Digital Signal Processing Reference
In-Depth Information
TABLE 4.1
Estimators and Corresponding Cost Functions Associated with the
Bussgang Algorithms
Algorithm
Estimator ψ
(
y
(
n
))
Cost Function
E
y
)
2
dec
y
dec
(
y
(
n
))
J
DD
=
(
n
)
−
(
n
DD
E
2
)
−
y
)
)
γ sign
y
)
γ sign
y
Sato
(
n
J
Sato
=
(
(
n
(
n
E
y
R
2
2
1
2
)
2
CMA
y
(
n
)
+
R
2
−|
y
(
n
)
|
J
CM
=
(
n
−
4.4 The Shalvi-Weinstein Algorithm
The work by Shalvi and Weinstein [269] was relevant not only as a theoret-
ical foundation to the problem of blind equalization, but also as the source
of important criteria and adaptive methods. The SWA is an implementation
based on the ideas presented in
Section 4.2.2
, and can be derived in two dif-
ferent versions: constrained and unconstrained. In the sequence we develop
both approaches.
4.4.1 Constrained Algorithm
The constrained algorithm can be seen as a direct interpretation of the
Shalvi-Weinstein theorem. First, we should notice that if the variance of the
input and output signals are the same, we should have
i
|
2
g
(
i
)
|
=
1, and
the following relations hold [269]:
•
i
|
4
g
(
i
)
|
≤
1
•
i
|
4
g
(
i
)
|
=
1 if and only if
g
corresponds to a perfect equalization
solution, i.e.,
g
(
n
)
=
...
...
[0,
,0,1,0,
,0]
Since equality only occurs if the ZF condition is attained, the theorem
naturally provides the following equalization criterion:
maximize
J
SW
c
(
)
c
4
y
)
w
(
n
subject to
c
2
y
)
=
(4.33)
(
n
c
2
(
s
(
n
))
Notice that the constraint is equivalent to
i
|
2
g
(
i
)
|
=
1, which is a require-
ment of the SW theorem.
The SWA is then derived employing a stochastic approximation
proposed in [41] to deal with constrained problems. In this kind of
