Arce G.R. Nonlinear Signal Processing: A Statistical Approach

John Wiley, 2005. — 484 p. — ISBN10: 0471676241, ISBN13: 978-0471676249.Linear filters today enjoy a rich theoretical framework based on the early and important contributions of Gauss (1795) on Least Squares, Wiener (1949) on optimal filtering, and Widrow (1970) on adaptive filtering. Linear filter theory has consistently provided the foundation upon which linear filters are used in numerous practical applications as detailed in classic treatments including that of Haykin, Kailath, and Widrow. Nonlinear signal processing, however, offers significant advantages over traditional linear signal processing in applications in which the underlying random processes are non-Gaussian in nature, or when the systems acting on the signals of interest are inherently nonlinear. Practice has shown that nonlinear systems and non-Gaussian processes emerge in a broad range of applications including imaging, teletraffic, communications, hydrology, geology, and economics. Nonlinear signal processing methods in all of these applications aim at exploiting the system’s nonlinearities or the statistical characteristics of the underlying signals to overcome many of the limitations of the traditional practices used in signal processing.
Traditional signal processing enjoys the rich and unified theory of linear systems. Nonlinear signal processing, on the other hand, lacks a unified and universal set of tools for analysis and design. Hundreds of nonlinear signal processing algorithms have been proposed in the literature. Most of the proposed methods, although well tailored for a given application, are not broadly applicable in general. While nonlinear signal processing is a dynamic and rapidly growing field, large classes of nonlinear signal processing algorithms can be grouped and studied in a unified framework. Textbooks on higher-and lower-order statistics, polynomial filters, neural-networks, and mathematical morphology have appeared recently with the common goal of grouping a "self-contained" class of nonlinear signal processing algorithms into a unified framework of study.
This book focuses on unifying the study of a broad and important class of nonlinear signal processing algorithms that emerge from statistical estimation principles, and where the underlying signals are non-Gaussian processes. Notably, by concentrating on just two non-Gaussian models, a large set of tools is developed that encompasses a large portion of the nonlinear signal processing tools proposed in the literature over the past several decades. In particular, under the generalized Gaussian distribution, signal processing algorithms based on weighted medians and their generalizations are developed. The class of stable distributions is used as the second non-Gaussian model from which weighted myriads emerge as the fundamental estimate from which general signal processing tools are developed. Within these two classes of nonlinear signal processing methods, a goal of the book is to develop a unified treatment on optimal and adaptive signal processing algorithms that mirror those of Wiener and Widrow, extensively presented in the linear filtering literature.
The current manuscript has evolved over several years while the author regularly taught a nonlinear signal processing course in the graduate program at the University of Delaware. The book serves an international market and is suitable for advanced undergraduates or graduate students in engineering and the sciences, and practicing engineers and researchers. The book contains many unique features including:
Numerous problems at the end of each chapter.
Numerous examples and case studies provided throughout the book in a wide range of applications.
A set of 60+ MatLAB software m-files allowing the reader to quickly design and apply any of the nonlinear signal processing algorithms described in the book to an application of interest.
An accompanying MatLAB software guide.
A companion PowerPoint presentation with more than 500 slides available for instruction.
The chapters in the book are grouped into three parts.
Part I provides the necessary theoretical tools that are used later in text. These include a review of non-Gaussian models emphasizing the class of generalized Gaussian distributions and the class of stable distributions. The basic principles of order statistics are covered, which are of essence in the study of weighted medians. Part I closes with a chapter on maximum likelihood and robust estimation principles which are used later in the book as the foundation on which signal processing methods are built upon.
Part II comprises of three chapters focusing on signal processing tools developed under the generalized Gaussian model with an emphasis on the Laplacian model. Weighted medians, L-filters, and several generalizations are studied at length.
Part III encompasses signal processing methods that emerge from parameter esti- The chapter sequence is thus assembled in a self-contained and unified framework mation within the stable distribution framework.
The chapter sequence is thus assembled in a self-contained and unified framework of study.Statistical Foundations
Non-Gaussian Models
Order Statistics
Statistical Foundations of Filtering
Signal Processing with Order Statistics
Median and Weighted Median Smoothers
Weighted Median Filters
Linear Combination of Order Statistics
Myriad Smoothers
Weighted Myriad Filters
Software Guide