Page 235 - Probability, Random Variables and Random Processes

CHAP. 6) ANALYSIS AND PROCESSING OF RANDOM PROCESSES

From Eq. (6.23) and expanding the exponential, we have

Since R,(-z) = RX(z), Rx(z) cos wz is an even function of z and RX(z) sin oz is an odd function of z, and
hence the imaginary term in Eq. (6.127) vanishes and we obtain

(6.2 28)

which indicates that Sx(o) is real. Since cos(-or) = cos(oz), it follows that

which indicates that the power spectrum of a real random process X(t) is an even function of frequency.

6.17. Consider the random process

where X(t) is a Poisson process with rate A. Thus Y(t) starts at Y(0) = 1 and switches back and
forth from + 1 to - 1 at random Poisson times q, as shown in Fig. 6-4. The process Y(t) is
known as the semirandom telegraph signal because its initial value Y(0) = 1 is not random.

(a) Find the mean of Y(t).
(b) Find the autocorrelation function of Y(t).
(a) We have
1 if X(t) is even
Y(t) =
- 1 if X(t) is odd
Thus, using Eq. (5.59, we have
P[Y(t) = 11 = P[X(t) = even integer]

P[Y(t) = - 11 = P[X(t) = odd integer]

Fig. 6-4 Semirandom telegraph signal.