The requirement of causality in system theory is inevitably accompanied by the appearance of certain mathematical operations, namely the Riesz proj- tion,theHilberttransform,andthespectralfactorizationmapping.Aclassical exampleillustratingthisisthedeterminationoftheso-calledWiener?lter(the linear, minimum means square error estimation ?lter for stationary stochastic sequences [88]). If the ?lter is not required to be causal, the transfer function of the Wiener ?lter is simply given by H(?)=? (?)/? (?),where ? (?) xy xx xx and ? (?) are certain given functions. However, if one requires that the - xy timation ?lter is causal, the transfer function of the optimal ?lter is given by 1 ? (?) xy H(?)= P ,?? (??,?] . + [? ] (?) [? ] (?) xx + xx? Here [? ] and [? ] represent the so called spectral factors of ? ,and xx + xx? xx P is the so called Riesz projection. Thus, compared to the non-causal ?lter, + two additional operations are necessary for the determination of the causal ?lter, namely the spectral factorization mapping ? ? ([? ] ,[? ] ),and xx xx + xx? the Riesz projection P .

Although adaptive filtering and adaptive array processing began with research and development efforts in the late 1950's and early 1960's, it was not until the publication of the pioneering books by Honig and Messerschmitt in 1984 and Widrow and Stearns in 1985 that the field of adaptive signal processing began to emerge as a distinct discipline in its own right. Since 1984 many new books have been published on adaptive signal processing, which serve to define what we will refer to throughout this book as conventional adaptive signal processing. These books deal primarily with basic architectures and algorithms for adaptive filtering and adaptive array processing, with many of them emphasizing practical applications. Most of the existing textbooks on adaptive signal processing focus on finite impulse response (FIR) filter structures that are trained with strategies based on steepest descent optimization, or more precisely, the least mean square (LMS) approximation to steepest descent. While literally hundreds of archival research papers have been published that deal with more advanced adaptive filtering concepts, none of the current books attempt to treat these advanced concepts in a unified framework. The goal of this new book is to present a number of important, but not so well known, topics that currently exist scattered in the research literature. The book also documents some new results that have been conceived and developed through research conducted at the University of Illinois during the past five years.

This textbook and reference for graduate level courses in digital signal processing can be used in a variety of courses. It includes details about deterministic signal processing, algorithms for convolution and DFT, multirate DSP, digital filter banks, wavelets and multiresolution analysis.

This book attempts to improve algorithms by novel theories and complex data analysis in different scopes including object detection, remote sensing, data transmission, data fusion, gesture recognition, and medical image processing and analysis. The book is directed to the Ph.D. students, professors, researchers, and software developers working in the areas of digital video processing and computer vision technologies.

A comprehensive and mathematically accessible introduction to digital signal processing, covering theory, advanced topics, and applications.