# Fast Fourier Transform and Convolution Algorithms

• Henri J. Nussbaumer
Book

Part of the Springer Series in Information Sciences book series (SSINF, volume 2)

1. Front Matter
Pages I-X
2. Henri J. Nussbaumer
Pages 1-3
3. Henri J. Nussbaumer
Pages 4-31
4. Henri J. Nussbaumer
Pages 32-79
5. Henri J. Nussbaumer
Pages 80-111
6. Henri J. Nussbaumer
Pages 112-150
7. Henri J. Nussbaumer
Pages 151-180
8. Henri J. Nussbaumer
Pages 181-210
9. Henri J. Nussbaumer
Pages 211-240
10. Back Matter
Pages 241-250

### Introduction

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.

### Keywords

Fourier Fourier transform algorithms boundary element method complexity computational complexity convolution filtering implementation information number theory presentation residue review techniques

#### Authors and affiliations

• Henri J. Nussbaumer
• 1
1. 1.IBM Centre d’Etudes et RecherchesLa Gaude, Alpes-MaritimesFrance

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-00551-4
• Copyright Information Springer-Verlag Berlin Heidelberg 1981
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-662-00553-8
• Online ISBN 978-3-662-00551-4
• Series Print ISSN 0720-678X