17.6 Describinr Discrete Ittilldoat Signals 43 I wc will forego exaniiriirrg them in clclail However we need to hc w a r e of tbe definitions of stat ionary and ergodicity as they apply to randorn secpienccs Stationaiy means in this ease that the second-order exp ~ ~ f ( z ~ ~ orily l ] ~drlwnd ~ [ on k integer ~ ] ~differences ~ K = k.1 - k.2 discrete t i m e variables kl and k ~ The time-average, necessarv for the tlefiiiitioit of ergodicity, i s givcii for sample r;mdoiri process by sequenccs of a cIiser (17.84) From here, the a u ~ o - ~ o r ~ e l a ~cifo~n ), s ~ - ~ o r r and ~ l ~covariiknces ~tio~ can bc considcrcd in the s m i P way as for continuous random processes Instead of autocorrelatitm aiid cross-correlation fimctions, we will be using auto-correlation and cross-correlation sequertces For the cross-power density spectrum of two discrete weak s~,a~ioiia~y faridam processes, we obtain from the ctiscret,e-timc Fourier transform the spectrum (17.85f periodic in 52 The power density spectrim of a discrete randorn process corresponds to the case where y = LT Example 19.12 A weak statioriary continuous randorn proccss Ic(-t> is charactcv-ised by ail exponentially dccayiag ACF ) zzz e-wol r1 , Samples arc taken at iritervals T z [ k ]= Li,.(kT) k t IIZ The a i ~ t ~ - c o r r e l ~ sequence t i ~ ) n is The power density spectrum of the discrete raiidorn process i s and i s slnown in Figure 17.12 1%is gositive, real arid even, like the power density spectrum of a continuom random process with real wfues hi addition, it is 2n periodic As the auto-correlation sequences vS.[ K ] is yielded simply by sampling