Digital Signal Processing Reference
In-Depth Information
−
)
=
y
1
(
−
)
T
y
(
n
n
−
)
···
y
K
(
n
(5.102c)
=−
2
,
,
2
,
...
(5.102d)
where
is the cross-correlation between the
i
th and
j
th outputs with
delay equal to
r
ik
,
(
n
)
, which corresponds to the
(
i
,
j
)
th element of the matrix
R
y
,
(
.
In terms of performance, the MU-CMA suffers from some aspects that are
related to the structure of the CMA, as discussed in [139, 223]. For instance
we may mention
n
)
,and
p
i
,
(
n
)
is the
i
th column of the matrix
P
(
n
)
•
Relatively low convergence speed.
•
The regularization factor must be chosen in order to take into
account the trade-off between steady-state error and number of erro-
neous recoveries (which is related to the number of non-recovered
sources) [65].
5.4.1.2 The Fast Multiuser Constant Modulus Algorithm
The algorithm proposed in [65] aims to improve the performance of the
MU-CMA at the cost of an increase in the associated computational complex-
ity. Its development is based on a recursive version of the MU-CMA, and the
technique is called fast multiuser constant modulus algorithm (FMU-CMA)
or least-squares with adaptive decorrelation constant modulus algorithm
(LSAD-CMA) [176].
The FMU-CMA also uses the decorrelation approach to force the recovery
of different source signals, and employs a recursive expression to optimize
a time-averaged version of Equation 5.94. The adaptive algorithm can be
described with the aid of the following equations:
R
−
1
w
k
(
n
)
=
xy
,
k
(
n
)
d
xy
,
k
(
n
)
(5.103a)
)
y
k
(
)
2
x
∗
(
x
T
R
xy
,
k
(
n
+
1
)
=
ζ
R
xy
,
k
(
n
)
+
(
1
−
ζ
n
n
)
(
n
)
(5.103b)
N
x
∗
(
d
xy
,
k
(
k
+
1
)
=
ζ
d
xy
,
k
(
n
)
+
(
1
−
ζ
)
ρ
2
y
k
(
n
)
n
)
−
γ
r
ik
(
n
)
p
i
(
n
)
, (5.103c)
i
=
1
i
=
k
are obtained from (5.97).
The recursive procedure in (5.103) improves the convergence speed of
the constant modulus approach at the cost of increasing the implementation
complexity of the algorithm, thus partially solving one of the points raised
where ζ is a smoothing term and
r
ik
(
n
)
and
p
i
(
n
)
