Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 490

Fig. 7.28 Filter in Exercise E.7.15

hðtÞ¼dðtÞdðt TÞ: (7.191)

The transfer function of the filter is obtained by taking the Fourier transform of
both sides of ( 7.191 ):

HðjoÞ¼ 1 e joT ; (7.192)

where

Fhðtfg¼HðjoÞ: (7.193)

The squared magnitude response of the filter is:

¼ 1 cos ðoTÞj sin ðoTÞ

2

2

2

¼ 1 e joT

j HðjoÞ j ½

¼ 21 cos ðoTÞ

2

þ sin 2 ¼ ½ 1 cos ðoTÞ ðoTÞ ½ : ð 7 : 194 Þ

The output power density spectrum is:

2

S YY ðoÞ¼ HðjoÞ j j S XX ðoÞ¼ 2 p 1 cos ðoTÞ ½ dðo þ o 0 Þþdðo þ o 0 Þ ½ : (7.195)

From ( 7.195 ), we conclude that

EYðtfg¼ 0 : (7.196)

The variance (mean power) is obtained in the following:

1

1
2 p

s YY ¼ R YY ð 0 Þ¼

2 p 1 cos ðoTÞ ½ dðo þ o 0 Þþdðo o 0 Þ ½ d o

1

¼ 1 cos ðo 0 TÞþ 1 cos ðo 0 TÞ¼ 21 cos ðo 0 TÞ ½ : ð 7 : 197 Þ