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 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 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) =