Tafeln und Aufgaben zur Harmonischen Analyse und Periodogrammrechnung

Gauss and the history of the fast fourier transform

“…This algorithm is not as general as the Cooley-Tukey FFT algorithm because it only allows doubling of the original sequence length, whereas the Cooley-Tukey approach efficiently computes the OFT for any multiple of the original length. The work of Runge also influenced Stumpff, who, in his book on harmonic analysis and periodograms [16], gives a doubling and tripling algorithm for the evaluation of harmonic series. Furthermore, on p. 142 of that book, he suggests a generalization to an arbitrary multiple.…”

“…This is also exactly the FFT algorithm derived by . Cooley Modeling underground temperature deviations C. Runge [7] 1903 2ftk All Harmonic analysis of functions K. Stumpff [16] 1939 2nk,3 n k All Harmonic analysis of functions Danielson and 1942 2 n All X-ray diffraction in lanczos [5] crystals l. H. Thomas [13] 1948 Any integer with All Harmonic analysis of relatively prime factors functions I. J. Good [3] 1958 Any integer with All Harmonic analysis of relatively prime factors functions Cooley and 1965 Any composite integer All Harmonic analysis of Tukey [1] functions S. Winograd [14] 1976 Any integer with All Use of complexity theory relatively prime factors for harmonic analysis…”

Historical notes on the fast Fourier transform

“…T o illustrate better how the indexing in the two algorithms differs, the mappings of n and j for the Thomas prime factor algorithm are given in Table I1 for comparison with the indexing described in Table I. The prime factor algorithm can be programmed very easily in a source language like FORTRAN and, therefore, can be used efficiently with a subroutine designed for a number of terms equal to a power of two. For example, if r is a power of 2 and s is any odd number, the subseries (9) can be computed by the power of 2 subroutine.…”

Historical Journey of the Fast Fourier Transform –From Enlightenment Europe via Bletchley Park to the Modern World

“…The Cooley-Tukey algorithm, which does not rely upon any specific factorization of the transform length, may thus be viewed as a simple generalization of this algorithm, as successive application of the doubling algorithm leads directly to the radix-2 version of the Cooley-Tukey algorithm which involves just O(N×log2N) arithmetic operations. Runge's influential work was subsequently picked up and popularized in publications by Karl Stumpff [6] in 1939 and Gordon Danielson and Cornelius Lanczos [7] in 1942, each in turn making contributions of their own to the subject. Danielson and Lanczos, for example, produced reduced-complexity solutions by exploiting symmetries in the transform kernel, whilst Stumpff discussed versions of both the doubling and the related tripling algorithm.…”