Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 476

Fig. 7.19 Filter with the input noise

Using ( 7.129 ), we find the PSD at the output of the filter

2

1
1 þ jo R

N 0
2 ¼

1 þ o R 2 N 0

1

S YY ðoÞ¼

2 :

(7.132)

The average power of the output process Y ( t ) is obtained by placing ( 7.132 ) into
( 7.130 ):

1 1

R 2
L 2

1
2 p

N 0
2

1
1 þ o R 2 d o ¼

N 0
4 p

1

P YY ¼

þ o 2 d o:

(7.133)

L 2

R

1 1

Finally, using the integral (7) from Appendix A , we arrive at:

N 0 R
4 L :

P YY ¼ (7.134)

7.5 Numerical Exercises

Exercise E.7.1 The autocorrelation function of a process X ( t ) is given as:

R XX ðtÞ¼Rð 0 Þ e ajtj cos o 0 t (7.135)

where a and o 0 are constants. Find the PSD of the process.
Answer According to the Wiener-Khinchin theorem, a PSD is a Fourier transform
of an autocorrelation function:

1

Rð 0 Þ e ajtj cos ðo 0 tÞ e jot d t: S XX ðoÞ¼FR XX ðtÞ f g ¼ (7.136)

1

We express cos( o 0 t ) in a complex form:

e jo 0 t þ e jo 0 t

2

cos ðo 0 tÞ¼ : (7.137)