FAST FOURIER TRANSFORM

Domain	Definition
Computing	Fast Fourier Transform (FFT) An algorithm for computing the Fourier transform of a set of discrete data values. Given a finite set of data points, for example a periodic sampling taken from a real-world signal, the FFT expresses the data in terms of its component frequencies. It also solves the essentially identical inverse problem of reconstructing a signal from the frequency data. The FFT is a mainstay of numerical analysis. Gilbert Strang described it as "the most important algorithm of our generation". The FFT also provides the asymptotically fastest known algorithm for multiplying two polynomials. Versions of the algorithm (in C and Fortran) can be found on-line from the GAMS server here (http://gams.nist.gov/cgi-bin/gams-serve/class/J1.html). ["Numerical Methods and Analysis", Buchanan and Turner]. (1994-11-09). Source: The Free On-line Dictionary of Computing.
Math	An algorithm to convert a set of uniformly spaced points from the time domain to the frequency domain. (references)
Source: compiled by the editor from various references; see credits.

Specialty Definition: Fast Fourier transform

A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. It is of great importance to a wide variety of applications, from digital signal processing to solving partial differential equations to algorithms for quickly multiplying large integers. This article describes the algorithms; see discrete Fourier transform for properties and applications of the transform.

Let x₀, ...., x_n-1 be complex numbers. The DFT is defined by the formula

Evaluating these sums directly would take O(n²) arithmetical operations (see Big O notation). An FFT is an algorithm to compute the same result in only O(n log n) operations.

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 as well.

The Cooley-Tukey algorithm

By far the most common FFT is the Cooley-Tukey algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size n = n₁n₂ into many smaller DFTs of sizes n₁ and n₂, along with O(n) multiplications by complex roots of unity traditionally called twiddle factors.

This method (and the general idea of an FFT) was popularized by a publication of J. W. Cooley and J. W. 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).

The most well-known use of the Cooley-Tukey algorithm is to divide the transform into two pieces of size n/2 at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey). These are called the radix-2 and mixed-radix cases, respectively (and other variants have their own names as well). Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley-Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below.

See also: Cooley-Tukey FFT algorithm.

Other FFT algorithms

There are other FFT algorithms distinct from Cooley-Tukey. For relatively prime n₁ and n₂, one can use the Prime-Factor (Good-Thomas) algorithm (PFA), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley-Tukey but without the twiddle factors. The Rader-Brenner algorithm is a Cooley-Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability. Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms. (The Rader-Brenner and QFT algorithms were proposed for power-of-two sizes, but it is possible that they could be adapted to general composite n. 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 zⁿ - 1, here into real-coefficient polynomials of the form z^m - 1 and z^2m + az^m + 1. Another polynomial viewpoint is exploited by the Winograd algorithm, which factorizes zⁿ - 1 into cyclotomic polynomials—these often have coefficients of 1, 0, or -1, and therefore require few (if any) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors. In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of prime sizes. 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). Another prime-size FFT is due to L. I. 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 jk = -(j-k)²/2 + j²/2 + k²/2.

FFT algorithms specialized for real and/or symmetric data

In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry

and efficient FFT algorithms have been designed for this situation. One approach consists of taking an ordinary algorithm (e.g. Cooley-Tukey) and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an even-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data), followed by O(n) post-processing operations.

It was once believed that real-input DFTs could be more efficiently computed by means of the Discrete Hartley transform (DHT), but this was subsequently disproved: a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs. Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular.

There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of ~2 in time/space and the DFT becomes the discrete cosine/sine transform(s) (DCT/DST). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via FFTs of real data combined with O(n) pre/post processing.

Accuracy and approximations

All of the FFT algorithms discussed so far compute the DFT exactly (in exact arithmetic, i.e. neglecting floating-point errors). A few "FFT" algorithms have been proposed, however, that compute the DFT approximately, with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate FFT algorithm by Edelman et al. (1999) achieves lower communication requirements for parallel computing with the help of a fast-multipole method. A wavelet-based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs (time/frequency localization) into account more efficiently than is possible with an exact FFT. Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al. (1995). 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.

Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small; most FFT algorithms, e.g. Cooley-Tukey, have excellent numerical properties. The upper bound on the relative error for the Cooley-Tukey algorithm is O(ε log n), compared to O(ε n^3/2) for the naive DFT formula (Gentleman and Sande, 1966), where ε is the machine floating-point relative precision. In fact, the average errors are much better than these upper bounds, being only O(&epsilon √log n) for Cooley-Tukey and O(ε √n) for the naive DFT (Schatzman, 1996). These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT (i.e. the trigonometric function values), and it is not unusual for incautious FFT implementations to have much worse accuracy, e.g. if they use inaccurate trigonometric recurrence formulas. Some FFTs other than Cooley-Tukey, such as the Rader-Brenner algorithm, are intrinsically less stable.

References

James W. Cooley and John W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Math. Comput. 19, 297–301 (1965).

Carl Friedrich Gauss, "Nachlass: Theoria interpolationis methodo nova tractata," Werke band 3, 265–327 (Königliche Gesellschaft der Wissenschaften, Göttingen, 1866). See also M. T. Heideman, D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform," IEEE ASSP Magazine 1 (4), 14–21 (1984).

P. Duhamel and M. Vetterli, "Fast Fourier transforms: a tutorial review and a state of the art," Signal Processing 19, 259–299 (1990).

W. M. Gentleman and G. Sande, "Fast Fourier transforms—for fun and profit," Proc. AFIPS 29, 563–578 (1966).

H. Guo, G. A. Sitton, and C. S. Burrus, "The Quick Discrete Fourier Transform," Proc. IEEE Conf. Acoust. Speech and Sig. Processing (ICASSP) 3, 445–448 (1994).

H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus, "Real-valued fast Fourier transform algorithms," IEEE Trans. Acoust. Speech Sig. Processing ASSP-35, 849–863 (1987).

A. Edelman, P. McCorquodale, and S. Toledo, "The future fast Fourier transform?" SIAM J. Sci. Computing 20, 1094–1114 (1999).

H. Guo and C. S. Burrus, "Fast approximate Fourier transform via wavelets transform," Proc. SPIE Intl. Soc. Opt. Eng. 2825, 250–259 (1996).

O. V. Shentov, S. K. Mitra, U. Heute, and A. N. Hossen, "Subband DFT. I. Definition, interpretations and extensions," Signal Processing 41 (3), 261–277 (1995).

James C. Schatzman, "Accuracy of the discrete Fourier transform and the fast Fourier transform," SIAM J. Sci. Comput. 17 (5), 1150–1166 (1996).

Links to FFT code and information online.

Commercial Usage: FAST FOURIER TRANSFORM

Domain	Title
Books	Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms (reference) Fast Fourier Transform and Convolution Algorithms (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Frequency of Internet Keywords: FAST FOURIER TRANSFORM

Modern Translation: FAST FOURIER TRANSFORM

Language		Translations for "FAST FOURIER TRANSFORM"; alternative meanings/domain in parentheses.
Danish		FFT (fast Fourier transformation), fast Fourier transformation (fast Fourier transformation). (various references)

Dutch		FFT (fast Fourier transformation), snelle Fourier-transformatie (fast Fourier transformation). (various references)

Finnish		nopea Fourier-muunnos (fast Fourier transformation). (various references)

French		transformation de Fourier rapide (fast Fourier transformation), TFR,fournit la transformée de Fourier rapide (fast Fourier transformation), TFR. (various references)

German		schnelle Fourier-Transformation (fast Fourier transformation). (various references)

Greek		γρήγορος μετασχηματισμός Φουριέ (fast Fourier transformation). (various references)

Italian		FFT, trasformata veloce di Fourier, trasformata rapida di Fourier (fast Fourier transformation), traformata rapida di Fourier. (various references)

Pig Latin		astfay ourierfay ansformtray

Portuguese		transformação de Fourier rápida (fast Fourier transformation). (various references)

Spanish		transformación de Fourier rapida, transformación de Fourier rápida (fast Fourier transformation), TFR (fast Fourier transformation). (various references)
Source: compiled by the editor from various translation references.

Anagrams: FAST FOURIER TRANSFORM

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "a-a-e-f-f-f-i-m-n-o-o-r-r-r-r-s-s-t-t-u"
	-4 letters: autotransformers.
	-5 letters: autotransformer.
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