Wiley signals and systems e book TLFe BO 437

17. Describing. hndoxn Signals

422

Next wc extend the idea of stationary from Defiirition 23 t o cover two randonr
processes. We call two random procescies jornt ~ ~ u ~ if ztheir
o joint
~ ~ second~ ,
older exxprctccl V ~ L I only
~ S depend on tlie difference z = tl - tz. For joint stationary random processes the cross-corrrlntion function then takes a form simifar to
(17.15):
c p & ) = E ( . x . ( f t Z).yft))=:E{z(t)-yi-d- z)) '
(17.46)

Finally WP introduce the second-ordw joznt tzrne-average

and call two random processes for which thr @hit cq)cct.ed valws agree with
thc joint timwiverages joznt crgodzc. There ?axe also weak forms for joint, stai
Liona.ry atid joint crgodic randovn p r o c ~ s s ~whcre
s,
tlic corresponding conditions,
are only fulfilleci for ~ ~ ~ ( ~= ~ ) ~. ~~~ { (~ ~~~ ~ ~
=~
) 3;(t1)
{ ( a~ d~ ~ ~ ~

?dfz).

.~~.~(~~~
=, ~ ~ ~ 2 ) }

Thc cross-correlation function performs it sinrilar fiirictiori for two random processes Illat the auto-correlation ftinrtion does for one random process. It is a
measure for the rF~atio~sliip
of valnes frorn the t~ 7 or a r i ~ o praccss
i~
at two timcs
separated by z. The extension to two random processes cat1 some (jiffererices
t o the anto-correlatioii function.
Firs! of till, two rimtforn pror.es~scan be uncorrefateti not only for large timespaiis hilt also for all ~Tlfuesof r. Their cross-correlation fiinctioii is then. the
protlurt of the linear cxpcc%Xlvaluos 1-1~and puyof the individual random processts:
Y.&j

= i*L

Ply

v r.

(17.48)

-

TIicre i h also the caiw that two riuidorn processes art? riot uxit:orielated for all
r, brit at least for 174 30:

values of

F t ~ r ~ , ~the
e rcros4
~ n corxelatiori
~ ~ ~ ~ ~function clors not have the even s ~ ~ ~ ~ e

of the auto-cmreiation functioa, as from (17.46) and s-cvagping n: and g, wv omly
obtain

The ai~to-rorrelat,ioir function pTJ( z) can be obtained from tlie crosscorrektion €ianc+ion pfy(7)% a special case y ( t ) = .it). Then using (17.50),
we cmt find Ihe syiiimotry propprty (1736) of the auto-correlation function.