Correntropy: Implications of nonGaussianity for the moment expansion and deconvolution

Accepted versio

An Examination of Some Signi cant Approaches to Statistical Deconvolution

We examine statistical approaches to two significant areas of deconvolution - Blind Deconvolution (BD) and Robust Deconvolution (RD) for stochastic stationary signals. For BD, we review some major classical and new methods in a unified framework of nonGaussian signals. The first class of algorithms we look at falls into the class of Minimum Entropy Deconvolution (MED) algorithms. We discuss the similarities between them despite differences in origins and motivations. We give new theoretical results concerning the behaviour and generality of these algorithms and give evidence of scenarios where they may fail. In some cases, we present new modifications to the algorithms to overcome these shortfalls. Following our discussion on the MED algorithms, we next look at a recently proposed BD algorithm based on the correntropy function, a function defined as a combination of the autocorrelation and the entropy functiosn. We examine its BD performance when compared with MED algorithms. We find that the BD carried out via correntropy-matching cannot be straightforwardly interpreted as simultaneous moment-matching due to the breakdown of the correntropy expansion in terms of moments. Other issues such as maximum/minimum phase ambiguity and computational complexity suggest that careful attention is required before establishing the correntropy algorithm as a superior alternative to the existing BD techniques. For the problem of RD, we give a categorisation of different kinds of uncertainties encountered in estimation and discuss techniques required to solve each individual case. Primarily, we tackle the overlooked cases of robustification of deconvolution filters based on estimated blurring response or estimated signal spectrum. We do this by utilising existing methods derived from criteria such as minimax MSE with imposed uncertainty bands and penalised MSE. In particular, we revisit the Modified Wiener Filter (MWF) which offers simplicity and flexibility in giving improved RDs to the standard plug-in Wiener Filter (WF)

An Analysis Of Unsupervised Signal Processing Methods In The Context Of Correlated Sources

In light of the recently proposed generalized correlation function named correntropy, which exploits higher-order statistics and the time structure of signals, we have, in this work, two main objectives: 1) to give a new interpretation - founded on the relationships between the constant modulus (CM) and Shalvi-Weinstein criteria and between the latter and methods for ICA based on nongaussianity - to the performance of the constant modulus approach under dependent sources and 2) to analyze the correntropy in the context of blind deconvolution of i.i.d. and dependent sources, as well as to establish elements of a comparison between it and the CMA. The analyses and simulation results unveil some theoretical aspects hitherto unexplored. © Springer-Verlag Berlin Heidelberg 2009.54418289Godard, D., Self-recovering equalization and carrier tracking in two-dimensional data communication systems (1980) IEEE Trans. on Communications, 28 (11), pp. 1867-1875Shalvi, O., Weinstein, E., New criteria for blind deconvolution of nonminimum phase systems (channels) (1990) IEEE Trans. on Information Theory, 36 (2), pp. 312-321Axford, R., Milstein, L., Zeidler, J., The effects of PN sequences on the misconvergence of the constant modulus algorithm (CMA) (1998) IEEE Trans. on Signal Processing, 46 (2), pp. 519-523Santamaria, I., Pokharel, P., Principe, J., Generalized correlation function: Definition, properties, and application to blind equalization (2006) IEEE Trans. on Signal Processing, 54 (6), pp. 2187-2197Attux, R., Neves, A., Duarte, L., Suyama, R., Junqueira, C., Rangel, L., Dias, T., Romano, J., On the relationship between blind equalization and blind source separation Part I: Foundations and Part II: Relationships (2009) Journal of Communication and Information Systems, , to appearJohnson, C., Schniter, P., Endres, T., Behm, J., Brown, D., Casas, R., Blind equalization using the constant modulus criterion: A review (1998) Proceedings of the IEEE, 86 (10), pp. 1927-1950LeBlanc, J., Dogancay, K., Kennedy, R., Johnson, R., Effects of input data correlation on the convergence of blind adaptive equalizers (1994) IEEE International Conference on Acoustic, Speech and Signal Processing, 3, pp. 313-316Treichler, J., Agee, B., New approach to multipath correction of constant modulus signals (1983) IEEE Trans. on Acoustic, Speech and Signal Processing, ASSP-31 (2), pp. 459-172Regalia, P., On the equivalence between the Godard and Shalvi Weinstein schemes of blind equalization (1999) Signal Processing, 73 (1-2), pp. 185-190Principe, J., Xu, D., Fischer, J., Information Theoretic Learning (2000) Unsupervised Adaptive Filtering, , Haykin, S, ed, Wiley, New YorkHyvarinen, A., Karhunen, J., Oja, E., (2001) Independent Component analysis, , Wiley Interscience, Hoboke