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Topic: Cooley Tukey algorithm

	John Tukey - TheBestLinks.com - John W. Tukey, Algorithm, Bell Labs, Brown University, ...
		John Wilder Tukey (June 16, 1915 - July 26, 2000) was a statistician.
		Born New Bedford, Massachusetts, Tukey obtained a Bachelor of Science degree in 1936 and a Master of Science degree in chemistry in 1937 from Brown University before moving to Princeton University to study for his doctorate in mathematics.
		Retiring in 1985, Tukey died in New Brunswick, New Jersey.
	www.thebestlinks.com /John_W._Tukey.html (304 words)

	Cooley-Tukey FFT algorithm - Wikipedia, the free encyclopedia
		This algorithm, including its recursive application, was already known around 1805 to 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.
		FFTs became popular after J. Cooley of IBM and John W. Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer (including how to arrange for the output to be produced in the natural ordering).
		Cooley and Tukey's 1965 paper reported a running time of 0.02 minutes for a size-2048 complex DFT on an IBM 7094 (probably in 36-bit single precision, ~8 digits).
	en.wikipedia.org /wiki/Cooley-Tukey_FFT_algorithm (2254 words)

	Learn more about Fast Fourier transform in the online encyclopedia. (Site not responding. Last check: 2007-10-03)
		This method (and the general idea of an FFT) was popularized by a publication of J. Cooley and J. Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms).
		Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime n, expresses a DFT of prime size n as a cyclic convolution of (composite) size n - 1, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods).
		Only the Edelman algorithm works equally well for sparse and non-sparse data, however, since it is based on the compressibility (rank deficiency) of the Fourier matrix itself rather than the compressibility (sparsity) of the data.
	www.onlineencyclopedia.org /f/fa/fast_fourier_transform.html (1480 words)

	Cooley-Tukey FFT algorithm -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-03)
		This process is an example of the general technique of (additional info and facts about divide and conquer) divide and conquer algorithms; in many traditional implementations, however, the explicit recursion is avoided, and instead one traverses the computational tree in (additional info and facts about breadth-first) breadth-first fashion.
		Cooley and Tukey's 1965 paper reported a running time of 0.02 minutes for a size-2048 complex DFT on an (additional info and facts about IBM 7094) IBM 7094 (probably in 36-bit (additional info and facts about single precision) single precision, ~8 digits).
		The Stockham auto-sort algorithm (Stockham, 1966) performs every stage of the FFT out-of-place, typically writing back and forth between two arrays, transposing one "digit" of the indices with each stage, and has been especially popular on (additional info and facts about SIMD) SIMD architectures (Swarztrauber, 1982).
	www.absoluteastronomy.com /encyclopedia/C/Co/Cooley-Tukey_FFT_algorithm3.htm (2017 words)

	Fast Fourier transform (Site not responding. Last check: 2007-10-03)
		This article describes the algorithms, of which there are many; see discrete Fourier transform for properties and applications of the transform.
		Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime n, expresses a DFT of prime size n as a cyclic convolution of (composite) size n − 1, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods).
		In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errors growing as O(√n) for the Cooley-Tukey algorithm (Oppenheim and Schafer, 1975).
	www.tocatch.info /en/Fast_Fourier_Transform.htm (2016 words)

	Cooley-Tukey FFT algorithm Info - Encyclopedia WikiWhat.com (Site not responding. Last check: 2007-10-03)
		This algorithm, including its recursive application, was already known around 1805 to 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 Latin); Gauss did not analyze the asymptotic computational time, however.
		FFTs became popular after J. Cooley of IBM and J. Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer (including how to arrange for the output to be produced in the natural ordering).
		The simplest and most common form of the Cooley-Tukey algorithm (moreso in textbooks than in high-performance implementations, however) is called a radix-2 decimation-in-time (DIT) FFT: it divides the problem size into two interleaved halves with each recursive stage.
	www.wikiwhat.com /encyclopedia/c/co/cooley_tukey_fft_algorithm.html (2189 words)

	Cooley-Tukey FFT algorithm (Site not responding. Last check: 2007-10-03)
		cooley cooley volkswagen david lathrop cooley algorithm genetic algorithm algorithm triangle tristrip layout genetic algorithm
		Cooley, Ray - Memories of Life in Chinook, Alberta, Canada in the 1930's Contains personal recollections and photographs.
		Martha Cooley: Der Archivar Rezension von Daniela Ecker in der "Leselust".
	www.serebella.com /encyclopedia/article-Cooley-Tukey_FFT_algorithm.html (2491 words)

	The FFT: Making Technology Fly
		Cooley takes pains to praise the Gentleman-Sande paper, as well as an earlier paper by Sande (who was a student of Tukey's) that was never published.
		In fact, Cooley says, the Cooley-Tukey algorithm could well have been known as the Sande-Tukey algorithm were it not for the "accident" that led to the publication of the now-famous 1965 paper.
		As he recounts it, the paper he co-authored with Tukey came to be written mainly because a mathematically inclined patent attorney happened to attend the seminar in which Cooley described the algorithm.
	www.siam.org /siamnews/mtc/mtc593.htm (1915 words)

	FFT Algorithm Details (Site not responding. Last check: 2007-10-03)
		The algorithm takes advantage of the fact that the discrete Fourier transform (DFT) of a discrete time series with an even number of points is equal to the sum of two DFTs, each half the length of the original.
		For data lengths that are a power of 2, this algorithm is used recursively, each iteration subdividing the data into smaller sets to be transformed.
		For real input data of even lengths, the FFT algorithm also takes advantage of the fact that the real array can be packed into a complex array of half the length, and unpacked at the end, thus cutting the running time in half.
	idlastro.gsfc.nasa.gov /idl_html_help/signal11.html (274 words)

	[No title]
		Typically, it refers to the O(m log m) algorithms since these are the most well known, but that is not an absolute.
		It is the DFT operation that an algorithm implements.
		If they didn't know what algorithm to use, but knew that they wanted you to avoid the direct 'naive' approach, they would just say "use an FFT" and leave the specifics of the implementation up to you.
	www.math.niu.edu /~rusin/known-math/99/fft_what (885 words)

	Fast Fourier Transform : FFT (Site not responding. Last check: 2007-10-03)
		Since the inverse DFT is the same as the DFT, but with the sign of the exponent flipped and a 1/n factor, any FFT algorithm can easily be adapted for it as well.
		Bruun's algorithm was adapted to the mixed-radix case for even n by H. Murakami.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial z
		FFT algorithm">Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime n, expresses a DFT of prime size n as a cyclic convolution of (composite) size n - 1, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods).
	www.termsdefined.net /ff/fft.html (2550 words)

	Cooley-Tukey FFT algorithm (C) - LiteratePrograms
		The Cooley-Tukey FFT algorithm is a popular fast Fourier transform algorithm for rapidly computing the discrete fourier transform of a sampled digital signal.
		The radix-2 decimation in time algorithm uses a divide-and-conquer approach to improve efficiency.
		Finally, this implementation has considerable function call overhead and would ideally be replaced by a simpler algorithm on small subvectors.
	en.literateprograms.org /Cooley-Tukey_FFT_algorithm_(C) (565 words)

	[No title] (Site not responding. Last check: 2007-10-03)
		The most commonly-implemented FFT algorithm is Cooley-Tukey, which takes a DFT of composite size N = N1 * N2 and expresses it as N2 DFTs of size N1 followed by N1 DFTs of size N2 (with some multiplications by phase factors in between).
		The algorithm you are thinking of is just Cooley-Tukey in the special case of N2 = 2, which is known as the "radix-2" Cooley-Tukey algorithm.
		The most commonly-implemented algorithm of this sort is known as the "chirp-z" algorithm and is due to Bluestein.
	www.math.niu.edu /~rusin/known-math/97/fft (575 words)

	DSPCore:ASIC and IP for FFT (Site not responding. Last check: 2007-10-03)
		However, the Cooley-Tukey algorithm requires that it be run in a batch process, in which input samples must first be assembled in a memory.
		The Prime-Factor algorithm is based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley-Tukey but without the twiddle factors.
		The Rader-Brenner algorithm is again a Cooley-Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability.
	dspcore.com /en/technology (762 words)

	DUC: FFT FIltering (Site not responding. Last check: 2007-10-03)
		Bruun's algorithm applies to arbitrary even composite sizes.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial $z^n-1, here into real-coefficient polynomials of the form z^m-1 and z^{2m} + az^m + 1 .$
		Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime $n, expresses a DFT of prime size n as a cyclic convolution of (composite) size n-1, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods).$
		In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errors growing as O(and#8730;n) for the Cooley-Tukey algorithm (Oppenheim and Schafer, 1975).
	duc.digidesign.com /showflat.php?Number=710792 (2679 words)

	[No title]
		The Cooley-Tukey algorithm is perhaps the simplest and most widely used form of FFT.
		The algorithm is performed "non-in-place", since the internal DSP56000/1 Data RAM's are used as additional workspace storage.
		Additional algorithm details are included in the source file; however, more algorithm description would be required for complete understanding by typical users.
	galaxy.uci.agh.edu.pl /~rumian/DSP_stuff/dsp56k/56000/fft/fftr2e.hlp (779 words)

	The world's top bruun s fft algorithm websites
		Bruun's algorithm is a fast Fourier transform (FFT) algorithm based on an unusual recursive polynomial-factorization approach, proposed for powers of two by G. Bruun in 1978 and generalized to arbitrary even composite sizes by H. Murakami in 1996.
		Bruun's algorithm has not seen widespread use, however, as approaches based on the ordinary Cooley-Tukey FFT algorithm have been successfully adapted to real data with at least as much efficiency.
		Nevertheless, Bruun's algorithm illustrates an alternative algorithmic framework that can express both itself and the Cooley-Tukey algorithm, and thus provides an interesting perspective on FFTs that permits mixtures of the two algorithms and other generalizations.
	dirs.org /wiki-article-tab.cfm/bruun_s_fft_algorithm (995 words)

	Table of Contents (Long)
		The mission of this section is to show that it is possible to formulate FFT frameworks for general n, as long as n is highly composite.
		Applying an FFT algorithm to the columns or rows of a matrix is an important problem in many applications.
		Fast algorithms for various sine and cosine transforms are given in Section 4.4 and then used in Section 4.5 to solve the Poisson equation problem.
	www.cs.cornell.edu /courses/cs621/Books/FFT/TClong.htm (804 words)

	Fast Fourier transform (Site not responding. Last check: 2007-10-03)
		Bruun's algorithm applies to arbitrary even composite sizes.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial
		Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime
		Another prime-size FFT is due to L. Bluestein, and is sometimes called the chirp-z algorithm; it also re-expresses a DFT as a convolution, but this time of the same size (which can be zero-padded to a power of two and evaluated by radix-2 Cooley-Tukey FFTs, for example), via the identity
	www.worldhistory.com /wiki/F/Fast-Fourier-transform.htm (1640 words)

	Bizarre Haskell Problem (Site not responding. Last check: 2007-10-03)
		TODO: Algorithm for 2N-point real FFT^-1 computed with N-point complex FFT TODO: Algorithm for 2 N-point real FFT's computed with N-point complex FFT TODO: Lyon's book derived the 2N-point real FFT separating out the real and imaginary parts.
		I believe this is equivalent to a radix-2 decimation-in-time (DIT) FFT, which is a special case of the Cooley-Tukey algorithm for N=2^v.
		This algorithm was taken from Cormen, Leiserson, and Rivest's _Introduction to Algorithms, and we added the hardcodes.
	www.haskell.org /pipermail/glasgow-haskell-bugs/2003-January/002939.html (3451 words)

	Pseudo-random code (PRC) surveilance radar - Patent 4042925
		This algorithm replaces the 128 by 128 discrete Fourier Transform matrix with a series of seven 128 by 2 discrete Fourier Transform matrices.
		Other derivative FFT algorithms which use both Radix 4 and Radix 8 matrix operations have been developed which require even fewer multiplication operations than the Radix 2 algorithm for specific input sequences.
		While the Radix 2 algorithm is employed in the system of the present invention, it will be understood that the invention is by no means limited thereto.
	www.freepatentsonline.com /4042925.html (8208 words)

	Dr. Dobb's \| A Simple and Efficient FFT Implementation in C++, Part I \| May 10, 2007
		This algorithm should give only a first impression of the FFT construction.
		This recursion form is instructive, but the overwhelming majority of FFT implementations use a loop structure first achieved by Cooley and Tukey [2] in 1965.
		The Cooley-Tukey algorithm uses the fact that if the elements of the original length N signal x are given a certain "bit-scrambling" permutation, then the FFT can be carried out with convenient nested loops.
	www.ddj.com /dept/java/199500857 (843 words)

	How FFTEASY works (Site not responding. Last check: 2007-10-03)
		For an excellent discussion of Fourier transforms (continuous and discrete) in general and the Cooley-Tukey algorithm in particular see Numerical Recipes.
		The heart of the method is the Danielson-Lanczos (DL) formula that allows one to compute a discrete Fourier transform (DFT) of size N by separately computing two FTs of size N/2.
		Formulas are given here in terms of complex numbers and then implemented in the functions explicitly in terms of real and imaginary parts.
	physics.stanford.edu /gfelder/ffteasy/ffteasydocs/node9.html (417 words)

	Digital Image Warping
		It is also geared to students of image processing who wish to apply their knowledge of that subject to a well-defined application.
		The theory segment is comprised of proofs and formulas derived to motivate the algorithms and to establish a standard of comparison among them.
		Source code, written in C, is scattered among the chapters and appendices to demonstrate implementation details for various algorithms.
	www-cs.engr.ccny.cuny.edu /~wolberg/diw.html (1115 words)

	Co-developer of FFT algorithm dies - 9/1/2000 - EDN (Site not responding. Last check: 2007-10-03)
		Co-developer of FFT algorithm dies - 9/1/2000 - EDN
		Statistician John Tukey, who along with James Cooley developed the Cooley-Tukey algorithm for FFTs, died on July 26.
		Although that claim may be open to some anecdotal dispute (typical of these types of proclamations), there is no doubt about the immense impact of the FFT algorithm he co-developed in 1965.
	www.edn.com /article/CA47156.html (214 words)

	[No title]
		The upper panel has not used the FFT algorithm to compute the length-4 DFTs while the lower one has.
		Other "fast" algorithms were discovered, all of which make use of how many common factors the transform length N has.
		In over thirty years of Fourier transform algorithm development, the original Cooley-Tukey algorithm is far and away the most frequently used.
	cnx.rice.edu /content/m0528/2.7/source (655 words)

	ipedia.com: Cooley-Tukey FFT algorithm Article (Site not responding. Last check: 2007-10-03)
		The Cooley-Tukey algorithm is the most common fast Fourier transform algorithm.
		It re-expresses the discrete Fourier transform of an arbitrary composite size n = n 1 n 2 in terms of smaller DFTs of si...
		3 Data reordering, bit reversal, and in-place algorithms
	www.ipedia.com /cooley_tukey_fft_algorithm.html (2214 words)

	Re: Books to recommend to the unbearably light
		The algorithm was described by I.J. Good in the Journal of the > Royal Statistical Society, Series B, in 1958.
		Actually, although Cooley and Tukey cited it in their seminal 1965 paper, Good's algorithm is entirely different from the Cooley-Tukey algorithm; it is what is now typically called the "Prime-factor Algorithm" or "Good-Thomas algorithm", and unlike the Cooley-Tukey algorithm it is restricted to relatively prime factorizations.
		On the other hand, it turns out that Cooley and Tukey's algorithm was known to Gauss around 1805, and was rediscovered multiple times over the next 150 years, so you are correct that what CandT did was popularize it (re-inventing it as well, since they weren't aware of Gauss et al.
	www.usenet.com /newsgroups/rec.arts.books/msg00177.html (311 words)

	Fast fourier transform using balanced coefficients - Patent 5365469
		The Fourier transform operation is well-known and a discrete Fourier transform algorithm by Berglund and a fast Fourier transform algorithm by Cooley and Tukey are discussed at length in a book entitled The Fast Fourier Transform and Its Applications by E. Brigham, 1988 by Prentice-Hall.
		Additionally, implementation of FFT algorithms can be done on large mainframe computers or on the ubiquitous personal computers.
		It was found that by rearranging the coefficients and the terms of the fast Fourier transform algorithm in a manner to balance the number of coefficients being calculated, the number of coefficients required can be greatly reduced.
	www.freepatentsonline.com /5365469.html (2488 words)

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