Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 485

Answer We have:

1

e t 2 e jot d t ¼ 2 1
0

e t 2 cos ðotÞ d t: S XX ðoÞ¼ (7.159)

1

Using integral 3 from Appendix A to ( 7.159 ), we arrive at:

p p

p e o 2

e o 2

= 4 = 4

S XX ðoÞ¼ 2 ¼ :

(7.160)

2

Exercise E.7.9 Find the output PSD of the process Y ( t ) at the output of the RC filter
shown in Fig. 7.26 . Find PSD, mean value, and variance of the process Y ( t ).

Consider that a WS stationary process X ( t ) in the input of the filter has the
autocorrelation function

R XX ðtÞ¼ cos ðo 0 tÞ; (7.161)

where o 0 is a constant.
Answer The PSD of the process X ( t ) is (see ( 7.56 )):

S XX ðoÞ¼pdðo þ o 0 Þþdðo o 0 Þ ½ : (7.162)

The transfer function of the filter is:

joRC
1 þ joRC :

HðjoÞ¼ (7.163)

The squared magnitude response is:

2

ðoRCÞ

2

j HðjoÞ j ¼

2 ;

(7.164)

1 þðoRCÞ

or equivalently

2

ðoTÞ

2

j HðjoÞ j ¼

2 :

(7.165)

1 þðoTÞ

Fig. 7.26 RC filter in
Exercise E.7.9