Digital Signal Processing Reference
In-Depth Information
nonzero cumulant of order higher than 2 of
s
(
n
)
and
y
(
n
)
are equal, then the
channel will have been perfectly equalized.
The importance of the SW theorem is notorious as it assures that blind
equalization can be accomplished by simply comparing the cumulants of the
transmitted and equalized signals. In addition, to providing simplified con-
ditions for blind equalization, the theorem gives support to the proposition
of feasible algorithms, as shown further in this chapter.
4.3 Bussgang Algorithms
The Benveniste-Goursat-Ruget and SW theorems clearly state the need
for dealing with the information brought by higher-order statistics of the
involved signals in order to guarantee blind equalization. In this section, we
concentrate our attention on algorithms that make implicit use of higher-
order statistics. These algorithms, commonly referred to as Bussgang algo-
rithms [36,125], employ some sort of prior information about the pdf of the
transmitted sequence
s
to obtain an estimate of the transmitted symbol.
Such estimate plays the role of the “pilot signal” in the adaptive algorithm.
The general form of these algorithms is given by
(
n
)
μ
ψ
)
x
(
n
+
)
=
(
n
)
+
[
y
(
n
)
]−
y
(
n
(
n
)
w
1
w
(4.6)
where
w
(
n
)
is the parameter vector at instant
n
x
is the equalizer input vector
μ is the step-size
(
n
)
This is similar to the least mean square (LMS) update rule given in
(3.65), but the pilot signal is replaced by ψ
, which corresponds to a
memoryless nonlinear estimator for the transmitted signal
s
[
y
(
n
)
]
(
n
)
.
In order to obtain the optimal estimator, let us consider the usual assump-
tion that both channel and equalizer are modeled as linear and time-invariant
systems, so that the equalizer output is given in terms of the combined
channel
+
equalizer response
g
(
n
)
, i.e.,
y
(
n
)
=
w
(
n
)
∗
x
(
n
)
=
w
(
n
)
∗
h
(
n
)
∗
s
(
n
)
=
g
(
n
)
∗
s
(
n
)
(4.7)
