Mathematics Of Multidimensional Fourier Transform Algorithms

The main emphasis of this book is the development of algorithms for processing multi-dimensional digital signals, and particularly algorithms for multi-dimensional Fourier transforms, in a form that is convenient for writing highly efficient code on a variety of vector and parallel computers.

The main emphasis of this book is the development of algorithms for processing multi-dimensional digital signals, and particularly algorithms for multi-dimensional Fourier transforms, in a form that is convenient for writing highly efficient code on a variety of vector and parallel computers.

Developing algorithms for multi-dimensional Fourier transforms, this book presents results that yield highly efficient code on a variety of vector and parallel computers. By emphasising the unified basis for the many approaches to both one-dimensional and multidimensional Fourier transforms, this book not only clarifies the fundamental similarities, but also shows how to exploit the differences in optimising implementations. It will thus be of great interest not only to applied mathematicians and computer scientists, but also to seismologists, high-energy physicists, crystallographers, and electrical engineers working on signal and image processing.

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.

This reference presents a more efficient, flexible, and manageable approach to unitary transform calculation and examines novel concepts in the design, classification, and management of fast algorithms for different transforms in one-, two-, and multidimensional cases. Illustrating methods to construct new unitary transforms for best algorithm sele

Are some areas of fast Fourier transforms still unclear to you? Do the notation and vocabulary seem inconsistent? Does your knowledge of their algorithmic aspects feel incomplete? The fast Fourier transform represents one of the most important advancements in scientific and engineering computing. Until now, however, treatments have been either brief, cryptic, intimidating, or not published in the open literature. Inside the FFT Black Box brings the numerous and varied ideas together in a common notational framework, clarifying vague FFT concepts. Examples and diagrams explain algorithms completely, with consistent notation. This approach connects the algorithms explicitly to the underlying mathematics. Reviews and explanations of FFT ideas taken from engineering, mathematics, and computer science journals teach the computational techniques relevant to FFT. Two appendices familiarize readers with the design and analysis of computer algorithms, as well. This volume employs a unified and systematic approach to FFT. It closes the gap between brief textbook introductions and intimidating treatments in the FFT literature. Inside the FFT Black Box provides an up-to-date, self-contained guide for learning the FFT and the multitude of ideas and computing techniques it employs.

The author captures the interplay between mathematics and the design of effective numerical algorithms.