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Other Signal and Image Processing Methods
average of “(x(t) − m(t)) 2 .” This statistical averaging can be extended to any general
function “g(x(t)),” namely,
+∞
g t
( ) = E g x t =
( ( )) X ( ) x t
(6.19)
( ( ( )))
g x t p t d ( )
∫

−∞
Similarly, for the discrete processes, we have
+∞
g n = E g x w n
( ( ( , ))) =
( ( i , )) X ( i , )
(6.20)
( )
∑ g x w n p w n
i=−∞
Some other popular functions whose expectations are useful in practical applications are “moment functions.” The moment of degree k is defined as E(g(x(t))),
where g(x(t)) = x(t) k . As can be seen, the mean function is nothing but the moment
of degree 1, and variance is closely related to the moment of degree 2. The moment of
degree 2 is often considered as the statistical power of a signal.
6.4.2 STATIONARY AND ERGODIC STOCHASTIC PROCESSES
A subset of stochastic processes provides some practically useful characteristics. This
family that is referred to as “stationary processes in wide sense” in which the mean
and variance functions remain constant for all time, i.e., m(t) = m 0 , and σ(t) = σ 0 . In
other words, even though the probability function p X (t) of such processes can change
through time, the mean and variance of the process stay the same at all times. Such a
simplifying assumption helps processing a number of practically useful signals. For
instance, it is often the case that the mean and variance of signals such as ECG and
EMG does not change at least during a rather large window of time. Such an observation allows calculation of mean and variance for only one time point because the
assumption of stationarity in wide sense states that the mean and variance functions
for all times will be the same.
A subset of wide sense stationary processes are “stationary in the strict sense”
processes in which the probability function p X (t) is assumed to be independent of
time. This means that if one finds the PDF for one time step, the same exact PDF
can be used to describe all statistical characteristics of the process in all times. From
the definition, one can see that while all strict sense stationary processes are also
wide sense stationary, the opposite is not true. This means that the strict sense stationary assumption is a stronger assumption and therefore applicable to fewer real
applications.
An interesting and very popular family of wide sense stationary processes is the
set of wide sense stationary Gaussian processes. In such processes, since the PDF
has only two parameters, mean and variance, once these two parameters are fixed
in time, the PDF becomes the same for all time. This means that for Gaussian processes, strict sense and wide sense stationary concepts are the same. Since many