Gauss and the History of the Fast Fourier Transform


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describing the algorithm was not published in his 266 lifetime; it appeared only in his collected works [10] as an unpublished manuscript.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 …

Showing 1-10 of 40 references

The interaction algorithm and practical Fourier analysis

  • I J Good
  • 1958
Highly Influential
4 Excerpts

A Bibliography of Fast Transform and Convolution Algorithms II

  • M T Heideman, C S Burrus
  • 1984

Inversion of the discrete Gauss transform

  • I J Good
  • 1979

A History of Numerical Analysis from the 16th Through the 19th Century

  • H H Goldstine
  • 1977
2 Excerpts

Index mappings for multidimensional formulation of the OFT and convolut.ion

  • C S Burrus
  • 1977

A fast Gaussian method for Fourier transform evaluation

  • L Hope
  • 1975

The Man and the Physicist

  • J Herivel, Joseph Fourier
  • 1975

FFT pruning

  • J D Markel
  • 1971
1 Excerpt
Showing 1-10 of 73 extracted citations


Citations per Year

161 Citations

Semantic Scholar estimates that this publication has received between 102 and 250 citations based on the available data.

See our FAQ for additional information.