Fast Fourier Transform And Convolution Algorithms

Fast Fourier Transform and Convolution Algorithms

This book presents in a unified way the various fast algorithms that are used for the implementation of digital filters and the evaluation of discrete Fourier transforms. The book consists of eight chapters. The first two chapters are devoted to background information and to introductory material on number theory and polynomial algebra. This section is limited to the basic concepts as they apply to other parts of the book. Thus, we have restricted our discussion of number theory to congruences, primitive roots, quadratic residues, and to the properties of Mersenne and Fermat numbers. The section on polynomial algebra deals primarily with the divisibility and congruence properties of polynomials and with algebraic computational complexity. The rest of the book is focused directly on fast digital filtering and discrete Fourier transform algorithms. We have attempted to present these techniques in a unified way by using polynomial algebra as extensively as possible. This objective has led us to reformulate many of the algorithms which are discussed in the book. It has been our experience that such a presentation serves to clarify the relationship between the algorithms and often provides clues to improved computation techniques. Chapter 3 reviews the fast digital filtering algorithms, with emphasis on algebraic methods and on the evaluation of one-dimensional circular convolutions. Chapters 4 and 5 present the fast Fourier transform and the Winograd Fourier transform algorithm.

Algorithms for Discrete Fourier Transform and Convolution

This book is based on several courses taught during the last five years at the City College of the City University of New York and at Fudan University, Shanghai, China in the Summer, 1986. It was originally our intention to present to a mixed audience of electrical engineers, mathematicians and computer scientists at the graduate level, a collection of algorithms which would serve to represent the vast array of algorithms designed over the last twenty years for com puting the finite Fourier transform (FFT) and finite convolution. However, it was soon apparent that the scope of the course had to be greatly expanded. For researchers interested in the design of new algorithms, a deeper understanding of the basic mathematical con cepts underlying algorithm design was essential. At the same time, a large gap remained between the statement of an algorithm and the implementation of the algorithm. The main goal of this text is to describe tools which can serve both of these needs. In fact, it is our belief that certain mathematical ideas provide a natural lan guage and culture for understanding, unifying and implementing a wide range of digital signal processing (DSP) algorithms. This belief is reenforced by the complex and time-consumming effort required to write code for recently available parallel and vector machines. A significant part of this text is devoted to establishing rules and precedures which reduce and at times automate this task. In Chapter 1, a survey is given of basic algebra.

Fast Fourier Transform and Convolution Algorithms