Digital Signal Processing Reference
In-Depth Information
some statistical properties of the input and reference processes. Under the assump-
tion on the positive definiteness of
R
x
(so that it will be nonsingular), the solution to
(
2.15
)is:
R
−
1
x
w
opt
=
r
x
d
,
(2.16)
which is known as the
Wiener filter
. An alternative way to find it is the following.
Using the definitions (
2.14
)into(
2.4
) results in
E
2
2
r
x
d
w
w
T
R
x
w
J
MSE
(
w
)
=
|
d
(
n
)
|
−
+
.
(2.17)
In addition, it can be easily shown that the following factorization holds:
w
T
R
x
w
2
r
x
d
w
T
R
−
1
r
x
d
R
−
1
−
=
(
−
−
r
x
d
)
−
r
x
d
.
R
x
w
r
x
d
)
(
R
x
w
(2.18)
x
x
Replacing (
2.18
)in(
2.17
) leads to
E
2
w
r
x
d
T
R
x
w
r
x
d
r
x
d
R
−
1
R
−
1
x
R
−
1
x
.
(2.19)
Using the fact that
R
x
is positive definite (and therefore, so is its inverse), it turns out
that the cost function reaches its minimum when the filter takes the form of (
2.16
),
i.e., the Wiener filter. The minimum MSE value (MMSE) on the surface (
2.19
)is:
J
MSE
(
w
)
=
|
d
(
n
)
|
−
r
x
d
+
−
−
x
E
2
E
2
E
2
r
x
d
R
−
1
|
d
opt
(
J
MMSE
=
J
MSE
(
w
opt
)
=
|
d
(
n
)
|
−
r
x
d
=
|
d
(
n
)
|
−
n
.
(2.20)
)
|
x
)
−
d
opt
(
We could have also arrived to this result by noticing that
e
min
(
)
and using the orthogonality principle as in (
2.9
). Therefore, the MMSE is given by
the difference between the variance of the reference signal
d
n
)
=
d
(
n
n
(
n
)
and the variance of
d
opt
(
its optimal estimate
.
It should be noticed that if the signals
x
n
)
(
n
)
and
d
(
n
)
are orthogonal (
r
x
d
=
0), the
2
. This is reasonable
since nothing can be done with the filter
w
if the input signal carries no information
about the reference signal (as they are orthogonal). Actually, (
2.17
) shows that in this
case, if any of the filter coefficients is nonzero, the MSE would be increased by the
term
w
T
R
x
w
, so it would not be optimal. On the other hand, if the reference signal
is generated by passing the input signal through a system
w
T
as in (
2.11
), with the
noise
v
E
|
optimal filter will be the null vector and
J
MMSE
=
d
(
n
)
|
(
n
)
being uncorrelated from the input
x
(
n
)
, the optimal filter will be
E
x
x
T
R
−
1
x
R
−
1
x
w
opt
=
r
x
d
=
(
n
)
(
n
)
w
T
+
v
(
n
)
=
w
T
.
(2.21)
This means that the Wiener solution will be able to identify the system
w
T
with
a resulting error given by
v
E
|
2
=
σ
v
.
Finally, it should be noticed that the autocorrelation matrix admits the eigende-
composition:
(
n
)
. Therefore, in this case
J
MMSE
=
v
(
n
)
|
