Gauss and the History of the Fast Fourier Transform

Abstract

Introduction The fast FOURIER transform (FFT) has become well known as a very efficient algorithm for calculating the discrete FOURIER transform (D F T)-a formula for evaluating the N FOURIER coefficients from a sequence of N numbers. The DFT 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, the publication of the FFT algorithm as a means of calculating the DFT by J. W. COOLEY • 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 DFT, which was previously thought to require N 2 arithmetic operations, could be calculated by the new FFT algorithm using a number of operations proportional to N log N. This algorithm had a revolutionary effect on the way much of digital processing was done and the FFT remains the most widely used method of computing FOU-RIER transforms [2]. In their original paper COOLEY &TuKEY referred only to the work of I. J. GOOD [3] published in 1958 as influencing 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 COOLEY-TUKEY paper, RUDNICK [4] demonstrated a similar algorithm based on the work of DANIELSON & LANCZOS [5] which had appeared in 1942. This discovery prompted an investigation into the history of the FFT algorithm by COOLEY, LEWIS, & WELCH [6]. They discovered that the DANIELSON-LANCZOS paper referred to the works of RUNGE published at the turn of the century [7, 8]. While not influencing their work directly, the algorithm developed by COOLEY & TUKEY clearly had roots in 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describing the algorithm was not published in his 266 lifetime; it appeared only in his collected works [10] as an unpublished manuscript.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 coefficients of FOURIER series were developed at least a century earlier than …

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