Multiplicative Complexity, Convolution, and the DFT

This book is intended to be a comprehensive reference to multiplicative com plexity theory as applied to digital signal processing computations. Although a few algorithms are included to illustrate the theory, I concentrated more on the develop ment of the theory itself. Howie Johnson's infectious enthusiasm for designing efficient DfT algorithms got me interested in this subject. I am grateful to Prof. Sid Burrus for encouraging and supporting me in this effort. I would also like to thank Henrik Sorensen and Doug Jones for many stimulating discussions. lowe a great debt to Shmuel Winograd, who, almost singlehandedly, provided most of the key theoretical results that led to this present work. His monograph, Arithmetic Complexity o/Computations, introduced me to the mechanism behind the proofs of theorems in multiplicative complexity. enabling me to return to his earlier papers and appreciate the elegance of his methods for deriving the theory. The second key work that influenced me was the paper by Louis Auslander and Winograd on multiplicative complexity of semilinear systems defined by polynomials. After reading this paper, it was clear to me that this theory could be applied to many impor tant computational problems. These influences can be easily discerned in the present work.

Texte du rabat

This book introduces multiplicative complexity theory and applies it to the computation of discrete convolutions and the discrete Fourier transform. A mathematical framework is established that enables the multiplicative complexity of many common bilinear systems to be determined. Semilinear systems are introduced as a means of determining the multiplicative complexity of the discrete Fourier transform. Complexity results are derived for other discrete transforms such as the cosine and Hartley transforms. Complete proofs of all the major results are included - the book is essentially self-contained. Many examples are provided to demonstrate the application of the theory. TOC:Contents: Introduction.- Multiplicative Complexity of Linear and Bilinear Systems.- Convolution and Polynomial Multiplication.- Constrained Polynomial Multiplication.- Multiplicative Complexity of Discrete Fourier Transform.- Restricted and Constrained DFTs.- Appendix A. Cyclotomic Polynomials and Their Properties.- Appendix B. Complexities of Multidimensional Cyclic Convolutions.- Appendix C. Programs for Computing Multiplicative Complexity.- Appendix D. Tabulated Complexities of the One-Dimensional DFT.- Problems.- Bibliography.- Index.

Contenu

Contents: Introduction.- Multiplicative Complexity of Linear and Bilinear Systems.- Convolution and Polynomial Multiplication.- Constrained Polynomial Multiplication.- Multiplicative Complexity of Discrete Fourier Transform.- Restricted and Constrained DFTs.- Appendix A. Cyclotomic Polynomials and Their Properties.- Appendix B. Complexities of Multidimensional Cyclic Convolutions.- Appendix C. Programs for Computing Multiplicative Complexity.- Appendix D. Tabulated Complexities of the One-Dimensional DFT.- Problems.- Bibliography.- Index.