Gauss and the History of the Fast Fourier Transform

Abstract

THE fast Fourier transform (Fm has become well known . as a very efficient algorithm for calculating the discrete Fourier Transform (Om of a sequence of N numbers. The OFT is used in many disciplines to obtain the spectrum or . frequency content of a Signal, and to facilitate the computation of discrete convolution and correlation. Indeed, published work on the FFT algorithm as a means of calculating the OFT, by J. W. Cooley and J. W. Tukey in 1965 [1], was a turning point in digital signal processing and in certain areas of numerical analysis. They showed that the OFT, which was previously thought to require N 2 arithmetic operations, could be calculated by the new FFT algorithm using only N log Noperations. This algorithm had a revolutionary effect on many digital processing methods, and remains the most Widely used method of computing Fourier transforms [2]. In their original paper, Cooley and Tukey referred only to I. J. Good's work published in 1958 [3] as having influenced their development. However, It was soon discovered there are major differences between the Cooley-Tukey FFT and the algorithm described by Good, which is now commonly referred to as the prime factor algorithm (PFA). Soon after the appearance of the CooleyTukey paper, Rudnick [4] demonstrated a similar algorithm, based on the work of Danielson and Lanczos [5] which had appeared in 1942. This discovery prompted an investigation into the history of the FFT algorithm by Cooley, Lewis, and Welch [6]. They discovered that the Oanielson-Lanczos paper referred to work by Runge published at the tu rn of the centu ry [7, 8]. The algorithm developed by Cooley and Tukey clearly had its roots in, though perhaps not a direct influence from, the early twentieth century. In a recently published history of numerical analysis [9], H. H. Goldstine attributes to Carl Friedrich Gauss, the eminent German mathematician, an algorithm similar to the FFT for the computation of the coefficients of a finite Fourier series. Gauss' treatise describing the algorithm was not published in his lifetime; it appeared only in his collected works [10] as an unpublished manuscript. The presumed year of the composition of this treatise is 1805, thereby suggesting that efficient algorithms for evaluating

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@inproceedings{Heideman1985GaussAT, title={Gauss and the History of the Fast Fourier Transform}, author={Michael T. Heideman and Don H. Johnson and Sidney Burrus}, year={1985} }