Let *x*_{0}, ...., *x*_{n-1} be complex numbers. The DFT is defined by the formula

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.

Table of contents |

2 Other FFT algorithms 3 FFT algorithms specialized for real and/or symmetric data 4 Accuracy and approximations 5 References |

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* = *n*_{1}*n*_{2} into many smaller DFTs of sizes *n*_{1} and *n*_{2}, 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.*

There are other FFT algorithms distinct from Cooley-Tukey. For *n* = *n*_{1}*n*_{2} with relatively prime *n*_{1} and *n*_{2}, 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 *z*^{n} - 1, here into real-coefficient polynomials of the form *z ^{m}* - 1 and

*See also: Prime-factor FFT algorithm, Bruun's FFT algorithm, Rader's FFT algorithm, Bluestein's FFT algorithm.*

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 discredited: 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.

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(ε √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.

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).

- 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). - Funda Ergün, "Testing multivariate linear functions: Overcoming the generator bottleneck,"
*Proc. 27th ACM Symposium on the Theory of Computing*, 407–416 (1995). - Links to FFT code and information online.