B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

Because pe(0) = 1/2n over (0, 2n) and O outside this range,
Hence,
9 .2 Classification of Random Processes 463
1 12rr
cos (w e t + 8) = - cos (w e t + 0) d0 = 0
2n 0
x(t) = 0
(9.7a)
Thus, the ensemble mean of sample function amplitudes at any instant t is zero.
The autocorrelation function R x (t1 , t 2 ) for this process also can be determined directly
from Eq. (9.3a),
R x U1, t 2 ) =A 2 cos (w e t1 + 8) cos (w e t 2 + 8)
= A 2 cos (w e ll + 8) cos (w e t 2 + 8)
= 2 { cos [we (tz - t1 )] + cos [we(tz + t1 ) + 28] }
The first term on the right-hand side contains no RV. Hence, cos[we(tz - t1 )] is cos [w e U2 -
t1)] itself. The second term is a function of the uniform RV 8, and its mean is
Hence,
1 12rr
cos [w e (t 2 + t1 ) + 28] = - cos [we(tz + t1 ) + 20] d0 = 0
2n 0
A 2
R x (tl , t 2 ) = 2
cos [we (tz - t1)]
(9.7b)
or
r = t 2 - t1
(9.7c)
From Eqs. (9.7a) and (9.7b) it is clear that x(t) is a wide-sense stationary process.
Ergodic Wide-Sense Stationary Processes
We have studied the mean and the autocorrelation function of a random process. These are
ensemble averages. For example, x(t) is the ensemble average of sample function amplitudes at
t, and Rx(tJ, t 2 ) = x1x 2 is the ensemble average of the product of sample function amplitudes
x(t1 ) and x(tz).
We can also define time averages for each sample function. For example, a time mean x(t)
of a sample function x(t) is*
- 1
T/2
x(t)= lim - l x(t) dt
T--+oo T
-T/2
(9.8a)
* Here a sample function x(t, {;) is represented by x(t) for convenience.