The DFT
An Owners' Manual for the Discrete Fourier Transform

Explores the practical and theoretical aspects of the DFT, one of the most widely used tools in science and engineering.

William L. Briggs (Author), Van Emden Henson (Author)

9780898713428, Society for Industrial and Applied Mathematics

Paperback / softback, published 1 January 1987

450 pages
25.5 x 17.3 x 2.3 cm, 0.76 kg

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 and engineering.

Preface
1. Introduction. A Bit of History
An Application
Problems
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
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
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
5. Multi-dimensional DFTs. Introduction
Two-dimensional DFTs
Geometry of Two-Dimensional Modes
Computing Multi-Dimensional DFTs
Symmetric DFTs in Two Dimensions
Problems
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
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
8. Related Transforms. Introduction
The Laplace Transform
The z- Transform
The Chebyshev Transform
Orthogonal Polynomial Transforms
The Discrete Hartley Transform (DHT)
Problems
9. Quadrature and the DFT. Introduction
The DFT and the Trapezoid Rule
Higher Order Quadrature Rules
Problems
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.

Subject Areas: Mathematical modelling [PBWH]