Digital Signal Processing Reference
In-Depth Information
n
0
across realizations. For stationary processes, it may seem intuitive that
instead of averaging across realizations, we can average across successive
samples of the same realization. This is not true in the general case, how-
ever. Consider for instance the process
l
g
r
,
y
i
d
.
,
©
,
L
s
X
[
n
]=
α
where
is a random variable. Clearly the process is stationary since each
realization of this process is a constant discrete-time signal, but the value of
the constant changes for each realization. If we try to estimate the mean of
the process from a single realization, we obtain no information on the dis-
tribution of
α
α
; that can be achieved only by looking at several independent
realizations.
The class of processes for which it is legitimate to estimate expectations
from a single realization is the class of
ergodic processes
.Forergodicpro-
cesses we can, for instance, take the time average of the samples of a single
realization and this average converges to the ensemble average or, in other
words, it represents a precise estimate of the true mean of the stochastic
process. The same can be said for expectations involving the product of
process samples, such as in the computation of the variance or of the corre-
lation.
Ergodicity is an extremely useful concept in the domain of stochastic
signal processing since it allows us to extract useful statistical information
from a single realization of the process. More often than not, experimental
data is difficult or expensive to obtain and it is not practical to repeat an
experiment over and over again to compute ensemble averages; ergodicity
is the way out this problem, and it is often just assumed (sometimes without
rigorous justification).
Example: Gaussian Random Processes.
A Gaussian random process
is one for which any set of samples is a jointly Gaussian random vector. A
fundamental property of a Gaussian random process is that, if it is wide-
sense stationary, then it is also stationary in the strict sense. This means that
second order statistics are a sufficient representation for Gaussian random
processes.
8.4 Spectral Representation
of Stationary Random Processes
Given a stationary random process, we are interested in characterizing
its “energy distribution” in the frequency domain. Note that we have used
quotes around the termenergy: since a stationary process does not decay in
