Digital Signal Processing Reference
In-Depth Information
Unsupervis
ed Multichannel Equalization
The blind equalization criteria studied in Chapter 4 for the SISO scenario
consider either implicit or explicit use of higher-order statistics. In such
a framework, the resulting cost function generally presents local minima,
so that the performance of the optimization methods often depends on
appropriate initializations. Another limiting aspect in SISO equalization was
concerned to
equalizability
, in the sense that zero-forcing (ZF) conditions can-
not be perfectly attained in the usual assumptions of FIR channel and FIR
equalizer.
The works on unsupervised processing in multichannel scenarios have
been intensified from the beginning of the 1990s for both practical and
theoretical reasons. From a theoretical standpoint, the use of models with
multiple inputs and/or multiple outputs revealed the possibility of over-
coming the drawbacks of SISO case mentioned above. In fact, two instigating
results have been brought out:
•
Dealing with multichannel configurations, it is possible, under
certain conditions, to attain perfect (ZF) equalization even if both
channel and equalizer are FIR structures.
•
Also, optimal equalizer can be obtained in unsupervised mode from
optimization criteria based only on second-order statistics of the
involved signals.
This last and important result becomes conceptually clearer if we con-
sider, for instance, that in SISO case second-order based methods could be
effective if we disposed of any suitable prior information, e.g., the phase-
response of the channel. In the multichannel case, this additional information
is related to some kind of diversity in the system.
The multichannel structure itself is endowed with an inherent spatial
diversity, which can be in practice associated to the use of multiple trans-
mitter and/or receiver disposed in the space. As presented further in this
chapter, proceeding with the optimization of the parameters of the mul-
tichannel structure, we get to optimal solutions that only depend on the
autocorrelations and cross-correlations of the involved signals.
On the other hand, a multichannel model can be used to represent tem-
poral diversity, which may be provided by the process of oversampling.
Thus, the multichannel model will in fact describe a SISO communication
system, operating with sampling rate higher than symbol rate. As presented
141
