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Encyclopedia > Fast Fourier transform

The Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. FFTs are 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, of which there are many; see discrete Fourier transform for properties and applications of the transform. In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... In mathematics, the discrete Fourier transform (DFT) is a transform for Fourier analysis of finite-domain discrete-time signals. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... A multiplication algorithm is an algorithm (or method) to multiply two numbers. ... In mathematics, the discrete Fourier transform (DFT) is a transform for Fourier analysis of finite-domain discrete-time signals. ...

Let x0, ...., xN-1 be complex numbers. The DFT is defined by the formula In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... $X_k = sum_{n=0}^{N-1} x_n e^{-{2pi i over N} nk } qquad k = 0,dots,N-1.$

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. In general, such algorithms depend upon the factorization of N, but (contrary to popular misconception) there are O(N log N) FFTs for all N, even prime N. Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. ... In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

Many FFT algorithms only depend on the fact that $e^{-{2pi i over N}}$ is a primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms. In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... The number-theoretic transform is similar to the discrete Fourier transform, but operates with modular arithmetic instead of complex numbers. ...

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.

Main article: Cooley-Tukey FFT algorithm. The Cooley-Tukey algorithm, named after J.W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) 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 (after Gentleman and Sande, 1966). The Cooley-Tukey algorithm, named after J.W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. ... In computer science, divide and conquer (D&C) is an important algorithm design paradigm. ... See also Recursion. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... A twiddle factor, in fast Fourier transform (FFT) algorithms, refers to the trigonometric constant coefficients that are multiplied by the data in the course of the algorithm. ...

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). Dr. James Cooley (born 1926) is an American mathematician. ... John Wilder Tukey (June 16, 1915 - July 26, 2000) was a statistician born in New Bedford, Massachusetts. ... 1965 (MCMLXV) was a common year starting on Friday (the link is to a full 1965 calendar). ...   (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... 1805 was a common year starting on Tuesday (see link for calendar). ...

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 such as the split-radix FFT 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. The split-radix FFT is a fast Fourier transform (FFT) algorithm for computing the discrete Fourier transform (DFT), and was first described in an obscure paper by R. Yavne (1968) and subsequently rediscovered simultaneously by various authors in 1984. ...

## Other FFT algorithms

Main articles: Prime-factor FFT algorithm, Bruun's FFT algorithm, Rader's FFT algorithm, Bluestein's FFT algorithm. The Prime-factor algorithm (PFA), also called the Good-Thomas algorithm (1958/1963), is a fast Fourier transform (FFT) algorithm that re-expresses the discrete Fourier transform (DFT) of a size n = n1n2 as a two-dimensional n1 by n2 DFT, but only for the case where n1 and n2... Bruuns 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. ... Raders algorithm (1968) is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution. ... Bluesteins FFT algorithm (1968), commonly called the chirp-z algorithm (1969), is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of arbitrary sizes (including prime sizes) by re-expressing the DFT as a linear convolution. ...

There are other FFT algorithms distinct from Cooley-Tukey. For N = N1N2 with coprime 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 (1976) 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 zM − 1 and z2M + azM + 1. Coprime - Wikipedia /**/ @import /skins-1. ... The Prime-factor algorithm (PFA), also called the Good-Thomas algorithm (1958/1963), is a fast Fourier transform (FFT) algorithm that re-expresses the discrete Fourier transform (DFT) of a size n = n1n2 as a two-dimensional n1 by n2 DFT, but only for the case where n1 and n2... Several related results in number theory and abstract algebra are known under the name Chinese remainder theorem. ... Bruuns 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. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

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. Indeed, Winograd showed that the DFT can be computed with only O(N) irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers. In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of prime sizes. In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... Die of an Intel 80486DX2 microprocessor (actual size: 12Ã—6. ... A floating point unit (FPU) is a part of a CPU specially designed to carry out operations on floating point numbers. ...

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 nk = − (kn)2 / 2 + n2 / 2 + k2 / 2. Raders algorithm (1968) is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... This picture illustrates how the hours in a clock form a group. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ... Bluesteins FFT algorithm (1968), commonly called the chirp-z algorithm (1969), is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of arbitrary sizes (including prime sizes) by re-expressing the DFT as a linear convolution. ... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ...

## 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 $X_{N-k} = X_k^*,$

and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). 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 it was subsequently argued that 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. A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous applications in signal processing and related fields. ...

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 (roughly) two in time and memory 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. In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. ... 2-D DCT compared to the DFT The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. ... In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but with one additional property: If the input consists of only real numbers, so will the output. ...

## Bounds on complexity and operation counts

A fundamental question of longstanding theoretical interest is to prove lower bounds on the complexity and exact operation counts of fast Fourier transforms, and many open problems remain. It is not even rigorously proved whether DFTs truly require Ω(NlogN) (i.e., order NlogN or greater) operations, even for the simple case of power of two sizes, although no algorithms with lower complexity are known. In particular, the count of arithmetic operations is usually the focus of such questions, although actual performance on modern-day computers is determined by many other factors such as cache or CPU pipeline optimization. As a branch of the theory of computation in computer science, computational complexity theory describes the scalability of algorithms, and the inherent difficulty in providing scalable algorithms for specific computational problems. ... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ... Look up cache in Wiktionary, the free dictionary. ... Instruction scheduling on the Intel Pentium 4. ...

Following pioneering work by Winograd (1978), a tight Θ(N) lower bound is known for the number of real multiplications required by an FFT. It can be shown that only $4N-2log_2^{2}N-2log_2 N-4$ irrational real multiplications are required to compute a DFT of power-of-two length N = 2m. Moreover, explicit algorithms that achieve this count are known (Heideman & Burrus, 1986; Duhamel, 1990). Unfortunately, these algorithms require too many additions to be practical, at least on modern computers with hardware multipliers. See Fast Fourier transform#Bounds on complexity and operation counts for a general summary of this issue. ...

A tight lower bound is not known on the number of required additions, although lower bounds have been proved under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an Ω(NlogN) lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). Pan (1986) proved a Ω(NlogN) lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear. For the case of power-of-two N, Papadimitriou (1979) argued that the number Nlog2N of complex-number additions achieved by Cooley-Tukey algorithms is optimal under certain assumptions on the graph of the algorithm (his assumptions imply, among other things, that no additive identities in the roots of unity are exploited). (This argument would imply that at least 2Nlog2N real additions are required, although this is not a tight bound because extra additions are required as part of complex-number multiplications.) Thus far, no published FFT algorithm has achieved fewer than Nlog2N complex-number additions (or their equivalent) for power-of-two N.

A third problem is to minimize the total number of real multiplications and additions, sometimes called the "arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being considered). Again, no tight lower bound has been proven. Since 1968, however, the lowest published count for power-of-two N was long achieved by the split-radix FFT algorithm, which requires 4Nlog2N − 6N + 8 real multiplications and additions for N > 1. This was recently reduced to $sim frac{34}{9} N log_2 N$ (Johnson and Frigo, 2007). The split-radix FFT is a fast Fourier transform (FFT) algorithm for computing the discrete Fourier transform (DFT), and was first described in an obscure paper by R. Yavne (1968) and subsequently rediscovered simultaneously by various authors in 1984. ...

Most of the attempts to lower or prove the complexity of FFT algorithms have focused on the ordinary complex-data case, because it is the simplest. However, complex-data FFTs are so closely related to algorithms for related problems such as real-data FFTs, discrete cosine transforms, discrete Hartley transforms, and so on, that any improvement in one of these would immediately lead to improvements in the others (Duhamel & Vetterli, 1990). 2-D DCT compared to the DFT The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. ... A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous applications in signal processing and related fields. ...

## 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. This article or section is in need of attention from an expert on the subject. ... Parallel computing is the simultaneous execution of the same task (split up and specially adapted) on multiple processors in order to obtain results faster. ... In mathematics, wavelets, wavelet analysis, and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). ...

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 N3/2) for the naïve DFT formula (Gentleman and Sande, 1966), where ε is the machine floating-point relative precision. In fact, the root mean square (rms) errors are much better than these upper bounds, being only O(ε √log N) for Cooley-Tukey and O(ε √N) for the naïve 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. In the mathematical subfield of numerical analysis the approximation error in some data is the discrepancy between an exact value and some approximation to it. ... In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ... All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... Tables of trigonometric functions are useful in a number of areas. ...

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 (Welch, 1969). Moreover, even achieving this accuracy requires careful attention to scaling in order to minimize the loss of precision, and fixed-point FFT algorithms involve rescaling at each intermediate stage of decompositions like Cooley-Tukey. It has been suggested that Binary scaling be merged into this article or section. ...

To verify the correctness of an FFT implementation, rigorous guarantees can be obtained in O(N log N) time by a simple procedure checking the linearity, impulse-response, and time-shift properties of the transform on random inputs (Ergün, 1995).

## Multidimensional FFT algorithms

As defined in the multidimensional DFT article, the multidimensional DFT In mathematics, the discrete Fourier transform (DFT) is a transform for Fourier analysis of finite-domain discrete-time signals. ... $X_mathbf{k} = sum_{mathbf{n}=0}^{mathbf{N}-1} e^{-2pi i mathbf{k} cdot (mathbf{n} / mathbf{N})} x_mathbf{n}$

transforms an array $x_mathbf{n}$ with a d-dimensional vector of indices $mathbf{n}=(n_1, n_2, ldots, n_d)$ by a set of d nested summations (over $n_j = 0 ldots N_j-1$ for each j), where the division $mathbf{n} / mathbf{N}$, defined as $mathbf{n} / mathbf{N} = (n_1/N_1, cdots, n_d/N_d)$, is performed element-wise. Equivalently, it is simply the composition of a sequence of d one-dimensional DFTs, performed along one dimension at a time (in any order). In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. ...

This compositional viewpoint immediately provides the simplest and most common multidimensional DFT algorithm, known as the row-column algorithm (after the two-dimensional case, below). That is, one simply performs a sequence of d one-dimensional FFTs (by any of the above algorithms): first you transform along the n1 dimension, then along the n2 dimension, and so on (or actually, any ordering will work). This method is easily shown to have the usual O(NlogN) complexity, where $N = N_1 N_2 cdots N_d$ is the total number of data points transformed. In particular, there are N / N1 transforms of size N1, etcetera, so the complexity of the sequence of FFTs is: $N/N_1 O(N_1 log N_1) + cdots + N/N_d O(N_d log N_d) = O(N [log N_1 + cdots + log N_d]) = O(N log N).$

In two dimensions, the $x_mathbf{k}$ can be viewed as an $n_1 times n_2$ matrix, and this algorithm corresponds to first performing the FFT of all the rows and then of all the columns (or vice versa), hence the name. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

In more than two dimensions, it is often advantageous for cache locality to group the dimensions recursively. For example, a three-dimensional FFT might first perform two-dimensional FFTs of each planar "slice" for each fixed n1, and then perform the one-dimensional FFTs along the n1 direction. More generally, an asymptotically optimal cache-oblivious algorithm consists of recursively dividing the dimensions into two groups $(n_1, cdots, n_{d/2})$ and $(n_{d/2+1}, cdots, n_d)$ that are transformed recursively (rounding if d is not even) (see Frigo and Johnson, 2005). Still, this remains a straightforward variation of the row-column algorithm that ultimately requires only a one-dimensional FFT algorithm as the base case, and still has O(NlogN) complexity. Yet another variation is to perform matrix transpositions in between transforming subsequent dimensions, so that the transforms operate on contiguous data; this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data is extremely time-consuming. Look up cache in Wiktionary, the free dictionary. ... In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor worse than the best possible algorithm. ... In computing, the cache-oblivious model is an abstract machine similar to many modern machines, featuring random access and a two-level cache. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ... In computer science and applications, out-of-core refers to algorithms which process data that is too large to fit into a computers main memory at one time. ... Distributed memory is a concept used in parallel computing. ...

There are other multidimensional FFT algorithms that are distinct from the row-column algorithm, although all of them have O(NlogN) complexity. Perhaps the simplest non-row-column FFT is the vector-radix FFT algorithm, which is a generalization of the ordinary Cooley-Tukey algorithm where one divides the transform dimensions by a vector $mathbf{r}=(r_1, r_2, cdots, r_d)$ of radices at each step. (This may also have cache benefits.) The simplest case of vector-radix is where all of the radices are equal (e.g. vector-radix-2 divides all of the dimensions by two), but this is not necessary. Vector radix with only a single non-unit radix at a time, i.e. $mathbf{r}=(1, cdots, 1, r, 1, cdots, 1)$, is essentially a row-column algorithm. Other, more complicated, methods include polynomial transform algorithms due to Nussbaumer (1977), which view the transform in terms of convolutions and polynomial products. See Duhamel and Vetterli (1990) for more information and references.

## Other generalizations

Two O(N5/2 log N) and O(N2 log2 N) generalizations to spherical harmonics and on the sphere S2 were given by Martin J. Mohlenkamp (1999), who also provides its implementation as the libftsh library. In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...

Various groups have also published "FFT" algorithms for non-equispaced data, as reviewed in Potts et al. (2001). Such algorithms do not strictly compute the DFT (which is only defined for equispaced data), but rather some approximation thereof (a non-equispaced discrete Fourier transform, or NDFT, which itself is often computed only approximately).

The Cooley-Tukey algorithm, named after J.W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. ... The split-radix FFT is a fast Fourier transform (FFT) algorithm for computing the discrete Fourier transform (DFT), and was first described in an obscure paper by R. Yavne (1968) and subsequently rediscovered simultaneously by various authors in 1984. ... The Prime-factor algorithm (PFA), also called the Good-Thomas algorithm (1958/1963), is a fast Fourier transform (FFT) algorithm that re-expresses the discrete Fourier transform (DFT) of a size n = n1n2 as a two-dimensional n1 by n2 DFT, but only for the case where n1 and n2... Bruuns 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. ... Raders algorithm (1968) is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution. ... Bluesteins FFT algorithm (1968), commonly called the chirp-z algorithm (1969), is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of arbitrary sizes (including prime sizes) by re-expressing the DFT as a linear convolution. ... Data-flow diagram connecting the inputs x (left) to the outputs y that depend on them (right) for a butterfly step of a radix-2 Cooley-Tukey FFT. This diagram resembles a butterfly (as in the Morpho butterfly shown for comparison), hence the name. ... Spectral music (or spectralism) is a musical genre or movement originating in France in the 1970s and characterized by the use of computer analysis of sound wave components as the basis for composition. ... Results from FactBites:

 fast Fourier Transform (1664 words) The transform of an infinite train of delta functions spaced by T is an infinite train of delta functions spaced by 1/T. The transform of a cos function is a positive delta at the appropriate positive and negative frequency. The transform of a sin function is a negative complex delta function at the appropriate positive frequency and a negative complex delta at the appropriate negative frequency. For example the transform of a truncated sin function are two delta functions convolved with a sinc function, a truncated sin function is a sin function multiplied by a square pulse.
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