Abstract
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, whichmakes it suitable formodeling 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.
Highlights
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
We establish a statistical relationship between the generalized Gaussian distribution (GGD) and generalized Cauchy distribution (GCD) families
Experiments evaluating the robustness of Lorentzian BP in different impulsive sampling noises are presented, comparing performance with traditional compressed sensing (CS) reconstruction algorithms orthogonal matching pursuit (OMP) [38] and basis pursuit denoising (BPD) [34]
Summary
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. (It should be noted that the original authors derived the myriad filter starting from α-stable distributions, noting that there are only two closed-form expressions for α-stable distributions [12, 17, 18].) These estimators provide a robust framework for heavy-tail signal processing problems. In yet another approach, the generalized-t model is shown to provide excellent fits to different types of atmospheric noise [23].
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