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Fast Fourier transform

A Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies.[1] This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors.[2] As a result, it manages to reduce the complexity of computing the DFT from , which arises if one simply applies the definition of DFT, to , where n is the data size. The difference in speed can be enormous, especially for long data sets where n may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.

"FFT" redirects here. For other uses, see FFT (disambiguation).

Fast Fourier transforms are widely used for applications in engineering, music, science, and mathematics. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805.[1] In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime",[3][4] and it was included in Top 10 Algorithms of 20th Century by the IEEE magazine Computing in Science & Engineering.[5]


The best-known FFT algorithms depend upon the factorization of n, but there are FFTs with complexity for all, even prime, n. Many FFT algorithms depend only on the fact that is an n'th primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms. Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/n factor, any FFT algorithm can easily be adapted for it.

History[edit]

The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno. Gauss wanted to interpolate the orbits from sample observations;[6][7] his method was very similar to the one that would be published in 1965 by James Cooley and John Tukey, who are generally credited for the invention of the modern generic FFT algorithm. While Gauss's work predated even Joseph Fourier's 1822 results, he did not analyze the method's complexity, and eventually used other methods to achieve the same end.


Between 1805 and 1965, some versions of FFT were published by other authors. Frank Yates in 1932 published his version called interaction algorithm, which provided efficient computation of Hadamard and Walsh transforms.[8] Yates' algorithm is still used in the field of statistical design and analysis of experiments. In 1942, G. C. Danielson and Cornelius Lanczos published their version to compute DFT for x-ray crystallography, a field where calculation of Fourier transforms presented a formidable bottleneck.[9][10] While many methods in the past had focused on reducing the constant factor for computation by taking advantage of "symmetries", Danielson and Lanczos realized that one could use the "periodicity" and apply a "doubling trick" to "double [n] with only slightly more than double the labor", though like Gauss they did not do the analysis to discover that this led to scaling.[11]


James Cooley and John Tukey independently rediscovered these earlier algorithms[7] and published a more general FFT in 1965 that is applicable when n is composite and not necessarily a power of 2, as well as analyzing the scaling.[12] Tukey came up with the idea during a meeting of President Kennedy's Science Advisory Committee where a discussion topic involved detecting nuclear tests by the Soviet Union by setting up sensors to surround the country from outside. To analyze the output of these sensors, an FFT algorithm would be needed. In discussion with Tukey, Richard Garwin recognized the general applicability of the algorithm not just to national security problems, but also to a wide range of problems including one of immediate interest to him, determining the periodicities of the spin orientations in a 3-D crystal of Helium-3.[13] Garwin gave Tukey's idea to Cooley (both worked at IBM's Watson labs) for implementation.[14] Cooley and Tukey published the paper in a relatively short time of six months.[15] As Tukey did not work at IBM, the patentability of the idea was doubted and the algorithm went into the public domain, which, through the computing revolution of the next decade, made FFT one of the indispensable algorithms in digital signal processing.

Other generalizations[edit]

An generalization to spherical harmonics on the sphere S2 with n2 nodes was described by Mohlenkamp,[41] along with an algorithm conjectured (but not proven) to have complexity; Mohlenkamp also provides an implementation in the libftsh library.[42] A spherical-harmonic algorithm with complexity is described by Rokhlin and Tygert.[43]


The fast folding algorithm is analogous to the FFT, except that it operates on a series of binned waveforms rather than a series of real or complex scalar values. Rotation (which in the FFT is multiplication by a complex phasor) is a circular shift of the component waveform.


Various groups have also published "FFT" algorithms for non-equispaced data, as reviewed in Potts et al. (2001).[44] Such algorithms do not strictly compute the DFT (which is only defined for equispaced data), but rather some approximation thereof (a non-uniform discrete Fourier transform, or NDFT, which itself is often computed only approximately). More generally there are various other methods of spectral estimation.

fast large-integer and polynomial multiplication,

multiplication algorithms

efficient matrix–vector multiplication for , circulant and other structured matrices,

Toeplitz

filtering algorithms (see and overlap–save methods),

overlap–add

fast algorithms for or sine transforms (e.g. fast DCT used for JPEG and MPEG/MP3 encoding and decoding),

discrete cosine

fast ,

Chebyshev approximation

solving ,

difference equations

computation of .[47]

isotopic distributions

modulation and demodulation of complex data symbols using orthogonal frequency division multiplexing (OFDM) for 5G, LTE, Wi-Fi, DSL, and other modern communication systems.

The FFT is used in digital recording, sampling, additive synthesis and pitch correction software.[45]


The FFT's importance derives from the fact that it has made working in the frequency domain equally computationally feasible as working in the temporal or spatial domain. Some of the important applications of the FFT include:[15][46]


An original application of the FFT in finance particularly in the Valuation of options was developed by Marcello Minenna.[48]

Bit-reversal permutation

FFT-related algorithms:


FFT implementations:


Other links:

Brigham, E. Oran (2002). The Fast Fourier Transform. New York: .

Prentice-Hall

; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "Chapter 30: Polynomials and the FFT". Introduction to Algorithms (2 ed.). MIT Press / McGraw-Hill. ISBN 0-262-03293-7.

Cormen, Thomas H.

Elliott, Douglas F.; Rao, K. Ramamohan (1982). Fast transforms: Algorithms, analyses, applications. New York, USA: .

Academic Press

Guo, Haitao; Sitton, Gary A.; (1994). "The quick discrete Fourier transform". Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing. Vol. 3. pp. 445–448. doi:10.1109/ICASSP.1994.389994. ISBN 978-0-7803-1775-8. S2CID 42639206.

Burrus, Charles Sidney

Johnson, Steven G.; Frigo, Matteo (2007). (PDF). IEEE Transactions on Signal Processing. 55 (1): 111–119. Bibcode:2007ITSP...55..111J. CiteSeerX 10.1.1.582.5497. doi:10.1109/tsp.2006.882087. S2CID 14772428. Archived (PDF) from the original on 2005-05-26.

"A modified split-radix FFT with fewer arithmetic operations"

Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007). . Numerical Recipes: The Art of Scientific Computing (3 ed.). New York, USA: Cambridge University Press. ISBN 978-0-521-88068-8. Archived from the original on 2011-08-11. Retrieved 2011-08-13.

"Chapter 12. Fast Fourier Transform"

Singleton, Richard Collom (June 1969). "A Short Bibliography on the Fast Fourier Transform". Special Issue on Fast Fourier Transform. Vol. AU-17. IEEE Audio and Electroacoustics Group. pp. 166–169. :10.1109/TAU.1969.1162029. (NB. Contains extensive bibliography.)

doi

Elena Prestini: "The Evolution of Applied Harmonic Analysis", Springer, ISBN 978-0-8176-4125-2 (2004), Sec.3.10 'Gauss and the asteroids: history of the FFT'.

 – fast Fourier algorithm

Fast Fourier Transform for Polynomial Multiplication

, Connexions online book edited by Charles Sidney Burrus, with chapters by Charles Sidney Burrus, Ivan Selesnick, Markus Pueschel, Matteo Frigo, and Steven G. Johnson (2008)

Fast Fourier Transforms

 – FFT programming in C++ – the Cooley–Tukey algorithm

Fast Fourier transform — FFT

Online documentation, links, book, and code

Sri Welaratna, " Archived 2014-01-12 at the Wayback Machine", Sound and Vibration (January 1997, 30th anniversary issue) – a historical review of hardware FFT devices

Thirty years of FFT analyzers

 – a dual/GPL-licensed multilanguage (VBA, C++, Pascal, etc.) numerical analysis and data processing library

ALGLIB FFT Code

 – MIT's sparse (sub-linear time) FFT algorithm, sFFT, and implementation

SFFT: Sparse Fast Fourier Transform

 – a VB6 optimized library implementation with source code

VB6 FFT

 – a visual interactive intro to Fourier transforms and FFT methods

Interactive FFT Tutorial