Random Processes - Random Signals and Processes Primer with MATLAB - page 379

or equivalently:

1

R XX ðt 1 ; t 2 Þ¼ x 1 x 2 f X 1 X 2 ðx 1 ; x 2 ; t 1 ; t 2 Þ d x 1 d x 2 : (6.31)

1

Note that an autocorrelation function is generally dependent on time instants t 1
and t 2 . In many practical problems, solutions can be simplified if we can assume
that an autocorrelation function does not depend on time instants t 1 and t 2 but rather
on their difference t ¼ t 2 t 1 . As mentioned in Sect. 6.3 , a second-order station-
ary process will satisfy this condition. However, knowing that a mean value and
autocorrelation function are the most important characteristics of a random process
it is useful to define a less restrictive form of stationarity involving only those
characteristics in stationary conditions of a process.

6.5.2 WS Stationary Processes

A process is said to be wide sense (WS) stationary if the two following conditions
are satisfied: the mean value is constant and the autocorrelation function depends
only on the difference of the time instants, t ¼ t 2 t 1 :

(a Þ XðtÞ¼ const : (6.32)

1

(b) R XX ðtÞ¼XðtÞXðt þ tÞ¼ x 1 x 2 f X 1 X 2 ðx 1 ; x 2 ; tÞ d x 1 d x 2 : (6.33)

1

Example 6.5.1 Find whether a random process given by

XðtÞ¼X 1 cos o 0 t þ X 2 sin o 0 t (6.34)

is stationary in the wide sense, if o 0 is constant, the variables X 1 and X 2 are
uncorrelated and have the mean values 0, and equal variances:

X 1 ¼ X 2 ¼ 0 ;

s 1 ¼ s 2 ¼ s 2 : (6.35)

Solution In order to determine if the first condition ( 6.32 ) is satisfied, we must find
the mean value of the process:

XðtÞ¼X 1 cos o 0 t þ X 2 sin o 0 t ¼ X 1 cos o 0 t þ X 2 sin o 0 t ¼ 0 : (6.36)