Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 445

The inverse Fourier transformation finds the time domain signal f ( t ) from its
spectrum

1

1
2 p

FðoÞ e jot d o: f ðtÞ¼ (7.3)

1

The question that one might naturally ask is: “Can we also apply the Fourier
transformation to the random signals?”
The answer is provided in the following section.

7.1.2 How to Apply the Fourier Transformation
to Random Signals?

The direct application of the Fourier transform to a random process is not possible
for the reasons listed below:

• There is a problem of the existence of the Fourier transform ( 7.2 ), i.e., for a
majority of the realizations of the process, the condition ( 7.2 ) is not satisfied and
the Fourier transform may not exist.
• Even in a case in which the Fourier transform of a particular realization exists, it
is obvious that the result cannot represent the whole process. The Fourier
transform of the individual realizations would be different.

• In the majority of cases, realizations of a process have irregular forms and cannot

be represented in analytical forms and, consequently, there is a problem of
practical calculation of the Fourier transform.

Next we will describe how to overcome these difficulties by applying the Fourier
transform to a random process.

7.1.2.1 Problem of the Existence of the Fourier Transformation

For a particular realization x ( t ) of a random process X ( t ), which exists in the interval
[ 1 , 1 ], the condition

1

jxðtÞj d t<1

(7.4)

1

may not be satisfied and, thus, the Fourier transform does not exist. The mean
energy W of a particular realization would also be infinity

T= 2

ð

x 2 W ¼ lim

T!1

ðtÞ d t:

(7.5)

T= 2