Algorithms for Discrete Fourier Transform and Convolution

Transcription

1 Algorithms for Discrete Fourier Transform and Convolution Second Edition Springer Science+Business Media, LLC

2 Signal Processing and Digital Filtering Synthetic Aperture Radar J.P. Fitch Multiplicative Complexity, Convolution and the DFf M.T. Heideman Array Signal Processing S.U. Pillai Maximum Likelihood Deconvolution J.M. Mendel Algorithms for Discrete Fourier Transform and Convolution Second Edition T. Tolimieri, M. An, and C. Lu Algebraic Methods for Signal Processing and Communications Coding R.E. Blahut Electromagnetic Devices for Motion Control and Signal Processing Y.M. Pulyer Mathematics of Multidimensional Fourier Transform Algorithms Second Edition R. Tolimieri, M. An, and C. Lu Lectures on Discrete Time Filtering R.S. Bucy Distributed Detection and Data Fusion P.K. Varshney

3 Richard Tolimieri Myoung An Chao Lu Algorithms for Discrete Fourier Transform and Convolution Second Edition C.S. Burrus Consulting Editior Springer

4 Richard Tolimieri Department of Electrical Engineering City College of CUNY New York, NY 10037, USA MyoungAn A.J. Devaney Associates 52 Ashford Street Allston, MA 02134, USA Chao Lu Department of Computer and Information Sciences Towson State University Towson, MD 21204, USA Consulting Editor Signal Processing and Digital Filtering C.S. Burrus Professor and Chairman Department of Electrical and Computer Engineering Rice University Houston, TX , USA Library of Congress Cataloging-in-Publication Data Tolimieri, Richard, Algorithms for discrete Fourier transform and convolution / Richard Tolimieri, Myoung An, Chao Lu. p. cm. - (Signal processing and digital filtering) Includes bibliographical references (p. ) and index. ISBN (alk. paper) I. Fourier transformations- Data processing. 2. Convolutions (Mathematics)- Data processing. 3. Digital filters (Mathematics) I. An, Myoung. 11. Lu, Chao Title. IV. Series. QA403.5.T '.723-dc Printed on acid-free paper Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1997 Softcover reprint of the hardcover 2nd edition 1997 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone. Production managed by Anthony Battle; manufacturing supervised by Johanna Tschebull. Photocomposed pages prepared from the authors' ~1EJX files ISBN ISBN (ebook) DOI /

5 Preface This book is based on several courses taught during the years at the City College of the City University of New York and at Fudan University, Shanghai, China, in the summer of 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 that would serve to represent the vast array of algorithms designed over the last twenty years for computing 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 concepts 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 that can serve both of these needs. In fact, it is our belief that certain mathematical ideas provide a natural language and culture for understanding, unifying and implementing a wide range of digital signal processing (DSP) algorithms. This belief is reinforced by the complex and time-consuming effort required to write code for recently available parallel and vector machines. A significant part of this text is devoted to establishing rules and procedures that reduce and at times automate this task. In chapter 1, a survey is given of basic algebra. It is assumed that much of this material is not newj in any case, the facts are easily described. The tensor product is introduced in chapter 2. The importance of the tensor product will be a reoccurring theme throughout this text. Tensor product factors have a direct interpretation as machine instructions on many vector

6 vi Preface and parallel computers. Tensor product identities provide linguistic rules for manipulating algorithms to match specific machine characteristics. Inherent in these rules are certain permutations, called stride permutations, which can be impiemented on many machines by a single instruction. The tedious effort of readdressing, required in many DSP algorithms, is greatly reduced. Also, the data ßow is highlighted, which is especially important on supercomputers where the data ßow is usually the major factor that determines the efficiency of the computation. The design of fast FFT algorithms can be dated back historically to the times of Gauss (1805) [1]. The collected work of some unpublished manuscripts by Gauss contained the essentials of the Cooley-Tukey FFT algorithm, but it did not attract much attention. In 1965, when Cooley and Tukey published their paper [2], known as the fast Fourier transform algorithm, the computing science started a revolutionary era. Since then, many variants of the Cooley-Tukey FFT algorithm have been developed. In chapters 3 and 4 of this book, the Cooley-Tukey FFT algorithm, along with its many variants, are unified under the banner of tensor product. From one point of view, these algorithms depend on mapping a one-dimensional array of data onto a multidimensional array of data (depending on the degree of compositeness of the transform size). Using tensor product we need to derive only the simplest case of mapping into a two-dimensional array. Tensor product identities can then be used to derive the general case. An explicit description of the data ßow is automatically given along with rules for varying this data ßow, if necessary. In chapter 5, the Good-Thomas prime factor algorithm is reformulated by tensor product. In chapters 6 and 7, various linear and cyclic convolution algorithms are described. The Chinese Remainder Theorem (CRT) for polynomials is the major tool. Matrix and tensor product formulations are used wherever possible. Results of Cook and Toom and Winograd are emphasized. The integer CRT is applied in chapter 7 to build a large convolution algorithm from efficient small convolution algorithms (Agarwal-Cooley). The scene changes in chapter 8. Various multiplicative FFT algorithms (depending on the ring structure of the indexing set) are described. The prime size algorithms are due to Rader. Winograd generalized Rader's method to composite transform size. We emphasize a variant of the Rader Winograd method. Tensor product language is used throughout, and tensor product identities serve as powerful tools for obtaining several variants that offer arithmetic and data ßow options. In chapter 13, we consider the duality between periodic and decimated data established by the Fourier transform. This duality is applied in the computation of the Fourier transform of odd prime power transform sizes, say pk. The ring structure of the indexing set has an especially simple ideal structure (local ring). The main result decomposes the computation of the Fourier transform into two pieces. The first is a Fourier transform of transform size pk-2. The description of the second is the main objective

7 Preface vii of chapters 14 and 15, where we introduce the theory of multiplicative characters and derive formulas for computing the Fourier transform of multiplicative characters. The authors are indebted to the patience and knowledge of L. Auslander, J. Cooley and S. Winograd, who, Over the years at IBM, Yorktown Heights, and at the City University of New York, have taken time to explain their works and ideas. The authors wish to thank C. S. Burrus, who read the manuscript of the book and suggested many improvements. We wish to thank DARPA for its support during the formative years of writing this book, and AFOSR for its support during the last two years, in which time the ideas of this book have been tested and refined in applications to electromagnetics, multispectral imaging, and imaging through turbulance. Whatever improvements in this revision are due to the routines written in these applications. Richard Tolimieri Myoung An Chao Lu

8 Contents Preface v 1 Review of Applied Algebra Introduction The Ring of Integers The Ring ZjN Chinese Remainder Theorem (CRT) Unit Groups Polynomial Rings Field Extension The Ring F[xl/ f(x) CRT for Polynomial Rings References Problems Tensor Product and Stride Permutation Introduction Tensor Product Stride Permutations Multidimensional Tensor Products Vector Implementation Parallel Implementation References Problems

9 x Contents 3 Cooley-Tukey FFT Algorithms 3.1 Introduction 3.2 Basic Properties of FT Matrix 3.3 An Example of an FT Algorithm 3.4 Cooley-Tukey FFT for N = 2M 3.5 Twiddle Factors 3.6 FT Factors. 3.7 Variants of the Cooley-Thkey FFT 3.8 Cooley-Thkey FFT for N = M L. 3.9 Arithmetic Cost References. Problems 4 Variants of FT AJgorithms and Implementations 4.1 Introduction 4.2 Radix-2 Cooley-Thkey FFT Algorithm 4.3 Pease FFT Algorithm 4.4 Auto-Sorting FT Algorithm 4.5 Mixed Radix Cooley-Tukey FFT. 4.6 Mixed Radix Agarwal-Cooley FFT 4.7 Mixed Radix Auto-Sorting FFT 4.8 Summary References Problems 5 Good-Thomas PFA 5.1 Introduction 5.2 Indexing by the CRT 5.3 An Example, N = Good-Thomas PFA for the General Case 5.5 Self-Sorting PFA References Problems 6 Linear and Cyclic Convolutions 6.1 Definitions Convolution Theorem Cook-Toom Algorithm Winograd Small Convolution Algorithm Linear and CycHc Convolutions Digital Filters References Problems

10 Contents xi 7 Agarwal-Cooley Convolution Algorithm Two-Dimensional Cyclic Convolution Agarwal-Cooley Algorithm References Problems Multiplicative Fourier Transform Algorithm References MFTA: The Prime Case 9.1 The Field Z/p The Fundamental Factorization Rader's Algorithm Reducing Additions Winograd Small FT Algorithm Summary... References.... Problems MFTA: Product of Two Distinct Primes 10.1 Basic Algebra Transform Size: Fundamental Factorization: Variants: Transform Size: pq Fundamental Factorization: pq Variants Summary References.... Problems MFTA: Composite Size 11.1 Introduction Main Theorem Product of Three Distinct Primes Variants Transform Size: Transform Size: 4p, p odd prime Transform Size: References Problems. 12 MFTA: r? 12.1 Introduction 12.2 An Example:

11 xii Contents 12.3 The General Case: r? 12.4 An Example: 33.. References. Problems Periodization and Decimation 13.1 Introduction Periodic and Decimated Data FT of Periodic and Decimated Data The Ring Z/pm... Problems Multiplicative Characters and the FT 14.1 Introduction Periodicity Periodic Multiplicative Characters Periodization and Decimation 14.3 F(P) of Multiplicative Characters F(pm) of Multiplicative Characters Primitive Multiplicative Characters Nonprimitive Multiplicative Characters Orthogonal Basis Diagonalizing F(P) Orthogonal Basis Diagonalizing F(pm) Orthogonal Basis of W Orthogonal Diagonalizing Basis References. Problems Rationality 15.1 An Example: Prime Case An Example: Transform Size: p Exponential Basis 15.6 Polynomial Product Modulo a Polynomial 15.7 An Example: 33 References.... Index