Table of Contents

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.

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