Random

Signals and S

ystems

Chapter 5

JitendraK

Tugnait

James B

Davis P

rofessor

Departm

ent of Electrical &

Com

puter Engineering

Auburn U

niversity

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Random

Processes

•R

andom V

ariable»

can take a number of values

»a probability is assigned to each value or range of values

•R

andom P

rocess»

random function (usually of tim

e or space)

»can be any one of a num

ber of functions

»a probability is assigned to each function or range of functions

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Exam

ple

•R

andom variable

»R

andom num

ber generator on a calculator–

Generates a random

number

•R

andom process

»A

“random process”

generator would generate a

random function

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Random

Processes

•T

he collection of all possible functions is called the ensem

ble

•A

particular mem

ber of the ensemble is called a

sample function

•A

n arbitrary sample function is denoted X

(t)

•1

Xt

1 , fixed is a random

variablet

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Random

Processes

050

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500-5 0 5

X(t)

050

100150

200250

300350

400450

500-5 0 5

X(t)

050

100150

200250

300350

400450

500-5 0 5

X(t)

050

100150

200250

300350

400450

500-5 0 5

X(t)

time (t) in seconds

Sample Functions

X(t1 ) is an random

variable

White N

oise

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Types of R

andom P

rocesses

•C

ontinuous/Discrete

»C

ontinuous: X(t1 ) is a continuous R

V

»D

iscrete: X(t1 ) is a discrete R

V

»M

ixed: Both continuous and discrete

–E

x: output of a half-wave rectifier

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Continuous/D

iscrete

Sam

ple FunctionP

DF

of X(t1 )

Continuous

Discrete

Mixed

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Types of R

andom P

rocesses

•D

eterministic / N

on-Determ

inistic»

Non-D

eterministic

–Future values cannot be determ

ined exactly from

observed past values

–E

xample: w

hite noise

»D

eterministic

–Future values can be determ

ined exactly from observed

past values

–E

xample

cosX

tA

t

Uniform

on

0,2

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Non-D

eterministic R

andom P

rocess

050

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X(t)

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100150

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500-5 0 5

X(t)

050

100150

200250

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400450

500-5 0 5

X(t)

050

100150

200250

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500-5 0 5

X(t)

time (t) in seconds

White N

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Determ

inistic Random

Process

050

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-1 0 1

X(t)

050

100150

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500

-1 0 1

X(t)

050

100150

200250

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500

-1 0 1

X(t)

050

100150

200250

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500

-1 0 1

X(t)

time (t) in seconds

X(t1 ) is an random

variable

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Exam

ple

()

e0

tX

tA

t

Observe the values X

(1)=1.21036, X

(2) = 0.73576

a) Determ

ine Aand β

b) Determ

ine X(3.2189)

Consider the random

process

(1)(1

2)

(2)

0.4978(1) 1.210361.21036

0.49780.73576

0.73576

1.21036=

1.9911

Ae

eA

e

Ae

A

0.4978(3.2189)

1.99110.4011

e

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Types of R

andom P

rocesses

•S

tationary/Non-stationary

»Stationary: PD

F’sdescribing X

(t1 ), X(t2 ), …

(both joint &

marginal) do not depend on the choice of tim

e

»N

on-stationary: PD

F’s

depend on the choice of time

»D

ifficult to prove that a random process is stationary

•W

ide-Sense S

tationary (WS

S)

»A

weaker form

of stationarity

»E

{X(t1 )} is not a function of tim

e

»E

{X(t1 ) X

(t2 )} is only a function of t1 -t2–

Note: this m

eans that the variance, E{X

2(t)}, must be constant

stationary

WS

S

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Stationary R

andom P

rocess

050

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500-5 0 5

X(t)

050

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500-5 0 5

X(t)

050

100150

200250

300350

400450

500-5 0 5

X(t)

050

100150

200250

300350

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500-5 0 5

X(t)

time (t) in seconds

White N

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Non-S

tationary Random

Process

050

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500

-20 0 20

X(t)

050

100150

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500

-20 0 20

X(t)

050

100150

200250

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400450

500

-20 0 20

X(t)

050

100150

200250

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500

-20 0 20

X(t)

time (t) in seconds

Random

Walk

Mean is constant, but variance increases as tim

e increases

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Stationarity

•S

tationary implies the follow

ing:

•A

lso, if any of the above three parameters vary

with tim

e, the RP

is non-stationary.

2

2

constant

constant

constantX

t

Xt

Xt

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Exam

ple

()

e0

tX

tA

t

Com

pute the mean and variance of X

(t). Assum

e Aand β

are independent

Consider the random

process

()

{}

{}

{}

()

()

t

t

tA

Xt

EA

e

EA

Ee

afa

dae

fd

22

22

22

()

{()

}

{}

{}

()

()

t

t

tA

Xt

EA

e

EA

Ee

af

ada

ef

d

22

Var{

()}

Xt

Xt

Xt

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Exam

ple

()

cos()

uniform on [0,2

], ,

are constantsX

tA

tA

Is X(t) w

ide sense stationary?

Consider the random

process

20

20

20

()

{cos(

)}

cos()

()

1cos(

)2

cos()

2

cos()

2

cos(2

)cos(

)20

Xt

EA

t

At

fd

At

d

At

d

At

At

t

Result is not a function of t, so X

(t) is potentially W

SS

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Exam

ple Continued

12

12

22

12

0

22

21

21

2

00

2

12

{(

)(

)}{

cos()

cos()}

1cos(

)cos()

2

1cos

coscos(

)cos(

)2

1cos

2cos

4

cos2

EX

tX

tE

At

At

At

td

AB

AB

AB

At

td

tt

d

At

t

If result only a function of t1 -t2 , then potentially WSS

BO

TH

conditions are true, therefore WSS

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Types of R

andom P

rocesses

•E

rgodic/Non-ergodic

»E

rgodic: Mom

ents [E{X

(t)}, E{X

2(t)}, etc.] can be com

puted from tim

e averages

»If ergodic, then stationary, but reverse is not necessarily true.

1lim

2

Tn

nn

TT

Xx

fx

dxX

tdt

T

ergodic

stationary

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Exam

ple

cos,

independent

random

uniform at

0,2

stationary but not ergodic

Xt

Aw

tA

AXt

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500-2 0 2

X(t)

050

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500-2 0 2

X(t)

050

100150

200250

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500-2 0 2

X(t)

050

100150

200250

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500-2 0 2

X(t)

time (t) in seconds

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Measuring the M

ean of an Ergodic

RP

•If a random

process is ergodic, the mean can be

computed from

a time average

•In practice, w

e usually work w

ith nsam

ples of a random

process: Xi =

X(iΔ

t) and estimate the

mean w

ith

»T

he variance of this estimate is

0

1ˆ

()

T

XX

tdt

T

1

1ˆ

n

ii

XX

n

2Xn

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Measuring the V

ariance of an Ergodic

RP

•S

imilarly, the variance of a sam

pled, ergodicrandom

process can be estimated by

22

12

ˆ1

n

ii

X

xn

x

n

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Classify T

his Random

Process

•A

random process w

here the random variable is

the number of cars per m

inute passing a traffic counter»

Continuous/D

iscrete/Mixed?

»D

eterministic/N

on-Determ

inistic?

»S

tationary/Non-S

tationary?

»E

rgodic/Non-E

rgodic?

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Classify T

his Random

Process

•T

hermal noise generated by a resistor (w

hite noise)

»C

ontinuous/Discrete/M

ixed?

»D

eterministic/N

on-Determ

inistic?

»S

tationary/Non-S

tationary?

»E

rgodic/Non-E

rgodic?

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Classify T

his Random

Process

•A

random process w

hen white noise is passed

through an ideal half-wave rectifier

»C

ontinuous/Discrete/M

ixed?

»D

eterministic/N

on-Determ

inistic?

»S

tationary/Non-S

tationary?

»E

rgodic/Non-E

rgodic?

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Other H

andy Form

ulas

22

22

Alw

ays true

Alw

ays true

0O

nly for stationary

Xt

X

Xt

EX

tXt

R

22

22

2

Alw

ays true

Alw

ays true

0O

nly for stationary

Xt

X

EX

tX

t

Xt

Xt

RX

t

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•R

ecall that X(t) is a random

variable

•N

ote this is are ensemble averages

»V

alid for any random process

Mean and V

ariance of RP

’s

-

22

-

12

Mean:

Mean-S

quare Value:

Autocorrelation:

X

X

EX

tx

tf

xt

dxt

EX

tx

tf

xt

dxt

EX

tX

t

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Exam

ple

22

12

12

12

0

0

EX

t

EX

t

EX

tX

tE

Xt

EX

tt

t

•W

hite noise»

X(t1 ) and X

(t2 ) are independent

»X

(t) with tfixed is a G

aussian random variable

–M

ean = 0

–V

ariance = σ

2

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Ensem

ble Average -

Mean

050

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500-2 0 2

X(t)

050

100150

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500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

Mean

time (t) in seconds

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Exam

ple

cos

where

is uniform in

0,1X

tA

tA

1

coscos

cos2

EX

tE

At

EA

tt

2

22

22

1cos

coscos

3E

Xt

EA

tE

At

t

12

12

12

1cos

coscos

cos3

EX

tX

tE

At

EA

tt

t

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Ensem

ble Average -

Mean

050

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500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

Mean

time (t) in seconds

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Mean and V

ariance of RP

’s

cos

uniform

on 0,2

Xt

t

20

1cos

2E

Xt

td

20

11

sinsin

2sin

02

2t

tt

2

12E

Xt

2

12

12

0

1cos

cos2

EX

tX

tt

td

21

1cos

2t

t

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Ensem

ble Average -

Mean

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

X(t)

050

100150

200250

300350

400450

500-2 0 2

Mean

time (t) in seconds

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Autocorrelation F

unctions

•R

andom variables

»C

ompletely described by a probability density

function (PD

F)

•R

andom process

»N

eed a PD

F for each tand joint P

DF

’s for all com

bination of t’s.

»T

his is difficult to obtain in engineering applications and is often not needed

»O

ften, it is sufficient to use the autocorrelation function

to describe the random process.

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Autocorrelation F

unctions

•C

onsider a random process X

(t) at two different

times t1

and t2 .»

X(t1 ) and X

(t2 ) are random variables

»R

X (t1 , t2 )=E

{X(t1 ) X

(t2 )} is called the autocorrelation function

(AC

F) of the random

process

•T

he autocorrelation function describes how

rapidly a random process can change

»F

or example, w

e would expect a high correlation

between the tim

e instants that are close together

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Exam

ple

050

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500-4 -2 0 2 4

time, t, in seconds

X(t)

-300-200

-1000

100200

300-0.5 0

0.5 1

lag time, =

t2 -t1 , in seconds

R()

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Exam

ple

050

100150

200250

300350

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500-1

-0.5 0

0.5 1

time, t, in seconds

X(t)

-300-200

-1000

100200

300-0.02 0

0.02

0.04

0.06

lag time, =

t2 -t1 , in seconds

R()

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Exam

ple

2

12

1 2

01

0elsew

here

uniform on -12 to 12

480

,1

0elsew

here

At

Xt

A

EA

tt

EX

tX

t

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AC

F of a W

SS

Function

•In general,

•If X

(t) is stationary, however, the A

CF

is only a function of τ=

t1 -t2 , and

12

1 2

1

21

21

2,

,X

Rt

tE

Xt

Xt

xx

fx

xd

xdx

X

RE

Xt

Xt

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Exam

ple

2

1(

)(

)w

hereexp

XZ

tX

tX

tR

22

22

11

22

1

11

1

11

1

11

()

()

()

()

{(

)(

)}

{[(

)(

)][(

)(

)]}

{(

)(

)}{

()

()}

{(

)(

)}{

()

()}

()

()

()

()

2

Z

XX

XX

RE

Zt

Zt

EX

tX

tX

tX

t

EX

tX

tE

Xt

Xt

EX

tX

tE

Xt

Xt

RR

RR

ee

ee

ee

2

1(

)e

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Properties of (S

tationary and Ergodic) A

CF’s

21)

0 m

ean-squared valueX

RX

2) X

XR

R

22

(0){

()

()}

{(

)}X

RE

Xt

Xt

EX

tX

()

{(

)(

)}{

()

()}

()

XX

RE

Xt

Xt

EX

tX

tR

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Properties of (S

tationary and Ergodic) A

CF’s

3) 0

XX

RR

2

22

22

{[(

)(

)]}

0

{(

)(

)2

()

()}

0

{(

)(

)}2

{(

)(

)}

2(0)

2|

{(

)(

)}|

(0)|

()|

X

XX

EX

tX

t

EX

tX

tX

tX

t

EX

tX

tE

Xt

Xt

RE

Xt

Xt

RR

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Properties of (S

tationary and Ergodic) A

CF’s

4) If has a non-zero m

ean (DC

component) then

will have a constant com

ponentX X

t

R

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Properties of A

CF

’s cont’d

5) If has a periodic com

ponent, then

does too with the sam

e period.X X

t

R

2

Ex:

cos

uniform on

0,2

cos2

X

Xt

At

AR

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Properties of A

CF

’s cont’d

0

6) If is ergodic and zero m

ean, and has no periodic components

lim0

X

XtR

•F

or ergodic random processes, the M

AG

NIT

UD

E of the m

ean can be determ

ined by ignoring any periodic components and

letting τ→∞

.

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Properties of A

CF

’s cont’d

7)

If

,

then

0 for all jX

X

X

FR

Re

d

FR

•A

mong other

things, this means

»N

o flat tops

»N

o vertical sides

»N

o amplitude

discontinuities

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Exam

ple

•C

an this function be an AC

F?

»N

O!

–Flat top

–V

ertical sides/discontinuity in amplitude

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Exam

ple

An ergodic R

P has the follow

ing AC

F

425

16cos20

36X

Re

22

mean-square value=

025

1636

77

mean value=

366

variance=77

3641

XR

XX

Note: w

e can only determ

ine the m

agnitudeof the

mean from

the AC

F

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Exam

ple

An ergodic R

P has the A

CF

2

2

2536

6.254

XR

0

22 2

25lim

42

6.25

mean value=

42

36m

ean-square value=0

94

variance=9

45

X

X

R

R

XX

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Exam

ple

Determ

ine largest value of Afor the follow

ing functions to be a valid autocorrelation function.

a) 4

69e

Ae

»S

ymm

etric about 0?–

Yes

»A

ny point higher than RX (0)?

–N

eed to check

»F

ourier transform ≥0?

–N

eed to check

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Exam

ple

»A

ny point higher than RX (0)?

46

a)9e

Ae

46

46

64

2

90

0

366

0

636

61

as0

366

de

Ae

de

Ae

Ae

e

Ae

A

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Exam

ple

2

2

92

42

6

1636

X

AF

R

2

22

7212

2592192

1636

AA

7212

06

AA

2592192

013.5

6A

AA

»F

ourier Transform

≥0?

46

a)9e

Ae

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Exam

ple

4b)

10A

e

»S

ymm

etric about 0?–

Only if A

=0

»A

ny point higher than RX (0)?

–O

nly if A=

0

»F

ourier transform ≥0?

–O

nly if A=

0

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Exam

ple

c)20

cos5sin

5A

»S

ymm

etric about 0?–

Only if A

=0

»A

ny point higher than RX (0)?

–O

nly if A=

0

»F

ourier transform ≥0?

–O

nly if A=

0

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Measurem

ent of Autocorrelation F

unctions

In practice, the autocorrelation function must either be derived

from the physics of the problem

or measured.

Suppose w

e observe an ergodic random voltage x(t) from

0 to T sec.

One

Definition of the estim

ated AC

F is the time correlation:

0

1ˆ

0T

XR

xt

xt

dtT

T

N

ote: both and

are only available overthe range

0 to

xt

xt

T

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Measurem

ent of Autocorrelation F

unctions

•In practice, w

e usually have are samples of a

random process

»x

k =x(kΔ

t) where 0 ≤

k ≤N

•A

nd are interested in values of the AC

F at

discrete lags»

RX (nΔ

t) where 0 ≤

n ≤M

•In this case, the estim

ate of the AC

F is

0

1ˆ

=

0,1,2,...,

1

Nn

Xk

kn

k

Rn

tx

xn

MM

NN

n

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•F

or two jointly W

SS

random processes X

(t) and Y

(t), we define the cross correlation function

as:

»N

ote the order is important

Cross C

orrelation Functions

11

11

()

and

()

XY

YX

RE

Xt

Yt

RE

Yt

Xt

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Exam

ple

•T

wo jointly W

SS

random processes

»θ

is uniformly distributed betw

een 0 and 2π

()

2cos(5

) and (

)10

sin(5)

Xt

tY

tt

20

20

20

()

{(

)(

)

120

cos(5)sin(5(

))2

201

[sin(55

2)

sin(5)]

22

5sin(5

52

)10

sin(5)

10sin(5

)

XY

RE

Xt

Ytt

td

td

td

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Properties of C

ross-Correlation F

unctions

•R

XY (0) and R

YX (0) have no particular physical

significance and do not represent mean-square values,

but »

RX

Y (0) = R

YX (0).

•R

XY (τ) =

RY

X (-τ)

•C

ross-correlation functions do not necessarily have their m

aximum

value at τ=

0, but»

| RX

Y (τ) |≤[RX

(0) RY

(0)] 1/2

•If the tw

o random processes are independent, then

»(

)(

)X

YYX

RX

YR

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Application: R

adar

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Application: R

adar

•T

ransmitted signal X

(t)

•R

eceived signal»

Y(t)=aX

(t-T)+

N(t)

–a: am

plitude attenuation factor

–N

(t): noise –independent of and m

uch stronger than aX(t-T)

•C

ross-correlate the transmitted signal w

ith the received signal

()

{(

)(

)}

{(

)[(

)(

)]}

{(

)(

)(

)(

)}

()

()

()

XY

XX

N

X

RE

Xt

Yt

EX

taX

tT

Nt

EaX

tX

tT

Xt

Nt

aRT

R

aRT

0