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The Cooley–Tukey algorithm, named after J.W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size ''N'' = ''N''1''N''2 in terms of smaller DFTs of sizes ''N''1 and ''N''2, recursively, to reduce the computation time to O(''N'' log ''N'') for highly composite ''N'' (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below. Because the Cooley-Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors. The algorithm, along with its recursive application, was invented by Carl Friedrich Gauss. Cooley and Tukey independently rediscovered and popularized it 160 years later. == History == This algorithm, including its recursive application, was invented around 1805 by Carl Friedrich Gauss, who used it to interpolate the trajectories of the asteroids Pallas and Juno, but his work was not widely recognized (being published only posthumously and in neo-Latin).〔Gauss, Carl Friedrich, "(Theoria interpolationis methodo nova tractata )", Werke, Band 3, 265–327 (Königliche Gesellschaft der Wissenschaften, Göttingen, 1866)〕〔Heideman, M. T., D. H. Johnson, and C. S. Burrus, "(Gauss and the history of the fast Fourier transform )," IEEE ASSP Magazine, 1, (4), 14–21 (1984)〕 Gauss did not analyze the asymptotic computational time, however. Various limited forms were also rediscovered several times throughout the 19th and early 20th centuries.〔 FFTs became popular after James Cooley of IBM and John Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer. Tukey reportedly came up with the idea during a meeting of President Kennedy’s Science Advisory Committee discussing ways to detect nuclear-weapon tests in the Soviet Union by employing seismometers located outside the country. These sensors would generate seismological time series. However, analysis of this data would require fast algorithms for computing DFT due to number of sensors and length of time. This task was critical for the ratification of the proposed nuclear test ban so that any violations could be detected without need to visit Soviet facilities.〔Rockmore, Daniel N. , ''Comput. Sci. Eng.'' 2 (1), 60 (2000). (The FFT — an algorithm the whole family can use ) Special issue on "top ten algorithms of the century "()〕 Another participant at that meeting, Richard Garwin of IBM, recognized the potential of the method and put Tukey in touch with Cooley however making sure that Cooley did not know the original purpose. Instead Cooley was told that this was needed to determine periodicities of the spin orientations in a 3-D crystal of Helium-3. Cooley and Tukey subsequently published their joint paper, and wide adoption quickly followed due to simultaneous development of analog to digital converters capable of sampling at the rates as much as of 300KHz. The fact that Gauss had described the same algorithm (albeit without analyzing its asymptotic cost) was not realized until several years after Cooley and Tukey's 1965 paper.〔 Their paper cited as inspiration for work by I. J. Good on what is now called the prime-factor FFT algorithm (PFA);〔 although Good's algorithm was initially thought to be equivalent to the Cooley–Tukey algorithm, it was quickly realized that PFA is a quite different algorithm (only working for sizes that have relatively prime factors and relying on the Chinese Remainder Theorem, unlike the support for any composite size in Cooley–Tukey).〔James W. Cooley, Peter A. W. Lewis, and Peter W. Welch, "Historical notes on the fast Fourier transform," ''Proc. IEEE'', vol. 55 (no. 10), p. 1675–1677 (1967).〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cooley–Tukey FFT algorithm」の詳細全文を読む スポンサード リンク
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