Wiley signals and systems e book TLFe BO 441

17, Dcscribing Rarrciorn Signals

426

part itularly uscful, art1 often Ic5,zds to elegant aieL1iocls for aualysing LTX-systems.
We would therefore like to have a detemiinistic dcscription of raxidoni sig~ialsin
the frequcncy-domain izs well
Thc idea that Erst comes to mind i s describing sai-riplefianction
doru process as expected va1ut.s instead of considering them as random signals. but
this idea is actually unsuitable. Stat ionmy random signals can never be i&egrat,ed
at~solntelyj9.4), as they do not decay for ltj -+ 00. Therefore the Laplace integral
cmirot exist a i d thc Fouricr transform earl only exist in qpecial cases. Instead
of transforming into the frequenc;y-doi~iajuand then fornrhg rxp
fwirt the expcrtcd values iK1 the t i ~ ~ i e - ~ ~ ~and
) r ~ ithen
a i n transfer blre ~ e ~ e r ~ i i ~ i ~ s t , i c
quantities to the freqwricy-domain. This idea is the hack fox the dehition of thc
power density spectriirn.

We start with tlie auto-correlation function or t8hem.&o-covariancefunction of a
weak stationary random process arid form its Pourier transform:
(17.69)

It B olso C R I I C ~ thc poituer denszty s ~ ~ c t of‘
~ 7the
6 random
~ ~
prcicess. The power
density spec(ruin chararter ises statistiml dcpcndcricics of the signal amplitude at
two diffcvent poiiits in time. ~ o r ~ ~ s ~ ? o i i dthe
i n Fourier

~ ~ y , tr tmhrxn of a crosscorrelation fimt%ion can Be fornied giving t hc cyass-powcr ilertszty sprwYurrt:

Its h e a r average
F;{X(p) 1. = 0