a If 0 determine whether Y t is widesense stationary b If

(a) If ? = 0, determine whether Y (t) is wide-sense stationary.

(b) If ? is uniformly distributed between [0, ?], show whether Y(t) is wide-sense stationary.

(c) What is the average power of X(t) and Y(t), respectively for part (b)?

7.7 Consider a random process Y(t)-x(t) cos(mt + ) in which x(t) is a stationary random process with E{X(t)} 0 and autocorrelation function Rx (r) = 4+12.

Solution

Given

Y(t) = X(t)cos(w_ct + O) in which X(t) is a stationary process with E[X(t)] = 0 and auto correlation function

R_X(n) - 1/4 + n²

For a Y(t), the process is said to be wide sense stationary when it is satisfies the following two conditions

1. When the mean of Y(t) is independent of t i.e. E[Y(t)] is independent of t

2. When its autocorrelation function is independent of t

From , the given value of autocorrelation function, it can be deduced that it is independent of t

a. When O = 0, then Y(t) = X(t)cos(w_ct) and its average value will be equal to 0. Hence, it can be termed as independent of t.

Therefore, Y(t) is wide sense stationary

b.Given the autocorrelation func. for X, we can find out the autocorrelation function for Y(t)

R_Y(n) = E[(A +X(t))(A + X(t + n))]

= E[A²] + E[A]E[X(t)] + AE[X(t+n)] + E[X(t)X(t + n)]

Since, X(t) is a stationary process, E[X(t)] can be termed as 0.

= E[A²] + 2E[A]E[X(t)] + R_X(n)

= E[A²] + R_X(n)

Hence, autocorrelation for Y (t) is independent of t

Therefore, Y(t) is wide sense stationary.