Digital Signal Processing Reference
In-Depth Information
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
![](Random-Signals-and-Processes-Primer-with-MATLAB_images/img-gen6769.png)
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