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EE278 Statistical Signal Processing

Prof. B. Prabhakar Autumn 02-03

Strict and Wide Sense Stationarity

Autocorrelation Function of a Stationary Process

Power Spectral Density

Response of LTI System to WSS Process Input

Linear Estimation The Random Process Case

Copyright c20002002 Abbas El Gamal

EE 278: Stationary Random Processes 9-1

Stationary Random Processes

Stationarity refers totime invarianceof some, or all, the statistics of a

random process, e.g., mean, autocorrelation, nth order distribution,etc

We define two types of stationarity, strict sense(SSS) and wide sense

(WSS)

A random process X(t) (orXn) is said to be SSS ifallits finite order

distributions are time invariant, i.e., the joint cdf (pdf, or pmf) of

X(t1), X(t2), . . . , X (tk) is the same as for X(t1+ ), X(t2+ ),

. . . , X (tk+ ), for all k, all t1, t2, . . . , tk, and all time shifts

So for a SSS process, the first order distribution is independent oft, and

the second order distribution, i.e., the distribution of any two samples

X(t1) and X(t2), depends only on =t2 t1

To see this, note that from the definition of stationarity, for anyt, the

joint distribution ofX(t1) and X(t2) is the same as the joint distribution

ofX(t1+ (t t1)) and X(t2+ (t t1)) =X(t + (t2 t1))

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Wide Sense Stationary Random Processes

A random process X(t) is said to be WSS if its mean and

autocorrelation functions are time invariant, i.e., E(X(t)) =,

independent oft and RX(t1, t2) is only a function of(t2 t1)

SinceRX(t1, t

2) =RX(t

2, t

1), ifX(t)is WSS,RX(t

1, t

2) is only a function

of|t2 t1|

Clearly SSS WSS, the converse, however, is not necessarily true

For GRP, WSS SSS, since the process is completely specified by its

mean and autocorrelation functions

Random walk is not WSS, sinceRX(n1, n2) = min{n1, n2} is not timeinvariant in fact no independent increment process can be WSS

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Autocorrelation Function of WSS Processes

Let X(t) be a WSS process and relabel RX(t1, t2) as RX(), where

=t2 t1

1.RX() is real and even, i.e., RX() =RX() for all

2. |RX()| RX(0) =E(X2(t)), the average power ofX(t)This can be shown using the Schwartz inequality

For any t

(RX())2 = (E(X(t)X(t + )))2

E(X2(t))E(X2(t + )) = (RX(0))2

3. IfRX(T) =RX(0) for some T, then RX() is periodic with periodT and

so is X(t) (with probability 1)

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Which Functions can be an RX() ?

1. 2.

e

e||

3. 4.

sinc

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5. 6.

T2

T

1

1

2|n|

n4321 1 2 3 4

7.

T T

EE 278: Stationary Random Processes 9-11

Interpretation of Autocorrelation Function

IfRX() drops quickly with, this means that samples become

uncorrelated quickly as we increase, conversely, ifRX() drops slowly

with , samples are highly correlated

RX()

RX()

So RX() is a measure of the rate of change ofX(t) with time t, i.e.,

the frequency response ofX(t)

It turned out that this is not just an interpretation the Fourier

Transform ofRX() (the power spectral density) is in fact the average

power density over frequency

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Power Spectral Density

The power spectral density(psd) of a WSS random process X(t), is the

Fourier Transform ofRX(),

SX(f) = F[RX()] =

RX()e

j2 fd

For a discrete time processXn, the power spectral density is the discrete

Fourier Transform (DFT) of the sequence RX(n),

SX(f) =

n=

RX(n)ej2nf, for|f|