Digital Signal Processing Reference
In-Depth Information
Finally, the update rule is obtained by replacing (4.39) in (4.34) and
removing the expectation operator:
2
y
w
(
n
+
1
)
=
w
(
n
)
+
μsgn [
c
4
(
s
(
n
))
]
|
y
(
n
)
|
(
n
)
x
(
n
)
(4.40a)
⎛
⎞
1
i
⎝
⎠
w
(
n
+
1
)
=
w
(
n
+
1
)
(normalization)
(4.40b)
2
w
(
i
)
where
1 vector containing the present and past samples of the
prewhitened sequence.
Despite the simplicity of the constrained version of the SWA presented
in (4.40), the prewhitening requirement can be very stringent or even pro-
hibitive in some cases. Therefore, in the sequel, we describe an unconstrained
version that does not demand a prewhitening step.
x
(
n
)
is the
M
×
4.4.2 Unconstrained Algorithm
Another possibility, developed in [269], is to transform (4.33) into an uncon-
strained optimization problem. The main idea behind this approach is to
include a penalty term related to the constraint into the cost function. The
derivation starts by considering the following potential function:
f
2
φ
(
g
)
=
F
(
g
)
+
g
f
i
2
4
=
|
g
(
i
)
|
+
|
g
(
i
)
|
(4.41)
i
)
=
i
|
4
and
f
:
where
F
(
g
(
i
)
|
[
∞] →
g
0,
R is a measurable function so that
x
2
l
(
x
)
=
+
f
(
x
)
(4.42)
is monotonically increasing in 0
≤
x
<
1, monotonically decreasing for
x
>
1,
and has a unique maximum at
x
=
1.
has a maximum if and only if
g
represents the
ZF solution, we can notice that [269]
In order to verify if φ
(
g
)
f
i
2
4
φ
(
g
)
=
|
g
(
i
)
|
+
|
g
(
i
)
|
i
2
2
f
i
2
l
i
2
≤
|
g
(
i
)
|
+
|
g
(
i
)
|
=
|
g
(
i
)
|
(4.43)
i
