Abstract
In this work, we perform an analysis, in the context of channel equalization, of two criteria that can be considered central to the field of information theoretic learning (ITL): the minimum error entropy criterion (MEEC) and the maximum correntropy criterion (MCC). An original derivation of the exact cost function of these criteria in the scenario of interest is provided and used to analyze their robustness and efficiency from a number of relevant standpoints. Another important feature of the paper is an study of the estimated versions of these cost functions, which raises several aspects regarding parameters of the canonical Parzen window estimator. The study is carried out for distinct channel and noise models, both in the combined response and parameter spaces, and also employs as benchmarks crucial metrics like the probability of bit error. The conclusions indicate under what conditions ITL criteria are particularly reliable and a number of factors that can lead to suboptimal performance.
Highlights
E EQUALIZATION was one of the first applications of adaptive filtering [1], and still is a domain of great interest due to its importance in communication systems, audio and image signal processing, seismology, among others
After defining the signal models explored in the channel equalization problem, we present a general procedure to obtain the probability density functions (PDFs) of the error signal as a function of the coefficients of the equalizer and of the parameters that describe the scenario, which shall serve as the basis for determining the Renyi’s quadratic entropy of the error signal, as well as the correntropy function considering two types of noise models, viz., the additive white Gaussian noise (AWGN) and an additive white impulsive noise (AWIN)
An immediate difference to be remarked is related to the type of information that each version of the information theoretic learning (ITL) criteria has access to: while the theoretical expressions of the error entropy and correntropy explore information about the set of possible transmitted sequences si(n), i = 0, . . . , 2K+D−1 − 1, and require knowledge of properties related to the noise and to the source, the estimators are based on a set of T observations of the error signal and use a single parameter, viz., the kernel size, whose value must be adequately selected [11]
Summary
E EQUALIZATION was one of the first applications of adaptive filtering [1], and still is a domain of great interest due to its importance in communication systems, audio and image signal processing, seismology, among others. The equalizer adopted in this work consists of a linear finite impulse response (FIR) filter, as it constitutes the classical equalization structure, for which a solid theory has been developed [10], and due to its mathematical tractability, which is essential for enabling the derivation of the signal PDFs. The proposed theoretical development provides answers to important questions that hitherto, to the best knowledge of the authors, have not yet been fully addressed in the literature, like: are these criteria suitable to solve the problem of supervised channel equalization, that is, do their minima really present good solutions in terms of reducing ISI or minimizing the probability of errors at the adaptive equalizer output?
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