Filtering and estimation theory: First-order, polynomial and decentralized signal processing

Signal processing methods have historically been dominated by the assumption that the underlying noise and interference processes are Gaussian. This is usually justified theoretically by the central limit theorem and may be a reasonable assumption for well-conditioned systems. Nevertheless, the environments commonly seen in wireless communications and signal processing applications are often characterized by bursty and impulsive statistics that are decidedly non-Gaussian. The future deployment of global, personal and mobile communication systems vital to the society of the twenty-first century is limited by power and capacity constraints. The trend in communications is towards smaller devices with larger transmission rates and increased resistance to bursty interference and impulsive noise. Under these severe operating conditions, it is well-known that Gaussian-based methods breakdown, calling for the development and the use of robust signal processing techniques that perform efficiently in such environments.; In this dissertation, we first focus on the three noise models that are of interest in current signal processing and communications applications: mixture-densities, multiplicative noise and algebraic-tailed noise. Based on the intimate link between filtering and estimation theory, we exploit the rich theory of robust statistics to build powerful families of first-order and polynomial filters. Generalized Mean-Median, Adaptive Speckle and Robust Meridian filtering theories are proposed for the processing of signal corrupted by mixture densities, multiplicative speckle noise and algebraic-tailed noise, respectively. We utilize analytical tools borrowed from robust statistics to analyze and compare the performances of the developed techniques against the state-of-the-art approaches in the field. Also presented are applications of the developed techniques in varying areas of statistical signal processing. Generalized Mean-Median is successfully applied to image de-noising and sharpening, and frequency selective filtering problems. The developed adaptive speckle suppression filter is successfully applied to ultrasound image enhancement. Analytical analysis and simulation examples on real data show that the new proposed despeckling theory overcomes the drawbacks of the previous methods.; Next addressed is the problem of robust processing in environments characterized by algebraic-tailed processes. A comprehensive new theory of robust signal processing is derived based on a newly established class of densities. The developed theory subsumes previous literature based on the generalized Gaussian density as a limiting case, and is thus more general and powerful (or at least as powerful) as that previously known. Presented analytical analysis and simulations show that the developed theory overcomes many drawbacks of previously reported robust methods. In addition, the theory retains all the advantages of techniques developed from the generalized Gaussian distribution family. Following rigorous analytical analysis, the developed theory is applied to problems in baseband communication models, powerline signal enhancement and frequency selective operations. It is shown that the proposed theory outperforms current state-of-the-art methods.; The related problem of polynomial filtering in heavy-tailed environments is addressed next. Traditional polynomial filtering theory, based on linear combinations of polynomial terms, is able to approximate important classes of nonlinear systems. We develop and analyze a robust polynomial filtering theory which, as shown in this dissertation, has a direct impact on applications such as nonlinear system identification and human vision-based image sharpening. Presented applications show that the proposed methodology successfully address the robustness and accurateness issues associated with the previous techniques.; Following the usage of various noise models and the derived optimal first-order and pol...