Digital Signal Processing Reference
In-Depth Information
REMARK 3.4. The previous calculations establish the mean square consistency of
the estimator of the mean and provide an expression of
()
→∞
var
and
without making strong hypotheses on the signal. However, if
x
(
n
) - µ is a linear
process, that is to say if:
lim
N
N
µ
ˆ
∞
∑
()
( )
x n
−=
µ
γ
e n
−
k
k
k
=
0
where e(n) is a sequence of random independent variables of zero mean and
variance
2
,
∞
∞
σ
γ
<∞
and
γ
≠
0
, then [BRO 91, theorem 7.1.2]:
∑
∑
k
ek
=−∞
k
=−∞
k
∞
⎛
⎞
as
∑
(
)
( )
N
µµ
ˆ
−
∼
N
⎜
0,
c
m
⎟
xx
⎜
⎟
⎝
⎠
m
=−∞
which means that ˆµ µ
−
is asymptotically distributed according to a normal law.
Once the mean has been estimated, we can subtract it from the measurements to
obtain a new signal
()
ˆ
xn
− .
In what follows, without losing generality, we will
suppose that the process has zero mean. The asymptotic results presented later are
not affected by this hypothesis [POR 94]. Under these conditions, the two natural
estimators of the covariance (or correlation) function are given by:
Nm
−−
1
1
−
∑
()
()( )
[3.12]
cm
=
xnxnm
+
xx
Nm
n
=
0
Nm
−−
1
1
∑
()
()( )
cm
ˆ
=
xnxnm
+
[3.13]
xx
N
n
=
0
The mean of each of these estimators is calculated as follows:
Nm
−−
1
1
−
∑
{
}
{
}
()
()( )
()
cm
=
xnxnm cm
+
=
[3.14]
E
E
xx
xx
Nm
n
=
0
Nm
−−
1
1
m
⎛
⎞
∑
{
}
{
}
()
()( )
()
ˆ
[3.15]
E
cm
=
E
xnxnm
+
= −
1
cm
⎜
⎟
xx
xx
N
N
⎝
⎠
n
=
0
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