Wiley signals and systems e book TLFe BO 436

17.4. Correlatioii Fimctions

423

time can be absolutely integrated (see Chapter 9 2.2). t h i s is suficieiit condition
for the c3xistcnce of the cleterministic functions. This property is not giveri for
auto-correlation fimctions, how~ver,it the linear average p, is non-~ero.
In order to o'vercorrie I b e difficulty statcd a l m ~ e ,the linear average can be
rernovt-itl horn thc outsct and instead of the signal .c(t>,the zero mean srgrd
( r ( f) ~ 1 % )can be cousidereti. Its aiito correlation function is called t l i ~autocovurmrm functzon of r ( l ) and is drnoktl by V J , ~ ( r):
ui/?,(T)

= E{(z(t) - PTl(.V(l -

z)- p 5 ) ) .

(17.38)

IJsing the calculation rules from Section 17.2.3 WP obtain
( 7 )=

PJ'(4

- P:

3

(17.39)

just as in (17 8). The properties of the airto-rovariance function col-iespor~dto
those of the auto-wrrelation function for z e ~ omean signals.

17.4.1.3 Cross-Correlation Function

The ailto-correlation funrtion is given by tlie expected value of two sigrral va1ue.i
that arc take11 from 07ic Iandorri proccss at LIYO tliffercnt times. This idea can
lie extcndcd to signal value:, from different rsncloni pr
cnt i t s propertics
expecled vitluc is called tlie crosr;-correluhon finct/on.
correctly we have to extencl the earlicr defintions of se.cmdstatiomry and ergodic rnndoni processes to ded with two random pzoccsses. A
sccond-oidergoznt exper Led valur~is the expectd valiir of a function j ( ~ ( i ' ~
CL)).
),
formed with sigiials from two diffeient raztdoiii proce
(17.44)
For the cross-correlation function y,, / ( t l , t z ) . it holds in general !.hat (comparc
(17.10)):
(1 7.45)
i p l y G i l l z ! =- hi { d t l ) '

"!"U

-

The rf(i~s-e0rr~7131ioii
ftinction i.i denoted like the auto-correlation limrtion. but
the second random process i s irrdimtcd by another letter in the indcx.