Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 469

Comparing ( 7.89 ) and ( 7.90 ), we see that:

S YX ðoÞ¼S XY ðoÞ: (7.91)

P.3. From ( 7.88 ). we have:

S XY ðoÞ¼j s 2

2 dðo o 0 Þj s 2 2 dðo þ o 0 Þ;

¼j s 2

2 dðo þ o 0 Þþj s 2

2 dðo o 0 Þ¼S XY ðoÞ:

ð 7 : 92 Þ

7.4 Operation of Processes

7.4.1 Sum of Random Processes

In many problems, we need to find the spectral characteristics of a process Y ( t )
which is obtained as a sum of two processes X 1 ( t ) and X 2 ( t ), as shown in Fig. 7.15 .

YðtÞ¼X 1 ðtÞþX 2 ðtÞ: (7.93)

Consider that processes X 1 ( t ) and X 2 ( t ) are both at least WS stationary processes.
Consequently, their sum will also be WS stationary (i.e., the process Y ( t ) will also
be at least WS stationary).
The autocorrelation function of the process Y ( t ) is:

R YY ðtÞ¼EYðtÞYðt þ tÞ f g ¼EX 1 ðtÞþX 2 ðtÞ f ð Þ X 1 ðt þ tÞþX 2 ðt þ tÞ ð Þ g;

¼ EX 1 ðtÞX 1 ðt þ tÞ f g þEX 2 ðtÞX 2 ðt þ tÞ f g

þEX 1 ðtÞX 2 ðt þ tÞ f g þEX 2 ðtÞX 1 ðt þ tÞ f g: (7.94)

We can recognize the autocorrelation and cross-correlation functions in the
expression ( 7.94 ). Therefore, from ( 7.94 ), it follows:

R YY ðtÞ¼R X 1 X 1 ðtÞþR X 2 X 2 ðtÞþR X 1 X 2 ðtÞþR X 2 X 1 ðtÞ: (7.95)

The PSD of the process Y ( t ) is obtained using the Fourier transform of ( 7.95 )

Fig. 7.15 Sum of
two processes