A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform 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. 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. As a result, it manages to reduce the complexity of computing the DFT from ${\textstyle O(n^{2})}$, which arises if one simply applies the definition of DFT, to ${\textstyle O(n\log n)}$, 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.

As the FFT is merely an algebraic refactoring of terms within the DFT, the DFT and the FFT both perform mathematically equivalent and interchangeable operations, assuming that all terms are computed with infinite precision. However, 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. The best-known FFT algorithms depend upon the factorization of n, but there are FFTs with ${\displaystyle O(n\log n)}$ complexity for all, even prime, n. Many FFT algorithms depend only on the fact that ${\textstyle e^{-2\pi i/n}}$ is an nth 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.

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. In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime", and it was included in Top 10 Algorithms of 20th Century by the IEEE magazine Computing in Science & Engineering.

The development of fast algorithms for DFT was prefigured in Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno. Gauss wanted to interpolate the orbits from sample observations; 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. 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. While many methods in the past had focused on reducing the constant factor for ${\textstyle O(n^{2})}$ 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 ${\textstyle O(n\log n)}$ scaling. In 1958, I. J. Good published a paper establishing the prime-factor FFT algorithm that applies to discrete Fourier transforms of size ${\textstyle n=n_{1}n_{2}}$, where ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ are coprime.

James Cooley and John Tukey independently rediscovered these earlier algorithms 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 ${\textstyle O(n\log n)}$ scaling. 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. Garwin gave Tukey's idea to Cooley (both worked at IBM's Watson labs) for implementation. Cooley and Tukey published the paper in a relatively short time of six months. 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.

Let ${\displaystyle x_{0},\ldots ,x_{n-1}}$ be complex numbers. The DFT is defined by the formula