Digital Signal Processing Reference
In-Depth Information
a
AUTOCORRELATION FUNCTION
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−
5
−
4
−
3
−
2
−
1
0
1
2
3
4
5
τ
b
PSD
0.8
0.7
0.6
0.5
0.4
0.3
π
/2
δ
(
ω
)
0.2
0.1
0
−5
0
5
ω
Fig. 7.23
(
a
) ACF and (
b
) PSD for a nonzero mean process
where
l ¼
0.5.
Answer
The PSD is a Fourier transform of an autocorrelation function:
1
1
4
ð
1
þ
e
2
ljtj
Þ
e
jot
d
t;
S
XX
ðoÞ¼
1
2
4
3
5
;
1
ð
1
0
1
4
1
4
e
jot
d
t þ
e
ð
2
ljoÞt
d
t þ
e
ð
2
lþjoÞt
d
t
¼
1
1
0
p
2
dðoÞþ
1
4
1
2
l jo
þ
1
2
l þ jo
p
2
dðoÞþ
l
¼
¼
þ o
2
:
ð
7
:
149
Þ
4
l
2
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