Digital Signal Processing Reference
In-Depth Information
be based on searching for good equalization solutions as well as for a paral-
lel between them and CM minima. Over all, perhaps the main conclusion to
be drawn from the comparison between supervised and blind criteria is that
of not thinking of a local minimum of a blind criterion necessarily as a bad
minimum, especially for the CM criterion, which does not possess spurious
minima like those found in approaches like DD [99].
4.8 Concluding Remarks
In this chapter, we stated the problem of unsupervised signal processing,
in particular that of channel equalization, in the context of SISO systems.
This scenario of interest was introduced after an initial and more general
presentation on unsupervised deconvolution, where we discussed the role of
second-order statistics in order to pave the way to the subsequent theoretical
results.
The central theoretical results are essentially established by the theo-
rems of Benveniste-Goussat-Ruget and Shalvi-Weinstein, enounced in this
chapter. Such theorems clearly establish the need of higher-order statistics to
provide equalization in an unsupervised mode.
The chapter particularly focuses on the methods that make an implicit
use of the higher-order statistics, since our major interest is dealing with
adaptive techniques. The Bussgang algorithms constitute a large family of
techniques, among which we emphasize DD, Sato, and Godard/CMA algo-
rithms. Also, we took the SW theorem as a starting point to derive SW
constrained and unconstrained algorithms, as well as the super-exponential.
If the first part of the chapter was devoted to presentation and derivation
of the main algorithms, the second part was concerned with the analysis of
the corresponding criteria.
First, we exposed the most classical results of the literature about the
equilibrium points of DD and CM criteria. Later, we discussed equivalence
relationships between criteria. In addition to the elegance of some results
presented in the last sections, it is worth pointing out that such analyses
are important to provide an adequate understanding about the operation of
the algorithms and, as a consequence, to envisage a suitable practical use
of them.
