Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 474

Denoting

t 1 ¼ ta;
t 2 ¼ t þ tb; (7.115)

the expected value in the integrals of ( 7.114 ) is the autocorrelation function:

E XðtaÞXðt þ tbÞ f g ¼ E Xðt 1 ÞXðt 2 Þ f g ¼ R XX ðt 1 ; t 2 Þ: (7.116)

If a process X ( t ) is at least WS stationary, then the autocorrelation function
( 7.116 ) depends only on the time difference,

t 2 t 1 ¼ tb þ a; (7.117)

and the autocorrelation function ( 7.116 ) becomes

E XðtaÞXðt þ tbÞ f g ¼ R XX ðtb þ aÞ: (7.118)

Using ( 7.118 ), the output autocorrelation function ( 7.114 ) becomes:

1 1

R YY ðt; t þ tÞ ¼ R YY ðtÞ ¼ R XX ðtb þ aÞhðaÞhðbÞ d a d b: (7.119)

1 1

The output PSD is equal to the Fourier transform of the output autocorrelation
function ( 7.119 ).

1

R YY ðtÞ e jot d t:

S YY ðoÞ ¼ (7.120)

1

Placing ( 7.119 ) into ( 7.94 ), we get:

1 1 1

R XX ðtb þ aÞ e jot S YY ðoÞ ¼ hðaÞ hðbÞ d t d a d b: (7.121)

1 1 1

We introduce the following auxiliary variable into ( 7.121 ):

g ¼ tb þ a; (7.122)

Thus, the expression ( 7.121 ) becomes:

1 1 1

hðaÞ e joa d a hðbÞ e job d b R XX ðgÞ e jog d g: S YY ðoÞ ¼ (7.123)

1 1 1