Gauss and the history of the fast Fourier transform

  title={Gauss and the history of the fast Fourier transform},
  author={Michael T. Heideman and Don H. Johnson and C. Sidney Burrus},
  journal={Archive for History of Exact Sciences},
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… 

The techniques of the generalized fast Fourier transform algorithm

It is shown that the equalization of FFTs leads to results which are different from the widely used intuitive ones and the formulae of the method can be easily adapted for deriving algorithms for the cosine/sine DFT.

The Cooley--Tukey FFT and Group Theory

In this paper, some of the recent work on he “separation of variables” approach to computing a Fourier transform on an arbitrary finite group is surveyed, a natural generalization of the Cooley–Tukey algorithm.

50 Years of FFT Algorithms and Applications

A brief overview of the key developments in FFT algorithms along with some popular applications in speech and image processing, signal analysis, and communication systems are presented.

The quick discrete Fourier transform

  • Haitao GuoG. SittonC. Burrus
  • Computer Science
    Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing
  • 1994
An algorithm is developed, called the quick Fourier transform (QFT), that will reduce the number of floating point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths.

In-place Truncated Fourier Transform 3.1 Background 3.1.1 Primitive Roots of Unity and the Discrete Fourier Transform

  • Mathematics
The fast Fourier transform (FFT) algorithm is a crucial operation in many areas of signal processing as well as computer science. Most relevant to our focus, all the asymptotically fastest methods

Some applications of generalized FFT's

The purpose of this paper is to survey some of the applications of generalized FFTs and thereby (hopefully!) motivate further work in this direction.

Discrete Fourier Analysis

Fast diffraction computation algorithms based on FFT

Improvements were the generalization of the DFT to scaled DFT which allowed freedom to choose the dimensions of the output window for the Fraunhofer-Fourier and Fresnel diffraction, the mathematical concept of linearized convolution which thwarts the circular character of the discrete Fourier transform and allows the use of the FFT, and last but not least the linearized discrete scaled convolution.

Engineering a Fast Fourier Transform

It is concluded that mid-low level optimizations can significantly improve an FFT implementation and that a modern general-purpose compiler does not generate optimal code without significant support from the programmer.

Fast Fourier Transforms




An algorithm for the machine calculation of complex Fourier series

Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.

Note on the calculation of Fourier series

A small-computer program has been written in this laboratory which uses the Danielson-Lanezos method with one minor modification, described below, and yields the same results as the binary form of the Cooley-Tukey algorithm with a comparable number of arithmetical operations.

How the Fast Fourier Transform Got its Name

The time has now come to publish this historical account to quiet all rumors and hearsay about the true history of the Fast Fourier Transform.

FFT pruning

It is shown that for situations in which the relative number of zero-valued samples is quite large, significant time-saving can be obtained by pruning the FFT algorithm.

A New Method of Approximate Harmonic Analysis by Selected Ordinates

Assume with Fourier that the curve representing any periodic single-valued function of x may be expressed by the harmonic series; y=A1 sin x + A2 sin 2x + A3 sin 3x +..... + B1 cos x + B2 cos 2x + B3

The Fourier Transform and Its Applications

1 Introduction 2 Groundwork 3 Convolution 4 Notation for Some Useful Functions 5 The Impulse Symbol 6 The Basic Theorems 7 Obtaining Transforms 8 The Two Domains 9 Waveforms, Spectra, Filters and

On computing the Discrete Fourier Transform.

  • S. Winograd
  • Computer Science
    Proceedings of the National Academy of Sciences of the United States of America
  • 1976
New algorithms for computing the Discrete Fourier Transform of n points are described, which use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.

A fast Gaussian method for Fourier transform evaluation

  • L.L. Hope
  • Computer Science
    Proceedings of the IEEE
  • 1975
A Gaussian method for fast evaluation of approximations to Fourier integral transforms is presented. This method is faster than the FFT for transforms of functions that require considerable computer

The inversion of the discrete gauss transform

A study is made of the matrix , especially for the efficient calculation of its inverse and for the solution of the corresponding set of linear equations. Applications are mentioned, especially to

A History of Numerical Analysis from the 16th through the 19th Century.

1. The Sixteenth and Early Seventeenth Centuries.- 1.1. Introduction.- 1.2. Napier and Logarithms.- 1.3. Briggs and His Logarithms.- 1.4. Burgi and His Antilogarithms.- 1.5. Interpolation.- 1.6.