Page 249 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES 24 1

6.43. Let z(o) be the Fourier transform of a random process X(t). If %(o) is a white noise with zero
mean and autocorrelation function q(o,)6(co1 - a,), then show that X(t) is WSS with power
spectral density q(o)/27r.
By Eq. (6.91),

Then ~[X(w)]ej'~~ 0
do
=
Assuming that X(t) is a complex random process, we have

dw2
dw,
-- ~[~(o,)~*(o,)]ej(~~~-"~~)
1 " "
-- 4n2 [ 1 mq(col)b(w, - 02)e'(w1t-w2G do, do,
- oo
1 rm

which depends only on t - s = z. Hence, we conclude that X(t) is WSS. Setting t - s = z and o, = o in Eq.
(6.1 78), we have

in view of Eq. (6.24). Thus, we obtain Sx(o) = q(o)/2n.

6.44. Verify Eq. (6.104).
By Eq. (6.100),

m co
Then R&, Q2) = E[X(Q,)~*(Q,)] = x ~[X(n)X*(rn)]e-j("~~-~~~)

in view of Eq. (6.1 03).

6.45. Derive Eqs. (6.105) and (6.106).
If X(n) is WSS, then Rx(n, m) = R,(n - m). By Eq. (6.103), and letting n - m = k, we have