Digital Signal Processing Reference
In-Depth Information
we write:
2
j
X
T
ðoÞ
j
¼ X
T
ðoÞX
T
ðoÞ
T=
2
ð
T=
2
ð
xðt
1
Þ
e
jot
1
d
t
1
xðt
2
Þ
e
jot
2
d
t
¼
T=
2
T=
2
T=
2
ð
T=
2
ð
xðt
1
Þxðt
2
Þ
e
joðt
2
t
1
Þ
d
t
1
d
t
2
:
¼
ð
7
:
21
Þ
T=
2
T=
2
Using (
7.18
) and (
7.21
) and knowing that expectation is applied to a process
X
(
t
)
we have [PEE93, p. 206]:
T=
ð
2
T=
ð
2
1
T
X
T
ðoÞ
1
T
2
S
XX
ðoÞ¼
lim
T!1
j
j
¼
lim
T!1
Xðt
1
Þ Xðt
2
Þ
e
joðt
2
t
1
Þ
d
t
1
d
t
2
:
(7.22)
T=
2
T=
2
By interchanging the operations of expectation and integration, we have:
T=
2
ð
T=
2
ð
1
T
Xðt
1
ÞXðt
2
Þ
e
joðt
2
t
1
Þ
d
t
1
d
t
2
:
S
XX
ðoÞ¼
lim
T!1
(7.23)
T=
2
T=
2
The expected value in (
7.23
) presents an autocorrelation function:
Xðt
1
ÞXðt
2
Þ¼R
XX
ðt
1
; t
2
Þ;
(7.24)
resulting in:
T=
2
ð
T=
2
ð
1
T
R
XX
ðt
1
; t
2
Þ
e
joðt
2
t
1
Þ
d
t
1
d
t
2
:
S
XX
ðoÞ¼
lim
T!1
(7.25)
T=
2
T=
2
Expressing
t
1
and
t
2
as
t
and
t
+
t
, respectively (where
t ¼ t
2
t
1
), we get:
T=
2
t
ð
T=
2
ð
1
T
R
XX
ðt; t þ tÞ
e
jot
d
t
d
t;
S
XX
ðoÞ¼
lim
T!1
(7.26)
T=
2
t
T=
2
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