Fast Fourier transform

related topics
{math, number, function}
{rate, high, increase}
{system, computer, user}
{math, energy, light}
{work, book, publish}
{company, market, business}

A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. 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 e^{-{2\pi i \over 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.


Full article ▸

related documents
Continued fraction
Closure (computer science)
Axiom of choice
Tensor product
Algebraic geometry
Adjoint functors
Hilbert's tenth problem
Singleton pattern
Functional programming
Dedekind domain
Orthogonal matrix
Kolmogorov complexity
Scheme (programming language)
Fourier series
Class (computer science)
Numeral system
Limit (category theory)
Design Patterns
LR parser
Lie group
Elliptic curve cryptography
Binary search tree
Riemann integral
Recurrence relation
Μ-recursive function