Page 189 - Probability, Random Variables and Random Processes

CHAP. 51 RANDOM PROCESSES

5.19. Consider a random process X(t) defined by
X(t) = U cos t + V sin t - m < t <

where U and V are independent r.v.'s, each of which assumes the values -2 and 1 with the
probabilities 4 and 3, respectively. Show that X(t) is WSS but not strict-sense stationary.
We have

Since U and V are independent,

Thus, by the results of Prob. 5.18, X(t) is WSS. To see if X(t) is strict-sense stationary, we consider E[x3(t)].
E[X3(t)] = E[(U cos t + V sin t)3]
= E(U3) cos3 t + 3E(U2V) cos2 t sin t + 3E(UV2) cos t sin2 t + E(V3) sin3 t
Now

Thus E[X3(t)J = --2(cos3 t + sin3 t)
which is a function of t. From Eq, (5.16), we see that all the moments of a strict-sense stationary process
must be independent of time. Thus X(t) is not strict-sense stationary.

5.20. Consider a random process X(t) defined by
X(t) = A cos(wt + 0) - co < t < co
where A and w are constants and 0 is a uniform r.v. over (-71, n). Show that X(t) is WSS.
From Eq. (2.44), we have

(0 otherwise
Then cos(wt + 0) dB = 0

Setting s = t + .t in Eq. (5.7), we have

= A' !. [cos wr + cos(2wt + 28 + wr)] d8
27c -,
2
=- cos wz
2
Since the mean of X(t) is a constant and the autocorrelation of X(t) is a function of time difference only, we
conclude that X(t) is WSS.

5.21. Let (X(t), t 2 0) be a random process with stationary independent increments, and assume that
X(0) = 0. Show that