fft

fft
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For other uses, see FFT (disambiguation). A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse.
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. "The FFT has been called the most important numerical algorithm
of our lifetime (Strang, 1994)." (Kent & Read 2002, 61) There are
many distinct FFT algorithms involving a wide range of mathematics, from
simple complex-number arithmetic to group theory and number theory;
this article gives an overview of the available techniques and some of
their general properties, while the specific algorithms are described in
subsidiary articles linked below.

A DFT decomposes a sequence of values into components of different frequencies. This operation is useful in many fields (see discrete Fourier transform
for properties and applications of the transform) but computing it
directly from the definition is often too slow to be practical. An FFT
is a way to compute the same result more quickly: computing a DFT of N points in the naive way, using the definition, takes O(N²) arithmetical operations, while an FFT can compute the same result in only O(N log N) operations. The difference in speed can be substantial, especially for long data sets where N may be in the thousands or millions—in practice, the computation time can be reduced by several orders of magnitude in such cases, and the improvement is roughly proportional to N / log(N).
This huge improvement made many DFT-based algorithms practical; FFTs
are of great importance to a wide variety of applications, from digital signal processing and solving partial differential equations to algorithms for quick multiplication of large integers.

The most well known FFT algorithms depend upon the factorization of N, but (contrary to popular misconception) there are FFTs with O(N log N) complexity for all N, even for prime N. Many FFT algorithms only depend on the fact that 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 Fédération Française de Tennis |

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