Digital Signal Processing Reference
In-Depth Information
5.4.2.1 The Multiuser Kurtosis Algorithm
The multiuser kurtosis maximization (MUK) criterion, proposed in [224,225],
is based on the SW theorem, previously discussed in Chapter 4, and uses a set
of necessary conditions for the blind equalization of several source signals.
Such conditions are the following:
1.
s
l
(
n
)
is i.i.d. and zero mean
(
l
=
1,
...
,
N
)
.
2.
s
l
(
n
)
and
s
q
(
n
)
are statistically independent for
l
=
q
, with the same
pdf.
3.
c
4
y
l
(
)
= |
n
c
4
[
s
(
n
)
]
|
(
l
=
1,
...
,
N
)
.
4.
E
y
l
(
2
σ
s
n
)
=
(
l
=
1,
...
,
N
)
.
5.
E
y
l
(
)
=
n
)
y
q
(
n
0,
l
=
q
.
where
c
4
[
s
] and σ
s
are, respectively, the kurtosis and the variance (power) of
the source signals
c
4
[
(
n
)
·
]
is the kurtosis operator, as defined in Chapter 4
We can note that Condition 5 aims to ensure the same desired condition
in the decorrelation approach defined by (5.94).
Furthermore, the MUK technique does not employ the decorrelation
approach such as the MU-CMA. In order to attain correct identification
of the different signals, the MUK takes an additional criterion based
on the orthogonalization of the combined channel+equalizer matrix
G
in
such a way that
G
H
G
=
I
. In this context, the criterion can be written
as [224,225]
⎧
⎨
N
c
4
y
k
max
G
J
MUK
(
G
)
=
k
=
1
.
(5.122)
⎩
subject to:
G
H
G
=
I
We can then divide this algorithm into two stages:
Equalization step: kurtosis maximization, in the spirit of the SW theorem.
This stage is associated with a matrix
W
e
.
Separation step: is responsible for promoting the uncorrelation of the esti-
mates of the several users. This stage is associated with a matrix
W
.
In order to force the global response matrix to be orthogonal, a Gram-
Schmidt orthogonalization procedure is used in the matrix
W
e
[225]. This
