Page 251 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

6.51. Consider a WSS random process X(t) with E[X(t)] = p,. Let

The process X(t) is said to be ergodic in the mean if

Find E [(X(r )) ,I.

Ans. px
6.52. Let X(t) = A cos(o, t + 0), where A and w, are constants, O is a uniform r.v. over (-n, n) (Prob. 5.20).
Find the power spectral density of X(t).
A2a
Ans. Sx(o) = T [6(o - o,) + 6(o + w,)]

6.53. A random process Y(t) is defined by

where A and o, are constants, O is a uniform r.v. over (-n, a), and X(t) is a zero-mean WSS random
process with the autocorrelation function RX(z) and the power spectral density Sx(o). Furthermore, X(t)
and O are independent. Show that Y(t) is WSS, and find the power spectral density of Y(t).
A2
Ans. Sdo) = q [Sx(w - w,) + Sx(o + o,)]

6.54. Consider a discrete-time random process defined by

where a, and Ri are real constants and Oi are independent uniform r.v.'s over (- n, n).
(a) Find the mean of X(n).
(b) Find the autocorrelation function of X(n).
Ans. (a) E[X(n)] = 0
I m
(b) Rx(n, n + k) = 1 a: cos(Ri k)
2 ,=I
6.55. Consider a discrete-time WSS random process X(n) with the autocorrelation function
Rx(k) = 10e-0.51k1
Find the power spectral density of X(n).
6.32
Ans. S R -n<Q<n
X( ) = 1.368 - 1.213 cos Q
6.56. Let X(t) and Y(t) be defined by
X(t)= U cosoot + V sin oot
Y(t)= Vcos mot- Usino,t
where o, is constant and U and V are independent r.v.'s both having zero mean and variance c2.
(a) Find the cross-correlation function of X(t) and Y(t).
(b) Find the cross power spectral density of X(t) and Y(t).