Page 229 - Schaum's Outlines - Probability, Random Variables And Random Processes

ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

(b) From Eq. (6.1 7), the cross-correlation function of X(t) and X1(t) is

(c) Using Eq. (6.1 13, the autocorrelation function of X'(t) is

6.8. If X(t) is a WSS random process and has a m.s. derivative X'(t), then show that

For a WSS process X(t), Rx(t, s) = RX(s - t). Thus, setting s - t = z in Eq. (6.115) of Prob. 6.7, we
obtain aRAs - t)/as = dRX(z)/dz and

Now aRx(s - t)/at = -dRx(z)/dz. Thus, a2Rx(s - t)/at as = -d2~,(z)/dr2, and by Eq. (6.116) of Prob,
6.7, we have

6.9. Show that the Wiener process X(t) does not have a m.s. derivative.
From Eq. (5.64), the autocorrelation function of the Wiener process X(t) is given by

Thus

where u(t - s) is a unit step function defined by

and it is not continuous at s = t (Fig. 6-2). Thus d2R&, s)/& as does not exist at s = t, and the Wiener
process X(t) does not have a m.s. derivative.