Digital Signal Processing Reference
In-Depth Information
From the linearity of the mathematical expectation operator and using the
definition of the autocorrelation function:
NN
1
(
)
()
+
∑∑
(
)
−
j
2
πν
n
−
k
ν
S
=
lim
r
n
−
k e
xx
xx
21
N
N
→∞
nNkN
=−
=−
By changing the variable
κ
=
n
-
k
, the cardinal of the set {(n, k) |
κ
=
n
-
k
and
|
n
|
≤
N
and |
k
|
≤
N
} is 2
N
+ 1 - |
κ
|:
2
N
1
(
)
()
()
+
∑
κ
−
j
2
πνκ
ν
κ
S
=
lim
2
N
+ −
1
r
e
xx
xx
21
N
N
→∞
κ
=−
2
N
Finally we obtain:
2
N
⎛
κ
⎞
()
∑
()
−
j
2
πνκ
S
ν
=
lim
1
−
r
κ
e
⎜
⎟
xx
xx
21
N
+
N
→∞
⎝
⎠
κ
=−
2
N
2
N
1
()
∑
()
−
j
2
πνκ
=
r
ν
−
lim
κ
r
κ
e
xx
xx
21
N
+
N
→∞
κ
=−
2
N
Under the hypothesis [5.10], the second term mentioned above disappears, and
we obtain the formula [5.12].
These considerations can be taken up succinctly for continuous time signals. A
random discrete time signal
x
(
t
),
t
∈
ℜ
is said to be stationary in the wide sense if its
average
m
x
and its autocorrelation function
()
xx
r
τ
defined by:
(
( )
()
⎧ =
⎪
⎨
mEt
x
x
[5.13]
(
)
(
)
(
(
)
()
)
r
τ
=
E
x
t
−
m
*
x
t
+ −
τ
m
∀ ∈ℜ
τ
⎪
⎩
xx
x
x
are independent of the time
t.
For a characterization
x
(
t
),
t
∈
ℜ
of a random signal
x
(
t
), the time average
〈
x
(
t
)
〉
is defined by:
1
T
T
()
()
∫
x t
=
lim
x t dt
[5.14]
2
T
−
T
→∞
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