Digital Signal Processing Reference
In-Depth Information
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FIGURE 4.6
Contours of the CM cost function for the AR case.
4.7 Relationships between Equalization Criteria
Some of the previously discussed criteria used in blind equalization, even
though developed from very different motivations, exhibit a close relation-
ship. Interestingly enough, it is also possible to obtain some relationships
between blind equalization criteria and the minima found using the Wiener
criterion. In this section we expose some ideas related to these relationships.
4.7.1 Relationships between the Constant Modulus
and Shalvi-Weinstein Criteria
After having proposed the constrained criterion described in (4.36), Shalvi
and Weinstein analyze it in the combined channel
equalizer domain. They
conclude that the equilibrium points (aside from the trivial null vector) are
twofold: a family of ZF maxima and a set of saddle points. Later, when
unconstrained criteria are discussed, a new analysis leads once more to these
solutions. If we recall the discussion carried out in
Section 4.6.4
, we are
compelled to state that the results of Shalvi and Weinstein are essentially
equivalent to Foschini's analysis of the CM criterion. In other words, both the
CM and SW criteria give rise to equivalent solutions in the combined domain.
In the same paper, more is said about the equivalence between the CM
criterion and the SW methods. Under their fundamental assumptions, Shalvi
and Weinstein show that it is possible to obtain the CM criterion and the
CMA as particular cases of their methods.
These proofs of equivalence are compromised by the limitation of the
scenarios in which they were obtained. Five more years would pass before
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