Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 450

or alternatively

8
<

9
=

1

T= 2

ð

1
T

e jot d t: S XX ðoÞ¼ lim

T!1

R XX ðt; t þ tÞ d t (7.27)

: ;

1

T= 2

The expression in the brackets presents a time average of the autocorrelation
function of the process X ( t )

T= 2

ð

1
T

AR XX ðt; t þ tÞ f g ¼ lim

T!1

R XX ðt; t þ tÞ d t: (7.28)

T= 2

If we suppose that the process is at least WS stationary, its autocorrelation function
does not depend on absolute time but only of a time difference t , resulting in:

T= 2

ð

1
T

AR XX ðt; t þ tÞ

g ¼ lim
T!1

R XX ðtÞ d t ¼ R XX ðtÞ:

f (7.29)

T= 2

Placing ( 7.29 ) into ( 7.27 ), we finally get:

1

R XX ðtÞ e jot d t;

S XX ðoÞ¼ (7.30)

1

or equivalently

1

1
2 p

S XX ðoÞ e jot d t: R XX ðtÞ¼ (7.31)

1

The obtained expressions ( 7.30 ) and ( 7.31 ) are known as the Wiener-Khinchin
theorem , which states that a PSD and a autocorrelation function of a process, which
is at least WS stationary, are the Fourier transform pair.
The importance of this theorem is that one can obtain a spectral characteristic
(i.e., the PSD of a random process as the Fourier transform of an autocorrelation
function), which itself is a deterministic function that represents the process.
Consequently, the Wiener-Khinchin theorem solves all problems related to the
application of the Fourier transform to a random process, mentioned in Sect. 7.1.2 .

Example 7.1.1 Find a PSD for the process considered in Example 6.5.2, where the
autocorrelation is obtained in the expression ( 6.77 ).