Topic

# Rader's FFT algorithm

About: Rader's FFT algorithm is a research topic. Over the lifetime, 576 publications have been published within this topic receiving 11436 citations.

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TL;DR: A new implementation of the real-valued split-radix FFT is presented, an algorithm that uses fewer operations than any otherreal-valued power-of-2-length FFT.

Abstract: This tutorial paper describes the methods for constructing fast algorithms for the computation of the discrete Fourier transform (DFT) of a real-valued series. The application of these ideas to all the major fast Fourier transform (FFT) algorithms is discussed, and the various algorithms are compared. We present a new implementation of the real-valued split-radix FFT, an algorithm that uses fewer operations than any other real-valued power-of-2-length FFT. We also compare the performance of inherently real-valued transform algorithms such as the fast Hartley transform (FHT) and the fast cosine transform (FCT) to real-valued FFT algorithms for the computation of power spectra and cyclic convolutions. Comparisons of these techniques reveal that the alternative techniques always require more additions than a method based on a real-valued FFT algorithm and result in computer code of equal or greater length and complexity.

489 citations

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01 Aug 1984TL;DR: The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied.

Abstract: A fast algorithm has been worked out for performing the Discrete Hartley Transform (DHT) of a data sequence of N elements in a time proportional to Nlog 2 N. The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied. A new timing diagram (stripe diagram) is presented to illustrate the overall dependence of running time on the subroutines composing one implementation; this mode of presentation supplements the simple counting of multiplies and adds. One may view the Fast Hartley procedure as a sequence of matrix operations on the data and thus as constituting a new factorization of the DFT matrix operator; this factorization is presented. The FHT computes convolutions and power spectra distinctly faster than the FFT.

455 citations

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11 Mar 2005

TL;DR: In this article, the authors report on the work of I. I. Schoenberg and his students in the field of algebraic geometry, which is closely related to ours and has supplemented it in certain respects.

Abstract: Introduction. The material I am reporting on here was prepared in collaboration with I. I. Hirschman. It will presently appear in book form in the Princeton Mathematical Series. I wish also to refer at once to the researches of I. J. Schoenberg and his students. Their work has been closely related to ours and has supplemented it in certain respects. Let me call attention especially to an article of Schoenberg [5, p. 199] in this Bulletin where the whole field is outlined and the historical development is traced. In view of the existence of this paper I shall t ry to avoid any parallel development here. Let me take rather a heuristic point of view and concentrate chiefly on trying to entertain you with what seems to me a fascinating subject.

430 citations

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CNET

^{1}TL;DR: A new N = 2n fast Fourier transform algorithm is presented, which has fewer multiplications and additions than radix 2n, n = 1, 2, 3 algorithms, has the same number of multiplications as the Raderi-Brenner algorithm, but much fewer additions.

Abstract: A new N = 2n fast Fourier transform algorithm is presented, which has fewer multiplications and additions than radix 2n, n = 1, 2, 3 algorithms, has the same number of multiplications as the Raderi-Brenner algorithm, but much fewer additions, and is numerically better conditioned, and is performed ‘in place’ by a repetitive use of a ‘butterfly’-type structure.

412 citations

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Rice University

^{1}TL;DR: Two recently developed ideas, the conversion of a discrete Fourier transform to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms.

Abstract: Two recently developed ideas, the conversion of a discrete Fourier transform (DFT) to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms Three transform algorithms are compared in terms of the number of multiplications and additions Timing for a prime factor fast Fourier transform (FFT) algorithm using high-speed convolution, which was programmed for an IBM 370 and an 8080 microprocessor, is presented

331 citations