Digital Signal Processing Reference
In-Depth Information
Denote
=
ˆ
Q
(
ˆ
(
β
)
β
)
G
,K
,K
N
ˆ
|
W
i
|
(
β
−
sgn
(
W
i
)
X
i
)
=
2
=
0
.
(5.93)
K
2
ˆ
2
+|
W
i
|
(
sgn
(
W
i
)
X
i
−
β)
i
=
1
Following a similar analysis as in the weight update, we can develop a
K
update algorithm. However, two reasons make it more attractive to update
the
squared linearity parameter
=
K
2
instead of
K
itself. First, in myriad
filters,
K
always occurs in its squared form. Second, the adaptive algorithm
for
K
might have an ambiguity problem in determining the sign of
K
. Rewrite
Equation 5.93 as
K
N
ˆ
|
|
(
β
−
(
)
)
W
i
sgn
W
i
X
i
ˆ
(
β
K
)
=
=
.
G
,
2
0
ˆ
2
K +|
W
i
|
(
sgn
(
W
i
)
X
i
−
β)
i
=
1
Implicitly differentiating both sides with respect to
K
,wehave
∂
∂
ˆ
G
β
∂
K
∂
G
∂
K
.
+
=
0
,
(5.94)
ˆ
∂
β
thus,
∂
G
∂
K
∂
ˆ
β
∂
K
=−
∂
.
(5.95)
G
ˆ
∂
β
Finally, the update for
K
can be expressed as
1
2
µ
∂
J
(
W
,
K
)
K
(
n
+
1
)
= K
(
n
)
−
(
n
)
i
i
∂
K
E
e
ˆ
)
∂
β
K
∂
K
(
= K
(
n
)
−
µ
(
n
n
)
.
(5.96)
Figure 5.15 depicts a blind equalization experiment where the constellation
of the signal is BPSK, and the channel impulse response is simply [1 0.5].
Additive stable noise with
002 corrupts the transmitted data.
Figure 5.15a is the traditional linear CMA equalization, while Figure 5.15b
is the proposed myriad CMA equalization. It can be seen that, under the
influence of impulsive noise, the linear equalizer diverges, but the myriad
equalizer is more robust and still gives very good performance. Figure 5.16
shows the adaptation of parameter
K
in the corresponding realization.
α
=
1
.
5,
γ
=
0
.
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