Wiley signals and systems e book TLFe BO 428

17.3. Stationary Handoni Processes

413

That means that for s t a t i o ~ random
~ r ~ processes, the first-order expected wfiies
do not ctepeitd oil time. In partictilar, for the linear average and the va,riance:
/&(tj = ps;

0 2 2 ( 1 ) = o, z *

(17.14)

The auto-correlation filnction c m be expressed more simply with z = tl - tz:

E{Z(tl)s(t2)) f= E(zftl)z(tl,.- c')) = E { ~ ( t +
z ~ > ~ : f l:
a )pXz(.).
}

(17.15)

The a , u L o - c ~ ~ r ~ I afunction
, ~ i ~ n pZ2(z); that we introduced in (17.15j ;ls R orie~ ~ i ~ e fuiiction
n ~ ~ isonot
~ the
a ~same fimctitm as pzz(tl,tz) in (17,101,which is a
funcliori of two vttriables, t L and t z . The two pE1:
are linked for sta,tionaxyrandom
sigmls (17.15). If the stationary coiidition only holds for
(17.16)

(17.17)

however, tmd not for gewral funcLions f ( , .), then the random process ts called
weak s ~ ~ ~ ~Also
~ for
~ ~the( linear
~ r ~
avex
1 age,
. the vaxiaace iilid the ~ ~ u ~ o - ~ o r r e i a ~ i o ~
function of weak stationary proc PS (17.131, (17.14), arid (17.15) hold. We can
thus obttiin the following staternent: for a weak stationary process the linear average and the correlatioii properties coiitainecl in the auto-corrclation function are
is iisually used in
constant with time. The concept of weak stationary proces
the context of rnotlelling and analysis of random procesms, where it is often a,
prclcondition. when only the linear average and auLtw.orrelation fund ion itre
coxisidered, weal
Example 17.5
M7e take a random piocess with linear average E{ t ( f ) }

pzz(tl,
t,) = sijtl

0 and the ACF

- 62).

Thir r n ~ ~ ~proress
~ ( ~ risn definitely weakly ~ t ~ t i o n nasr ~we, can mite its AC'F ah

i F r z ( d = "i(

).

;+riditi linear merage p z = 0 i s constant. Thus (17.12) is fulfilled for the funchxis (17.16), (17.l.7). The vaiiuncv of the signal ir
cr; = E { 2 > = V L I( 0 ) =

I.

With t,lic given ~ I ~ ~ o-tver ~nniiot
~ ~ tell
~ t whtxthcr
~ o ~or not the r a n d ~ ~process
n
is
stationary in tlic strict mnse