Digital Signal Processing Reference
In-Depth Information
by gradually reducing the value of
until a desired level of robustness is
achieved. During the actual filtering process
σ
is fixed. Since this process
strengthens the median-like properties, while weakening the influence of the
FIR filter weights, this procedure is referred to as the
medianization
of a linear
FIR filter.
The median affine filter can also be adaptively optimized under the mean
square error (MSE) criteria in an approach that has been applied to related
filter structures, such as radial basis functions.
13
,
14
σ
Consider first the opti-
mization of
σ
for a fixed set of filter coefficients. To simplify the notation, let
2
. Then, under the MSE criteria, the cost function to be minimized is
γ
=
σ
=
d
E
[
e
2
]
2
]
,
J
(γ )
=
E
[
(
d
−
)
(2.39)
d
is the filtering error, and
E
[
where
e
] stands for the statistical expec-
tation operator. The optimization problem can be stated as the minimization
of
J
=
d
−
·
is restricted to nonnegative, real-valued numbers. Because of
the nonlinearity of the median affine estimate, this is a nonlinear optimization
problem.
Although finding a closed form solution is intractable, an iterative LMS-
type approach can be adopted.
11
,
15
(γ )
, where
γ
Under this approach,
γ
is indexed and
updated according to
)
−
µ
γ
∂
J
∂γ
(
γ(
n
+
1
)
=
γ(
n
n
)
,
(2.40)
where
µ
γ
is the appropriately chosen step size. For the case of positive valued
weights, differentiating
J
, substituting in the above, and
performing some simplification yields the update
(γ )
with respect to
γ
d
)
+
µ
γ
(
(
)
−
(
))
d
n
n
γ(
+
)
=
γ(
n
1
n
γ
2
(
)
n
N
2
R
i,
(δ)
(
d
×
1
w
x
i
(
n
)
−
(
n
))(
x
i
(
n
)
−
x
(δ)
(
n
))
.
(2.41)
i
i
=
At each iteration the positive constraint can be enforced by a simple maximum
operator,
.
Adopting a similar approach for the optimization of the filter weights,
in which
J
γ(
n
+
1
)
=
max
{
γ(
n
+
1
)
,
0
}
(γ )
is differentiated with respect to the filter weights, yields the
update
)
−
µ
w
∂
J
∂w
w
(
+
)
=
w
(
i
(
)
n
1
n
n
(2.42)
i
i
for
i
=
1
,
2
,
...
,N
, where
µ
w
is the step size and
R
i,
(δ)
(
N
J
∂w
∂
d
R
k,
(δ)
(
i
(
n
)
=−
(
d
(
n
)
−
(
n
))
n
)
1
w
n
)(
x
i
(
n
)
−
x
k
(
n
))
.
(2.43)
k
k
=
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