Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 455

Denoting a power, which belongs to a infinitesimal frequency band d f ,as S ( f )d f ,
we get:

ð

f k

P k ¼

Sð f Þ d f ; (7.45)

0

or generally

ð

f

Pð f Þ¼

Sð f Þ d f :

(7.46)

0

From here, we have:

d P ð f Þ
d f

Sð f Þ¼ : (7.47)

The obtained result, shown in Fig. 7.5a , is the spectral density of a random
signal, and is, naturally, defined only for the positive frequencies. However, the
negative frequencies are introduced in a frequency analysis for mathematical
reasons in order to present the Fourier series in exponential form.
Therefore, in order to obtain the mathematical spectral density, all values of S ( f )
are halved, and the top part of the plot is mapped to the left part, resulting in the
diagram shown in Fig. 7.5b .
Therefore,

f k

ð

f k

ð

P k ¼ Sðf Þ d f ¼ S xx ðf Þ d f : (7.48)

0 f k

Fig. 7.5 Physical ( a ) and mathematical spectral density ( b )