A Generalized Cauchy Distribution Framework for Problems Requiring Robust Behavior
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Research Article A Generalized Cauchy Distribution Framework for Problems Requiring Robust Behavior Rafael E. Carrillo, Tuncer C. Aysal (EURASIP Member), and Kenneth E. Barner Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA Correspondence should be addressed to Rafael E. Carrillo, [email protected] Received 8 February 2010; Revised 27 May 2010; Accepted 7 August 2010 Academic Editor: Igor Djurovi´c Copyright © 2010 Rafael E. Carrillo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Statistical modeling is at the heart of many engineering problems. The importance of statistical modeling emanates not only from the desire to accurately characterize stochastic events, but also from the fact that distributions are the central models utilized to derive sample processing theories and methods. The generalized Cauchy distribution (GCD) family has a closed-form pdf expression across the whole family as well as algebraic tails, which makes it suitable for modeling many real-life impulsive processes. This paper develops a GCD theory-based approach that allows challenging problems to be formulated in a robust fashion. Notably, the proposed framework subsumes generalized Gaussian distribution (GGD) family-based developments, thereby guaranteeing performance improvements over traditional GCD-based problem formulation techniques. This robust framework can be adapted to a variety of applications in signal processing. As examples, we formulate four practical applications under this framework: (1) filtering for power line communications, (2) estimation in sensor networks with noisy channels, (3) reconstruction methods for compressed sensing, and (4) fuzzy clustering.
1. Introduction Traditional signal processing and communications methods are dominated by three simplifying assumptions: (1) the systems under consideration are linear; the signal and noise processes are (2) stationary and (3) Gaussian distributed. Although these assumptions are valid in some applications and have significantly reduced the complexity of techniques developed, over the last three decades practitioners in various branches of statistics, signal processing, and communications have become increasingly aware of the limitations these assumptions pose in addressing many real-world applications. In particular, it has been observed that the Gaussian distribution is too light-tailed to model signals and noise that exhibits impulsive and nonsymmetric characteristics [1]. A broad spectrum of applications exists in which such processes emerge, including wireless communications, teletraffic, hydrology, geology, atmospheric noise compensation, economics, and image and video processing (see [2, 3] and references therein). The need to describe impulsive data, coupled with computational advances that enable processing
of models more complicated than the
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