Digital Signal Processing Reference
In-Depth Information
Let us consider the simple case where
x
(
k
)
is a complex exponential function at
the frequency
f
embedded in an additive, complex, centered white noise
b
(
k
)
of
exp
j
variance σ
2
.
Let
Ce
be the complex amplitude of this exponential, the signal is
written in the following vector form:
Ce
φ
j
X
=
E
+
B
[7.7]
f
exp
where
X
and
E
are defined as in equation [7.4] and:
f
exp
(
)
T
(
)
(
)
bk M
,
…
,
bk
1
B
=
−
−
The signal covariance matrix is written:
2
H
2
I
[7.8]
RE
=
C
E
+
σ
x
f
f
exp
exp
with
I
being the identity matrix of dimension
M
x
M.
Using the Sherman-Morrison formula [GOL 85], we deduce the inverse of
autocorrelation matrix:
⎛
2
2
⎞
C
⎜
⎟
1
σ
−
1
⎜
H
⎟
R
=
I
−
E
E
[7.9]
x
f
f
2
⎜
2
⎟
exp
exp
σ
C
1
+
M
⎜
⎟
2
⎝
σ
⎠
Let us assume:
2
2
C
σ
Q
=
[7.10]
2
C
1
+
M
2
σ
2
C
Let us note that
represents the signal to noise ratio. By substituting equations
2
σ
[7.9] and [7.10] into equation [7.5], we obtain the expression of the MV filter
impulse response:
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