,~-~
yt7-
~
jlil
y
~
~
-
.
= - - a 2 , l j y t 1 - ~- a2,2yt1-2,
-
a2,1
a2,2
ytI. ~
1 -’
~
ykF3 = -a2,2 y I I - -
+1
II
~
jTfI-2.
Chapter 9
252
LMS A
LMS 10. 10
LMS
LMS PARCOR
on
RLS PARCOR Eq.
Lattice Whitening Filter
T-&......................
I: ;
.
U:
~
.cj
....................
1st Stage
rDw .........................
............................
Mth Stage
2nd Stage
LMS Whitening Filter z4
Y“
*
Figure 10
FIR J’,,.
6-
E-
,
r-
253
Recursive Least-Squares Techniques
i=
n
M,,l
M.
,tf%l
by y,,
Rif c,~~,,,
jytI
-,
>0 L - ~ , -~j ,
As
=[
Rif 0 0 ... 0 ... 0 0 RL on
-,
Chapter 9
254
by
Eq.
p+
M by
UM+I,k
=aM,k
+ As
LIM+I,~+I
no
M+
R 1%
R.I'
...
Eq.
... ...
Rl:
M
+
255
Recursive Least-Squares Techniques
i= 1
ALtI by
H by
PARCOR
Eq.
Eqs
A:tr+l,
Eq.
Chapter 9
256
by 1
c ~ ) ,= , ~rO,,l
AI,
=)in
rO,,r-i .
I:~),,,.
I?;,
,
1)
-K‘ -U;.,,. (11.1 = ~ ~ . ~ - K ~ U ~ on. , ~ A,
-K;=U~,~ = -K(
by
11
p
E
Figure 11
+
Recursive Least-Squares Techniques
257
by N,
1/ ( 1 - a )
N
Ap+l,tl 2 N E ( E , , ,r~,,,, -
1..
Eq.
up
+1
do on by
8.2,
up on on 9.4
up
N
THE ADAPTIVE LEAST-SQUARES LATTICE ALGORITHM
As
8.2
11,
PARCOR
258
Chapter 9
by
on by 8.2 by
by
?.'n-N:n-l
= LYn-N
* *
.Yn-2Yn-11
N
p,
n -N+l
n
1).
Ep,n-N+l : n
= ?'n-N+I :,?(I- P Y p J
PY,,,= Y:N( Yp,NY:v)- Yp,,v n by
x = [ 0 0 ... 0 1
1.
Recursive Least-SquaresTechniques
259
8.2
by
p
by
i f p -I , n -
by
yp -
1 - y,,
n:
-
-
1 - y p - I,n- I
-1
1 - y p - I,t,(lip - I , n
-1
no
8.2.
V
by by
WH,
by
Yr.,, S.
V ( I - ~ { Y , . , V + S )W ) H= V ( I - PY,.,J W H - V ( I - PYP,h)s[SH(I -
PYp,.v)s]-’sH(r - PY,.,,) W H
260
Chapter 9
on by
V = yll-.~+I
Yp,N Yp,N.
S = j$Th+l ,,f W = n,
(9.4.15)
V= W=n
1
- I !
,p,tf - I
-
on
by y,,
up 1 - yI, - I , r , - I
Recursive Least-Squares Techniques
261
Yp,N
by
by
V = yLT,L+
:
W=
Y N V = y,l-N+l PQi
S W = n,
-
:n ,
1) V= W=n
A
9
6
on
(9.4.2
N + l by N+l
N by N
Py,,.N+l by
Chapter 9
262 Y,,N
by n 8.3,
V ( I - Pi Y f , N
+ ,))
WH
V(I - P Y f . N W )" by
S=
7t,
V=y,
-
W = yiTk+l:n
+ I:n
A
(x
on
S=n
R;.,] = q
, * - 1
S =n
V= W=J*,~-~+~:,,
+ &p,n[l V = W =J
If
-1
H
(9.4.27)
'p,n
~
~
~
~
,
+
~
:
~
Recursive Least-Squares Techniques
263
1 - y,,,?
12
50
7. x
RLS
P-’ 13. 13
5
12
13 2 1.5
1
.cI
0.5
C
g
o
8 air
-0.5 -1 -1.5 -2
0
10
20
30
50
40
Iteration
Figure 12
RLS 7.
50 Hz
264
Chapter 9
1 II
1.5
.w=
-
LMS
*
Lattice
-
0.5
-l -1.5
-2
L
0
20
10
30
40
50
Iteration
Figure 13
RLS,
on by on by 2
9 6 1
on
up
M 3. As
Recursive Least-Squares Techniques
265
Table 2
n =0 ,..., M CO.11
= rO,rl = ?’,I
.
.for p = 0, 1 . 2 , . . . M Q
= ( N - 1)/N
Table 3 t
ap,O = u ; , ~= 1 %+l/J+I
ap+1.r Clp+l.p-,+l
s,, s,,
9.2. do
=
q
2. . . . , M
for. p = 0, -
-q+1
= -up.,r - K;,+P;./J-,*I j = 1 , 2. . . . , p - ap,p-/+1 - ~ / , + P P . I
J’,,
s,,
14.
266
Chapter 9 .
..... _....._._._.___.._..__._.-....-.---. ..... .. .....
...
....
..__..........
'lr
8
, ,
2nd Stage
1st Stage
Mth Stage
Figure 14 y,,.
A-,,
Eq. ... 1 0 ...
'10
cr:
I
L
0
,:
-
Eq.
by
Rb = c h g (RbR; . . . & }
R', = LRX,LH
Rif = E(#q&},
[Rjt,]-'= L"R:,]-'L
Rb
L
L ~"
~
Eqs ~
3
:
= g),H = HTL-'F&, A t t , t f . = [ K i K r .. . ~
Ki,LI
K:,]
J ' ~ ~
$,tf
= ?'n
-
qrpJ
p = 1, . . . , M
14.
Recursive Least-Squares Techniques
267
K;,,, = A:,tl/RL,t,= A;,!!;;?. 14.
H
H = LHK;::,
hp = K j
+
3
M
aLrpKF p = 0, 1 , 2 , . . . , M i=p+ 1
15 2
25 Hz
2.5
5.
1
2-
1.5 I 0.5
0 -0.5
c "
-1 -1.5 -2
I
I
Figure 15
LMS,
I
1
on
RLS, 2.
Chapter 9
268
15 do
by on
16 by
by
u priori ~ipostc~riori
2.5 2
1.5
7 C
.P
g
8
c”
0.5
o -0.5
-7 -7.5
-2.5
‘
0
1
I
700
200
300
1 -
400
500
Iteration
Figure 16 f 1 e ng ;I1gor h
the
to the RLS
269
Recursive Least-Squares Techniques
(9.4.36)
PARCOR - + Fp++l,I?
- %I?
+
(9.4.37)
- K;;+l,,l-lrp,,?-l
(9.4.38) (9.4.39)
[+
p s 1 .I1
rp+ I . I 1
- $,I2
-
q+,,h+
(9.4.41 )
,I?
(9.4.42)
= rp,,,- I - K;+ I ,,&I1
by by
PARCOR
Eq. (9.4.36). Eq. (9.4.43).
PARCOR
RLS 4.
etnbodded PARCOR PARCORs
due
Chapter 9
270
Table
n =0 ( . . . (M
A
A ,..., M
for IX
= ( N - I)/N
A
A
A A
9.4.2 p = 0. 1 , 2 , . . . , A4
16 4 8
20 13 MACs 11
3
271
Recursive Least-Squares Techniques
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
9.5
N
M N M ( 1+ N M )
N
M’ 100
10
1,002,000
50 49 Hz. exponentially-forgetting
M’+2M2+2M 10, DSP
2 1220
2
50 40,000
13 4
3
10 16 630 79,000
50
M2 M 2 - A4
RLS
272
Chapter 9
3
3
19
3M 10, 50 1
LMS
PROBLEMS
1. 3
I .
M?
3 3. M = 64
4. M=64
5.
6.
( -2
7.
+1 0
5
7
8. /i
= 0.1.
11 ,U =
9.
10.
v
n
5.
-1
1,
Recursive Least-Squares Techniques
273
BIBLIOGRAPHY 1988. S. 1891, pp. 1979, p. 1326. 2nd
S.
1988. 1985.
S.
B. 63,
1975, pp. 64, 1976, pp.
REFERENCES 1.
39,
1,
1991, pp.
2. 1974. 3.
70, 4. S.
1982, pp. 1988
162.
This page intentionally left blank
10 Recursive Adaptive Filtering
9
book
9
on
10.1 gun on by
+6 -6 J’cL)
1 ljo ( - 6
on an analog cornputer 275
Chapter 10
276
a
on a
(1)
1958 by
&
du
by
do do
do.
10.1
8
9.
ov
10.2 3.1
as
3.2.
on on a
Recursive Adaptive Filtering
277
5.4
good
book. 10.1
ADAPTIVE KALMAN FILTERING
on
17,500
50
25,000
6500 geosj~nchronous,
on on up
on up
by
All
278
Chapter 10
(GPS) bond
rz:
n,
t,
z(t),
H(t),
n=
x(t),
n, H(t)
n,
:(t) = H ( r ) . y ( t )
+ bc(t) H s (t )
z( t )
by
w(t).
s(t )
by
11
(
N
J( N ) ( N n , x Nn,),
.Y
z’” Nnz ( r z , x 1).
by 1 RA‘
), H” (Nri, x E{itv(t)w(/)H;
( Nnz x
ti,),
R”
Recursive Adaptive Filtering
279
8.1, z(t).
N
good 4.1.
x by on N
{
s'(N)= [HNIH[Ry-"HN]
s'(N),
[H"]"[R"]-'z" (
x
(10.1.7)
(10.1.2)-(
N
+
280
Chapter 10
=
[ H J \ ' ~ H ( N
[T+
+
H(i'L
R(N+I)
+ l ) H R ( N+ = P ( N ) - ' + H ( N + I)"R(N + I ) - ' H ( N + 1 ) = [H"']"[R1']-l[H1v] H ( N
1)
H(N (
P(N+ P(N
+
+ I ) [ R ( N + 1) + H ( N + I ) P ( N ) l f H ( . 2 +' 1
= P ( N )- P ( N ) H H ( N
+ I)P(N)
x H(N
(10.
by (10.1.5).
;(N+ I)=z(N+ 1 =:(N+1 =H(N+ =H(N
+
=H(N+
S(N
+ 1 ),
as
Eqs
1)
Eq. (
281
Recursive Adaptive Filtering
P(N+ K(N+
x'(N
+
= .Y'(N)
+K(N +
=[I- K(N+
P(N)
+ 9.1. Eq.
As N
on
good
A The Kalman filter
N+ 1 .Y(N
1.
N.
+
= F(N
+
+ v ( N )+ G ( N ) u ( N )
282
Chapter 10
Table 1
RLS
H(N K(N
1
u
n
su
+ 1)
N + 1) + 1) = R ( N + 1) +P(HN)H”( ( N + I ) P ( N ) H ” ( N+ = P ( N ) H f ’ ( N+ I ) [ S ( N + 1
on
n
+ 1) = z ( N + - z ’ ( N + 1 ) = z ( N + 1) - H ( N + I).Y’(‘V) N + ) = .v’(N)+ K ( N + 1(); h‘ + 1 )
;(N
.Y‘(
process noise G(N) u(N)
v(N)
F ( N + 1) As
tlw sfuft’ trunsition rmitrix H(N+
N
+ 1,
N .Y’(N
+ 1 IN) = F ( N ) s ( N J N )
T
A
Eq. ( E [ v ( N ) , ~ “ ( N=c,’. )~ T,
by by
!AT2
Recursive Adaptive Filtering
Q(N)
283
Eq. ( 10.1 . 1
F
Q,
H
Q
P’(N
+ 1IN) = F ( N ) P ( N I N ) F H ( N+) Q ( N )
S ( N + 1) = H ( N
K(N
+ I)P’(N +
+ 1) = P’(N +
+ 1) + R ( N +
+ z(N+
+
(10.1.2 1 )
N + 1,
Eqs
x(N
+ 1IN + 1)
x’(N
1
+ 1IN) + K ( N +
+
-H(N
+
book.
30 by by
+ 1 IN))
Chapter 10
284
;I
1991.
If
2 H ( N + 1) R(N+ Q(N+
F(N+
.-:,
As
H , R , Q , F.
CT:.
As
2,
0'
+ l ) H ( N + l ) ] P ( N + 1 IN)[I - K ( N + I ) H ( N + 1 +K ( N + l)R(N + + 1)"
P(iV + 1 I N ) = [ I - K ( N
(10.1.24)
3,
4.2
x-/&iy
As
zt-/&;f
100 13 13
3
Recursive Adaptive Filtering
285
Table 2
A
A A
+ 1) = P’(N + 1 IN)H”(N+ 1 )[S(h‘+ 1 i ( N + 1) = z ( N + 1) - H(h’ + s ( N + 1IN + 1 ) = s ’ ( N + 1”) + K ( N + I)<(iV + 1 ) K(N
I).Y‘(“
on
(o,,
on
R
cc-p-y cr-p-;!
U-p-7
1
2
4.2.
2 0.3741 0.27461 1.7545 3.04991.
x-p-y
286
Chapter 10
alpha-beta-gamma Tracker
60 I
1
Measurements
40
E c 0 P
20
8 Q
o -20
2
0
4
6
a
I
U
10
Seconds 150
r
I
100 -*
-
50 0
s
v s
s
.
-
C..
.*
.
b-
- . . b *
,* -
. I
*
a
v
-50 -100
'
-1500
.
- -Tracked I
I
4
2
.
S .
1
6
a
10
6
a
10
Seconds
-
I
J
0
2
4
Seconds
Figure 1
cc-P-7
3
13.
4.2)
1/ T 1 20 2001
[1
T = 100 x-,!l-;t
3
1 x-/j--y
0.3851 0.28711, 3
287
Recursive Adaptive Filtering
Kalman filter tracker
-20 0
2
4
6
8
1
L
10
Seconds
50
0 -50 -100 -150
I
,
1
0
2
4
6
8
J
10
Seconds
20
0 -20 I
-40 0
2
6
4
1
8
I
10
Seconds
Figure 2 z-p-;~
13. a-p-y
288
Chapter 10
Kalman filter tracker
60 40
E
E 0
20
n8
0
C
-ZU
2
0
6
4
10
8
Seconds 100
.
0 b
b
0 .
*.
..-
0
-
*
0 b
.
.
.
0 .
0. b
b 1
-1 00
0
6
4
2
8
10
8
10
Seconds 10
5
4 /2
/\ 11
L
0
'
\
\
-5 -10
-15
2
0
6
4
Seconds
Figure 3
as
1.
%-/)-;I
4
5. 4.2
ii
1 2.
289
Recursive Adaptive Filtering
alpha-beta-gamma Tracker
60
1
I
1
Measurements
40
E
=0
20
P
8
P
O I
I
-20 0
2
6
4
8
10
Seconds 20 10
0 -10
-20
'
-30 0
4
2
I
I
1
6
8
10
6
8
10
Seconds
'
I
-20 0
4
2
Seconds
~-p-;l
Figure 4 13.
(x-p-y
on
on
290
Chapter 10
Kalman filter tracker
60
r
1
'
-20 0
6
2
6
4
8
10
Seconds
20 r
r
I
10
s o
8
-10
-20 -30 0
1
I
1
L
2
4
6
8
10
6
a
10
Seconds 10
5
9
c 2
0 -5
A2
er
a
-10 -1 5
0
4
2
Seconds
Figure 5
0.1
1
u-/j-;b
13.
good
8u-/&;t
Recursive Adaptive Filtering
291
The a-b-y tracking filter 4.2 cc-fl-y (3) a, /I,
a-b-y
y by
-y
T.
ct
p
<
G , ~
a
4.5 a-b-y
P
by
cc
p
1-kl
[-‘T
0 11
by
(1
Chapter 10
292
( 10.1.31 )
pi2
Eqs (
1)
2
k l =r
k 2=
7‘.
(
p’ +x p +=0 4
x2 x
(
/j ( [j
ix
(
-12
lx-/j-;~
/?h.
pi2.
;I=
( (
Eqs a-1j-y 0,.
T.
(T,,
x , /j,
7
293
Recursive Adaptive Filtering
a-/&?
10.2
IIR FORMS FOR LMS AND LATTICE FILTERS
8
9
by 3.1
MA, on 3.2
ARMA
by
6
B[s]lA[z]
B[z]
A[z]
by e[n],
by .r[n]
y[n].
294
Chapter 10
Figure 6 s[n]
y[n].
ARMA P (P
ARMA
Eq.
j*[rz]
+ +
E[=] = hb +hi=-’ . . . hbz-0 ,4721 = a’,=-’ uiz-? + . . . + cl+-’
+
P
AR A[:].
A’[:]
ARMA
Recursive Adaptive Filtering
295
P
P
P
n
by 6
by
j*’[n]
So
A’[=]
s[r2]
y[n]
pb
po,
Error signal bootstrapping y[n]
by eh[n]= y[n]- y”[n] = y[n]- eh[n-
lib', - . . . - eh[n- Q]bb+ y[n - 110;
+ ... +~ x[n]
~-[P nI 4
Chapter 10
296
by
rh[n]. rh[n],
jq"[ti]
on
ARMA MA on on
LMS
ARMA
4n],
ARMA tJ[12] = j*[n] -
y"n3
= j.[tZ] - &z]h;, +)'[I?
ARMA
1;.
-
-
c q n - l]h; - . . . - &z
+ ...+
-
- Q]hQ
PI,;,
by
.t-[12]
AR
c?[n].
MA MA
ARMA MA
by
by
ARMA by
RLS
ARMA ARMA
ARMA by
MA
ARMA ARMA LMS
MA RLS
ARMA ARMA MA
7
8
f 200 f400 1024 Hz on on
A R M A LMS ARMA 0.98 0.95. f 1.227
3 5.0
A z
'
A
ARMA 2000
+
B[z]= 5.0 A[:] = 1
~
-
'
ARMA
'+
'.
297
Recursive Adaptive Filtering
200
100 150
I 1
I
-Lm.. *...a.
mP
0 -
-50
-100 -
-150 -200
I
I 60 -
1
I
40 -
%
20 1.......*.....*'. 0 -20 1
I
100
0
200
300
400
500
Hz Figure 7
.\-[n]
ARMA
Eq. , I ~ [ H ]=
~ . O S [ U ]+ -
-
+ 4.51254n - 21 +
-
- 21
7 by 6.2
on ARMA
by
298
Chapter 10
2ol1 I
9
loo0
500
0
1500
2000
Iteration
Figure 8
LMS A
p
3
4
3.1
-'*'I
7 = 5.0000
' by
+ 7.34372 + 4.5 1262 ' -
-
A'[:] = 1 .0000 - 0.66032
200
400 MA
I
+
Recursive Adaptive Filtering
299
ARMA
RLS on
LMS 8.
A LMS
ARMA
by
7
8.
100
LMS
ARMA
8
AR
MA
7
8
ARMA
good
on
by
LMS AR
AR by ARMA 2000 -
by
MA
MA
ARMA LMS
+
B’[z]= 6.73 1 1
+ 3.327
-
As
The embedding technique by
ARMA - 2.498 1z AR
‘
A’[:] = 1
RLS
9.1
ARMA ARMA AR
RLS
9.1.
ARMA
300
Chapter 10
h ct, = j',,- 4 ; f f
1)
on ARMA
t;:
c,, =
-
*\I),
= j',,-
Eq.
cp,, H
[i:si:fi-,
( 102.12)
...
An
RLS A R M A
9 10 LMS A R M A
RLS A R M A
as
7
8.
RLS
LMS
ARMA
MA AR MA = 5.0 + 7.34361 + 4.51251 RLS ARMA ,4[1]= 1 .0 - 0.66032 + 0.96042 '. = 8. 1+ + '. RLS a weak ARMA
'
0.96042 '. 4.5 1252 ARMA
' '+
~
ARMA
RLS
'
ARMA AR
A[:] = 1 .0 -
MA
ARMA Normalizing the bootstrapped error signal
MA
+ +
A[:] = 1 .0 -
ARMA
301
Recursive Adaptive Filtering
Frequency Responses
200 r
,
- FFT Response - - Polynomial
150 1
*
j
100
I
I
a
-RLS System ID A...'"
-50 -
I
-100 -150 -200
0
300
200
100
1
I
400
500
400
500
I
60 40
20 0 -20 1
J
I
4
200
100
0
300 Hz
Figure 9 A R M A
RLS
ARMA
PARCOR by
ARMA MA
by MA
by
MA LMS
RLS A R M A
good M A
ARMA MA
ho #
302
Chapter 10
..
I
500
1
, 1500
1
-2 0
1000
.-
2000
1.5 1
0.5
-RLS
c1
$
0
*
-
* *
System ID
,ARMA Bootstrap
-0.5 -1
500
0
1000 Iteration
2000
1500
Figure 10 A R M A ARMA
MA
AR
4;
c: = $kH,
Eq.
oh
RLS
Recursive Adaptive Filtering
RLS
303
Eq. ( 10.2.1 4)
RLS A R M A 12. RLS MA
11
bo = 5
=1
ho = 1
RLS MA
RLS
ARMA by
LMS 200 I
-
'
FFT Response
- - Polynomial
-RLS System ID 8
0'
U
-
-100
'
-200
0
J
I
100
200
100
200
300
400
500
300
400
500
60 40
%
20
0 -20 0
Hz
Figure 11
= 5?
304
Chapter 10
...................................................,............................................................
I
' 0.5
- - Polynomial -RLS System ID 1 --
4
-
-0.5:
.....
'
ARMA NBootstrap
1
,
- ........
. * ......
I
-1
I
1000 Iteration
500
0
1500
2000
Figure 12 t\
A
t
p
saj ~
1
,
~
good a s
'
= 5.01 14
+ 7.5 1781 ' + 4.32441 ~
good Embedding an A R M A filter into a lattice structure a
8
H
= 1 .0000 -
'+
Recursive Adaptive Filtering
305
9
by
ARMA
di = ui/ho
do = 1 / h o ,
uo =
= 1.
ci = h,/ho
= 1,2, ..., M .
ARMA
ARMA (
p
ARMA
+
13.
Figure 13 ARMA ARMA
ARMA
306
Chapter 10
14
ARMA ARMA ARMA ARMA ARMA
c;.,~ t;;
-v,,
E;,,]
. 15
good.
ARMA ARMA
16 ARMA
ARMA ARMA M
up
M ho
P
A[:] Q, ARMA P
do = 1 / h o ) . ho (10.2.18)
Figure 14 A R M A 1 ng .
307
Recursive Adaptive Filtering
P -1 00
..=/
.......... -200 0
n U
1 I-_ ~ d 1
-20
- -PO1 'FFT Response nomial '
'*
'*
Lattice System ID 'ARMA Lattice Bootstrap
0
300
400
500
300
400
500
200
100
I
200
100
Hz
Figure 15
ARMA
bo
by B'[z](b6
B[z] ARMA
by
( 10.2.1
by
CI
=1-
As
ARMA
15,
308
Chapter 10
20
15
N
10
Y
m
5
0
500
1000
1500
2000
500
1000 Iteration
1500
2000
1 0.5
9
0
-0.5
0
Figure 16 A R M A
3
by good
by good 16
iq
200 200
100 pa a b 1
on v g
.
NW
Recursive Adaptive Filtering
309
Table 3 ho
RLSN
h2
U1
N2
N
5.0000
7.3436
4.5 125
0.9604
Y
5 5.0000 5.0047
7.3437 7.3436 7.5809
4.5 126 4.5125 4.8011
0.9604 0.9604 0.9678
100 200 200
6.73 1 8.1381 5.01 14 5.3203
3.3271 1.6342 7.5178 7.2554
1 0.4648 4.3244 4.4 180
0.9537 0.9786 0.9 149 0.988 1
100 200 200 200
1
C
C
-FII = [IV,!J-~.. . J $ ] ~ .
-
ip+l.ll =
-
K~+l~l~F~,,ll+~, I rp.11?
1
$1-
1)
1
( 10.2.2 1)
17. 17
310
Chapter 10
Figure 17
block first
by
E;+l,,l by 90
18.
18
book
311
Recursive Adaptive Filtering
Figure 18 at
10.3
FREQUENCY DOMAIN ADAPTIVE FILTERS
(FDLMS)
31 2
Chapter 10
by no 50%.
50%
(50'%,
50% O"iI
do
U
A good ;in
6.3
6.4
15.4. on as
on h k , , , . k = 0.
2. . . . , 'Z.I.
ti
(
p,llc,, x f 1 .As
pmc,lu = 1 / [of
pfC+
9
,LI,~.~
r~:
\
10.3.1 )
Recursive Adaptive Filtering
313
M+ 1 p r C J l =1 / ( M +
N+ 1
M+ 1 No no
1
No (No
-
(
M
+ 1 by N + 1
1 by
5.4
on As
5.4, M+ 1
31 4
Chapter 10
2M+ 2
M+ 1
a
pmLiX(co),
p,.,,/
5.4 10.3.7),
Eqs
.\I+
1
by
9
1.40 ‘ 5 0 . 3 5 ~0.90 ~
0.99, 1024 384
U
307.2
0.95 109.4
179.2
0.001. 10
250 Hz,
30
80
20
390
18 36
36 36
no As
19
20
22
good.
315
Recursive Adaptive Filtering
Figure 19 9
Figure 20
Hz
450
Chapter 10
316
Figure 21 ;i
0 512 H z
Figure 22
0 512
it
by on
450
t
450
Recursive Adaptive Filtering
317
on
LMS 23 on
by X*(w) on
X(cu) A
Hop,(co)
E(co). on
I
1 II
loo 80 60
0.500
5
10
15
20
5
10
15
20
FFT Bin
Figure 23
Chapter 10
318
80
250
5.0
0.00 1
390
24, bound 80 Hz, 250
bound. 390
24
24. 6.1
6.2,
on.
FDLMS
10.4
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
12
I
I
10
8
6 4
2
0
0
100
200
300
400
500
600
Hz
Figure 24
on
Recursive Adaptive Filtering
319
LMS ARMA by book 9
RLS
LMS
on
by
RLS 10.1
a-B-7
(
3
x-p-jl
by
x-[$)t
or-p-7
320
Chapter 10
on 10.2 a
(M
M
2 by 2 4.
good
10.2
18 a
A4 17.
10.3
by
a
up ii
Recursive Adaptive Filtering
321
do
PROBLEMS
1.
0, 270 10
90,
200 180,
a-/I
1
1
2.
s
j’
1.
3. 4.
pro/= 0.01
100 x
0.99.
5. dl=u,lbo 6.
ci=b,lbo
Eq. i = 1, 2, ..., M , do= Ilbo, bo
7.
8. 9.
2M
M
ao=cO= 1.
322
Chapter 10
10.
BIBLIOGRAPHY Y.
1993. 30,
5,
1981,
111,
pp. 70, 1982, pp. 8
867. 1974.
W . S.
29,
3,
111,
1981, pp. on
10-12. 1980. pp. D.
B. 29. 111,
3,
1981, pp. 1988.
S.
29,
3,
111,
1981, pp.
on
1993. pp. 562 573. 1, 1992, pp.
REFERENCES R.
80,
2.
Y.
3.
Y. pp.
1958, pp. 6, 1993.
Part IV Wavenum ber Sensor Systems
k = culc, o k=2n/R,
c i
body body by
body no
on on
(1 on
323
324
Part I V
by
as
by by a
by
by
by
nlutd~Ldfi’1ter
on
you 11
Wavenumber Sensor Systems
325
12,
do
go by
13 on
This page intentionally left blank
11 Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
P,/,
PI,,
on
by on A
by
Phi.
P,/,
Pd
on
PrCl.
Chapter 11
328
11.2
(CFAR)
on by
1 1.3.
on
on a
1
co
11.1
1
on
on,
n
.
THE RlClAN PROBABILITY DENSITY FUNCTION
P,,,
P,,,,
3.5
1 2
by do
2 10
a
/
M.
7.5
10
(5
2, 10 r(t).
1 R(-r) = 7E(s(t)r(t "7
+ 7))
( 1 1.1.1) ~ ( t )
0;
(1
azmhdjlter,
329
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
I
0
Transmit, Receive, & Cross-Correlation with 1 Ave 1
5
10
15
1
2 0 -2 4 0
5
2
20
25
I
10
15
20
1
I
1
8
I
1
25
0
-2
5
0
10
15
20
25
usec
Figure 1
3.6
7.5
1500
Transmit, Receive & Cross-Correlation Magnitudes with 1 Ave 2
1
1
1
0
I
0
5
1
10
II
Figure 2
15
20
25
Chapter 11
330
2
1
256 n/2,
T
10 128,
Time synchronous averaging
M,
.I-
0;
(1
J* = U.Y.
j?
.Y?,
..., . Y , ~
=.f(s),
Transmit, Receive, & Cross-Comlation with 10 Ave 1
4
0 r
-2
1
0
5
10
15 I
I
L
10
15
20
I
5
usec
Figure 3 10
20
1
25
i 25
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
4' = ax.
y l a = x, d,uldy = 1 / a
a;, = a2at.
331
p(y)
M y=aM,
Eq. ( 1
6.1 z =x I
+ x2 + ... + x N .
N
z
xl, x2, ..., x N . z
A p(x) QY(
U).
--oo
+cc
-m
x
z = x1 + x2 + ... + x N , p(z)
x i ; i = 1,2,
... ,N .
z,
Chapter 11
332
Eq. ( 1
1.1.8)
As
( 1 1.1.8),
Eqs z,
1.1.9)
is
1 := ( l / N ) X ! ,
10 .Y,.
by a
128
N.
1 4, 1 / 128,
by
by
1 / 11.3. a
0
0 4[
0
0
1.5
0 0
5
10
15
5
15
10
1
,
5
10
I
15
usec
Figure 4 ;I
;I
20 1
I
20
25
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
333
SNR. 1,
3
10
4
10
5 Envelope Detection of a Signal in Gaussian Noise
So
6,.
Eqs (6.1.7)
(11.1.2)
6.1, (11.1.3)
y=x2
Chi-Syucrre I degree of freedom density function
( 1 1.1.10)
E { J *= ] E ( s ' J = a:
x
ZMG
(6.1.11)
E()*2)
2af (6.1.12)-( 16. 13).
Chi-Square probabilitj7 c h 6.2
sit)' function
Dotaction Threshold
1
Recehrer r(t)
Matched Average
(XCor)
Envolop. Dotactor Nob
Figure 5 on
334
Chapter 11
y =xi
+ x:,
2 exponential probability density function. 1.1.1
1.1.1
y
z = y'
=( x i
+ x~)".
(dy/dz)=
Rayleigh probability density function.
2
4. eiC"*.
ZMG o.v
0.:
by
0
1 +I+;)
=
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
335
z -
-
a2 = 2 2 - -2 =
z
- :)o:.
6
x.
on .xR
no
0
x2
xl
02.
sI,
xI
.yR
s1
.xz
.xl
o:., -x2
sI
Eq.
8 1
I
0.0
-
0.8
-
1
I
1
I
I
I
I
i
Y
Figure 6
I
x2
I’
Chapter 11
336 r
0
.yl
x7).
(1
Eq. ( 1 1.120)
0 0 up.
Eq.
(
So << o,.,
So >> o,, by ( 1 1.1.14)
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
337
(1
I'
= SO
7
of,
by
So
of.
(1 .Y J'=.Y'
j?'
Rician Densities for Various SNRs
0.7 i Noise Only SNRl SNR2 --SNR 10 +
10 Received Waveform Magnitude
Figure 7
15
2, to
10
Chapter 11
338 4, = A-'
~ 7 " ~ .
Eq.
z =j"' -O.v
c$,=2zdz.
1
(11.1.28)
6
Eq. ( I
z'
4.''
z'
z - So,
z = - z ' - So z
8
z
z
Gtrzrs.sitrrz r~ugnitirdc~ pmhcrhilitjt densitj*fitnctiorz. ( 1 1.1.29)
Gaussian Densities for Various SNRs 0.8
1
0.7
+
SNRl SNR2
0.6
0.5 0.4
0.3 0.2 0.1
n
-0
Figure 8
5 10 Received Waveform Magnitude
15
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
11.2
339
CONSTANT FALSE ALARM RATE DETECTION
to on
by
P
Pji,.
11 9
8).
02,
o . ~A.
A
11
50%). A
0.5
Pr; 1
I
1
I
1
0.9
0.8
0.7
Figure 9
T =2 4.4%
340
Chapter 11
1000 by
A
cx:
P,,,=
I
p(.Y)cis = 1 -
A
I
1.2.1)
p(S)dY
0
I
N
( / m c ,
0,
A =T J m o ,
T A. As
9 A
cx: 9
Tso 0).
9 A
A
T , / ~ o , ,
cc
no
twcitr
A T. T ~proh(rhilitj3 P of’cr.fii1.w~ l u r n is rlicwfbrc~cotistcirit, hcrckgrorrrid rroise l c ~ r . l~ h ~ ~ i g e s .
J
~
G
by,
c ’ w i [ f ’ t / i t >u.stitii(ited
by on
P,,=0.02, 100
A
2 0.1‘%,o n a
15 10 U\.
2
341
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
I I I
0.9 -
Figure 10 T =2
on
ZMG
Pfi1 T, A=T
J ~ o ,
A = To,
12
T
x
(1
(1
342
Chapter 11
1.2
1
i
1
1
0.8
E
0.6 0.4
0.2
0
0
2 3 Relative Threshold T
1
4
5
4
5
Figure 11 T.
1.2 1'
0.8
if
0.6 0.4
0.2 0
0
Figure 12
2 3 Relative Threshold T
1 P,,,
P,,,
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
343
12.
PJ;
1.2.5) (1
by
(1
jfrz]
=A
Rorz + 129[rz]
(1
= 27rfo/.f,
1c1[n]
o-:.
no
1/20:,.
by k , 2.0
5.3
(8/3)12
by
0.8165
50% by
344
Chapter 11
by
by A =T,/mo,, (1
no A. T
P,/
Eq. A,
13
o,,.,
P,, 50'1/;,
P,/,
Eq. by 14,
P,/ by
P,,
0.7
-
0.6
-
--SNR6dB -. --.,SNRlOdB * . * * SNR20dB
0.5 0.4
-
0.3 0.2 -
0' 0
Figure 13
2
4
a s ii
6 0 10 Relative Threshold T
12 T
14
16
345
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
1 0.9 0.8
0.7
-SNR OdB -SNR 6 dB -**SNR 10 dB * - * * Q S N20dB R
-
0.6
z
0.5 0.4
0.3 0.2 0.1 0
-4
Figure 14
-3
-2 by
-1
1
0
T - SNR
2
4
3
SNR
on
0
=
20
=
15
T,
SNR.
on
P,X T , S N R )
T
-
SNR -
1 2 SNR
~
by
346
Chapter 11 1
0.0
0.8 0.7 0.6
z
0.5 0.4
Y
4
1
-3
0 T SNR
-
2
1
3
4
Figure 15 A T, SNR.
book.
by
P,,, P,,
As
11.3,
347
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
11.3
STATISTICAL MODELING OF MULTIPATH
A 6.3.
-
Multi-source multipath 13
on
Chapter 11
340
13.4
by
Coherent multipath on
16
11 by
R
on 16
on
(1
R
A
(k=
=2 n / i ) ,
k
R, 17
(o
Boundary
hl Receiver
Source R 4
Figure 16
I +
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
349
-30 -35
Y
1
1
l i
I
... .
40
%
1
1.
45 I
-50 I
-55
-60
0
100
200
300
400
500
U7
Figure 17
by 100
30
R = 100
l1=30,
345
18
A
(c/f=i).
350
Chapter 11
Image Source \
\ \
\ \
\ \
\ \ \
.
Boundary
Source R 4
Figure 18
7.1. Statistical representation of multipath
on on book. by by by
by
351
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
19
2000 345 by 100
on
dl=0.3
30 2000
16.
0 500 Hz,
19
on 20. on
on
(1
R?,,,
R N = 2000
dl= h / R
h = 30 m. ,
-30
I‘ ‘
-60 0
1
t I
t:
1
I
1
100
200
300 Hz
Figure 19
1
400
500
352
Chapter 11
Boundary Standard Deviation 0.25m -30-
1
1
I
-35-
1 -40
-
-45
-
- 1
-50-
1
’
-55 0
1
100
I
1
I
200
300
400
500
Figure 20 Hz
20 0.25
on
j*= h,
+ <.
j’
j*
20 200 Hz
300 Hz on
(1
353
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
R; h, Rf,
Eq.
no
c.
a;
Eq.
by
p’ -40 7r/2
20,
-34
4hlRl
7c/4,
R = 100
h = 30,
k on
7c
170 Hz.
20,
h
>> R.
h
h h l R = 0.5.
Random variations in refractive index
At
AM
20
k
354
Chapter 11
by
by
by
good
on
on
by
on on
5
by
21
355
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
c ( z ) = CO
+ z -dc(z) dz z
R 21,
R=-
c ( z ) = 0.
by
dc(z ) Id,x
21,
s
1.3.11 )
s = OR
Refracted Ray s
\
DirectRay x
0
Figure 21 a
Chapter 11
356
/I=R
[
m]
As A-=
1
by
= 345
R=3450
cic'/cl,- =
/1=36.4 3.5
+ 0.1 1003.5
i s - -.y = 3.5
+
+ 350;'
As
1000
by
11.4
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
11
on
357
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
statistical conjidence
data situatioriul
fzision, awareness.
P,,, Pf,,
A sentient by
by
on on
body
(3) by
good do
body by
1 1.1
Tinge-sjwchronous uveruging is one of the wost siniple and effctive r7ietliod.s for. signal-to-noise iniprovement in signal processing arid should he esploittd ~tiliel.4~ierpossihle.
5.4
11.2.
358
Chapter 11
up
11.3
by
on on
on
PROBLEMS
1
50 Hz
1
400 Hz,
1024
3.2 kHz, 8192
If
8192
no
on
50 1x ‘/U
you
by
359
Narrowband Probability of Detection (PD) and False Alarm Rates (FAR)
5.
by 6.
by by
7.
1%
on no 0.1%)
8.
p(r) p(s).
BIBLIOGRAPHY
26,
I. 1972. 2nd
S. 1991. 1971. 1,
1977.
S. 7,
1986. 3,
K.
1979. 1968. 7, 1996. REFERENCES 1.
K. 103(3),
1998.
This page intentionally left blank
12 Wavenumber and Bearing Estimation
7.1 ).
by
on
361
Chapter 12
362
by
by
12.1
bound
11
8
12.2 by
12.3
12.1
THE CRAMER-RA0 LOWER BOUND
bound
Wavenumber and Bearing Estimation
363
on As
on N by
N+ 1
N
1
N
on N
m
02.
A=[n?02]
N
I;( Y J ) ,
MF
R 8
364
Chapter 1 2
F ( 12.1.4)
F( Y , j , ) .
Eq.
Eq. ( 12.1.4)
]:[
=E -
+ E[F$]
Eq. (
A,
F. F F = 1, E [ 4 ] = O .
F=$
365
Wavenumber and Bearing Estimation
[3
-E - = E[$$T]
E
J
J
Fisher Infomatiori A4atr.i.u.
J
J (1
Eq,
A( Y ) ;I(Y)
R( Y).
$( Y)
8.1
by
/?
p.
E[p]= E[R$7]E[$$‘]-’
$,
Eq.
E[$]
E[e]=E[;I].
A)” = R - E [ i ] Ae = e
-
E[e]
A) A4 = E [ i @ T = ] E[Ah,hT] M
bius of theparmietcv estinicitiorz
(
366
Chapter 12
E[AeAeT]= E[(AR - MJ-'$)(A/ZT - ~ T J - ' M T ) ] = EIAI.AAT] - MJ-'E[$AAT] - EIAI.$']J-'MT = E[AAAAT]- M J - ' M T - M J - ' M T
+ MJ-'$$'J-'MT
+ MJ-'MT
= E[AiAAT]- M J - ' M T ( 12.1.18)
qAeAe'] 2 0
a2(i)=
bound on
E[AAAAT]2 M J - ' M T
( -
on
M = E[A't,bT]= --[A']
-E
= 1 for the unbiased case
Eq.
;I'
no
on
A, A
2.1.2
a2(i) = E[AAAAT]2 J-'
N
Eq.
CRLB efjcient bound.
N
(
Wavenumber and Bearing Estimation
367
i= [m. ' ] a
Y,i)
L.
i= 1
I
-
I)( Y ,
Y , A) =
=
- 172)
N 1 ' -- 4 202 20
CO,- In)'
+
Y
r Y ( Y , E,) =
1
Y , A) =
ai
J = E ( Y (Y , A)} =
1;
0 N 2a4
]
[
E ( A ~ H A ~E{AniAa') ~] a-(i) = '* E(AniAa') E { A a 2 A a 2 )
25
[ ] N
2
9. 1% 1%
1
9 0.25
3 / a . N
144
0.09, N .6
0.09, N
145,800
Chapter 12
368
12.2
BEARING ESTIMATION AND BEAMSTEERING
by by
on by on
by by
1 tI
on
2 3
on
tl
1
0,
= $3
9,. &,
=
-
(
I$~ (o
k = t o / c , c’ 0
Eq. ( (
369
Wavenumber and Bearing Estimation
Sensor 3
d
Sensor 2
Sensor 1
0
Figure 1
by
7.1
(1
(12.2.5)
Chapter 12
370
(r, (I-,,
0, - r/ OA - r,
OJ)
(r,
0,)
(I-,,
8, - r, 0~ - r/
0,) 0,) L
J (
by
1
2
Figure 2
371
Wavenumber and Bearing Estimation 04,
c4 =
{ J=
S
{ -S1N R 1
(1
oNis
x
y
by kd
0 3 by kd 0.01
dli
t
COS
Figure 3 by kd
372
Chapter 12
0.5,
(//A
0.05,
(
a,<
M 1,
d
A/2
M (
/j
on
hi' 1
cc
~
T
S
N T?R
(1
Wavenumber and Bearing Estimation
373
1'
.Y
on
-Y
.Y
0, by
Oh
d (
(Ocl 2 (I,,),
374
Chapter 1 2
no
by
13.
4.
I I
9
I
I
I I I
I I
I I
I
I I
I I
I I I
1
I
I I I I I
I
Distant Source
II I I
I
Wave at Element dx Arrives xsin 0 early
Linesensor
-- 4
g-x=i
x=-u2
-X
Figure 4
x=uz ,U b
+
--L,
dx
+ *
0, s = 0.
Wavenumber and Bearing Estimation
375
on
on (
k = 5.385
-
c = 350
'.
/
L =2
300 Hz, 6
5 8'
300 Hz
0'
.u= -L j 2
7
30 by 30 on. no 7.2 256
32
by 32 128,
by 256 2
16 8
(x,
)I)
2
1.5 -
\
1-
0.5 -
0
-150
Figure 5
-100
0
-50 2
50
100 300 Hz
150
376
Chapter 12
0
180 Figure 6
2
0
180 Figure 7
300 ).
2
30
377
Wavenumber and Bearing Estimation
Figure 8
Figure 9
2
8
378
Chapter 12
9 on
k = 5.385
- I.
5.
k = 5.385 Hz,
300 Hz,
by k on
1
1 1024 Hz 1024 - 512 Hz + 512 32 by 32 k, 8
1024
1 Hz on
2
256 R=klk, +nk,. -8n
300
k= - n 5 SZ 5 + n , +8n.
- nk, 5 k 5
-f , / 2 5 f 5 +f,/2. 30
10
7.
30
s
y
k, = k,. = kcos 8
(
k
Table 1
Xmax N k, = --TI 5 R 5 +n, =k/k, k = 27112. = w / c , (o = 27rf -xk, 5 k 5 +nk,
379
Wavenumber and Bearing Estimation
Figure 10
30
2 k = 5.3856
2
32
300
256
16
11 2
by 300
12
30 270
350
300
2
1.167
0.5
by 4 600
k = 10.771
’.
13 600
0.5 14
2
600 30
8
by 8
2
30
on 15 16
300
Chapter 12
380
Figure 11
2
30
300 Hz
0.5
Figure 12
2
300 Hz
30
Wavenumber and Bearing Estimation
Figure 13
381
600
2
Figure 14
3 15 30
225
600
4
Chapter 12
382
Figure 15
8 by 8
2 up
300 Hz
Figure 16
8 by 8 30
2
by 2
30 by
by 2
300
Wavenumber and Bearing Estimation
383
15. 15.
up
0 180
up,
90
270
on
k, = k k,.
0 -k
0’
k
0’
0 do
(
(
by (
3 by 0
0’.
on
( 12.2.2
Chapter 12
384
A L =U 2
(cl[= 1 )
0
dl=2L/i..
L
on Nxve
by by 112. by
.Y)/.Y
= 0.707
.Y
= 1.4
A
(
up
2Lli
L iL 1
(
2.2.1 1)
L > i., good
M
Eq.
no
Wavenumber and Bearing Estimation
385 U
hoirnd by go
12.3
loic~cv.
on As
FIELD RECONSTRUCTION TECHNIQUES
on
on
by )
by
Chapter 12
386
dS
D,s
y,
dv
X = ( s , y ,z )
X’ = ( X ‘ , J ~ ’ ,z ‘ ) on
p(X) g (XlX’)
X, X’
X
no
on
on p(X)
F(X’). by on on Eq.
X
X. on by As
on
by
on
(ap(X)/an)
by on
Wavenumber and Bearing Estimation
387
on on
on
on
do
on
by on 2
X = ( x , y ,z )
X‘ = (x’, y’, z’)
Chapter 12
388
17 k = 12.65
700 Hz
z = 0.001 10
by 10
X’= 0, z= 1 z = 50.00 on 64 by 64
= 350
A
on
700 Hz 17 G(k\-, kJ.,z),on
on
Figure 17 3
700 Hz.
Wavenumber and Bearing Estimation
389
6 17
k’
2
k:. + k;
k,
2
0.
k = 12.65. 0.5 0.5 in,
=
z = 0.001
z = 1.00
by As z ( z = 50
100
6.3
100 17 2
k,,, 6
-
‘(‘/zL/z)
L
10%
4
z
17 2
z=50
10 z
A-)’
by 10 1.00
J = z,,?.
= 0.001
z =1
1 90
5.7
50
Chapter 12
390 z =z,.
P ( k , , k , , ,z,,,),
(
z,,,
z,
k, +k,,n.
-k,n
L,
L,
21
18 As
holographically-reconstructed on 4
18, 700
1 z =1
z
H ( k , , k , , z , , z,J 1
18.
(I,
( - 1,
( - 1, -
-
19
1 18
19
1.3), ( - 1.8, -
(-1
1 20
4
5
-
Wavenumber and Bearing Estimation
Figure 18
391
64 by 64
1
10
by 10
700
5
21
1050 Hz,
5 30
10
A by
A’ =
8=
8
As
1’
“
p=
A=
z
8
392
Chapter 12
Figure 19
700
1
A
N by
LIN, N 64
N
z on
zo LIN zo.
Wavenumber and Bearing Estimation
Figure 20
393
700 Hz
5
zo
N
good
22 32 by 32
700 Hz
5 64 by 64
20. by
700 Hz 1.5
32 by 32
32 by 32
zo
64 by 64 2.75
zo
3.2
f 10.05 on
0.3125 22
0.5
Chapter 12
394
Figure 21
1050 Hz ( L = 5
d = Jz,
-
zil by
(12.3.1
G(k,, k y ,d ) = ,
g
J
d
( 12.3.1
m
d
395
Wavenumber and Bearing Estimation
Figure 22
5
700 Hz
}-'
G(k,, k,,, d)-' =
jd
,/-~ Eq.
by 23
24
a 32 by 32
396
Chapter 12
Figure 23 700
32 by 32
5 good
by
d=zi-z,,,
on
12.4
WAVE PROPAGATION MODELING
tomographic measurement of the media inhomogeneity.
Wavenumber and Bearing Estimation
Figure 24
397
15
by
by
8
+3 + k2(r,z)$ = 0 r-
-
k(r,z)
o/c(r,z)
r
z
z
Chapter 12
398
r.
r-z
r
$2D
= fi
by $3D.
Q = (#/az2) + k 2 ( z ) (5).
k(r,z)
by r + Ar
r
12.3.
e2tu'
Ar, $(r
+ Ar, z ) =
pJArfi+(r,
z)
Q r "(r
r
+ Ar
+ Ar, k,) = e J k r A r Y ( k:) r,
Y(r + Ar, k:) = dAr-Y(r,
kz)
Ar. Q
Wavenumber and Bearing Estimation
399
Ar.
kz
z
r + Ar
z by
on
up k,.(z)
=
+ Sk2(z)
(1
(
Eq. $(r
+ Ar, z ) = d A r f i $ ( r , z )
by
r
r + Ar
Ar.
k,. z
(Sk2(z)/2k,(0)).
Chapter 12
400
on. on Ar
z = 0.
Z,q(r).
--w f M
P ( r ) =k,.(O)lZ,(r)
R(k,) = [k,(r)Z,(r) - k , . ( O ) ] / [ k , ( r ) Z , (+k,.(O)]. r) pc = 41 5 As Zcq+1, R ( k = -) +0 no P ( T ) - +k,(O), no AT. Z,(T)-+ 1 Eq. on Eq. (
Y(r,/j)
Y(r,z)
/j
Z,
/) by 2jP z.
c’
””,
Wavenumber and Bearing Estimation
401
p,
25
on AM
26
Figure 25 2, = 12.81
100
100 Hz
1 c(z) = 340 + 0.1
Chapter 12
402
Figure 26
100 Hz
c(z) = 340 - 0.1
on
26 no
no 27
on
up by
Wavenumber and Bearing Estimation
Figure 27
100 Hz
good
25
26
403
Chapter 12
404
(
by
12.5
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
by
12.1
bound
on 12.2.
12.3 a a
12.4
p
1 p p1
on
Wavenumber and Bearing Estimation
405
As
12
to
PROBLEMS
1.
A
100
10
1 2. A 5 Hz
345
3. A (c=
500
24
2
16
1500 1
90. 3b,
4.
-3 2 kHz, 1 kHz
5.
200 Hz
2 6.
by 120
8x8
2 7. A
by 2 100
1
2
406
Chapter 12
1000 40 by BIBLIOGRAPHY 1991.
S.
1951 , 32.3.
X.
“A
1993.
H. 1 ),
1969. 2nd
S J. C
2nd
1973. 1971 .
1981.
S
E. l ) , 1980.
1998.
REFERENCES 1951,
1.
32.3.
2.
2nd 1973.
3.
E. I),
4.
pp. E. 23.5.
1980,
1971,
5. 1993, pp. 6. X . 1993,
pp.
720.
13 Adaptive Beamforming
by
by
by on no
A
407
408
Chapter 13
cohertwf,
on
13.
by
13.2. 8.2 a
10 ii
it1 - 1
Adaptive Beamforming
409
13.3
A
13.1
A R R A Y “NULL-FORMING”
by
on by
no
on
A
by
O1
AI
A2
41
42.
(1 3.1
41 0
Chapter 13
s = ci,
Xo
90
X,
X,
fz.
dl cf,
0,
U?) by
by
go
do
1 Hz
10
100
10 10 Hz 1 Hz
Adaptive Beamforming
411
A
13.2,
1
M
1
0. by
N
X3,l
...
x3,t-
...
x3.1-2
... ...
-x2,I -N + 1
I
.
.
(13.1.5)
Array
Array
Figure 1 A
Chapter 13
41 2
-
C = So - X H
3.
(
by
(1
r t
A(z)
z
(,-
/kclt,tl
k
('
dtIlI 0
on
on A(:)
by z = c / ' ~ ' ~ . z
by a
by (1
-
+n
by k d k
0.
Adaptive Beamforming ?i
= d,,,,
413
= c/,~,,~
(13.1.1 1)
k by k’ = k
x-j*
)I,
k
( k /c=27r/i). ( 7 = 90
on
k’ = k .
0
7
s-ji
Eq. ( 13. ; !
A(:)
by
on by
3.
k (I)
N (1 3.1.5)-( 13.1.9). Eq. (13.1.12) 0 < 8 < 360
0 < 2‘ < 180 A
by bound. 2
100 Hz
20, 80,
140 100 Hz
10
10 0
27r
8
10
100 10
Chapter 13
414
10
I
I
1
1
I
1
I
0-
-20-
-30-40 -
'
-50
0
1
I
1
I
20
40
60
80
1
1
120
140
I
100
1
1
160
Figure 2
10 20, 80,
140
on
8
SNR 100 Hz
10 m
M+ 1
M
M xM
3
2 bound
10
10 by
13.2
EIGENVECTOR METHODS OF MUSIC AND MVDR
8 good.
10
415
Adaptive Beamforming
20
1
1
1
1
I
Figure 3
10 20, 138,
1
I
140
1
100 Hz
SNR, 10 10 8
A by by
no
by
8
k 4
150 Hz
60
90
0 by
on
Chapter 13
416 150 HZ at 90 degrees bearing ................................................ .
,
. . . ..
..............................................
L
.
.
.............
c
*
1
*
. . . - . . . . . . -.......... . - . . . . . . . . . . . . . . . . . . . . . . . . ;;
1-;
/
0
0.2
0.6
0.4
0.8
A
1
/
1.2
/
1.4
/
/
1.6
1.8
-1
Really1
0 degrees bearing
. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
0
0.2
0.6
0.4
0.8
1
1.2
1.4
1.6
1.8
Really1 Line Array Position D
Figure 4
150 Hz
( c = 345
60
by
5 4
by Eq. (13.1.3)
a
3.2.1 )
417
Adaptive Beamforming
x
z
0--’
. ..
90
.- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- ............................................ I
-
3
.
-
U)
?!
CR
D
o/ -90
I
-180 L
Figure 5
by 150 Hz
60
N
xi,,
N As
N
by
N by
Eq.
by M + 1
M
M
M+ 1 13.1,
by M on
RA on
+1
418
Chapter 13
by
by
N
orthonornd.
by 13.1
do
on by no
13.1.
look
Adaptive Beamforming
419
on by -
Rv,= IT;& i = 0, 1, . . . , M Ri
v,
as
(1
A
A H A=I,
A
-
’ =A H .
“H’
is by
A A H R A = A H A A= A
(1
since A H A = I
A (1).
a
A;, on
by
420
Chapter 13
(kJ)
8
(I,k)
(1
0 RIl = R22
no
by on, A
N (1
(
M+ 1 A good
by
on 9.4
book.
Ri,.
-'
[RX,,] = LH[R',]
-
L
Adaptive Beamforming
421
by
L
: l=I m t 1
""
r.\1 - I
RLl
10
10
I1
Y M ,t ,
LH
L
L
by
LH
3.2.1 1) L U Choksk?~Juctorizution, L 01
Also,
8
100 Hz
90
1.25
( c = 343
137 Hz). A
+ 7r.
-7r~ni
+
by 8
U,
5.5 d'
Hz by 6
by 1024 by
on
8
by kd
A
-I
(7rlkd)
500 1x
3 by
8
9.000006,
422
Chapter 13
Eigenvalue is 9.000006 (
O
*
f?
‘021-
45
0
90
180
135
Figure 6
100 Hz
3
90
0.0001
SNR on
1 x 10 “signul eigenvalues”
“noise eigenvalues”,
(AIC)
[VOVI
* * *
1.’,I.I]=
As
+ AN = [vs,vs2 . *
\is,,
*
vs,,,Ii”, 1.”>
.* VNJ
= 1, 2, ..., N ,
vN,, j = 1, 2 , ..., Nn N n + N s = A4
+ 1,
noise subspace
A N
(1
signul suhspacc. A s by
7 7 The noise eigenvectors all have the same ntdl(s) ussociuteci Lrqith the signal eigenvector(s). MUSIC
8
423
Adaptive Beamforming
Figure 7
7 on
1x
90
20 0
-20 -40
-60 -80 -1 00 -1 20
45
0
Figure 8
90 MUSIC
90
180
135 by
424
Chapter 13
MUSIC (
2, 3,
1,
45, 90,
135
1x
9
9.2026, 3.8384,
0.9737 9
135
90, 45, on 500
is
by 8x
14.0147
14. do
Signal Eigenvector Responses 20 [
I
1
I
-60
pJ
-80
- -vs2
-100 -120 0
I
45
I
I
90
135
degrees
Figure 9 2, 3.
100 Hz
45, 90,
135
-
180
Adaptive Beamforming
425
10 45, 90,
135
1 1
2
3
vsl,
0
0’
(1 32.15) Cii i = 1,2, ... , M
“H”
(
20
-120‘ 0
I
I 1
1
45
90
Figure 10 MUSIC
135 100
180 45, 90.
135
426
Chapter 13
9.1.
S(0’) 11 112.5 45, 90,
135 112.5
8
8
MUSIC Eq. (
20 0 -20 40
E8
J
-80 I
-100
-
ILU
0
45
135
Figure 11
180
ii
112.5
Adaptive Beamforming
427
z=e
z
ik‘’cOsi’ .
’
S ( z )= [ 1 z - z - z
...
-
=
M]fi
/kdcoSl).
-
good
1. 2. 2.
on
3.
on
4.
-n
-kd
+n +kd,
by
5.
by
on 12
by go
1. 2. z.
by
428
Chapter 13
2o
-120
90
45
0
180
135
Figure 12 by 45
135
90
on
on
-n
+n
n. you
on by
13
45
135
150
30 on 0 on
' (d,/kd)
do
' 0
TC.
(d,i'kt/)
429
Adaptive Beamforming
20
-120
'
I
45
0
90
180
135
Figure 13 135, on
90,
45
by
look
$',
(@/kd).
by
13 13
by 13.1. N N=M
As,
Ns AN,
Eq.
+ 1 -Ns A" =A
(
by
Y
A4
+1
-
'
Chapter 13
430
Ns Y=AL&, Y H Y = QlAcAsQs = QtQs.
Ns
by
Y
do W=A N Q N ,
p y + p w = Y ( Y HY ) - ’ Y H + U’( W H W ) - ’W H= I
by 8.2, 9.3
Py=
I
-
Pw.
9.4
by by
up
SNR
13.3
COHERENT MULTIPATH RESOLUTION TECHNIQUES
by
(1
Adaptive Beamforming
431
1 N
Hz
do by
do
As
8
( M=
d, i = 1,2, ..., M
ki i = 1,2,3 on
si i = 1,2,3.
B
A
C =AB. (AB)
good good
432
Chapter 13
N by
up
no
Maximal Length Sequences (MLS)
by
'
2 -1
N a
2
-1 - 1)
on
2 young
N.
by
a
by
by
Adaptive Beamforming
433
1
-1
A
14
2N - 1
[N,i,j,k,
i,j , k ,
7 1
7, 3, 2, 1 6
1
2
7,
7 1,
1 by
7,3,2,
A
1 by
book on
Modulo-2 Additions (exclusive OR)
4
6
6
7
Figure 14
b
7 7, 3, 2,
1
-
1,
127
434
Chapter 13
Table 1
MLS
4 5 6 7 8 9 10 11 12 13 14 15 16 18 24 32
31 63 127 255 51 1 1.023 2,047 4,095 8,191 16,383 32,767 65,535 262,143 16,777,215 4,294,967,295
Spread Spectrum Sinusoids
/
2N- 1
by
by 1 rnodulariorz
by 0
-
MLS
1.
hi-pIime
. 15
64
1024 33 16, 32, 64,
good
64 on
16 64 33 Hz,
Hz) by
size 16
200
Adaptive Beamforming
435 Spread Spectrum Time Signal I
I
4' 0
21
1
I
0.1
0.05 I
I
I
I
0.15
1
I
L
,
1
1
I
1
0.2
0.25
I
1
I
I
0.35
0.3
I
I
I
I
,
,
0.4
?iI -
iil -:-
-
I
I?
,
2 $
0
8
-2 4' 0
2
I
I
I
0.05
0.1
t
I
I
0.2
0.15
0.25
I
0.35
0.3
0.4
88c
Figure 15 A 64 Hz
105
55(!l
I
510
by
1
I
160
l;O
1
260
I
250
1
360
[7,1] MLS
33
1
I
3kO
460
I
450
I
56;
Figure 16 15
Hz).
Chapter 13
436
100
M
Resolving Coherent Multipath
350 500 go
on 35
465
535 c/.j; = 3 5 0 / 3 5
10
( 8
10 c( 1
f1
/v;T’])
f;
T[,
bound (
5.3) 33 100
38
205
MLS 17. 18
on
350
3.5
(a/./;
5
by
The Channel Impulse Response
by
437
Adaptive Beamf orming
Figure 17 1,
200 = 1024
MLS
100 16
Hz).
ia
: 1Oi0 1
I
1
I
60
80
1
16
14
12 10
a 6 4 2 I
0’ 0
1
20
40
100
Figure 18 100
by [ 1 1.11
33
305 38
Chapter 13
438
19 20 33
38 c/f, = 3501 1024
0.342 34
38 100
205 Hz
by
MLS)
21
22
350
205 Hz 33
23
16 Hz 34 512
175 100
1024 Hz
3.5
16 Hz
21.875
no no
As
4 175
87.5
Channel Transfer Fcn
I
1
100
P
0
U
-100
0
100
200
300
400
500
100
200
300
400
500
20
E! 0 Figure 19 is
439
"0
40
20
100
80
60
Figure 20 33
25 Figure 21
38
30 33 1024 Hz
40
35 34 vs.
45
Chapter 13
440
65
I
60 0
Figure 22
1
I
I
I
I
I
100
200
300
400
500
a 16 Hz
205 Hz
34
33
Hz).
I
1
1
1
1
1
28
30
32
34
36
38
Figure 23
33
34
16 H z
40
Adaptive Beamforming
441
24 4 by 11 4
31.82
25
26 87.5
350
no 27 11
4
31 As
87 SNR
24
20
Figure 24 no
25
30
35
40 33
34
45 4
Chapter 13
442
266
MLS by on
601
0
Figure 25
x 10” 3.5
I
MLS
I
I
I
I
1
100
200
300
400
500
11
I
by
4
I
I
I
I
1
1.5 1-
0.50, “0 0
10
1
1
20
30
40
50
Figure 26
11 33
60
70
80
90
4
[ 1 1.13 MLS
443
Adaptive Beamforming
Channel Transfer Fcn 100 0
hp
U
-100 0
100
200
300
400
500
100
200
300
400
500
~~
0
Figure 27
1 1 Hz
4 Hz Hz).
30 Figure 28
33
32
30
36
34 1 1 Hz
87.5
34
31.8
Chapter 13
444
by
on
(NASA’s
// as
13.4
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
13 13.1
13.3.
a
by do 13.2 Eq.
Adaptive Beamforming
445
by by
by
13.2
But, As
good 13.2
13.3
by on
Chapter 13
446
on
MLS by
13.3
no
PROBLEMS
1.
3 10,000
60
0
90 2. 2
30
1500 no
. 3. 1
LMS
4.
do
5.
6.
500
c‘f
1500 10,
16
2
7. 8. 9.
by [3,
MLS 7
-1
20
Adaptive Beamforming
447
10.
BIBLIOGRAPHY
14,
W. S. 1991.
3.7, 1994. 6,
S. W.
S. 7.4
1988.
W. 1 1.1,
1986. on
1985,
pp.
1998. A
1988. REFERENCES 1.
W.
W. 1986.
1 1.1,
2. on
1985.
3. on
1967.
4. 3.6,
1994.
This page intentionally left blank
Part V Signal Processing Applications
449
This page intentionally left blank
14 Intelligent Sensor Systems
by
1.
2.
3.
4.
5.
on
by 451
Chapter 14
452
irzjbrrmtiorz
information:
on
diugr1o.si.s of’ tlic c’iirro)it progriosis of’ tlir c’.\.PL’c’ti~i/.firti~rr
stirtr stirte.
on
book,
book A
on up
A
on no
by
A
1.1,
Intelligent Sensor Systems
453
upon
by on by
by by
11.2
by
by
Chapter 14
454
on 14.1 no
14.2 14.3
14.1
AUTOMATIC TARGET RECOGNITION ALGORITHMS
on
by
features. by
truining dutu set. aduprive neural netiiwrk
syntacticfuzz)? logic
dutu Jitsion by
Intelligent Sensor Systems
455
Statistical Pattern Recognition by by A
by
12
36,500 on
ok
V?k
tj’k.
x wk
Mahalanohis distance
Eq.
.U
by
by good
x,
wk.
x
q.
Chapter 14
456
upriori u posteriori
s
1 - 2.0, 0.0,
+ 3.0,
1.25, 1
1.50. Eq.
15.0
- 2.00, 1
1
- 2.00
1 - 2.00
*
ATR 2
1
1
2, 3.
0.4
P(wk).
I
I
I
P
I
1
I
I1
Feature Value
Figure 1 A, B,
pdf
Intelligent Sensor Systems
457
Table 1
A =
1
2 3 1
2 3
1 2 3
0.8066 0.1365 0.0569 0.8234
0.0000 4.0000 4.9383 0.3886
0.0000 2.0000 2.2222 0.6234
1
0.1071 0.5130 0.3799 1 .oooo
3.6864
2
0.7901 0.0000
1.9200 1 .oooo 0.8888 0.0000
118.37 225.00 28.444 8.4628
10.879 15.000 5.3333 2.909 1
1 .oooo
0.0000 0.0000
.oooo
0.0145
1
15.0,
3 by
0.0145
by
tt’k
it’,,
458
Chapter 14
o;,,~
i
~ ‘ k .
\i*k
Ck
(SVD) Ck
by
Sk
CL
SL
no CL A
1-0
on 1
m 1= [7.5 81
459
Intelligent Sensor Systems
&=[
01
01
-
1.2412 1.3681
8.7588
-
Ol = nzz=[7 71 c2
02
=
][
3.00 0.001 0.00 1.00
[
01
01
1
1 2
sl
[
021 2.00 0.001 0.00 1.00
-
[
02
02
3.6491
1
1.3509 02=
on
Eqs
on on 1-0 bound on
4
on 2
50 on do
do Eqs
X.
wk
do
on
( 14.. 1 .1
Eq.
2
460
Chapter 14
X
X
X
10
08 0 0
N
x
0
0
X
5-
0 xx
X
X X
x
x
x X
X
X
X
0 0
I
I
5
10
Figure 2
1-0
0.01
3-0
--
3
Eqs
4
on
on
2,
1,
Adaptive Neural Networks
on
sJ*ncrp.sr.s
Intelligent Sensor Systems
Figure 3
Figure 4
461
Chapter 14
462
1,000
10,000
20
no
on on
generalized delta rule
book. 5
by netj.
N
q,k
N
(14.1.15)
netj
Figure 5
N J’(net,)
q, Oj.
bias
h,
Intelligent Sensor Systems
463
no netj
0 by no
( 14.1 .1
5
o, iJk,
b,.
rijh,
c
As E
net,
net,
E,
up
(LMS) 9.2.
LMS
s1
x2,
w1
6
w2,
5.
o4
1,
01 2.
os,
11
by by
o,!’ tp.
p
464
Chapter 14
X Figure 6 A back
p,
p
( 14.
by
Wc. hive no of‘knoii’iiigidwtIier the netirwrk is the optiiiirrrii r 1 t w o r k , ~>/ic>tlier U c i i l f c w n t nirnihtv- oj’notles cincl hiclilen lei?*er,s\\-ill pecforrn hut tor, or \thc-lther. tlw training U / ’ the nc)t\r*ork is cwmplutr. on
465
Intelligent Sensor Systems
( 14.1.1
(1
(
(Eq. (
(Eq. atl(.t; a’ljh
(14.1.22)
- ‘IX
on (
. illxI,
= illx,
+ 2116;i!; p
(
LMS 0.10,
f
by
Chapter 14
466
by
neti,
by
( 14.1
N I = 37
15
N 2 = 42 15‘s
on
M
k
A4 by by
1.
2. no by
3. by (
(Eqs
Intelligent Sensor Systems
467
4. (Eq.
5. (Eqs 6.
(Eq. (Eqs
7.
1
7 y
x 1
2 6
1
1 2 2
1
2 1
7 good
Figure 7 1
A
1 (x) 2 (0)
Chapter 14
468
good 1
8
2 by
A
do go
by
Figure 8
1 (x)
Intelligent Sensor Systems
469
Syntactic Pattern Recognition
sjwt~i.~,
good
thc scieti1ific nzcthod
The niciiri diffim/tj7 i s tlicit
oiic
riiust kiiow the s j v i t ~ ~ .
no
s e t nimiber.sliip
@zjy
firnctiori
Eq.
p(s). do on s it 14 a t io I z (i 1 n
no on
r en
11y
by
1g
h .
o
470
Chapter 14
= h,.y 5
U
[d + b + (d - h )
P ( a , h, c‘, d , -U) =
= d,.v 2
c‘
9
Eq.
h = 1.5, c = 4.5
cz = 0.9,
d= 0.1.
a
Eq. Eq.
10 body
afuzzy A N D function
by
q,?/x
Property Value
Figure 9 A
471
Intelligent Sensor Systems 1.
I
0.9
-
0.8
-
0.7
-
0.6
-
0.5
-
0.4
-
0
1
1
1
I
I
I
U= C
*
0.2 0.3
0.1
J
0
0
1
1
1
I
I
I
I
10
20
30
40
50
60
70
80
Figure 10 A
qlk
AND
(14.1.28)
ZI
0.1.
472
Chapter 14
( 14.1.29)
no
fl.
(
N 5 4. 0.5
U/.
all iis
by by on 11s
do
as
A
n,
4
Intelligent Sensor Systems
473
11
on /?(- 1,2 1, 0, ds,,),
d ~ ~ ,
on on
15 finite state granmar. natural language 16
0
0.1
0.2
0.4 0.5 0.6 0.7 Property Confidence Difference
0.3
Figure 11 A
by the
0.8
0.9
1
Chapter 14
474
Figure 12 2nd,
by
Figure 13 up 1.5
by
1 5
1.5
5
by
475
Intelligent Sensor Systems
Figure 14 1
Figure 15
3
476
Chapter 14
Cf(T-3, Not Quiet)
1 .o
Cf(T-3, Up Fast)
Cf(T-2, Not Quiet) Cf(T-2, Up Fast)
\
7 O'*
Cf(T-1, Down Fast)
Cf(T, Constant)
1 .o
Cf(T, 2k)
Cf(T, Screaming Phiha)
0.5
Figure 16 13.
T, C.'f(T, S P ) ,
[
C f (T , S P ) = 0.8 * Cf'(T , constant) A 0.9*
0.5 * C'f( T , I k ) v 0.5 * C(' T , 2k) A
0.7 *
cf'(T - 2 , UpfhSt)
A
12
1 , t i o jirst) ~ ~
v
Cf(T, BJ),
by
C f ; ( T ,M W )
-
Cj'(T - 3, LcpfU.st
AND
A
C f (T
477
Intelligent Sensor Systems
Cf(T , M W ) = C f (T , consturzt) A V
C f ’ ( T , I k ) A [ l - C f ( T - l , I k ) ] A [ I- C f ’ ( T - 2 , I k ) ]
cf(T ,
A
- Cf( T
-
A
-
Cf( T
-
0.5,
on
LISP
hidden M ~ i r k o vm o&/. on on
on
SNR by on
Chapter 14
478
book,
on book.
14.2
SIGNAL AND IMAGE FEATURES uiz
irlfbrniution c-oncentrutor,
by no
by
by on
by
6.1.
Intelligent Sensor Systems
479
on
by
Basic statistical measures of signals 1 crest.fuctor,
by
Signal characterization by derivative/integral relationships
17
on.
18
1
480
Chapter 14 dF[ t)/dt 0.5 r
0 -0.5 L 100
0
200
300
400
0
1 100
200
300
: ’1 l 100
-5 0
200
300
-1 0
100
200
300
400
100
200
300
400
1
1000,
:
400
400
I
-5 0
C% 10- 100
200
300
400
o
200
300
400
t
1
1000,
0
100
200
300
400
t
Figure 17
Periodic signals have unique harmonic patterns
on
Eq. (
T,
12
by t = ) I T ,
481
Intelligent Sensor Systems
Pop Sound
E
.E
/
0-
-
-
U
(C U
-1
I
I
,
,
0.02
0.025
-5
0.01
0.015
I
I
I
I
0.03
0.035
sec
Figure 18
N,
bp.
on
N,,.
13/1
19
pp
19
odd odd N, N,
N,,,.
up do
pp pp
N,,,
bp
13,= 1
As P,
2. N,. = 10, /3 - 10, ”7 1000 odd N , = 100 N,,.= 11.1 1 1 1, A odd The Fourier transform provides the spectral envelope N,,, N , = 100
by
N,. 2 19
odd
by
Chapter 14 Spectrum
Nr-100 Nw-10 Fa=lOOO
0
’ 1
-20
-4 0
m -60 -80 -100
0
20
40
60
s oc
HZ
Nr-100 N w = l l 11 Fs-1000
Spectrum
80
100
120
1
Ol-----I 0 5
0
-05 1
-1
4
0
02
01
03
0 4
05
-1001
0
1
20
.
’
40
60 HZ
Sec
Figure 19
odd
’
80
.
100
J
120
by
by
20
2
‘‘0”
3.5 on
plzonenie.
40
up
483
Intelligent Sensor Systems eeee Sound Spectrum 100,
I
-150 0
1
I
I
1
1
I
I
I
500
1000
1500
2000
2500
3000
3500
4000
1
I
1
I
0000
1
1
Sound Spectrum
100
50 0 m U
-50
’ 0
’‘
I”“
’
I
I
I
I
I
,
I
I
I
500
1000
1500
2000
2500
3000
3500
4000
Hz
Figure 20
“0”
on
by Distortions in signals can be characterized
Chapter 14
484
A 6.1, Eq.
on
c;(cu,,(!h)= E(X(tol)X(cuZ)X*(cu, + COZ)}
(
by
X * ( m l + w2 )
Eq.
X(col)
As
+02) +
. ~ ( t= )
+ 02)
(
Eq. .I!( t ) = s(t )
+ EXZ( t ) -v(t)
by
Eq.
(
Eq. 2toI
Eq.
no do
ml
Intelligent Sensor Systems
485
Amplitude and frequency modulation of signals
300
on
on
on
no
1
Chapter 14
486
by
by up on on
A AM by
A M signals are common in vibration
5 t ) = [1
+
100
o!
r
100
w,~,,
5
w,
21 20 on,
Eq.
o!
= 1.
4
(
AM
y=
m=
Q =4
22
350
M= 5 A,
on
22 1@
up
x'~,,,~
487
Intelligent Sensor Systems AM Response
2 1
a C U)
go a
a
-1
-2 0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Sec
0.81
I
1
1
1
I
1
I
4700
4800
4900
5000
5100
5200
5300
a
C U)
g
0.4
U)
a
0.2 0 4600
5400
Hz
Figure 21
5 kHz
AM
100 Hz
AM
AM Response
30
I
I
I
I
I
1
I
1
1
I
I
1
I
I
1
1
1
I
20 a
C U)
g
10
U)
a
0
0
I
0.001
I
'11'
-I.
-10I
I
0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Sec
3v) a
2v)
a
1 -
0
0.5
1.5
1
2
HZ
Figure 22 5 kHz AM
2.5 10'
350 Hz.
Chapter 14
488
up
FM signals are more complicated
on
1
up CO(.
jft)
=
+ [j
[j
phuse modulation,
2nl(o,,,
( 14.2.10) ,,=-cc
(14.2.1 1)
Intelligent Sensor Systems
489
on 23
J,,(p). 5
p
100
0.5.
20 kHz, 24
p
4f
-=
.fi
40.
p- z .A 4 9
24
by
FM Response beta=.5 1 0.5 0
rn C
xal o
a
-0.5 -1
0
0.001
0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 sec
1
4600
4700
4800
4900
5000
5100
5200
5300
5400
Hz
Figure 23 AM
490
Chapter 14 FM Response beta-40 1
0.5 Q C a
x o 0)
a
-0.5 -1
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Sec
0.2
1
I
1
1
I
1
1
I
I
1
0.151 v)
2
0.1
v)
a
0.05
0 0
1000
2000
3000
4000
5000 Hz
6000
7000
8000
9000 10000
Figure 24
40 by
Rotational vibration signal often contain both AM and FM signals
!A
!A
Intelligent Sensor Systems
491
+
‘/2
U.
by - 1
- cu
no up
‘/z
up
25 100 Hz 40 Hz 5 kHz.
2
= 0.9
p = 50 ( f2 kHz
y;i/(t),
by ( 14.2.1
FM=40Hz beta=50 AM=100Hz alpha=.9
2,
...
I
1 a,
c
v)
g o a, [I
-1
-2 0
0.005
0.01 5
0.01 sec
0.15
-
: U) Q)
0.1
v)
a
0.05
n 2000
3000
4000
5000
6000
7000
8000
Hz
Figure 25 A
40 Hz FM
p = 50
100 Hz A M
a = 0.9
492
Chapter 14
2nj.i.= U,,,
An)
26
on
no by
fiuws
on
FM
Q)
*
1
1
0.005
0.01
1
1
0.5
0
n $
a
0 -0.5 -1
0
0.01 5
Sec
Figure 26
AM
0.02
0.025
[),
Intelligent Sensor Systems
493
N , = 20
N , = 400
20,000
25
odd
up
475 Hz.
1
27 28
p
29,
10 20
5
30
by
n
Figure 27 A 1 kHz
/3
1
Chapter 14
494
Figure 28
p = 1.
Figure 29
p = 10.
1 kHz
495
Intelligent Sensor Systems
U
-20 -25
1 I
J I
I
1
I
1
I
J
Figure 30
/j =
10
1 kHz
26. 40 26
p=SO
2
0.771
fl
in[r1]
n,
p = 10 475
/?
(5 15 40, Two-dimensional features can be extracted from image data on
on
.foca/-p/anc~ .f~atures on
texture features.
by by
Chapter 14
496
good on
24 A
A
on.
(4). do
do,
by
2
cones rods
on 1 10 Hz
30
= 60
100 Hz
30
on
a
At
Intelligent Sensor Systems
497
on
horizoiitul cells, the hipolur cells, the uniacrine cells, und tlw gnngliori ct>lIs. niicjget hipolur w1l.s
by vernierucuitj*
6
good -
good
0.06
up
0.1
0.01
Y
-
X
Y a
.first iind sc~conddiflc>rence 4.3
2.5
498
Chapter 14
by
no
. on
upon no
on
31 31 by
do
by box A
Intelligent Sensor Systems
499
Figure 31
image segmentation
A A
graph. 32
500
Chapter 14
Segmentation
Window
3
1
7
4 Figure 32
A
iis ii
a
odd
a
A
it
on
a
Intelligent Sensor Systems
501
on
by by
I X - ~
up
on strategy
14.3
DYNAMIC FEATURE TRACKING AND PREDICTION
prognostics.
on
do by
on
502
Chapter 14
A body
up A
on
body So,
10
on
good on A
body
on by 3,000
you
3 3
time-bused muintenunce
...” on
A
do
good
Intelligent Sensor Systems
503
on $1
$100 you
A
by
on
by by by on Condition-Based Mairztenance,
on
on on
you
you up,
do on
on
Chapter 14
504
by
-
hub,
on
-
by
by
on
on
/
on
a
on a
a
as
good a a as
Intelligent Sensor Systems
505
by on.
do
33
on
do on on
33
400 350
300 250 0
2
200
0
150
100
50
0 0
Figure 33
20
40
60
80
100 Hours
120
140
160
180
200
506
Chapter 14
no on on on
on 30
33, 30
130
AHM
X 50%
#
10 17 17 17 Y
Y 2
Z (50%).
#
f5(% 1OO"h.
good
34
10.1 predicted
updated
507
Intelligent Sensor Systems 400
350 300
250 0
g
200
n 150
100
50
0 0
60
40
20
Figure 34
80
100 Hours
120
140
160
180
200
40 on
d
+
Y ( N +dlN) = F(N d)x(NJN) 1 Td $ T 2 d 2 = 0 1 0 0
[
‘p
][
~ N J N
T = Td,
34
on
by by
508
Chapter 14
40 (
40 100
34 240 80
ii
140
65 90 140
35.
24 200
400 300 210
180 175
do 36
150 160
10
220 155
400
I
180
1
I
I
1
I
I
1
1
350 Failure
300
+
--
250
.....
0
g
200
-.-
n
Measurements Tracked Data Resistor Fault Regulator Fault Linear Predicted
..
Repair
150
/
,
100
50
0
-0
20
40
60
80
100
120
Hours
Figure 35
140
no
140
160
180
200
+)
509
Intelligent Sensor Systems 40C
35c /
: 1
-
25C
..... -.-
0
p
/
,*
Failure
30C
200
Measurements Tracked Data
.’
/.
*,
Regulator Fault Linear Predicted
.’/
.’
,4 ’
n
,
Repair
150
/
: / / :/ . I /
100 50
0 0
20
40
60
100
80
120
140
160
Hours
Figure 36 At 150 30
20
170
180 As
37
do Eq.
on
GT,,
180
200
510
Chapter 14
2 200 0
400
CR
I -10 1
I
I
I
1
1
40
60
80
100
120
140
I
I
1
I
I
20
40
60
80
1
10-320 2
0
1
I
100
120
I
140
1
I
180
200
1
I
1
160
180
200
I
160
Time Hours
Figure 37
CT,,
P’(N
+ 1 J N )= F ( N ) P ( N I N ) F H ( N+) Q ( N ) 2 by 2 I )-( 14.1,12)), on
by
A
6.1. tF.
2.
Intelligent Sensor Systems
s
511
a;, z = x/ji.
.:a
by z < 2,
r
z
P(z 5 Z), s 2j-2 j’ < 0.
s 5 j7.Z
j*2 0,
(1
by s
j j ,
z S=JY
0
-cc
IF.
38 a = a F / a R= 10.
by f 1.836 1.836 39
aF/aR.
on 180 140
170
512
Chapter 14 0.04
0.035 0.03
0.025
5n 0.02 0.015 0.01
0.005
-100
-80
-60
-40
-20
0
20
40
60
80
100
2
Figure 38 (TF/(TR =
10
CJ
= 10.
140
160
300
250
200
E
150
3
I -I
2
100
50
0
-50 0
Figure 39 on
20
40
60
80
100 120 Time Hours
180
200
Intelligent Sensor Systems
513
39
40
on
170
good.
up
160
As
As by
41
PDF and SuMvor CURBS
200
RUL Hours
Figure 40 A
Time Hours
Chapter 14
514
50%).
41
50%
As
180 At
do
as
by
by
H(t)
1
0.9
I.....
0.8 0.7
~~
Survivor Probability
Normalized Hazard Rate
0.6 0.5
0.4
0.3 0.2 0.1
0 20
40
60
80
100
120
Time Hours
Figure 41
140
160
180
200
515
Intelligent Sensor Systems
( t - tF)2
t - tF
by on
42
on 1O-e 1
-3.5
I
1
I
I
I
1
I
I
I
Time Hours
Figure 42 A
I
I
1
1
r
I
1
1
516
Chapter 14
no
good
14.4
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
book As
14.1
on A
on
A by
AND, on As
book.
Intelligent Sensor Systems
517
14.2, on
comwi-
on
trute
14.3
on on by
by you
50% on
6 go
book
518
Chapter 14
PROBLEMS
1.
A
A
A
2
5, 1
5.5
1
2.5
1
1.75
9,
- 3.2.
2
bound 2. A 2,
2 1
6.5
5, A
2
you
3.
1
4.35 5.36
A
2.
4. 0
1 A
> 0.5
4
5.
on 6. A
25. 100 7. 94 100
1
117 17
0.1 200
217
0.05
”/o
234
0
8.
9.
U,,
T 10.
9.
9
on
519
Intelligent Sensor Systems
BIBLIOGRAPHY
1993. 1983. 1991. 1996.
S. Z. 1988. 1997.
I. 1992. 2nd
1992.
I: 1986. 1992. 1992. 1974. REFERENCES 1. 2.
1992. 1974.
3.
E. 1710,
4. 1992.
1992, pp.
This page intentionally left blank
15 Sensors, EIect ro n ics, and Noise Reduction Techniques
Intelligent Serisor Sjtstcwi
on
sewticvit processing. self-(iH’(ire~i24.s,s,
situcitiorzcil
ci~~ir~vic~.s.s.
no
on
521
522
Chapter 15
by
15.1
ELECTRONIC NOISE
A As ttzerniul popcwrrz
.slzot
c‘orittrct
523
Sensors, Electronics, and Noise ReductionTechniques
by Thermal Noise by
by by
good
no on
on Eq.
v,= J4kTBR
(15.1 .l)
5.1 x
(Q)
524
Chapter 15
up
&E 01
=
J4kTBR
( 1 5.
As
1 10
10
10 (1 g 0.001 g. 100 pg
by 10 0.1
9.81
10
1 10 Hz
1000 jig
30 pg
on
a
Eq. as ‘/z
Shot Noise
A
by
It,(.
B x 10.-
y ii
’’)
by a
525
Sensors, Electronics, and Noise Reduction Techniques
by 1/
1/ 2 N 1
Contact Noise
(1 5 .
K
Thermal or Shot Noise 4
7
1
-
0
1
I
0.2
0.3
I
I
1
1
I
I
0.6
0.7
0.8
0.9
~ ~
~~
0.1
0.4
0.5 sec
1
-20 m U
-40
-601 0
1
50
1
I
1
I
1
1
1
1
I
100
150
200
250
300
350
400
450
500
Hz
Figure 1
526
Chapter 15
by
by
by
R,
o,.(f) = RKI‘/$
/f”
by j ; / N ,
Hz !4
ZMG
-1
0
k
N-k,
k N -k
05k<
by 2.
1 lf
Contact Noise 4,
1
1
I
I
1
1
1
1
1
I
2 0
r . 0 -2 -A
0
I
1
1
0.1
0.2
0.3
1
I
I
1
I
1
0.4
0.5
0.6
0.7
0.8
0.9
I
I
I
1
400
450
500
1
sec
0,
I
1
1
1
1
-20 -40
-60
-80I 0
Figure 2
1
1
I
1
1
1
I
50
100
150
200
250 Hz
300
350
by
I
“1
10
527
Sensors, Electronics, and Noise ReductionTechniques
Popcorn Noise
by
n
1 lf”,
2.
3 Electronic Design Considerations 1.
2. 3. 4.
Popcorn Noise
6-
I
I
I
1
1
1
I
I
I
U
z
2-
X
I
-21 0
I
0.1
1
I
1
1
I
1
I
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
I
1 1
Sec
20
0
Figure 3
1
50
I
I
1
1
1
I
100
150
200
250 Hz
300
350
400
1
I
450
500
528
Chapter 15
by on
SNR.
f
by
SNR by
SNR
1 50. 12.65 (
20 160 (
+ 22
20 SNR by
Signal Shielding
A
ZlC. = 20 =
/?
(
= x
x
good
>> j w z ,
17 (T
(
529
Sensors, Electronics, and Noise ReductionTechniques
11,.
0,. =
=p / p o
= 5.82 x 107
11,.
100 Hz,
148 1 MHz,
on good
108
300 x 106
60 Hz
5
6.3
Hz) 10 100 MHz 100
(r-c
by 1
= 271fEOr
(1
r
(3
RE = 321 +
-
dB
-
(r <
RH
= 14.6
+ 10 by
(3+ 2
10
dB
530
Chapter 15
RH
t,
RH
Eq. (
R H= 0 6,
0
=
J-
2
(1
Eq.
(1
9
100 Hz,
1
by
box.
by
by
Ground Loops
50 Hz
Hz
you 120, 240, 360 Hz,
t,tvicsc)
you
531
Sensors, Electronics, and Noise ReductionTechniques
box on 1
on
As
by
4
hut this cable shield must not he connected to an?’ C‘OHductor at the sensor.
(
+)
2 - W i r e Sensor
Common Ground
/-
Cable Shields
3 - Wire Sensor
FET
t v supply (-)
\
I
1
Common Ground 0
Figure 4 no
Chapter 15
532
(-)
( -)
3
(
+)
( -)
by by ( +)
4
15.2
ACOUSTICS AND VIBRATION SENSORS
by on
do book. book,
on
Sensors, Electronics, and Noise ReductionTechniques
533
reciprocitji
X Y/X,
Y, Y / X by
X
Y I
by
V
I on
electromagnetic mechanical anical
electrostatic mech-
micro-electro-mechanical system ( M E M S )
The electromagnetic mechanical transducer on
5 U
f
f = Bli
i
( 1 5.2.1)
e = Blu
1
B
i
1
I
1 e
U
Chapter 15
534
Electromagnetic - Mechanical Transducer
Z,=(BI)* / Z,
Ekl
Electrical Impedance
Mechanical Impedance
ElectricaI Impedance
Mechanical Mobility
(D)
(C)
Figure 5
by
by
flux
flux
by
by
f
1 1 2,v = - = @I)- 7 = (BI)2 -
5.2.2)
e
ZE
U
ZM ZE.
7
6
Re
on
Le
6
on
7. on
1930s no
6
535
Sensors, Electronics, and Noise ReductionTechniques
%zi-+Impedance
R
Le
Md
fElectrica;-+Impedance
+--Ezi+ Impedance
Acoustical Impedance
Impedance
d'
RA CJS'
Impedance
Mobility
Electrical Impedance
Figure 6
a
Md.
Cd, U
S
Rd. U = Su p
Chapter 15
536
Figure 7
by S. C, = Vb/pc2,
V, 345
p = 1.21
no Z, Z u = pc T ( Tk2a2 +j0.6ka
M , = 0.6pa/S
(1
S = za2 by S2 6. C,/S2
C, C’ -
on
CA,. c dc u
da - S2CA
(1
+ c,
6 by
537
Sensors, Electronics, and Noise Reduction Techniques
i by
Bfi
U
by = Blu.
6
8 30 R d = 1.5 55 Hz.
box
R,
Re = 6 0,Le = 3 Vb 1
Bl= 13.25, M d = 60
8
1R Re + j u L e
+ (B02
by 8
OHz
Re. Le.
Loudspeaker Input Electrical Impedance
100
I
I
1
I
1
I
1
I
I
n
-1 00
0
'
100
-
I
1
1
I
1
1
I
1
1
50
100
150
200
250
300
350
400
450
I
v)
E
50-
Figure 8 57.7 3
5
500
538
Chapter 15
57.7 Hz. 55 Hz
1 by
QT,
Z,.,,s
Z(f;,ff) = -/,. QT
ff
Qnls
QT
Qes.
,/a.
by
M’
A
by
“Bl
by (
Rll
Rd = ( ( U , M D R E / Q T Z , . ~ ~ ) . by QT.
QT
is QT= !h
QT= 0.707 9.
P
Bl S R ,
(
539
Sensors, Electronics, and Noise ReductionTechniques 20
0
-20
>
-40
\
a n -60
-80
-1 00
-1 20
50
0
100
150
200
250
300
350
400
450
500
Hz
Figure 9
Qr = 0.707
2nd
QT by
fs,
by QT
0.707, QT
1 Joirrizal oftlze Azrdio (3).
Engineering Society
by on
F = Sp, by
S e = Blu,
on BlS
e
10
540
Chapter 15 -25
-30 -35
-40
-
6
n >
-45
m U
-50
-55 -60
-65
0
50
100
150
Figure 10 Voltage response
200
250 Hz
300
350
400
450
500
the loudspeaker when used as a “Dynamic Microphone”.
QT
0.707
500 Hz 3
good
1930s
11
by As
3
6.3)
541
Sensors, Electronics, and Noise ReductionTechniques
i
~
Acoustic Baffle
uspended Ribbon
M Figure 11 A
by
on joU/S.
by
(15.2.10)
12
by A
13. The electrostatic transducer
piezen
542
Figure 12
Figure 13 A
Chapter 15
543
Sensors, Electronics, and Noise ReductionTechniques
on
on
F= k F=qE,
E
on
f =$I4
e=
$I i =d q / d t )
5. e = -$I-
1
U
jw
J" e = Blu)
(f= Bli Piezoceranzic
Piezoelectric
by no
on on
Chapter 15
544
by
on
p,
Mason equivalent circuits S, I, k,, Q I C .
14
1 14. M1 14
M1
14
on
545
Sensors, Electronics, and Noise ReductionTechniques
M,
Ml
M2
M , = jac,S tan(k,l/2)
M 2 = jpmc,S sin(kml)
C, = -jRc,S / sin(kml)
C2 = -jpmcmS / tan(kJ2)
Figure 14 c,,,,
pnl,
1
S,
14 on
on
15, on
15
1 : CO4.
CO
Ro 75 x
i2 1600
Ro
1 by 15
f= (u=O).
i i/jcuCo, i/jcuCo = 4i/jcu, on CO
U,.
-uCo$.
Chapter 15
546
Figure 15 a
e = -uCo 4/jcoCo=4 iljco, on CO 42/jco.
4 1000
d31
A4
C
15 1750 1/4”
1 x 107
1/4”. 3.5192 x 7750
1.1458 x 7000 3.8963 x 10-4 400 Cl
1/ 4
1x
1/4”
1
2.112 x
2.5
1 A As by
547
Sensors, Electronics, and Noise Reduction Techniques
ZM.
15
2,.=
1
($2co 1 +-+-+R’+jcc,
JZ
JCL)
1
JCL)c,
Z.\, =jw U,
Eq.
by 9810
16
g. by Eq.
p = 1025
c = 1500
on ZL.
on on on
548
Chapter 15 Typlcal Accrlerornefer Voltage Senrlfivlty
1oz
10’
to‘
1OS Hz
1oa
10’
10’
Figure 16 mass
1 M Z M = -+j w JWC 2
+R
(1
M
C
PZT 15
i,
( 1 5.2.2 1)
.4
I-
(1
fL-17
ZL
(1
1 Kg
As
549
Sensors, Electronics, and Noise Reduction Techniques Force Actuator Response
- 1 Kg Load
20 0
-20
!! -80 -1 00
Figure 17 Force response relative to applied current for the PZT driving a 1 Kg mass.
ZL
Za/S2
by Co.
Rl
P
R, 18 on
PZT PZT by
by
550
Chapter 15 Hydrophone Transmited Pressure Response
60
50
40
> a
30
Q
20
10
0 1 o2
10'
10'
1 o8
1OS
Hz
Figure 18
on
by S
4
e
(
19
to',
on
0 Hz
1 /CO
551
Sensors, Electronics, and Noise ReductionTechniques PZT Hydrophone Receiving Sensitivity
-30I
1o2
1
1o3
1
10‘
1
1
I
1
1 o5
1o6
1 O7
1on
1
1O @
10
HZ
Figure 19
up 0
good
on 10
1 on
The condenser microphone
by
Chapter 15
552
As up 19 101,300
1
94
1
20 good
20
Figure 20
553
Sensors, Electronics, and Noise Reduction Techniques
low
on S,lp,
p
S,/ = 4 x 10 - 6 U,/,
S,B,
U,.,
S,.=m,,
14,
c‘
C,,, = I / / ( p c 2 S i )
V
5 x 10
~
R,, = 100
’
20
M,,,. =
C,/ = 6.3326 x 10 M,l= 1 S,I= nai
S,, = m,,,
(1
11 = 18 1 ,L(
on
S,,p on
S(/p by
S,.p. (
by
360,000 36
by 0.1
0.354 4x
21 1
100 10 by
100
Chapter 15
554 Condenser Mic Responses vs Shunt Resistance R,
200,
1
I
1
1
1
100
g
o
U
-100
-200
1 o1
1oo
40
30
1o2
1
1
1 o3
1 o5
10'
-
m
% >
2
0
..
0 r4
r
/
E
10
-
1
.:
/ / 0
0 - - - - + 7
1
1
Figure 21
10
(MEMS) 22 1
Ro 21
22
1 x 10" Q
1 kHz.
up
by A
555
Sensors, Electronics, and Noise ReductionTechniques
200
k.-.-
z
-100
..
* . .
I
1
.- . ..
.2mm w n t
\
- \ - - -
-
-200 ~
40
30
-.-......
-----.
o?
U
1
1
I
1
I
1
I
I
I
I
10’
1 o2
-
0
L
\
> E
1 oo
Hz
Figure 22
on
on
As
on
Micro Electro-Mechanical Systems (MEMS) 20
556
Chapter 15
on
by by on
by
on
a
by by 23
as
as
good
Sensors, Electronics, and Noise Reduction Techniques
557
Figure 23 A IC
body
on Electronic sensor interface circuits
15.1,
Chapter 15
558
by
by
by
By
up
by
24
CO
Figure 24
Eo, C,
Vo = -EoAC/C,.
559
Sensors, Electronics, and Noise ReductionTechniques
1 fL
=27cR1(1/ Cl
V,,/Aq= e = Aq/ CO, V , / e = - CO/Cl .
e
-
1 / Cl. RI
1 fH
RI
T
8 = COT.
CO,
R1 The Reciprocity Calibration Technique by
25
B B
3 B, do
A4
J= M/S,
S
= B/i
= Blzr @/.j(o
e=
- bu/j(o,
“B/”
by ‘‘4/.j(o’’
560
Chapter 15
Free Field Reciprocity B n \
Setup A
A
Cavity Reciprocity
Figure 25 B A.
piezoelectric/electrostatic pS,,,
p,
S,,
ir
by by
ZAlf. ii
P = Blir.
( 15.2.3 1 )
(
Sensors, Electronics, and Noise ReductionTechniques
561
by by
Q
do ( 1 5.2.33)
k
(colc)
i
f= Bli, U
by
=Bli/ZM.
11
S,,
Q
(1
(1 5.2.35)
25, CA
C,%,= C,4/Sj.
Vlyca2,
by
(1
i,
(1
CA
M J = - =joCdq S
(1
Eqs
do (V
do ),
(p
Chapter 15
562
A
A
25
B E,4
EH
A.
A
B
Is
B
B.
SB
M,4, E,:
A,
Is
(1
A’s (
J
on
B good
(fixed by
563
Sensors, Electronics, and Noise ReductionTechniques
15.3
NOISE CANCELLATION TECHNIQUES
noise
sigual
3.3
by
by
on
by Adaptive Signal Enhancement A good
62
60
62
60 Hz. 60
564
Chapter 15
10.2.
by
c w w l u t i o n c*anceIer
on Adaptive Signal Whitening
by innowfiorz.
Y [ z ]=
E[:]
(1
+ U,:-’ + (Qz-2 + . . . + C l M z - M ]
E[z]
(1
J“.
i= I
jil,
ulz
~
I
+
U?: - 2
+ . . .+U
A[:] = 1 jir
~ Z -
A[z] j i I
+
565
Sensors, Electronics, and Noise ReductionTechniques jfl
5.3.4)
"tit -
t
up
t
-
1.
by
A[=],
10.2. LMS (1 a, ; i =
1, 2,. . ., M . its
26
Eq.
5.3.5) p
Eq.
N
p,.(,/
As
10.2,
p,.(,/
sigrial r?~o&l
60 Hz
Figure 26
FIR
-Pf~I-~
566
Chapter 15
Adaptive Signal Enhancement by
27 120 Hz 350 Hz
non 10 Hz.
by
27 on
LMS 200 28 60 70 120 Hz
50
20 29
Adaptive Noise Cancellation 30
Figure 27
120
567
Sensors, Electronics, and Noise Reduction Techniques
.
-a-
.
w,.-r-,.*
*,,.,.
I
.
,
,
..,
'
.
1
* , . I
*.
-0
-30-
41
50
I
I
1
1
I
I
1
I
I
1
I
I
I
I
I
Figure 28 120 Hz
I
0
I
I
1
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 6 0 4 0 0 4 5 0 5 0 0 k
Figure 29 by
568
Chapter 15
Desired
Reference Signal
\d+
I
.I Xt
’
Unknown System
,
nt
Signal Plus Noise Interference
&t
+
J
Error Signal
J Et
-1
/ Predicted Noise Interference Figure 30
10.2 A
A
on
30 by n,,
569
Sensors, Electronics, and Noise Reduction Techniques
t - 1.
xi,
k=O
5.3.7) d,
LMS (1
by on on by
on (6).
by
KVAC
570
Chapter 15
31.
100
300
100 20
30.1 40
50.1 105 300 Hz
10
20 10,000
1024 105 32
105
Active Noise Cancellation on
Figure 31
on
100 Hz
571
Sensors, Electronics, and Noise ReductionTechniques
-20
. .. :.;,?: ..:,.3.:; :*. 1:; ;-, ;:: :* , :-.: :’:-.:* ;: ..
’a
*-,.,
..
I
I*,
-30
-
-40
-
*
-50
.
I
I
I
I
I
I
1
Figure 32
1
1
I
1 .
31
Hz
Active Noisc Cancellation
Adaptive Noise Cancellation 33
active on
33, “S[z]”
572
Chapter 15
Adaptive Noise Cancellation dt
xt
’
Unknown System
Active Noise Cancellation
Figure 33
ANC on
by
“flz].”
33, ANC “S[z],” by “E(:].” “SE“
“S[z]E(z],”
573
Sensors, Electronics, and Noise Reduction Techniques
by
SE by
SE SE ANC
34. ANC,
A(f),
A[,-]
A z
A=
-P
(1
S( 1 - FP)
“E’ ANC
II,
Figure 34
“r,”
Chapter 15
574
Eq. ‘bP”
“S”
“F’ S
A
F IFPI< % . W e call the requirement for IFP(
ASF < ‘/z the “passivity condition”
by
SE. SE
(8).
35
SE
Figure 35 SE.
575
Sensors, Electronics, and Noise Reduction Techniques
SE ANC PE
36 SE no by
PE
ANC SE PE
E;,
ANC
SE
SE ANC
SE ANC 4
ANC
k
A
ps4
ANC
m
.
Figure 36
m
I
by
576
Chapter 15
(k > q), ( q> k ) ,
(15.3.10)
S qsk
S
q.vy
S on
S S
S
2 go by by
k
q
p
p q.vk S E
p
(r,
by p>y
p
to k>y y
k
Active Noise Attenuation Using Feedback Control
p.vq
577
Sensors, Electronics, and Noise Reduction Techniques
37
(H[z]
”
A
tit-1
Figure 37
ARMAX
Ho
z.
Chapter 15
578
(G[z]
H(s)= Hoe-’‘’’‘.
by G(s) =
up
1
+
1 jo/q
1 m . =‘ RC
(15.3.1 1 )
(11,
( - 12
38 300 Hz
300 Hz Low-Pass Filter Responses 200
I
1
1
100
g
U
=
o
. A -
-1 00
...........-..... -200 1 O0
0-
-20
rn U
1
10’
1o2
-
-1 00
1 oo
1
1 o5
10‘
I
I
-
-60 -80
I
\ \
.....
2-pole
-.-
8-pole
* .
-
\
\
1
1
10’
1 o2
\
i 1 .
‘
\
1o3
.
I
1 o5
10‘
Hz
Figure 38
2nd,
300 Hz.
low
579
Sensors, Electronics, and Noise Reduction Techniques
H(s)G(s)
"n"
Open-loop stability implies that there are no poles of in the right haw s-plane while closed-loop stability inzplies that there are no zeros of 1 + in the right haws-plane.
(1
characteristic equation
Eq.
1
2nd,
39.
-100
* /
-
;!
-200
1
1
30
1
1
m U
-10
. ....
2-pole
-.-
8-pole
t
1
1
1
1
I
i I '
. '
. '/ <. <
-20 1 oo
1
10'
1
1
1
1o2
1o3
10'
Hz
Figure 39
3
100 p s
1 10'
580
Chapter 15
no
no
H(s)G(s)
s=
+ .joo
s
on
=j0 jw
40. (11 =
+jw.As H(s)G(.v)
s =jw
[ - 1, jO], cwc*ir*c~lc~.s
40
Open Loop H[s]G[s] Response
3
I
1
1
1
1
2
.....
1-pole 2-pole
-- -
8-pole
1
-
(3
r
- 0 0,
.-E
!
r
1
;'\ :
-1
-2
-3 -3
-2
-1
0
1
real [HG]
Figure 40
H[s]G[.s] [- 1.01
2
3
581
Sensors, Electronics, and Noise ReductionTechniques
39
no no 3.3
271 39
40
40
1
on [ - 1, jO]. z.
3.0 12
5
up
no
“d“ G(s)
H(s) 21, +ell,
(1
~
582
Chapter 15
ARMAX ARMA
ARMA
by
wf,
12,.
K
M
k=O
n1= 1
d
t
B k l ,-[I H [ z ]= U[+-&. A[z]
y= 1
p=o
ARMAX
Eq.
y,
Eq.
a
ARMAX
G[z]
~ I z ] w[:]
ARMAX
--
G[Z]= 4 4 ,+d ( C b l B[z] D[z] do
ARMAX MA
by W[z]C[z]A[z].
Eq.
by
583
Sensors, Electronics, and Noise ReductionTechniques
B[z]D[z] U[-] =
AR
U,.
W [ z ] C [ z ] A [ z-] Y [ z ] A [ z ] D [ z ] B[z]D[z] U,
GCA[z].
B[z]
Wz],
B[z]D[z] B[z]D[z]
ARMAX 1
+ H [ z ] G ~ , ~=[ zA][231D[z] ~
by
no (A[z] (D[z]
persistently exciting
no
C[z]
on
Mz]
41 50
ARMAX LMS
up
ARMAX
15.4
SUMMARY, PROBLEMS, AND BIBLIOGRAPHY
15.1
on 15.2
584
Chapter 15 .5
1
0.5
c-
0
-0.5
-1
-1.5
1
I
I
I
100
200
300
400
1
1
500 600 Iteration
I
700
I
800
1
900
Figure 41
1000
1024
10
15.3
by
by
Sensors, Electronics, and Noise ReductionTechniques
585
on
ANC ANC on ANC
Concluding Remarks
book
book.
by on book body
do book.
PROBLEMS
1.
2.
by on 0.1
120
1 60
586
Chapter 15
3. “
4.
on 10
0.1
5. A MEMS 10
SNR
g 0.001 g 5 Hz
10 6.
5
A A
10
20
A 7.
no
0.90
8.
ANC
9.
gun
10. A
on on
200 3
gun Hz.
1Hz
BIBLIOGRAPHY
3,
1986.
I5 I.
1992.
1993. S. 1996. S. 1984. &
S. 1997. W.
1974. V. -
..., 10165.
I,
60
42nd
Sensors, Electronics, and Noise Reduction Techniques
587
1992. 1976.
S. G. 60
42nd
pp. 1976. 1998. REFERENCES 1. 32, 2.
1928, pp. , 32, 1928, pp.
3.
...,
-
60
I,
42nd
10165. 3,
4. 5.
1986.
G. 60
42nd
pp. S. 6.
35( 7.
S.
pp.
K. 1 ),
1999, pp. 15.
8.
1992. 9. S. 1991, pp. 10.
C. 2nd
K.
1993, pp. 11. K .
1. 1989.
This page intentionally left blank
Appendix Answers to Problems
CHAPTER 1
0.25 3.758. -1 - 213, 8191 0.000305 x 8 192 = 2.5 10 19 20.5 8.9125.
+
1 .5
= 20
=
+ 213
- 8192.
f 3080. 20
20.5 19 on
69.8 SNR
2. 25
f 32767 12.5
96 100 48
77.3
1 42 48 90.3
.j?
50
is
50
k -.I;.)/.fc
= 25
25,765 3.
1.5 4.99667 5.00334
6666.67 /
666.67
20000
4.
1.41
1 .S
20 500, 589
590
Answers to Problems
1
525
1.9 A
3.8 1/8"
1
odd
0.771
0.0916 0.2
0.2
0.4 39.0625 1.445 32767,
32768
odd. -
65535
1
CHAPTER 2
1. 2.
2.1.
z = cJT
3.
zero on Ix 1 = 1 ,
s =0
+jco.A on on u
4.
N
by
-
z-'/T).
=
=
- 2-I)
+
-z-')~).
5.
-
CHAPTER 3
on
1.
f65.3
f (n 2.
bo = 1, bl = - 0.8357, +x(n 10
b2 = 1
100 50
3.
=x ( n )-
-
5000
591
Answers to Problems
4. 5.
CHAPTER 4
xl[k
1.
+ 11 =
x2[k
+ 11 =
2.
5 x 108. T 3.
-
-
+ hox2[k]
y [ k ]= T< T
i=
Eq. 2500,
=
10
1
0.10 0.0002774.
4.2.8
c(
1'
4.
0 -v+45
=
2
0 1
0
5. 4.4.3
4.4.6.
CHAPTER 5
1.
72 1
31.25
36 100,000
3125 2. N(2
+
1152, 12,288, 25,000,000
N = 128, 1024, 8 192,
122,880, 21,701, 2,034,
128, 1024, 46 493
203,
4.93 on on
-
3.
8192,
.U( t )
25
Answers to Problems
592
by by 4.
by N * ( u ) =1 - n / N 12
=0
%N,
N
N 2.00.
by
312
1.732.
CHAPTER 6
1.
CJ;
d/( N ’M)
M = 15,
/N
2.3 14
f 1.4
2 jiv4 1024
43
53.81
2.
1.65 by
1
5.1.
1/( 19.765
3. 0.0025.
4.
67
rzormul
5.
by
CHAPTER 7
1.
1 2 rj
r2
3 U
Oj. k
c,
(rx
01,
rl U‘,
-
Ox).
.j
/
k
Answers to Problems
593
1
[z] [ (1’1
=
2
3,
1
(1’1
81 - 1’2 01 - 1’3
02) 03)
(1’1 (t-1
01 - 1-2 01 - 1’3
02)
A
2.
1 = 0.1778.
1
1. 3.
I’
=1
2
O1 = 180, Q l 3 = 270, ... ,
1 13
22.5‘. O2 = 157.5, ... , 9 16 O I 6 = 337.5
O9 = 0, .... ,
0
k /V
N
i= 1
i= 1
4. W s
3x3 by
I(!??) =
-
-
-5
7
A
0
(2,2 [
+1
7 -9
m = fN / 2 .
-7
+ 13
594
Answers to Problems
CHAPTER 8
T
4' =
1.
0.0 -0.005
H(1)
-0.02
-0.005 1.1
p=
60 I [ 1
120 1
180 240 1 1
1
'4.15 2.95 1.75
,0.55
To=e5.356=212"'C.
-
2. by
do
3. ''
H
4.
+
C H C = .VH-F - H H ( X H H ) H o- H , H ( X H H ) H H ( X H H ) H = .PH.V
-
(2Hf
-
HH)(XHH)H
Ho H = Ho
by N
5. 4 1824.25.
595
Answers to Problems
2
[::::] [ 1257
[
=
1200
1 4
:2]
[PO
1.5 0.2
ho = 1251.5
1
hl = - 15.5.
1174.
1200
1 1 1
0.5878
[
[:
2”5
1
0.25 0 0.4
ho = 1212.8
tq = 17.9.
1212. on
.
6. 7. 8. 9.
$104 by 2,
1, 4,
3, - H -
-1-H-
H=(X X) X y
11 ,1 3 1 4 11
107.5 -0.7’5 = 104.
105
1
596
Answers to Problems
CHAPTER 9
1. 2. /
3.
+ 65
2
270,464
+4
+ 193 LMS
4.
+1
+
4,832
64 1
3 65 MACs
1
2
195
LMS
+1
9220
257
64 = 64.
5.
6. by
g;(k)=RA, g i ( k ) = 0,
by
Rk=E{y,, y n - k ) , g,+(k)
k = 0, 1,2,..., p - 1. g f r ( k )= g;(p
- k)”.
j’,,
k = 1,2, . . . , p ,
597
Answers to Problems
g;+,(P
K;+, =
+
=0
g;+l(0) = 0.
g;(P+ g; (P)
g t ( k )= R k , k ...,M - 1
,...,M
p 5k 5 M p+ 15 k 5 M
g;+,(k) g;+,(k)
7.
2.0.
2 1 0
0
6
2
7 7 8
7 7
12
8
1
-1
0 -1
2
2 6
0 2
0
0
2
217 217 218
0
0
216 217 21 7 0
.-3
2.0,
8.
2
2
0
0
1 -0.5 -1
1 0 -1
0
2 1 0 -1
1.75
-0.5 -0.25
2
0.25
0 217 217 218
0
2.4
5
p,,,
p = 5.0
2 1 -10 -1 -38
0 0 -10 +40
2 1
0 -1 2
+40 p
0 2 1 0 -1
p = 0.5
0 217 217 218 0
= 0.4
Answers to Problems
598
9.
10.
CHAPTER 10
0.10. 0.0739
1. 0.3475
P, 2.
a
110 34.20
do
= 3.3
3. 4. A
1.2 0.01 = 1 / 100
c(
=
-
0.99
100
100.
5.
AR
MA by 6.
bo
do.
Answers to Problems
599
7.
-
y,,J
8. 9.
by
Xi(w){Ei(w))*
{Xi(w))*Ei(w),
10. on
no
CHAPTER 11
1.
SNR
30.1
on no
3600
2.
52,560,000
T z [-
SNR
24 no
8192
30.417 1 Pfu
z 6.042 6.042 6.042 x 10
3.
~
SNR
Answers to Problems
600
4.
~ e - ' ~ i / ,=x - fe-".
5. 1/ M .
by by 6.
by T 1% 0.1%
7.
2.597
T
3.346. by
1
0.776,
by
10.
8.
by
by
CHAPTER 12
1.
100 100
04/50 2. ( 12.2.1
0.03 105
10 ( 10 0.1779 0.3i.
3.
i/i/> 2.
f
< 1909.859 Hz
8 rz = 0, 1,2, . . . , 7
Answers to Problems
601
15, 3
14,
on, 16
1
1 9
8
16
by
2
on
on 0.1493 1.5 0.6254
1
35.83
16 15
up
1.4, 1
fmax
k 2
1909.859 828
1500 345 >
8.66 120
5 28.79 - 20 - 10
-6
R R
on X = 103 = 1000 CHAPTER 13
1.
Plane Wave @ 60 deg 2
I
)i
3
3
2,
2
3, 1
d
1
3
0
60
60 60 3
Answers to Problems
602
3
s = [ 1e-jkd
Ue-j2kd
60 X = [X~(~)X~(CU)X~(W)]
3
S = [e+j2kdcosn
0 T
e+jkdcoso
on
XS. up
k = m/lO,OOO 5 kHz,
d=1
2.
1
on 9
< 1 , 0 >, >,
.c
5 on
< 0, 1 >,
0
If 1
16,
3
by 2
on.
15,
0 =0
0
by 16
0
0, S(c0,) = [e
16)
01 -jk[cos( n/ 16)
e
...
e-jk[cos([n-I]n/16)cosn+sin([n-l]n/16)sin 01 e-jk[cos(
-n <
<
...
16)cos
+n,
0 k 5 (o/c k >u/c
upon
k = co/c.
3.
4.
16)
LMS LMS
3
01 T
1
0 - 0’. CO
Answers to Problems
603
by
5.
d=
6.
<< d 00
=
-
-7)
A
=
+
good - . . .).
fad d
7.
-1 on you 001 1101,
0
you 11 110100, 0.
MLS 100,
MLS
1
0,
110 1, 011 001
0,
1, 101
11 1
1, 010
1.
001 1101, -
+ 1.
1+1+1+1
7 Oth
2N - 1
on
8.
50 40 Hz
9.
10.
2R
604
Answers to Problems
CHAPTER 14
1.
2x2
by
1
2.5
5.5
0.00
0.4
0= O.4
+
O,4(5.5 -
0,4
= -57.56''
=1
2
+
a:, =
0 5 ~=
0,d
0.4
0.4
(1.4 -
2.
+ (5.5) + (5.5)
f3,4 =
7.5341
= 0.4659
B 2.24
/
\
I
\ I I
\ \
/
\
/
\ %
-
'
1
I
1
1
1
1
1
1
2
3
4
5
6
7
1.31,
Answers to Problems
605
3.
ii'k
S,
S
it'k
= 1.8735
3.3634. = 0.0097296.
0.0044664.
no
by
4. Task 5. 6. 25
70.7 1.767 v.
7.
200 17
8.
234 217 Hz no
14.2.16.
up
9. 20 1
20
5%.
20
606
Answers to Problems
aN)= [ 20
400 20 1
1 2 0
0
2 5 0
o
400 20 20
825 40 40
28.7 CTF/CTR=
I.
28.7 1.836
28.7
52.7
20 432.7
10.
A
3
1
50%
20
607
Answers to Problems
0.026
0.024
1
I
I
I
I
-
0.022-
0.02
-
0.016
-
0.01
0
I
1
10
20
I
30
1
I
So
50
0
Min
30 20 on
CHAPTER 15
1. 2.
15.1.10,
287 8.517
po =
1
- 1.2
0
on
120
10.95 d 10 AWG
3.15 24 AWG
83 65
1
'
Answers to Problems
608
60
24.6
100 0.3 15/ 24600,
12.8 x
20
1 1
3.
“B”
4.
“Bl”
no
on
on BI
by by
5.2.1
0.001 g
5.
56.56
100 pV. 387 565 pV 10
0.006 20
SNR
J =jcoC,d.
6. CA
M.4 =
7.
,/
V,d/(pc’),
x
VA
-
= 5.22 n z V / P a
“SE” 6.2
0.9,
13 12.7
8.
-
SE
-
Answers to Problems
609
9.
S
“S”
pp. 2
IEEE Tram Speech and Audio Proc., 10.
200
3.77 )
3 3
no
6
3 0.1
1
200 1
1.257 2
E
2
X 0.1
1
+
This page intentionally left blank
Index
215, 241 460 4 13 22 1 223
536 169 160 385, 387 535 158, 162, 163 347 158 397 536 532 535
237 522, 566 241 50, 241, 246, 252 218, 265 160 422 283 a-P-7
2, 26 62, 291
576 591 175 11, 158
485 5 8, 16 5 188, 192, 195
407 21 57 1,576 241
159, 184 296 312
582
2 17 252, 257 566 565
140 220 253 61 1
612
Index
460 148
127 454 47, 293
:
430
147 47 350 203 250
1
266, 421 25 1 218, 224 455 210 186, 324, 373 361, 368 1 1, 190 bound, 362 373 189, 368 167 177 178 28, 32
pu
, 49 7
55 1 136 328. 343 345 525 24 1 24, 29 564 457 140 bound, 362 479 30, 329 1 1. 147
141
2 72 333
130
78
559 553 21, 50
47, 293 32
63, 131 132, 135, 139 436 579 139 558 558 549 54 1 550 536
44 11 13 44, 47 1 113 186
32 72, 196 72 74
555 76 77 78
55 1 555 540 553
75 296 80 14
135, 333 266, 42 1 1 1 1 , 3 14
80
613
Index
189 383
165 462 95
20 141
433 251 434
11, 158 54 1
268
/
501
277 419 256
372 243, 248
190, 381
414 422 425, 427 429 417
142 350. 386 477 138
430 534, 575 533 54 1 299 145, 346 248
222 252 223 223 222, 429430 222 13, 44, 105. 490
bound, 362
549
237
344
132, 367 128, 132 133, 367 389 334
196 19, 186 325, 347, 409 335, 409, 410 295, 298, 582 45 1
339 164, 325
158
101
164
365 193
165
25 1
296
25 1 378
419
497
62, 64, 65
488
335
331 51 1
282 278
333 334
501
284 217
239, 280 338,
34
278, 280 283
614
Index
422 284 278, 280 282
485 132 164
279, 281 68, 277
457
139
400 45, 349, 373 418 425
21 252 261 26 1
333, 337 485
307 266
164, 325, 529
262 268
522 522
304 260 260
525 531 527 524 523
309 263 254
156
252 241 243 on 31 1
566 145, 166 427 580 9, 10
139
259 224, 259
254 94 259 260 458
266 88, 94 418
19 1 16-1 18
534 365
266, 421 25 1 30 163 544
544 1 1-1 13, 324,
328, 333
159 139 158, 384 186 14
455 536, 553
527 144
555
127 129 62
615
Index
5
131 131, 135 327 506
8, 7 13 6,16 39 13
222 240 259 378
223
250 541 357
397 347 398 145 403 158
23 524 1 1 1 , 192 427 496
483 478 495 522, 570 528
206
134
332 134 112 132 132
94 134 by 139
131 330 333 132 331 132 334
251
378, 391 127 100 159 5 16 19, 49
175 128 327 350
559 237
455 241
238
4 13
427 137
469
254 242
351, 400
523 253 62, 68, 277
391 372 378 94, 96
545 541 283 328, 335
148, 156 152, 157 145
490 580
616
Index
524, 527 141
on LMS
243-248
145-147
323, 328
50, 241, 246, 252, 563 485 LMS 241 242, 563 101, 106 19 1
1 15 132
21 20
108
43 1, 446
109 108
132
106
171-173 219, 238-243 176
107 108
108
253 323, 349 323
62
158
44, 47, 293-294 186
159
11
19, 2
