Digital Signal Processing Reference
In-Depth Information
ˆ
Recall that the output of the weighted myriad filter is
β
(
W
,
X
)
=
K
arg min
β
Q
(β)
, where
Q
(β)
is given by
N
log[
K
2
2
]
Q
(β)
=
+|
W
i
|·
(
sgn
(
W
i
)
X
i
−
β)
.
(5.80)
i
=
1
The derivative of
ˆ
with respect to the weight
W
i
, holding all other
quantities constant, can be shown to be
30
β
(
W
,
X
)
K
K
2
sgn
ˆ
(
W
i
)(
β
−
sgn
(
W
i
)
X
i
)
−
K
2
ˆ
2
2
(
+|
W
i
|·
(
β
−
sgn
(
W
i
)
X
i
)
)
∂
ˆ
N
j
.
β
K
(
W
,
X
)
=
(5.81)
∂
W
i
ˆ
K
2
−|
|·
(
β
−
(
)
)
2
W
j
sgn
W
j
X
j
1
|
W
j
|
ˆ
(
K
2
+|
W
j
|·
(
β
−
sgn
(
W
j
)
X
j
)
2
)
2
=
Using Equation 5.78, the following adaptive algorithm is obtained to update
the weights
N
i
{
W
i
}
1
:
=
ˆ
)
∂
β
W
i
(
n
+
1
)
=
W
i
(
n
)
−
µ
e
(
n
W
i
(
n
)
,
(5.82)
∂
ˆ
with
given by Equation 5.81.
Considerable simplification of the algorithm can be achieved by just re-
moving the denominator from the update term above; this does not change
the direction of the gradient estimate or the values of the final weights. This
leads to the following computationally attractive algorithm:
(∂
β/∂
W
i
)(
n
)
K
2
sgn
ˆ
(
W
i
)(
β
−
sgn
(
W
i
)
X
i
)
W
i
(
n
+
1
)
=
W
i
(
n
)
−
µ
e
(
n
)
.
(5.83)
K
2
ˆ
2
2
(
+|
|·
(
β
−
(
)
)
)
W
i
sgn
W
i
X
i
It is important to note that the optimal filtering action is independent of
the choice of
K
; the filter only depends on the value of
w
K
2
.Inthis context,
one might ask how the algorithm scales as the value of
K
is changed and how
the step-size
/
µ
and the initial weight vector
w
(
0
)
are changed as
K
is varied.
=
To answer this, let
g
o
w
o,
1
denote the optimal weight vector for
K
=
1.
K
2
. Now consider
two situations. In the first, the algorithm in Equation 5.83 is used with
K
K
2
2
or
g
o
=
Then, from Equation 5.29,
w
o,K
/
=
g
o
/(
1
)
w
o,K
/
=
1,
step-size
. This
is expected to converge to the weights
g
o
.Inthe second, the algorithm uses a
general value of
K
, step-size
µ
=
µ
1
, weights denoted as
g
i
(
n
)
, and initial weight vector
g
(
0
)
µ
=
µ
K
, and initial weight vector
w
K
(
0
)
. Rewrite
Equation 5.83 by dividing throughout by
K
2
and writing the algorithm in
K
2
. This is expected to converge to
w
o,K
K
2
because
terms of an update of
w
/
/
i
K
2
, the above
two situations can be compared and the initial weight vector
w
K
Equation 5.83 should converge to
w
o,K
. Because
g
o
=
w
o,K
/
(
0
)
and the
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