Page 250 - Probability, Random Variables and Random Processes

ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

From the Fourier transform pair (Appendix B) x(n) = 1 ++ 27rs(Q), we have
Q)
1 e-jrn('l +n2) = 27cS(R, + a,)
m=-w
Hence &(%, n2) = 2&(%)6(fil + n2)
Next, from Eq. (6.1 04) and the above result, we obtain
MQ,, fi,) = Rx(fi,, 4,) 2nsx(Q,)s(Q, - Q,)
=

Supplementary Problems

6.46. Is the Poisson process X(t) m.s. continuous?
Hint: Use Eq. (5.60) and proceed as in Prob. 6.4.
Ans. Yes.
6.47. Let X(t) be defined by (Prob. 5.4)
X(t) = Y cos ot t 2 0
where Y is a uniform r.v. over (0, 1) and w is a constant.

(a) Is X(t) m.s. continuous?
(b) Does X(t) have a m.s. derivative?
Hint: Use Eq. (5.87) of Prob. 5.12.
Ans. (a) Yes; (b) yes.

6.48. Let Z(t) be the random telegraph signal of Prob. 6.18.
(a) Is Z(t) m.s. continuous?
(b) Does Z(t) have a m.s. derivative?
Hint: Use Eq. (6.132) of Prob. 6.18.
Ans. (a) Yes; (b) no.

6.49. Let X(t) be a WSS random process, and let X1(t) be its m.s. derivative. Show that EIX(t)X1(t)] = 0.
Hint : Use Eqs. (6.1 3) [or (6.1 4)] and (6.1 17).
2 t+T/2
6.50. Let at) = 7 1 X(a) da
where X(t) is given by Prob. 6.47 with w = 2nlT.
(a) Find the mean of Z(t).
(b) Find the autocorrelation function of Z(t).
1
Ans. (a) - - sin ot
7t
4
(b) R,(t, s) = - at sin us
sin
3n2