Random Processes - Random Signals and Processes Primer with MATLAB - page 386

P.6 If a process does not have a periodic component and its mean value is zero,
then its autocorrelation function approaches to zero for high values of lag t ,

lim R XX ðtÞ¼ 0 ;

(6.69)

jj!1:

However, if a process has a nonzero mean value, then its autocorrelation
function approaches a squared mean value, for high values of

t

,

2

lim R XX ðtÞ¼X ;

(6.70)

jj!1:

The proof is a self-explanatory knowing that the similarity of the amplitudes
decreases as the distance of the time instants t increases.

Example 6.5.2 A random process X ( t ) consists of rectangular pulses with
amplitudes U and U and random lengths, as shown in Fig. 6.9 . The process has
the following characteristics:

• The number of zero crossing in any interval is independent of the zero crossing
in any other interval if the intervals are not overlapped either totally or partially.

Fig. 6.9 Random process
in Example 6.5.2