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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
Chapter 7. Random Process - Spectral Characteristics2
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
Chapter 7. Random Process - Spectral Characteristics3
7.1 Power density spectrum and its properties
( ) ( ) ( )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
Chapter 7. Random Process - Spectral Characteristics4
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics5
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics6
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics7
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics8
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics9
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics10
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics11
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics12
7.1 Power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics13
7.1 Power density spectrum and its properties
2
2rms
( )100
( )
XX
XX
S dW
S d
rms BW
2 2
10( )
[1 ( /10) ]XXS
rms 10 rad/secW
Chapter 7. Random Process - Spectral Characteristics14
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
Chapter 7. Random Process - Spectral Characteristics15
7.2 Relationship between power spectrum and autocorrelation function
( ) [ ( , )] 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
Chapter 7. Random Process - Spectral Characteristics16
7.2 Relationship between power spectrum and autocorrelation function
( ) ( ) 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
Chapter 7. Random Process - Spectral Characteristics17
7.2 Relationship between power spectrum and autocorrelation function
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
Chapter 7. Random Process - Spectral Characteristics18
7.2 Relationship between power spectrum and autocorrelation function
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
Chapter 7. Random Process - Spectral Characteristics19
7.2 Relationship between power spectrum and autocorrelation function
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
Chapter 7. Random Process - Spectral Characteristics20
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
Chapter 7. Random Process - Spectral Characteristics21
7.3 Cross-power density spectrum and its properties
( ),( )
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
Chapter 7. Random Process - Spectral Characteristics22
7.3 Cross-power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics23
7.3 Cross-power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics24
7.3 Cross-power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics25
7.3 Cross-power density spectrum and its properties
(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
Chapter 7. Random Process - Spectral Characteristics26
7.3 Cross-power density spectrum and its properties
* ( ) 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
Chapter 7. Random Process - Spectral Characteristics27
7.3 Cross-power density spectrum and its properties
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
)
Chapter 7. Random Process - Spectral Characteristics28
7.3 Cross-power density spectrum and its properties
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
Chapter 7. Random Process - Spectral Characteristics29
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
Chapter 7. Random Process - Spectral Characteristics30
7.4 Relationship between cross-power spectrum and cross-correlation function
( ) [ ( , )] 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
Chapter 7. Random Process - Spectral Characteristics31
7.4 Relationship between cross-power spectrum and cross-correlation function
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
Chapter 7. Random Process - Spectral Characteristics32
7.6 Some noise definitions and other topics
white noise & colored noise ( )N t
colored noise
1( ) unrealizable
2 NNS d
white noise
Chapter 7. Random Process - Spectral Characteristics33
7.6 Some noise definitions and other topics
sin( )( )NN
WR P
W
lowpass type
bandpass type
0
sin( / 2)( ) cos( )
( / 2)NN
WR P
W
band-limited white noise
Chapter 7. Random Process - Spectral Characteristics34
7.6 Some noise definitions and other topics
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
Chapter 7. Random Process - Spectral Characteristics35
7.6 Some noise definitions and other topics
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
Chapter 7. Random Process - Spectral Characteristics36
7.6 Some noise definitions and other topics
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
Chapter 7. Random Process - Spectral Characteristics37
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
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