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Crystal Cooley Biography & Facts

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=N1N2{\displaystyle N=N_{1}N_{2}} in terms of N1 smaller DFTs of sizes N2, 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 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 DFTs due to the 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. Another participant at that meeting, Richard Garwin of IBM, recognized the potential of the method and put Tukey in touch with Cooley. However, Garwin made 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 the simultaneous development of Analog-to-digital converters capable of sampling at rates up to 300 kHz. 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 only the 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 (working only for sizes that have relatively prime factors and relying on the Chinese remainder theorem, unlike the support for any composite size in Cooley–Tukey). The radix-2 DIT case A radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley–Tukey algorithm, although highly optimized Cooley–Tukey implementations typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT of size N into two interleaved DFTs (hence the name "radix-2") of size N/2 with each recursive stage. The discrete Fourier transform (DFT) is defined by the formula: Xk=∑n=0N−1xne−2πiNnk,{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}e^{-{\frac {2\pi i}{N}}nk},}where k{\displaystyle k} is an integer ranging from 0 to N−1{\displaystyle N-1}. Radix-2 DIT first computes the DFTs of the even-indexed inputs (x2m=x0,x2,…,xN−2){\displaystyle (x_{2m}=x_{0},x_{2},\ldots ,x_{N-2})} and of the odd-indexed inputs (x2m+1=x1,x3,…,xN−1){\displaystyle (x_{2m+1}=x_{1},x_{3},\ldots ,x_{N-1})}, and then combines those two results to produce the DFT of the whole sequence. This idea can then be performed recursively to reduce the overall runtime to O(N log N). This simplified form assumes that N is a power of two; since the number of sample points N can usually be chosen freely by the application (e.g. by changing the sample rate or window, zero-padding, etcetera), this is often not an important restriction. The radix-2 DIT algorithm rearranges the DFT of the function xn{\displaystyle x_{n}} into two parts: a sum over the even-numbered indices n=2m{\displaystyle n={2m}} and a sum over the odd-numbered indices n=2m+1{\displaystyle n={2m+1}}: Xk=∑m=0N/2−1x2me−2πiN(2m)k+∑m=0N/2−1x2m+1e−2πiN(2m+1)k{\displaystyle {\begin{matrix}X_{k}&=&\sum \limits _{m=0}^{N/2-1}x_{2m}e^{-{\frac {2\pi i}{N}}(2m)k}+\sum \limits _{m=0}^{N/2-1}x_{2m+1}e^{-{\frac {2\pi i}{N}}(2m+1)k}\end{matrix}}}One can factor a common multiplier e−2πiNk{\displaystyle e^{-{\frac {2\pi i}{N}}k}} out of the second sum, as shown in the equation below. It is then clear that the two sums are the DFT of the even-indexed part x2m{\displaystyle x_{2m}} and the DFT of odd-indexed part x2m+1{\displaystyle x_{2m+1}} of the function xn{\displaystyle x_{n}}. Denote the DFT of the Even-indexed inputs x2m{\displaystyle x_{2m}} by Ek{\displaystyle E_{k}} and the DFT of the Odd-indexed inputs x2m+1{\displaystyle x_{2m+1}} by Ok{\displaystyle O_{k}} and we obtain: Xk=∑m=0N/2−1x2me−2πiN/2mk⏟DFTofeven−indexedpartofxn+e−2πiNk∑m=0N/2−1x2m+1e−2πiN/2mk⏟DFTofodd−indexedpartofxn=Ek+e−2πiNkOk for k=0,…,N2−1.{\displaystyle {\begin{matrix}X_{k}=\underbrace {\sum \limits _{m=0}^{N/2-1}x_{2m}e^{-{\frac {2\pi i}{N/2}}mk}} _{\mathrm {DFT\;of\;even-indexed\;part\;of\;} x_{n}}{}+e^{-{\frac {2\pi i}{N}}k}\underbrace {\sum \limits _{m=0}^{N/2-1}x_{2m+1}e^{-{\frac {2\pi i}{N/2}}mk}} _{\mathrm {DFT\;of\;odd-indexed\;part\;of\;} x_{n}}=E_{k}+e^{-{\frac {2\pi i}{N}}k}O_{k}\qquad {\text{ for }}k=0,\dots ,{\frac {N}{2}}-1.\end{matrix}}}Note that the equalities hold for k=0,…,N−1{\displaystyle k=0,\dots ,N-1}, but the crux is that Ek{\displaystyle E_{k}} and Ok{\displaystyle O_{k}} are calculated in this way for k=0,…,N2−1{\displaystyle k=0,\dots ,{\frac {N}{2}}-1} only. Thanks to the periodicity of the complex exponential, Xk+N2{\displaystyle X_{k+{\frac {N}{2}}}} is also obtained from Ek{\displaystyle E_{k}} and Ok{\displaystyle O_{k}}: Xk+N2=∑m=0N/2−1x2me−2πiN/2m(k+N2)+e−2πiN(k+N2)∑m=0N/2−1x2m+1e−2πiN/2m(k+N2)=∑m=0N/2−1x2me−2πiN/2mke−2πmi+e−2πiNke−πi∑m=0N/2−1x2m+1e−2πiN/2mke−2πmi=∑m=0N/2−1x2me−2πiN/2mk−e−2πiNk∑m=0N/2−1x2m+1e−2πiN/2mk=Ek−e−2πiNkOk{\displaystyle {\begin{aligned}X_{k+{\frac {N}{2}}.... Discover the Crystal Cooley popular books. Find the top 100 most popular Crystal Cooley books.