Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 459

Fig. 7.9 Colored noise: ( a ) PSD ( b ) ACF

7.2.3 Nonzero Mean Process

Consider a process X ( t ) which has a nonzero mean value c . Similarly, as in ( 6.59 ),
we can consider the process X ( t ) to be a sum of the constant c and a zero-mean
process X 0 ( t ). In this case, the autocorrelation function of a process X ( t ) has a
constant term which corresponds to a squared mean value denoted as c 2 (see
( 6.60 )). Let us find the PSD for this constant term A ¼ c 2 of the autocorrelation
function, shown in Fig. 7.10 .

1

R XX ðtÞ e jot d t ¼ A 1
1

e jot d t ¼ Adðf Þ: S XX ðf Þ¼ (7.54)

1

This result shows that a spectral component in a PSD, which corresponds to
a nonzero mean, is a delta function at f ¼ 0, as shown in Fig. 7.10b . This is also
in agreement with a duality time-frequency in which a constant PSD corresponds
to an autocorrelation function which is a delta function and vice versa.
A constant autocorrelation function has a corresponding PSD which is a delta
function at f ¼ 0.
An example of the autocorrelation function and the corresponding PSD is shown
in Fig. 7.11 .