Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 461

Fig. 7.12 Autocorrelation function and PSD of a sinusoidal random process: ( a ) ACF ( b ) PSD

The corresponding PSD is obtained using the Fourier transform fromAppendix E :

A 2
2 pdðo o 0 Þþdðoþo 0 Þ

A 2
4

S XX ðoÞ¼ ½ ¼ ½ dð f f 0 Þþdð f þ f 0 Þ : (7.56)

The autocorrelation function and the corresponding PSD are shown in Fig. 7.12 .
As expected, a delta function in the PSD indicates that all power is concentrated at
the frequency f 0 and that the power is finite at that frequency. This is because the
random process has only one frequency which is f 0 . On the other hand, this confirms
that a spectral density shows how the power of a random signal is distributed over
various frequencies. This process has only one frequency, f 0 , and thus all power
belongs to this frequency. The other delta function at f 0 is used for mathematical
reasons, explained in Sect. 7.1.5 .

7.2.5 Bandwidth of Random Process

There are different definitions of a bandwidth of a random process that are given
below. The definition, which can be used, mainly depends on the application.
However, in all cases, a measured bandwidth is only measured for positive
frequencies [MIL04, p. 380].