Digital Signal Processing Reference
In-Depth Information
Answer
The PSD is:
2
4
3
5
;
1
ð
1
0
e
3
jtj
e
jot
d
t ¼
5
e
tð
3
joÞ
d
t þ
e
tð
3
þjoÞ
d
t
S
XX
ðoÞ¼
5
1
1
0
1
3
jo
þ
1
3
þ jo
30
9
þ o
2
:
¼
5
¼
ð
7
:
151
Þ
The PSD (Fig.
7.24b
) does not have a delta function for
o ¼
0, which indicates
that the process does not have a finite power at the frequency for
o ¼
0, i.e., it does
not have a DC component.
The process has not delta function at a particular value of
o
,
o 6¼
0
.
This shows
that process does not have a periodic component.
Exercise E.7.7
Find the mean, variance, and autocorrelation function of the output
process
Y
(
t
) from Example 7.4.1.
Answer
The PSD is found to be:
N
0
2
1
1
þ o
R
2
:
S
YY
ðoÞ¼
(7.152)
The autocorrelation function of
Y
(
t
) is the inverse Fourier transform of
S
YY
(
o
)
R
YY
ðtÞ¼F
1
f
S
YY
ðoÞ
g:
(7.153)
In order to use the inverse Fourier transform given in Appendix
E
,
2
k
F
1
¼
e
kjtj
;
k>
0
:
(7.154)
k
2
þ o
2
we rewrite the PSD of (
7.152
):
2
N
0
2
1
1
þ o
R
2
¼
N
0
2
ð
R=L
Þ
N
0
R
4
L
2
R=L
ð
Þ
S
YY
ðoÞ¼
þ o
2
¼
þ o
2
:
(7.155)
2
2
ð
R=L
Þ
ð
R=L
Þ
From here, using (
7.154
), we get:
N
0
R
4
L
e
L
jtj
:
R
YY
ðtÞ¼
(7.156)
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