Random Processes - Random Signals and Processes Primer with MATLAB - page 383

or equivalently,

X 1 þ X 2 2 X 1 X 2 0 : (6.47)

From here, we have:

:

X 1 þ X 2 2 X 1 X 2 (6.48)

Both left terms of ( 6.48 ) are the autocorrelation functions for t ¼ 0, as shown
below:

X 1 ðt 1 ÞX 1 ðt 1 þ 0 Þ¼X 1 ¼ R XX ð 0 Þ;
X 2 ðt 2 ÞX 2 ðt 2 þ 0 Þ¼X 2 ¼ R XX ð 0 Þ:

(6.49)

Knowing that the right-side termfrom( 6.48 ) is equal to | R XX ( t )|, where t ¼ t 2 t 1 ,
and using ( 6.49 ), we obtain the desired result:

2 R XX ð 0 Þ 2 R XX ðtÞ j j: (6.50)

P.2 An autocorrelation function is an even function,

R XX ðtÞ¼R XX ðtÞ: (6.51)

For a WS process, we can write:

R XX ðtÞ¼EXðt 1 ÞXðt 2 Þ f g ¼ EXðt 3 ÞXðt 4 Þ f g; (6.52)

where

t 1 ¼ t; t 2 ¼ t þ t: (6.53a)

t 3 ¼ t; t 4 ¼ t t: (6.53b)

Placing ( 6.53b ) into ( 6.52 ), we get:

R XX ðtÞ¼EXðt 3 ÞXðt 4 Þ f g ¼ EXðtÞXðt tÞ f g ¼ R XX ðtÞ; (6.54)

P.3 The value of an autocorrelation function in t ¼ 0 is a mean squared value of a

process, that is a power of the process,

R XX ð 0 Þ¼X 2 ðtÞ: (6.55)