Digital Signal Processing Reference
In-Depth Information
where H
L
is defined in (5.44). We can note that (5.60) is equivalent to a ZF
condition. Also, from (5.60), it is possible to notice that we have
L
+
K
−
1
equations and
LP
unknowns. In this system,
L
1
is the number of posi-
tions in the combined channel+equalizer response and
KP
is the length of
the equalizer. Since the number of unknowns should be at least equal to the
number of equations, we find a first constraint on the length of the equalizer,
given by [229]
+
K
−
L
−
1
KP
≥
L
+
K
−
1
⇒
K
≥
(5.61)
P
−
1
where
·
denotes the ceiling function.
5.3 Methods for Blind SIMO Equalization
Previous discussions confirm the possibility of carrying out blind equaliza-
tion in multichannel scenarios by dealing only with second-order statistics.
Nevertheless, it does not discard the possible interest in using higher-order
techniques due, for instance, to their characteristics of robustness. This option
is discussed next, and in the sequence we present the two main second-order
methods, those of subspace decomposition and linear prediction.
5.3.1 Blind Equalization Based on Higher-Order Statistics
The idea of FS equalization was originally conceived as an alternative
to reduce the sensitivity to sampling timing errors and noise amplifica-
tion [184], for both supervised and blind equalizers. In the latter case, a
special attention was paid to the CMA, which is commonly termed FS-CMA.
However, the relationship between oversampling and the multichannel
model opened a new perspective to the study of FS-CMA.
The main characteristic associated with the FS-CMA is the possibil-
ity of effective global convergence. In fact, in the absence of noise, if all
subchannels share no common zeros and the condition about the length of
the equalizer is respected, then all minima of the CM cost function corre-
spond to ZF solutions [184]. In other words, in contrast with the baud-rate
equalization case, the ZF condition is blindly attainable with finite length
equalizers.
Different works contributed to the study of the behavior of the FS-CMA.
In [184], the authors prove the global convergence behavior of FS-CMA
under the aforementioned conditions. Moreover, a brief discussion is pre-
sented about the influence of noise, which violates the ideal conditions, in
the minima location. The discussion about convergence is extended in [95],
which analyzes the equilibrium points of cost functions of other Bussgang
