Digital Signal Processing Reference
In-Depth Information
Using these two results in (4.70) and (4.71), we finally get to
R
−
1
b
w
(
n
+
1
)
=
(4.74a)
w
(
n
+
1
)
(
n
+
)
=
w
1
(4.74b)
w
T
(
n
+
1
)
R
w
(
n
+
1
)
where matrix
R
is square with a dimension equal to the equalizer length,
having elements given by [270]
c
2
x
)
cov
x
)
(
n
−
i
)
;
x
(
n
−
j
(
n
−
i
)
;
x
(
n
−
j
r
ij
=
=
(4.75)
c
2
(
s
(
n
))
var
(
s
(
n
))
and
b
is a column vector with dimension equal to the equalizer length, its
elements being given by
c
y
)
,
y
∗
(
,
x
∗
(
(
n
)
,
y
(
n
)
n
)
n
−
l
b
i
=
(4.76)
c
4
(
s
(
n
))
In the algorithm shown in (4.74a) and (4.74b), we only need to know the
cumulants of the input and to compute the joint cumulant of the input and
output signals. This can be done by means of empirical cumulants, as shown
and discussed in [70,270,271]. In this sense, the property of convergence of
the SEA shows some advantages in terms of speed, and there are no sig-
nificant hypotheses about the signal model, an exception being the usual
non-Gaussianity assumption.
4.6 Analysis of the Equilibrium Solutions of Unsupervised
Criteria
In Chapter 3 we presented and discussed the Wiener theory, founded on
an MSE criterion, the importance of which can be justified on at least two
bases: conceptual soundness and mathematical tractability. Therefore, this
approach establishes a benchmark against which any other solution could
be compared. Now, if the delay is considered an additional free parameter
in the search of the optimal equalizer, the Wiener procedure acquires a mul-
timodal character, as typically occurs with the unsupervised criteria. In order
to well explain such idea, we take the following example.
Let us consider a binary (
1) i.i.d. signal transmitted by a noiseless
FIR channel whose transfer function is
H
+
1/
−
1.5
z
−
1
. We are interested in
(
z
)
=
1
+
