The DFT
An Owner's Manual for the Discrete Fourier Transform
Will be published approx. on 30. April 1995
Book
Paperback/Softback
449 pages
978-0-89871-342-8 (ISBN)
Description
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.
This book 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.
This book 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.
More details
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Product notice
Paperback (trade)
Dimensions
Height: 255 mm
Width: 173 mm
Thickness: 23 mm
Weight
760 gr
ISBN-13
978-0-89871-342-8 (9780898713428)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
Preface
List of Frequently Used Terms
Chapter 1: Introduction. A Bit of History
An Application
Problems
Chapter 2: The Discrete Fourier Transform . 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: Multidimensional 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. Introduction
Splitting Methods
Index Expansions (One ---> Multi-dimensional)
Matrix Factorizations
Prime Factor and Convolution Methods
FFT Performance
Notes
Problems
Appendix: Table of Discrete Fourier Transforms
Bibliography.
List of Frequently Used Terms
Chapter 1: Introduction. A Bit of History
An Application
Problems
Chapter 2: The Discrete Fourier Transform . 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: Multidimensional 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. Introduction
Splitting Methods
Index Expansions (One ---> Multi-dimensional)
Matrix Factorizations
Prime Factor and Convolution Methods
FFT Performance
Notes
Problems
Appendix: Table of Discrete Fourier Transforms
Bibliography.