Hypercubes, Kronecker Products & Sorting in Digital Signal Processing

This book covers recent advances and contributions to the area of digital signal processing. It starts with a revisit and a rewriting of the sampling theorem in the context of signals containing discontinuities. The approach of impulse invariance for converting continuous-time domain filters to discrete-time domain filters is questioned in light of the Mittag-Leffler expansion. Higher quality digital filters than those obtained using the present day approach are studied. General base perfect shuffle transformations are shown to be basic operations prevalent in transform factorisation and parallel processing. Hypercube transformations, Kronecker products and sorting formalism have had a major impact on transformations of generalised spectral analysis, processor architecture, optimal parallel, massively parallel processing and parallel sorting. The objective of the present book is to render some of the authors previously and recently published papers in the domain of digital signal processing and the architecture of parallel digital signal processors into a simpler format for all to read. In the topics covered in this book, matrix formalism is often employed. Hypercubes, the Kronecker product of matrices and matrix operators such as the general base perfect shuffle matrix are powerful mathematical tools that effectively convert sequential information into matrices. Matrix formalism is a powerful mathematical tool. In fact, it may be said that if a picture is worth a thousand words, a matrix is worth a thousand equations. Chapter One deals with a recent paper which reveals an age-old mathematical error in the literature that has, until today, produced nefariously inferior digital filters. The error, which has been shown to erroneously apply Shannons sampling theorem for decades, exists to date in Matlab©. The error is part of the well-known technique of impulse invariance, which transforms analogue continuous-domain filters into digital filters. A correction of the error is proposed, producing a vastly superior digital filter than obtained using the present day impulse invariance approach. Chapter Two deals with radix-2 fast Fourier transform (FFT) factorisation. A unique approach is presented in which the authors alternate between equations and corresponding matrices to develop the factorisation of the discrete Fourier transform (DFT) matrix. In Chapter Three, a generalisation is applied to obtain FFT factorisations to a general radix r. The subject of generalised spectral analysis, including generalised Walsh transform are studied in Chapter Four. Chapter Five presents parallelism in Generalised Spectral Analysis and in particular the Generalized Walsh Chrestenson transform. Optimal parallel and pipelined processors are considered in Chapter Six. Generalised transform factorisation for massive parallelism is covered in Chapter Seven. In Chapter Eight, the authors study Hypercube transformations for massive parallelism. Chapter Nine introduces a generalisation of the Dirac-delta function. Chapter Ten relates to a generalisation of the theory of distributions. New Laplace, Z and Fourier-related transforms, which are results of the proposed generalisation of the Dirac-delta impulse, are presented in Chapter Eleven. Chapter Twelve relates to a Z domain counterpart to Pronys method. Chapter Thirteen presents an approach to Massively Parallel and Comparison-Set Minimized Sorting.