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The DFT: An Owners' Manual for the Discrete Fourier Transform

William L. Briggs and Van Emden Henson



Just as a prism separates white light into its component bands of colored light, so the discrete Fourier transform (DFT) is used to separate a signal into its constituent frequencies. Just as a pair of sunglasses reduces the glare of white light, permitting only the softer green light to pass, so the DFT may be used to modify a signal to achieve a desired effect. In fact, by analyzing the component frequencies of a signal or any system, the DFT can be used in an astonishing variety of problems. Among the applications of the DFT are digital signal processing, oil and gas exploration, medical imaging, aircraft and spacecraft guidance, and the solution of differential equations of physics and engineering.

The DFT: An Owner's Manual for the Discrete Fourier Transform explores both the practical and theoretical aspects of the DFT, one of the most widely used tools in science, engineering, and computational mathematics. Designed to be accessible to an audience with diverse interests and mathematical backgrounds, the book is written in an informal style and is supported by many examples, figures, and problems.

Conceived as an "owner's" manual, this comprehensive book covers such topics as the history of the DFT, derivations and properties of the DFT, comprehensive error analysis, issues concerning the implementation of the DFT in one and several dimensions, symmetric DFTs, a sample of DFT applications, and an overview of the FFT.

Audience

This book will appeal to engineers, scientists, and applied and computational mathematicians, and could be used as a textbook for courses in numerical analysis, signal processing, and Fourier analysis. It is recommended that readers have a familiarity with complex numbers, calculus, and linear algebra.

Contents

Preface; Chapter 1: Introduction. A Bit of History; An Application; Problems; Chapter 2: The Discrete Fourier Transform (DFT). Introduction; DFT Approximation to the Fourier Transform; The DFT-IDFT pair; DFT Approximations to Fourier Series Coefficients; The DFT from Trigonometric Approximation; Transforming a Spike Train; Limiting Forms of the DFT-IDFT Pair; Problems; Chapter 3: Properties of the DFT. Alternate Forms for the DFT; Basic Properties of the DFT; Other Properties of the DFT; A Few Practical Considerations; Analytical DFTs; Problems; Chapter 4: Symmetric DFTs. Introduction; Real sequences and the Real DFT (RDFT); Even Sequences and the Discrete Cosine Transform (DST); Odd Sequences and the Discrete Sine Transform (DST); Computing Symmetric DFTs; Notes; Problems; Chapter 5: Multi-dimensional DFTs. Introduction; Two-dimensional DFTs; Geometry of Two-Dimensional Modes; Computing Multi-Dimensional DFTs; Symmetric DFTs in Two Dimensions; Problems; Chapter 6: Errors in the DFT. Introduction; Periodic, Band-limited Input; Periodic, Non-band-limited Input; Replication and the Poisson Summation Formula; Input with Compact Support; General Band-Limited Functions; General Input; Errors in the Inverse DFT; DFT Interpolation - Mean Square Error; Notes and References; Problems; Chapter 7: A Few Applications of the DFT. Difference Equations - Boundary Value Problems; Digital Filtering of Signals; FK Migration of Seismic Data; Image Reconstruction from Projections; Problems; Chapter 8: Related Transforms. Introduction; The Laplace Transform; The z- Transform; The Chebyshev Transform; Orthogonal Polynomial Transforms; The Discrete Hartley Transform (DHT); Problems; Chapter 9: Quadrature and the DFT. Introduction; The DFT and the Trapezoid Rule; Higher Order Quadrature Rules; Problems; Chapter 10: The Fast Fourier Transform (FFT). Introduction; Splitting Methods; Index Expansions (One ---> Multi-dimensional); Matrix Factorizations; Prime Factor and Convolution Methods; FFT Performance; Notes; Problems; Glossary of (Frequently and Consistently Used) Notations; References.

1995 / xvi+434 pages / Softcover / ISBN-13: 978-0-898713-42-8 / ISBN-10: 0-89871-342-0 /
List Price $70.00 / SIAM Member Price $49.00 / Order Code OT45
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