Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 473

An at least WS stationary process, which is multiplied by a sinusoidal signal of
the frequency o 0 , has a PSD which has the two scaled original PSDs around the
frequencies o 0 and o 0 .

7.4.3 Filtering of Random Process

We consider here an output autocorrelation and a PSD of an input process X ( t )
filtered by a LTI (linear time invariant) filter with an impulse response h ( t ), as
shown in Fig. 7.18 .
The autocorrelation function of the output signal Y ( t ) is:

R YY ðt; t þ tÞ¼EYðtÞYðt þ tÞ f g: (7.111)

The output Y ( t ) and the input X ( t ) are related by a convolution operation

1

YðtÞ¼XðtÞhðtÞ¼ hðaÞXðt aÞ d a: (7.112)

1

Similarly,

1

Yðt þ tÞ¼ hðbÞXðt þ t bÞ d b: (7.113)

1

From ( 7.112 ) and ( 7.113 ), the autocorrelation function ( 7.111 ) becomes:

8
<

9
=

1 1

R YY ðt; t þ tÞ¼E hðaÞXðt aÞ d a hðbÞXðt þ t bÞ d b

: ;

1 1

1 1

¼ EXðt aÞXðt þ t bÞ f ghðaÞhðbÞ d a d b: (7.114)

1 1

Fig. 7.18 LTI filtering
of process X ( t )