Signal Processing for Intelligent Sensor Systems

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 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

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
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’).


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

Pu.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,

+ > 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, 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
> 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
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

py y

k 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