William L. Briggs

The DFT : an owner's manual for the discrete Fourier transform

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.

Wavelets and multigrid

This note explores the suggestive similarities between wavelet (multiresolution) and multigrid approaches to general operator equations. After presenting the essentials of wavelet and multigrid methods, it can be shown that classical multigrid methods use near-orthogonal basis functions to provide the same orthogonal decomposition of the fine grid space

Paper: FFTs and three-dimensional Poisson solvers for hypercubes

{\bf \Omega} ^h

The FFT as a multigrid

that multiresolution methods generate for the maximum resolution space

A table of analytical discrete Fourier transforms

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A multigrid tutorial

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A multigrid tutorial: second edition

Abstract This paper investigates the implementation of fast direct methods for solving the three-dimensional Poisson equation on loosely coupled hypercube multiprocessors. As a preliminary step, the problem of computing multiple FFTs is considered and two different algorithms are compared. These algorithms are then used to implement two FFT-based fast Poisson solvers. Proceeding both with experiments and with performance models, these two solvers are studied and compared. No single algorithm is superior in all ranges of the parameters and the best choice depends upon problem size, number of processors and communication costs. In closing, two additional parallel algorithms for solving the Poisson equation are proposed, both of which appear to be competitive.

A multigrid tutorial (2nd ed.)

Apart from the fact that they are both ingenious and remarkably efficient, there would appear to be little kinship between fast Fourier transform (FFT) algorithms and basic multigrid methods. The fast Fourier transform is a powerful direct method for computing the discrete Fourier transform (DFT) of a sequence of complex numbers, whereas multigrid methods are highly refined iterative methods that can be used to approximate the solution of systems of equations. However, on closer examination, there are features of both algorithms that suggest at least a conceptual similarity. This note is written in a heuristic spirit and is intended to offer a motivation for the FFT that allows it to be viewed as a multigrid process. The outcome of this exercise is certainly not a more efficient FFT algorithm. But hopefully, it will serve to demonstrate the way in which multilevel thinking can be used to interpret algorithms and it may lead the way to new algorithms.

Paper: Bluestein's FFT for arbitrary N on the hypercube

Abstract While most people rely on numerical methods (most notably the fast Fourier transform) for computing discrete Fourier transforms (DFTs), there it is still an occasional need to have analytical DFTs close at hand. Such a table of analytical DFTs is provided in this paper, along with comments and observations, in the belief that it will serve as a useful resource or teaching aid for Fourier practitioners.