Digital Signal Processing Reference
In-Depth Information
procedure, carried out iteratively, forces the equalizer outputs to be
uncorrelated.
Consequently, the stochastic gradient of the MUK criterion is given by
[224,225]
N
y
k
(
.
)
2
y
k
(
x
∗
(
∇
J
MUK
(
G
)
=
4sign
(
c
4
[
s
(
n
)
]
)
n
n
)
n
)
(5.123)
k
=
1
We can observe the similarity of the above equation with Equation 4.37.
Hence, at the first step (equalization), an adaptation of
W
is performed
in the direction of the instantaneous gradient, in a very similar way to the SW
algorithm, leading to
(
n
)
W
e
(
n
+
1
)
=
W
(
n
)
+
μ sign
(
c
4
(
s
(
n
)))
x
(
n
)Y(
n
)
,
(5.124)
and μ is a step-size.
Once the equalization stage is carried out, the constraint related to the
orthogonalization of
G
must be respected. In addition, the receiver data must
be whitened, in the space or in both space and time domains, which means
that the matrix
H
must be unitary. The goal of such hypothesis is also to
ensure a constant variance (power) of the transmitted data, hence respecting
the conditions for signal recovery.
The orthogonalization step, for the
k
th user, is given by
y
1
(
y
K
(
2
y
1
(
2
y
K
(
Y(
n
)
=
where
n
)
n
)
···
n
)
n
)
1
w
l
(
)
w
l
(
)
−
k
−
1
l
w
k
(
w
k
(
n
+
1
n
+
1
)
n
+
1
n
+
1
)
=
w
k
(
n
+
1
)
=
, (5.125)
)
−
k
−
1
l
1
w
l
(
)
w
l
(
w
k
(
w
k
(
n
+
1
n
+
1
)
n
+
1
n
+
1
)
=
stands for the
l
2
-norm of the vector.
To carry out this step, it is usual to employ the Schur algorithm [90-92,
134] to the covariance matrix of the noise-free signal
R
·
where
x
, which, due to its
symmetry, provides the following decomposition:
LDL
H
,
R
=
(5.126)
x
where
x
represents the noise-free signals
L
is a unitary matrix
D
is a diagonal matrix with real entries
A noise-free estimation of
E
xx
H
R
x
=
(5.127)
