Digital Signal Processing Reference
In-Depth Information
⎧
⎨
T
B
2
x
(
t
);
|
t
| ≤
x
T
(
t
)
=
(4.5)
⎩
0; otherwise
where,
T
is large but not infinite. Now the Fourier Transform can be defined as
∞
x
T
(
t
)
e
−
j
ω
t
dt
=
X
T
(
f
)
(4.6)
−∞
If Eq. (
4.6
) exists, then we can move towards the use of the property of auto-
correlation mentioned before. Considering the integral, we get from Eq. (
4.6
),
∞
∞
2
e
j
ωτ
df
x
T
(
t
)
x
T
(
t
−
τ
)
dt
=
|
X
T
(
f
)
|
(4.7)
−∞
−∞
If
T
, the integration has no meaning, but if the average is taken and then the
limit is applied, we can define
→∞
ξ
(
τ
)as
∞
1
T
2
e
j
ωτ
df
ξ
(
τ
)
=
Lt
|
X
T
(
f
)
|
(4.8)
T
→∞
−∞
Hence, we can have the auto-correlation function, interchanging the order of
integration as
∞
2
E
|
X
T
(
f
)
|
e
j
ωτ
df
R
x
(
τ
)
=
Lt
(4.9)
T
T
→∞
−∞
According to W-K theorem, auto-correlation function and PSD (
G
x
(
f
)) are Furier
Transform pair, i.e.,
R
x
(
∞
G
x
(
f
)
e
j
ωτ
df
. Therefore, from Eq. (
4.9
),
τ
)
=
−∞
2
|
|
E
X
T
(
f
)
=
G
x
(
f
)
Lt
T
→∞
(4.10)
T
If there are samples expressed with
t
=
mT
B
, with
m
=
0,
±
1,
±
2,
...
,
±
M
, i.e.
(2M
+
1) number of samples,
T
=
(2
M
+
1)
T
B
. Then Eq. (
4.10
) becomes
2
|
|
E
X
T
(
f
)
=
G
x
(
f
)
Lt
M
→∞
(4.11)
(
2
M
+
1
)
T
B
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