Page 243 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

Taking the Fourier transform of Eq. (6.1 57), we obtain

(b) Similarly, if X(t) is WSS, then by Eq. (6.156), Eq. (6.153) becomes

which indicates that Rdt, s) is a function of the time difference z = s - t only. Hence

Taking the Fourier transform RAT), we obtain

Note that from Eqs. (6.154) and (6.155), we obtain Eq. (6.63); that is,

6.32. Consider a WSS process X(t) with autocorrelation function RAT) and power spectral density
Sx(o). Let Xf(t) = dX(t)/dt. Show that

(a) If X(t) is the input to a differentiator, then its output is Y(t) = X'(t). The frequency response of a
differentiator is known as H(o) = jo. Then from Eq. (6.1 54),

Taking the inverse Fourier transform of both sides, we obtain

(b) From Eq. (6.155),