Digital Signal Processing Reference
In-Depth Information
Considering the previous discussion, in [248] it is shown that the iterative
procedure composed of
g
<
3
>
)
ν
=
P
T
(
(4.66)
ν
||
ν
||
g
=
(4.67)
where
g
<
3
>
is defined componentwise as
g
<
3
>
)
k
=|
2
g
(
g
(
k
)
|
(
k
)
(4.68)
will converge to the same stationary points of
f
4
(
g
)
for an initial choice of
a unit-norm vector
g
. It is also demonstrated that this kind of iteration
may converge to the ZF solution at a super-exponential rate, and, hence, the
name of the algorithm.
Since we are dealing with finite length equalizers, it is necessary to look
for a solution in the equalizer parameter space
∈
T
w
be as close as
possible to the solution obtained by the iteration in (4.66), i.e.,
w
such that
H
g
<
3
>
||
2
min
||
H
w
−
(4.69)
w
The solution of this optimization problem is given by
−
1
H
H
g
<
3
>
w
=
H
H
H
(4.70)
and the normalization step in the iteration in (4.66) becomes
1
w
=
√
w
w
(4.71)
w
H
H
H
H
The procedure still depends on the unknown channel and
g
. There-
fore, it is necessary to express these quantities in terms of the cumulants of
the equalizer input and output signals. Employing the cumulant properties
described in Chapter 2, we may obtain the following relationships:
ij
c
x
)
=
,
x
∗
(
T
(
n
−
j
)
n
−
i
c
2
(
s
(
n
))
H
H
(4.72)
where
(
·
)
ij
is the
(
i
,
j
)
th element of the matrix in the argument, and
c
y
)
,
y
∗
(
,
x
∗
(
(
n
)
,
y
(
n
)
n
)
n
−
i
)
−
1
c
s
,
s
∗
(
,
x
∗
(
H
H
g
<
3
>
=
(
n
)
,
s
(
n
)
n
)
n
−
i
H
H
H
(4.73)
