1.8 Sampling and random sequences

The usual Sampling Theorem applies to random processes, with the spectrum of interest beign the power spectrum. If stationaryprocess $X(t)$ is bandlimited - ${}_X()=0$ , $\left|\right|> W$ , as long as the sampling interval $T$ satisfies the classic constraint $T< \frac{\pi }{W}$ the sequence $X(lT)$ represents the original process. A sampled process is itself a random process defined over discrete time. Hence, allof the random process notions introduced in the previous section apply to the random sequence $\stackrel{}{X}(l)\equiv X(lT)$ . The correlation functions of these two processes are related as $$R_{\stackrel{}{X}}(k)=(\stackrel{}{X}(l)\stackrel{}{X}(l+k))=R_X(kT)$$

We note especially that for distinct samples of a random process to be uncorrelated, the correlation function $R_X(kT)$ must equal zero for all non-zero $k$ . This requirement places severe restrictions on the correlation function (hence the powerspectrum) of the original process. One correlation function satisfying this property is derived from the random processwhich has a bandlimited, constant-valued power spectrum over precisely the frequency region needed to satisfy the samplingcriterion. No other power spectrum satisfying the sampling criterion has this property . Hence, sampling does not normally yield uncorrelated amplitudes,meaning that discrete-time white noise is a rarity. White noise has a correlation function given by $R_{\stackrel{}{X}}(k)=^2(k)$ , where $()$ is the unit sample. The power spectrum of white noise is a constant: ${}_{\stackrel{}{X}}()=^2$ .