Non-uniform discrete Fourier transform

About: Non-uniform discrete Fourier transform is a research topic. Over the lifetime, 4067 publications have been published within this topic receiving 123952 citations.

Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations

Abstract: We review our efforts to apply the nonuniform fast Fourier transform (NUFFT) and related fast transform algorithms to numerical solutions of Maxwell's equations in the time and frequency domains. The NUFFT is a fast algorithm to perform the discrete Fourier transform of data sampled nonuniformly (NUDFT). Through oversampling and fast interpolation, the forward and inverse NUFFTs can be achieved with O(N log/sub 2/ N) arithmetic operations, asymptotically the same as the regular fast Fourier transform (FFT) algorithms. Using the NUFFT scheme, we develop nonuniform fast cosine transform (NUFCT) and fast Hankel transform (NUFHT) algorithms. These algorithms provide an efficient tool for numerical differentiation and integration, the key in the solutions to differential equations and volume integral equations. We present sample applications of these nonuniform fast transform algorithms in the numerical solution to Maxwell's equations.

Fractional Fourier transform of tempered distributions and generalized pseudo-differential operator

Abstract: A brief introduction to the fractional Fourier transform and its basic properties is given. Fractional Fourier transform of tempered distributions is studied. Generalized pseudo-differential operators involving two classes of symbols and fractional Fourier transforms are investigated. An application of the fractional Fourier transform in solving a generalized heat equation is given.

Dilating Gabor transform for the fringe analysis of 3-D shape measurement

Abstract: In order to overcome the limitations of conventional Fourier transform and Gabor transform analyzing nonstationary signals, dilating Gabor transform is applied to analyze the optical fringes of 3-D shape measurement. The dilating Gabor transformation is introduced by using a changeable window of Gaussian function in a conventional Gabor transform. This phase analysis method provides more accurate results than Fourier transform and Gabor transform. Simulation and experimental results are presented that demonstrate the validity of the principle.

The Cooley--Tukey FFT and Group Theory

Abstract: In 1965 J. Cooley and J. Tukey published an article detailing an efficient algorithm to compute the Discrete Fourier Transform, necessary for processing the newly available reams of digital time series produced by recently invented analog-to-digital converters. Since then, the Cooley– Tukey Fast Fourier Transform and its variants has been a staple of digital signal processing. Among the many casts of the algorithm, a natural one is as an efficient algorithm for computing the Fourier expansion of a function on a finite abelian group. In this paper we survey some of our recent work on he “separation of variables” approach to computing a Fourier transform on an arbitrary finite group. This is a natural generalization of the Cooley–Tukey algorithm. In addition we touch on extensions of this idea to compact and noncompact groups. Pure and Applied Mathematics: Two Sides of a Coin The Bulletin of the AMS for November 1979 had a paper by L. Auslander and R. Tolimieri [3] with the delightful title “Is computing with the Finite Fourier Transform pure or applied mathematics?” This rhetorical question was answered by showing that in fact, the finite Fourier transform, and the family of efficient algorithms used to compute it, the Fast Fourier Transform (FFT), a pillar of the world of digital signal processing, were of interest to both pure and applied mathematicians. Mathematics Subject Classification: 20C15; Secondary 65T10.

A virtual instrument for time-frequency analysis

Abstract: A virtual instrument for time-frequency analysis is presented. Its realization is based on an order recursive approach to the time-frequency signal analysis. Starting from the short time Fourier transform and using the S-method, a distribution having the auto-terms concentrated as high as in the Wigner distribution, without cross-terms, may be obtained. The same relation is used in a recursive manner to produce higher order time-frequency representations without cross-terms. Thus, the introduction of this new virtual instrument for time-frequency analysis may be of help to the scientists and practitioners in signal analysis. Application of the instrument is demonstrated on several simulated and real data examples.