Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 448

7.1.2.2 How Can We Obtain PSD of the Process from the PSD
of a Particular Realization?

In order to obtain the power spectral density of the process S XX ( o ), we need to find
the average value of power spectral densities over the whole ensemble. In other
words, we have to find an average value of power spectral densities S xx ( o ) for all
realizations:

S XX ðoÞ¼ES xx ðoÞ f g: (7.17)

Therefore, we have:

n o

2

EX T ðoÞ j j

1
T X T ðoÞ

2

S XX ðoÞ¼E lim
T!1

j j ¼ lim

T!1

: (7.18)

T

The mean power of the process is given below:

1

1
2 p

P XX ¼ S XX ðoÞ d o: (7.19)

1

7.1.2.3 How CanWe Find the Spectral Density ( 7.18 ) If the Realizations Have
an Irregular Form and Cannot Be Represented in Analytical Form?

Obviously, in this case, there is a problem related to the practical calculation of the
PSD. The solution is given in the following.

7.1.3 PSD and Autocorrelation Function

As mentioned before, the obtained expression ( 7.18 ) for the PSD is not convenient
for a practical application, because it is necessary to calculate the Fourier transform
of the squared amplitudes of a random process, which usually cannot be expressed
in an analytical form. Consequently, it is of interest to express the PSD in a more
convenient form, as is explained below.
From the Fourier transform of a signal x T ( t ), defined in ( 7.8 ):

1

T= 2

ð

x T ðtÞ e jot d t ¼ xðtÞ e jot d t; X T ðoÞ¼ (7.20)

1

T= 2