Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 475

In the expression ( 7.123 ), we can recognize the PSD S XX ( o ):

1

R XX ðgÞ e jog d g: S XX ðoÞ¼ (7.124)

1

Similarly, the second integral in ( 7.123 ) is a Fourier transform of the impulse
response h ( t ) of the filter, which is a transfer function of a filter H ( o ):

1

hðbÞ e job d b: HðoÞ¼ (7.125)

1

Similarly, the first integral in ( 7.123 ) is:

1

hðaÞ e joa d a: HðoÞ¼ (7.126)

1

Finally, placing ( 7.124 )-( 7.126 ) into ( 7.123 ), we arrive at:

S YY ðoÞ¼HðoÞHðoÞS XX ðoÞ: (7.127)

Knowing that a magnitude characteristic of the filter can be expressed as:

2

HðoÞ ¼ HðoÞHðoÞ;

j j (7.128)

we get the final result for the output PSD

2

S YY ðoÞ¼ HðoÞ j j S XX ðoÞ: (7.129)

Note that the final result ( 7.129 ) is real because the PSD is a real function.
From ( 7.129 ), the average power of the output process is:

1 1

1
2 p

1
2 p

2

P YY ¼ S YY ðoÞ d o ¼ j HðoÞ j S XX ðoÞ d o: (7.130)

1 1

Example 7.4.1 Find the mean power at the output of the filter shown in Fig. 7.19 ,if
in the input is a white noise with the PSD equal to N 0 /2.
Solution The transfer function of the filter is:

1
1 þ jo R :

HðoÞ¼ (7.131)