Topics on Generalized Convolution and Fourier Transforms: Theory and Applications in Digital Signal Processing and System Theory

Abstract

<p>This thesis investigates some aspects of the theory of convolution and Fourier transforms on q-adic and multiplicative abelian groups, as well as their applications for solving various signal processing and system problems.</p> <p>A brief introduction to the importance and basic theory of convolution and Fourier transforms on locally compact abelian groups is given, followed by four major section:</p> <p>1. Subsequent to a comprehensive introduction of generalized Walsh functions, Walsh-Fourier analysis and harmonic differentiation on q-adic groups, a presentation is made of the theory of q-adic translation invariant linear systems from the point of view of both input-output and state-space description. This is followed by an analysis of the structure of Walsh transforms, so that is becomes possible to point to (and critically review) those engineering problems for which Walsh functions are suited to bring an optimal solution, as well as those problems for which they may bring suboptimal but efficient solutions.</p> <p>2. Signal processing in spaces of finite field-valued functions on finite abelian groups is investigated, emphasis being placed on the study of those linear operators whose eigenfunctions are the group characters. A harmonic differential calculus in finite fields is introduced.</p> <p>3. A study of the concept of frequency is undertaken with the objective of generalizing it to function spaces other than that of complex-valued functions on the real line. A generalized concept of frequency is proposed. An analysis of the relationship between the concepts of sequency and frequency proves unfounded the claims that the former is a generalization of the latter.</p> <p>4. The problem of analyzing signals formed of linear combinations of components having the same shape and location but different amplitude and widths parameters is investigated with the objective of providing a technique for its numerical solution. It is shown that this problem can be modelled as a convolution transform on a multiplicative abelian group. A brief introduction to the theory of Fourier transforms on multiplicative groups is presented, followed by the description of an efficient algorithm for performing the analysis. The problems pertaining to the practical implementation of this algorithm are discussed both in general terms and with reference to the analysis of multi-component exponential decays.</p>