Random Processes - Random Signals and Processes Primer with MATLAB - page 382

Fig. 6.8 Autocorrelation
functions for slow (R XX ( t ))
and fast (R YY ( t )) processes

1

R YY ðtÞ¼YðtÞYðt þ tÞ¼ y 1 y 2 f X 1 X 2 ðy 1 ; y 2 ; tÞ d y 1 d y 2 : (6.44)

1

From ( 6.43 ) and Fig. 6.7 , we can note that the product x 1 x 2 will be positive in the
majority of realizations, because the amplitudes of x 1 and x 2 will have the same sign
in the majority of cases.
On the contrary, amplitudes of y 1 and y 2 for the process Y ( t ) will sometimes be
with the same and sometimes with a co ntrary sign.
As a consequence, the average value X 1 X 2 will be larger than that of Y 1 Y 2 , for the
same value of t .
Additionally, the random variables X 1 and X 2 still have a high correlation for
higher values of t in contrast with the variables Y 1 and Y 2 where the correlation is
lost for small values of t .
In this way, an autocorrelation function gives us information about whether
process changes slowly or quickly, i.e., it gives us information about the rate of
change of a process.

6.5.4 Properties of Autocorrelation Function
for WS Stationary Processes

An autocorrelation function of a WS stationary process has the following
properties:

P.1 The maximum value of the autocorrelation function is at t ¼ 0,

j R XX ðtÞ j R XX ð 0 Þ: (6.45)

Consider the random variables X 1 and X 2 of the process X ( t ) in the time instants
t 1 and t 2 , respectively. We have:

2

ðX 1 X 2 Þ 0 ; (6.46)