chapter 7. random process –spectral characteristics 7-ho.pdf · 3. cross-power density spectrum...

1

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

1

7.1 Power density spectrum and its properties

1( ) ( )

2

j tx t X e d

ω

ω ω

π

∞

−∞

= ∫Inverse FT

Fourier transform ( ) ( )j t

X x t e dtω

ω

∞−

−∞

= ∫

Goal: Find the frequency component of RP X(t).

For deterministic signal x(t) it is simple

You can also get the time domain representation of the signal using

This is not the case with RP X(t). Why?

How to find the frequency spectrum of RP X(t)?

2

2


22 2 1( ) ( ) ( ) ( )

2

T

T TT

E T x t dt x t dt X dω ω

π

∞ ∞

−∞ − −∞

= = =∫ ∫ ∫

( ) ( ) ( )T

j t j t

T TT

X x t e dt x t e dtω ω

ω

∞− −

−∞ −

= =∫ ∫

( ),( )

0, /T

x t T t Tx t

o w

− < <=

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

Assume ( ) , for all finite .T

TT

x t dt T−

< ∞∫

Energy saving property

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

2

2( )1 1

( ) ( )2 2 2

TT

T

XP T x t dt d

T T

ω

ω

π

∞

− −∞

= =∫ ∫Power=Energy/time

3

For a RP X(t), we find the average power using the same equation as

but with two modifications:

1. We take the expectation of X(t) to include all possible sample values.

2. We let T� infinity to cover the whole time domain

Thus we can write the average power of RP X(t) as


2{ [ ( ) ]}

XXP A E X t=

power density spectrum

2

2[ ( ) ]1 1

[ ( ) ]2 2 2

lim limT

T

XXT

T T

E XP E X t dt d

T T

ω

ω

π

∞

− −∞→∞ →∞

= =∫ ∫1

( )2

XX XXP S dω ω

π

∞

−∞

= ∫2

[ ( ) ]

2lim

T

XX

T

E XS

T

ω

→∞

=

Power Density Spectrum

Power Spectral Density

PSD4

3


2 2

2 2 2 0 0

0 0 0

2 2 2 2

0 0 0 02 20 0 00

2 2

0 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

π π

θ

ω ω

ω θ θ ω θπ π

ωπ

=

= +Θ = + + Θ

= + + = + +

= −

∫

2 2 2

2 0 0 0

0

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

2 2 2lim

T

XXT

T

A A AP A E X t t dt

Tω

π−→∞

= = − =∫

2{ [ ( ) ]}

XXP A E X t=

w.s.s. (0)XX XX

P R⇒ =

Example 7.1-1:0 0

( ) cos( )X t A tω= +Θ -- uniformly distributed on (0, )2

π

Θ

5


power density spectrum

Example 7.1-2: 0 0( ) cos( )X t A tω= +Θ

0 0

0 0

0 0 0

( ) ( )0 0

0 00 0

0 0

1( ) cos( ) [ ]

2

2 2

sin[( ) ] sin[( ) ]

( ) ( )

T Tj t j tj t j j j t

TT 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

ω ωω ω

ω ω ω ω

ω ω

ω ω ω ω

ω ω ω ω

−− Θ − Θ −

− −

− − +Θ − Θ

− −

Θ − Θ

= +Θ = +

= +

− += +

− +

∫ ∫

∫ ∫

1 sin( )2

j T j TT T

j t j t

t TT

e e Te dt e T

j j T

β ββ β β

β β β

−

=−−

−

= = =∫2

[ ( ) ]

2lim

T

XX

T

E XS

T

ω

→∞

=

6

4


2 22 * 2 2 20 0

0 2 2 2 2

0 0

2 2 2 2 0 0

0

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

ω ω ω ω

ω ω ω

ω ω ω ω

ω ω ω ω

ω ω ω ω

Θ − Θ

− += = +

− +

− ++ +

− +

0 0

0 0

0 0

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

( ) ( )

j j

T

T TX A Te A Te

T T

ω ω ω ω

ω

ω ω ω ω

Θ − Θ− +

= +

− +

2 2 / 22

00

2 2[ ] [2cos2 ] 2cos2 sin 2 0j j

E e e E d

π

π

θ θ θπ π

Θ − Θ+ = Θ = = =∫

22 2 2

0 0 0

2 2 2 2

0 0

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

2 2 ( ) ( )

TE X A T TT T

T T T

ω π ω ω ω ω

π ω ω π ω ω

− +

= +

− +

* 0 0

0 0

0 0

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

( ) ( )

j j

T

T TX A Te A Te

T T

ω ω ω ω

ω

ω ω ω ω

− Θ Θ− +

= +

− +

7


22

0

0 0

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

2 2

T

XXT

E X AS

T

ω πω δ ω ω δ ω ω

→∞

= = − + +

2

2

sin (C-54)

xdx

xπ

∞

−∞

=∫

2

2

, if 0sin ( )lim (b)

0, if 0( )T

T T

T

αα

απ α→∞

∞ ==

≠2

2

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

( )T

T T

T

αδ α

π α→∞

⇒ =

2 2

2 2

sin ( ) sin 11 (a)

( )

T T T xd dx

T x T

α

α

π α π

∞ ∞

−∞ −∞

= =∫ ∫

2 2

0 0

0 0

1 1( ) [ ( ) ( )]

2 2 2 2XX XX

A AP S d d

πω ω δ ω ω δ ω ω ω

π π

∞ ∞

−∞ −∞

= = − + + =∫ ∫

8

5


Properties of the power density spectrum:

(1) ( ) 0XX

S ω ≥

(2) ( ) real ( ) ( )XX XX

X t S Sω ω⇒ − =

(3) ( ) is realXX

S ω

21(4) ( ) { [ ( ) ]}

2XX

S d A E X tω ω

π

∞

−∞

=∫

( ) ( )T

j t

TT

X X t e dtω

ω−

−

= ∫

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

2 2

T T T T

XX XXT T

E X X E X XS S

T T

ω ω ω ω

ω ω→∞ →∞

− −

− = = =

PF of (2):

* *( ) ( ) ( ) ( )

T Tj t j t

T TT T

X X t e dt X t e dt Xω ω

ω ω− −

= = = −∫ ∫

2

[ ( ) ]( ) lim

2

T

XXT

E XS

T

ω

ω→∞

=

9


Properties of the power density spectrum

2(5) ( ) ( )

XXXXS Sω ω ω=ɺ ɺ

22 2

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

( ) lim lim lim ( )2 2 2

T T T

XXXXT T T

E X E j X E XS S

T T T

ω ω ω ω

ω ω ω ω→∞ →∞ →∞

= = = =ɺ ɺ

ɺ

PF of (5):

0

( ) ( )( ) lim

d X t X tX t

dt ε

ε

ε→

+ −=

0

( ) ( )lim ,

( )

0, o/wT

X t X tT t T

X t ε

ε

ε→

+ −− < <

=

ɺ

FT

0

( ) ( )( ) lim = ( )

j

T TT T

X e XX t j X

ωε

ε

ω ω

ω ω

ε→

−←→

ɺ

( ) ( )FT j af t a F e ω

ω−

− ←→

10

6


Bandwidth of the power density spectrum

( ) real ( ) evenXX

X t S ω⇒

( ) lowpass form XX

S ω ⇒

2

2

rms

( )

( )

XX

XX

S d

W

S d

ω ω ω

ω ω

∞

−∞

∞

−∞

=

∫

∫

2

02 0

rms

0

4 ( ) ( )

( )

XX

XX

S d

W

S d

ω ω ω ω

ω ω

∞

∞

−

=

∫

∫

root mean square Bandwidth

mean frequency

rms BW

( ) bandpass form XX

S ω ⇒0

0

0

( )

( )

XX

XX

S d

S d

ω ω ω

ω

ω ω

∞

∞=

∫

∫

11


Example 7.1-3: ( ) lowpass formXX

S ω2 2

10( )

[1 ( /10) ]XX

S ω

ω

=

+

/ 22

2 2 2 2/ 2

/ 2 / 2 / 22

2/ 2 / 2 / 2

10 10( ) 10sec

[1 ( /10) ] [1 tan ]

100 1 cos2100cos 100 50

sec 2

XXS d d d

d d d

π

π

π π π

π π π

ω ω ω θ θω θ

θθ θ θ θ π

θ

∞ ∞

−∞ −∞ −

− − −

= =

+ +

+= = = =

∫ ∫ ∫

∫ ∫ ∫

2

2

rms

( )100

( )

XX

XX

S d

W

S d

ω ω ω

ω ω

∞

−∞

∞

−∞

= =

∫

∫rms

10 rad/secW =rms BW

12

7

7.2 Relationship between power spectrum and autocorrelation function

(6)

1( ) [ ( , )]

2

( ) [ ( , )]

j

XX XX

j

XX XX

S e d A R t t

S A R t t e d

ωτ

ωτ

ω ω τ

π

ω τ τ

∞

−∞

∞−

−∞

= +

= +

∫

∫1 2

1 2

1 2

*

1 1 2 2

( )

1 2 2 1

( )

1 2 2 1

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

2 2

1lim [ ( ) ( )]

2

1lim ( , )

2

T Tj t j tT T

XXT TT T

T Tj t t

T TT

T Tj t t

XXT 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

ω ω

ω

ω

ω ω

ω−

− −→∞ →∞

−

− −→∞

−

− −→∞

= =

=

=

∫ ∫

∫ ∫

∫ ∫

1 2

1 2

( )

1 2 2 1

( )

1 2 2 1

1 1 1( ) lim ( , )

2 2 2

1 1lim ( , )

2 2

T Tj t tj j

XX XXT TT

T Tj t t

XXT TT

S e d R t t e dt dt e dT

R t t e d dt dtT

ωωτ ωτ

ω τ

ω ω ω

π π

ω

π

∞ ∞−

−∞ −∞ − −→∞

∞+ −

− − −∞→∞

=

=

∫ ∫ ∫ ∫

∫ ∫ ∫13


( ) [ ( , )]j

XX XXS A R t t e d

ωτ

ω τ τ

∞−

−∞

= +∫

1 2 1 2 2 1

1 1 1

1 1( ) lim ( , ) ( )

2 2

1 1lim ( , ) lim ( , )

2 2

[ ( , )]

T Tj

XX 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

ωτ

ω ω δ τπ

τ τ

τ

∞

−∞ − −→∞

− −→∞ →∞

= + −

= + = +

= +

∫ ∫ ∫

∫ ∫

( ) 1FT

tδ ←→

( ) 1

1( )

2

j t

j t

t e dt

t e d

ω

ω

δ

δ ωπ

∞−

−∞

∞

−∞

=

=

∫

∫

[ ( , )] ( )FT

XX XXA R t t Sτ ω+ ←→

14

8


( ) ( )j

XX XXS R e dωτ

ω τ τ

∞−

−∞

= ∫

1( ) ( )

2

j

XX XXR S e dωτ

τ ω ω

π

∞

−∞

= ∫

( ) ( )FT

XX XXR Sτ ω←→

( ) w.s.s. [ ( , )] ( )XX XX

X t A R t t Rτ τ⇒ + =

15


2

0

0 0( ) [ ( ) ( )]

2XX

AS

πω δ ω ω δ ω ω= − + +

1 2 ( )FT

πδ ω←→

Example 7.2-1:0

( ) cos( )X t A tω= +Θ

2

0

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

2XX

AR τ ω τ=

( ) ( )FTj t

x t e Xα

ω α←→ −

0 0

2

0( ) ( )4

j j

XX

AR e e

ω τ ω τ

τ−

= +

16

9


0 00 0

( ) (1 )( ) 2 (1 )cos( )T T

j j

XXS A e e d A d

T T

ωτ ωττ τ

ω τ ωτ τ−

= − + = −∫ ∫

0

0 00

( ) ( ) (1 ) (1 )T

j j j

XX XXT

S R e d A e d A e dT T

ωτ ωτ ωττ τ

ω τ τ τ τ

∞− − −

−∞ −

= = + + −∫ ∫ ∫

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

0[1 ] ,

( )

0 , elsewhere

XX

A T TR T

τ

τ

τ

− − < <

=

1cos( ) sin( )u uωτ ωτ

ω

′ = ⇒ =1

1 v vT T

τ −′= − ⇒ =

0 0

0

(1 ) (1 ) ( ) (1 )T

j j j

T Te d e d e d

T T T

ωτ ωα ωττ α τ

τ α τ−

−

+ = − − = −∫ ∫ ∫

17


0

0

0

0

0 0

2 2

0

2 2

0 02 2

2

0

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

τ ωτ

ω

ω

ωτ

τ

ω

ωτ

ω

ω ω

ω ω

ω ω

ω

= −

+

−= = −

= =

=

∫

18

10

7.6 Some noise definitions and other topics

white noise & colored noise ( )N t

colored noise

1( ) unrealizable

2NN

S dω ω

π

∞

−∞

= ∞ ⇒∫

white noise

19


sin( )( )

NN

WR P

W

τ

τ

τ

=

lowpass type

bandpass type

0

sin( / 2)( ) cos( )

( / 2)NN

WR P

W

τ

τ ω τ

τ

=

band-limited white noise

20

11


3

( )NN

R Peτ

τ−

=

3

0(3 ) (3 )

0

0

(3 ) (3 )

0

2

( )

1 1

3 3

1 1 6[ ]3 3 9

j

NN

j j

j j

S Pe e d

Pe d Pe d

P e P ej j

PP

j j

τ ωτ

ω τ ω τ

ω τ ω τ

ω τ

τ τ

ω ω

ω ω ω

∞− −

−∞

∞− − +

−∞

∞

− − +

−∞

=

= +

−= +

− +

= + =

− + +

∫

∫ ∫

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

21


0 0( ) ( ) cos( )Y t X t A tω=

2

0 0 0 0

2

0

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

τ τ τ ω ω ω τ

τ ω τ ω ω τ

+ = + = + +

= + + +

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

2

0

0 0( ) [ ( ) ( )]

4YY XX XX

AS S Sω ω ω ω ω= − + +

2

0

0[ ( , )] ( )cos( )

2YY XX

AA R t t Rτ τ ω τ+ =

22

12


2

0 0 0

2

0 0

2

0 0 0

/ 8, 2 / 2

/ 4, / 2( )

/8, 2 / 2

0, o/w

RF

RF

YY

RF

N A W

N A WS

N A W

ω ω

ω

ω

ω ω

+ <

<=

− <

2

0 0/ 4, / 2

( )0, o/w

RF

YY

N A WS

ω

ω

<=

0 0( ) ( ) cos( )Y t X t A tω=

Example 7.6-2: 0 0

0 0

/ 2, / 2

( ) / 2, / 2

0, o/w

RF

XX RF

N W

S N W

ω ω

ω ω ω

+ <

= − <

After lowpass filtering,

2 2/ 2

0 0 0 0

/ 2

1output noise power =

2 4 8

RF

RF

WRF

W

N A N A Wdω

π π−

=∫23

chapter 7. random process –spectral characteristics 7-ho.pdf · 3. cross-power density spectrum...

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