Fast Fourier Transform

A Fast Fourier Transform is an numerical algorithm for rapid computation of the coefficients in a finite Fourier transform.

• AKA: FFT.
• See: Discrete Fourier Transform, Fourier Analysis, Frequency Domain, Matrix Decomposition, DFT Matrix.

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References

2016

• (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/Fast_Fourier_transform Retrieved:2016-4-13.
  • A fast Fourier transform (FFT) algorithm computes the discrete Fourier transform (DFT) of a sequence, or its inverse. Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. 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 [math]\displaystyle{ O(n^2) }[/math] , which arises if one simply applies the definition of DFT, to [math]\displaystyle{ O(n \log n) }[/math] , where [math]\displaystyle{ n }[/math] is the data size. Fast Fourier transforms are widely used for many applications in engineering, 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 journal Computing in Science & Engineering.

1965

• (Tukey & Cooley, 1965).

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