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Random Processes and Spectral Analysis Chap. 6
Comparing Eq. (6–9) with Eq. (6–11) and Eq. (6–10) with Eq. (6–12), we see that the time
average is equal to the ensemble average for the first and second moments. Consequently, we suspect
that this process might be ergodic. However, we have not proven that the process is ergodic,
because we have not evaluated all the possible time and ensemble averages or all the moments.
However, it seems that the other time and ensemble averages would be equal, so we will assume
that the process is ergodic. In general, it is difficult to prove that a process is ergodic, so we will
assume that this is the case if the process appears to be stationary and some of the time averages
are equal to the corresponding ensemble averages. An ergodic process has to be stationary, since
the time averages cannot be functions of time. However, if a process is known to be stationary, it
may or may not be ergodic.
At the end of the chapter in Prob. 6–2, it will be shown that the random process
described by Eq. (6–8) would not be stationary (and, consequently, would not be ergodic) if u
were uniformly distributed over (0, p2) instead of (0, 2 p).
Correlation Functions and Wide-Sense Stationarity
DEFINITION.
The autocorrelation function of a real process x(t) is
(6–13)
where x l = x(t 1 ) and x 2 = x(t 2 ). If the process is stationary to the second order, the autocorrelation
function is a function only of the time difference t = t 2 - t 1 .
That is,
R x (t) = x(t)x(t + t)
(6–14)
if x(t) is second-order stationary. †
DEFINITION.
R x (t 1 , t 2 ) = x(t 1 )x(t 2 ) =
L
q
-q L-q
A random process is said to be wide-sense stationary if
1. x(t) = constant and
(6–15a)
2. R x (t 1, t 2 ) = R x (t)
(6–15b)
where t = t 2 - t 1 .
A process that is stationary to order 2 or greater is wide-sense stationary. However, the
converse is not necessarily true, because only certain ensemble averages, namely, those of
Eq. (6–15), need to be satisfied for wide-sense stationarity. ‡ As indicated by Eq. (6–15), the
mean and autocorrelation functions of a wide-sense stationary process do not change with a
q
x 1 x 2 f x (x 1 , x 2 ) dx 1 dx 2
† The time average type of the autocorrelation function was defined in Chapter 2. The time average autocorrelation
function, Eq. (2–68), is identical to the ensemble average autocorrelation function, Eq. (6–14), when the
process is ergodic.
‡ An exception occurs for the Gaussian random process, in which wide-sense stationarity does imply that the
process is strict-sense stationary, since the N → q-dimensional Gaussian PDF is completely specified by the mean,
variance, and covariance of x(t 1 ), x(t 2 ), . . . , x(t N ).