Chapter 7. Random Process - Spectral Characteristics 1 Chapter 7. Random Process – Spectral Characteristics 0. Introduction 1. Power density spectrum and.

Chapter 7. Random Process - Spectral Characteristics1

Chapter 7. Random Process – Spectral

Characteristics

0. Introduction

1. Power density spectrum and its properties

2. Relationship between power spectrum and autocorrelation function

3. Cross-power density spectrum and its properties

4. Relationship between cross-power spectrum and cross-correlation function

5. Power spectrums for discrete-time processes and sequences

6. Some noise definitions and other topics

7. Power spectrums of complex processes


7.1 Power density spectrum and its properties

1( ) ( )

2j tx t X e d

1( ) [ ( ) ]

2j j tx t x e d e d

Fourier integral

Inverse FT

Fourier transform ( ) ( ) j tX x t e dt



( ) ( ) ( )Tj t j t

T T TX x t e dt x t e dt

( ),( )

0, /T

x t T t Tx t

o w

22 2 1( ) ( ) ( ) ( )

2

T

T TTE T x t dt x t dt X d

Energy contained in ( ) in the interval ( , )x t T T

Assume ( ) , for all finite .T

TTx t dt T



average power in ( ) over the interval ( , )x t T T

( ) ( ), take expectation, let .x t X t T

2{ [ ( ) ]}XXP A E X t

power density spectrum

2

2 ( )1 1( ) ( )

2 2 2

T T

T

XP T x t dt d

T T

average power in r.p. ( )X t2

2 [ ( ) ]1 1[ ( ) ]

2 2 2lim limT T

XX TT T

E XP E X t dt d

T T

1

( )2XX XXP S d

2[ ( ) ]

2lim TXX

T

E XS

T



2 22 2 2 0 0

0 0 0

2 2 2 20 0 0 02 2

0 0 00

2 20 0

0

[ ( ) ] [ cos ( )] [ cos(2 2 )]2 2

2cos(2 2 ) sin(2 2 )

2 2 2 2

sin(2 )2

A AE X t E A t E t

A A A At d t

A At

2{ [ ( ) ]}XXP A E X t

w.s.s. (0)XX XXP R

Example 7.1-1:0 0( ) cos( )X t A t -- uniformly distributed on (0, )

2

2 2 22 0 0 0

0

1{ [ ( ) ]} [ sin(2 )]

2 2 2limT

XX TT

A A AP A E X t t dt

T



power density spectrum

Example 7.1-2: 0 0( ) cos( )X t A t

0 0

0 0

0 0 0

( ) ( )0 0

0 00 0

0 0

1( ) cos( ) [ ]

2

2 2sin[( ) ] sin[( ) ]

( ) ( )

T T j t j tj t j j j tT T T

T Tj t j tj j

T T

j j

X A t e dt A e e e e e dt

A Ae e dt e e dt

T TA Te A Te

T T

1 sin( )2

j T j TT Tj t j t

t TT

e e Te dt e T

j j T

2[ ( ) ]

2lim TXX

T

E XS

T



0 00 0

0 0

sin[( ) ] sin[( ) ]( )

( ) ( )j j

T

T TX A Te A Te

T T

2 22 * 2 2 20 0

0 2 2 2 20 0

2 2 2 2 0 00

0 0

sin [( ) ] sin [( ) ]( ) ( ) ( ) [ ]

( ) ( )

sin[( ) ] sin[( ) ]( )

( ) ( )

T T T

j j

T TX X X A T T

T T

T TA T e e

T T

2 2 / 2200

2 2[ ] [2cos 2 ] 2cos 2 sin 2 0j jE e e E d

2 2 2 20 0 0

2 2 2 20 0

[ ( ) ] sin [( ) ] sin [( ) ][ ]

2 2 ( ) ( )TE X A T TT T

T T T

* 0 00 0

0 0

sin[( ) ] sin[( ) ]( )

( ) ( )j j

T

T TX A Te A Te

T T



2 20

0 0

[ ( ) ]( ) lim [ ( ) ( )]

2 2T

XX T

E X AS

T

2

2

sin (C-54)

xdx

x

2

2

, if 0sin ( )lim (b)

0, if 0( )T

T T

T

2

2

sin ( )(a) & (b) lim ( )

( )T

T T

T

2 2

2 2

sin ( ) sin 11 (a)

( )

T T T xd dx

T x T

2 20 0

0 0

1 1( ) [ ( ) ( )]

2 2 2 2XX XX

A AP S d d



Properties of the power density spectrum:

(1) ( ) 0XXS

(2) ( ) real ( ) ( )XX XXX t S S

(3) ( ) is realXXS 21

(4) ( ) { [ ( ) ]}2 XXS d A E X t

( ) ( )T j t

T TX X t e dt

* *[ ( ) ( ) ] [ ( ) ( )]( ) lim lim ( )

2 2T T T T

XX XXT T

E X X E X XS S

T T

PF of (2):

* *( ) ( ) ( ) ( )T Tj t j t

T TT TX X t e dt X t e dt X

2[ ( ) ]

( ) lim2T

XX T

E XS

T



Properties of the power density spectrum2(5) ( ) ( )XXXX

S S

2 2 2

2 2[ ( ) ] [ ( ) ] [ ( ) ]

( ) lim lim lim ( )2 2 2T T T

XXXX T T T

E X E j X E XS S

T T T

PF of (5):

0

( ) ( )( ) lim

d X t X tX t

dt

0

( ) ( )lim ,

( )0, o/w

T

X t X tT t T

X t

FT

0

( ) ( )( ) lim = ( )

jT T

T T

X e XX t j X

( ) ( )FT j af t a F e



Bandwidth of the power density spectrum

( ) real ( ) evenXXX t S

( ) lowpass form XXS 2

2rms

( )

( )

XX

XX

S dW

S d

202 0

rms

0

4 ( ) ( )

( )

XX

XX

S dW

S d

root mean square Bandwidth

mean frequency

rms BW

( ) bandpass form XXS 00

0

( )

( )

XX

XX

S d

S d



Example 7.1-3: ( ) lowpass formXXS 2 2

10( )

[1 ( /10) ]XXS

/ 2 22 2 2 2/ 2

/ 2 / 2 / 222/ 2 / 2 / 2

10 10( ) 10sec

[1 ( /10) ] [1 tan ]

100 1 cos 2100cos 100 50

sec 2

XXS d d d

d d d

210 tan 10secd d

2 3 2/ 22 2

2 2 2 2/ 2

4 2/ 2 / 2 / 24 2 4

2/ 2 / 2 / 2

10 10 tan( ) 10sec

[1 ( /10) ] [1 tan ]

10 tan 1 cos 210 sin 10 5000

sec 2

XXS d d d

d d d



2

2rms

( )100

( )

XX

XX

S dW

S d

rms BW

2 2

10( )

[1 ( /10) ]XXS

rms 10 rad/secW


7.2 Relationship between power spectrum and autocorrelation function

(6)

1( ) [ ( , )]

2

( ) [ ( , )]

jXX XX

jXX XX

S e d A R t t

S A R t t e d

1 2

1 2

1 2

*

1 1 2 2

( )1 2 2 1

( )1 2 2 1

[ ( ) ( )] 1( ) lim lim [ ( ) ( ) ]

2 21

lim [ ( ) ( )]21

lim ( , )2

T Tj t j tT TXX T TT T

T T j t t

T TT

T T j t tXXT TT

E X XS E X t e dt X t e dt

T T

E X t X t e dt dtT

R t t e dt dtT

1 2

1 2

( )1 2 2 1

( )1 2 2 1

1 1 1( ) lim ( , )

2 2 21 1

lim ( , )2 2

T T j t tj jXX XXT TT

T T j t tXXT TT

S e d R t t e dt dt e dT

R t t e d dt dtT



( ) [ ( , )] j

XX XXS A R t t e d

1 2 1 2 2 1

1 1 1

1 1( ) lim ( , ) ( )

2 21 1

lim ( , ) lim ( , )2 2

[ ( , )]

T TjXX XXT TT

T T

XX XXT TT T

XX

S e d R t t t t dt dtT

R t t dt R t t dtT T

A R t t

( ) 1FTt ( ) 1

1( )

2

j t

j t

t e dt

t e d

[ ( , )] ( )FTXX XXA R t t S



( ) ( ) jXX XXS R e d

1( ) ( )

2j

XX XXR S e d

( ) ( )FTXX XXR S

( ) w.s.s. [ ( , )] ( )XX XXX t A R t t R



20

0 0( ) [ ( ) ( )]2XX

AS

1 2 ( )FT

Example 7.2-1: 0( ) cos( )X t A t 20

0by Ex 6.2-1, ( ) cos( )2XX

AR

( ) ( )FTj tx t e X

0 0

20( ) ( )

4j j

XX

AR e e



0

0 0 0( ) ( ) (1 ) (1 )

Tj j jXX XX TS R e d A e d A e d

T T

Example 7.2-2: ( ) -- w.s.s.X t

0[1 ] ,( )

0 , elsewhereXX

A T TR T

0 00 0( ) (1 )( ) 2 (1 )cos( )

T Tj jXXS A e e d A d

T T

1cos( ) sin( )u u

1

1 v vT T

0 0

0(1 ) (1 ) ( ) (1 )

Tj j j

T Te d e d e d

T T T



00

0

0

0 02 2

0

2 2

0 02 2

20

sin( )( ) 2 (1 )

2 sin( )

2 2cos( )[1 cos( )]

sin ( / 2) sin ( / 2)4

( / 2)

Sa ( / 2)

T

XX

T

T

S AT

Ad

T

A AT

T T

T TA A T

T T

A T T


7.3 Cross-power density spectrum and its properties

( ) ( ) ( )W t X t Y t

( , ) [ ( ) ( )] {[ ( ) ( )][ ( ) ( )]}

( , ) ( , ) ( , ) ( , )WW

XX YY XY YX

R t t E W t W t E X t Y t X t Y t

R t t R t t R t t R t t

( ) ( ) ( ) { [ ( , )]} { [ ( , )]}WW XX YY XY YXS S S F A R t t F A R t t



( ),( )

0, /T

x t T t Tx t

o w

Cross Power contained in ( ), ( ) in the interval ( , )x t y t T T

( ),( )

0, /T

y t T t Ty t

o w

FT( ) ( )T Tx t X FT( ) ( )T Ty t Y

Parseval's theorem

Assume ( ) & ( ) , for all finite .T T

T TT Tx t dt y t dt T

*( ) ( )1 1 1( ) ( ) ( ) ( ) ( )

2 2 2 2

TT T

XY T T T

X YP T x t y t dt x t y t dt d

T T T



average Cross Power contained in ( ), ( ) in the interval ( , )X t Y t T T

cross-power density spectrum

total average Cross Power contained in ( ), ( )X t Y t

*[ ( ) ( )]1 1lim ( , ) lim

2 2 2

TT T

XY XYTT T

E X YP R t t dt d

T T

*[ ( ) ( )]( ) lim

2T T

XY T

E X YS

T

*[ ( ) ( )]1 1( ) ( , )

2 2 2

TT T

XY XYT

E X YP T R t t dt d

T T



Total cross power = XY YXP P

( ), ( ) orthogonal 0XY YXX t Y t P P

1( )

2XY XYP S d

*[ ( ) ( )]( ) lim

2T T

YX T

E Y XS

T

*1( )

2YX YX XYP S d P



Properties of the cross-power density spectrum:

*(1) ( ) ( ) ( )XY YX YXS S S

( ) ( )T j t

T TX X t e dt

* *[ ( ) ( )] [ ( ) ( ) ]( ) lim lim ( )

2 2T T T T

YX XYT T

E Y X E Y XS S

T T

PF of (1):

( ), ( ) realX t Y t

* **[ ( ) ( )] [ ( ) ( ) ]

( ) lim lim ( )2 2

T T T TYX YXT T

E Y X E Y XS S

T T

* *( ) ( ) ( ) ( )T Tj t j t

T TT TX X t e dt X t e dt X



(2) Re[ ( )] & Re[ ( )] -- evenXY YXS S

(6) [ ( , )] ( )

[ ( , )] ( )

FTXY XY

FTYX YX

A R t t S

A R t t S

(4) ( ) & ( ) orthogonal ( ) ( ) 0XY YXX t Y t S S

(5) ( ) & ( ) uncorrelated & have constant mean ,

( ) ( ) 2 ( )XY YX

X t Y t X Y

S S XY

(3) Im[ ( )] & Im[ ( )] -- oddXY YXS S

( ) & ( ) orthogonal ( , ) 0 A[ ( , )] 0XY XYX t Y t R t t R t t



* ( ) 2 ( ) ( )XY YXS XY S

PF of (5): ( , ) [ ( , )]XY XYR t t XY A R t t XY

( ), ( ) -- jointly w.s.s. X t Y t FT( ) ( )XY XYR S

FT( ) ( )YX YXR S



Example 7.3-1: ( ), ( ) -- jointly w.s.s.X t Y t

/ ,( )

0, o/wXY

a jb W W WS

2

2

2

1 1 1( ) ( ) ( )

2 2 2

1 ( ) ( ) 1[ ]

2 2

1 1[ ( )]

2 2

sin( ) cos( ) sin(

WjW Wj j

XY W WW

WjW jW j

W

jW jW j jjW jW

b b e bR a j e d a j e d

W W j W

a jb e a jb e be

j j jW

a e e b b e ee e

j W j

a b bW W W

W

)



Example 7.3-2: ( ), ( ) -- jointly w.s.s.X t Y t

3

8( ) 0

( )XYS j

2( ) 4 ( )XYR u e

23

2( )

( )FTu e

j


7.4 Relationship between cross-power spectrum and cross-correlation function

(6)1

( ) [ ( , )]2

( ) [ ( , )]

jXY XY

jXY XY

S e d A R t t

S A R t t e d

1 2

1 2

1 2

*

1 1 2 2

( )1 2 2 1

( )1 2 2 1

[ ( ) ( )] 1( ) lim lim [ ( ) ( ) ]

2 21

lim [ ( ) ( )]21

lim ( , )2

T Tj t j tT TXY T TT T

T T j t t

T TT

T T j t tXYT TT

E X YS E X t e dt Y t e dt

T T

E X t Y t e dt dtT

R t t e dt dtT

1 2

1 2

( )1 2 2 1

( )1 2 2 1

1 1 1( ) lim ( , )

2 2 21 1

lim ( , )2 2

T T j t tj jXY XYT TT

T T j t tXYT TT

S e d R t t e dt dt e dT

R t t e d dt dtT



( ) [ ( , )] j

XY XYS A R t t e d

1 2 1 2 2 1

1 1 1

1 1( ) lim ( , ) ( )

2 21 1

lim ( , ) lim ( , )2 2

[ ( , )]

T TjXY XYT TT

T T

XY XYT TT T

XY

S e d R t t t t dt dtT

R t t dt R t t dtT T

A R t t

( ) 1FTt ( ) 1

1( )

2

j t

j t

t e dt

t e d

[ ( , )] ( )FTXY XYA R t t S



0 0 0 0( ) [2 ( ) 2 ( )] [ ( ) ( )]4 2XY

AB j ABS

j

0 0( , ) {sin( ) cos[ (2 )]}2XY

ABR t t t

Example 7.4-1:

0 0

0 0

0

1[ ( , )] lim ( , )

21

sin( ) lim cos[ (2 )]2 2 2

sin( ) [ ]2 4

T

XY XYTT

T

TT

j j

A R t t R t t dtT

AB ABt dt

TAB AB

e ej


7.6 Some noise definitions and other topics

white noise & colored noise ( )N t

colored noise

1( ) unrealizable

2 NNS d

white noise



sin( )( )NN

WR P

W

lowpass type

bandpass type

0

sin( / 2)( ) cos( )

( / 2)NN

WR P

W

band-limited white noise



3( )NNR Pe

3

0 (3 ) (3 )

0

0

(3 ) (3 )

0

2

( )

1 1

3 3

1 1 6[ ]3 3 9

jNN

j j

j j

S Pe e d

Pe d Pe d

P e P ej j

PP

j j

Example 7.6-1: w.s.s. ( )N t



0 0( ) ( ) cos( )Y t X t A t

20 0 0 0

20

0 0 0

( , ) [ ( ) ( )] [ ( ) ( )cos( )cos( )]

( , )[cos( ) cos(2 )]2

YY

XX

R t t E Y t Y t E A X t X t t t

AR t t t

( ) -- w.s.s. ( ) -- NOT w.s.s. X t Y t

20

0 0( ) [ ( ) ( )]4YY XX XX

AS S S

20

0[ ( , )] ( )cos( )2YY XX

AA R t t R



20 0 0

20 0

20 0 0

/8, 2 / 2

/ 4, / 2( )

/8, 2 / 2

0, o/w

RF

RFYY

RF

N A W

N A WS

N A W

0 0( ) ( ) cos( )Y t X t A t

Example 7.6-2: 0 0

0 0

/ 2, / 2

( ) / 2, / 2

0, o/w

RF

XX RF

N W

S N W

After lowpass filtering,2

0 0 / 4, / 2( )

0, o/wRF

YY

N A WS

2 2

/ 20 0 0 0

/ 2

1output noise power =

2 4 8

RF

RF

WRF

W

N A N A Wd


7.7 Power Spectrums of Complex Processes

n,Z jointly w.s.s. mZ

( ) ( ) jZZ ZZS R e d

1( ) ( )

2j

ZZ ZZR S e d

( ) ( )m n m n

jZ Z Z ZS R e d

1( ) ( )

2m n m n

jZ Z Z ZR S e d

Example 7.7-1: ( ) of Example 6.7-1V t

0 2

1

( ) [ ]N

jVV n

n

R e E A

( ) . . . Z t w s s

20

1

( ) 2 ( ) [ ]N

VV nn

S E A

Chapter 7. Random Process - Spectral Characteristics 1 Chapter 7. Random Process – Spectral Characteristics 0. Introduction 1. Power density spectrum and.

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