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Fast Fourier transform - Wikipedia
A Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis 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 … See more
The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno.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 … See more
Let $${\displaystyle x_{0},\ldots ,x_{n-1}}$$ be complex numbers. The DFT is defined by the formula
$${\displaystyle X_{k}=\sum _{m=0}^{n-1}x_{m}e^{-i2\pi km/n}\qquad k=0,\ldots ,n-1,}$$
where See moreBounds on complexity and operation counts
A fundamental question of longstanding theoretical interest is … See moreAn $${\textstyle O(n^{5/2}\log n)}$$ generalization to spherical harmonics on the sphere S with n nodes was described by Mohlenkamp, along with an algorithm conjectured (but … See more
1805Carl Friedrich Gauss's unpublished work on the orbits of asteroids Pallas and Juno.1932Frank Yates published his version of FFT algorithm called interaction algorithm.1942G. C. Danielson and Cornelius Lanczos published their version to compute DFT for x-ray crystallography.1965Cooley and Tukey independently rediscovered the earlier algorithms and published a more general FFT.1994Gilbert Strang described the FFT as 'the most important numerical algorithm of our lifetime'.2005Frigo and Johnson published a paper on FFT algorithms.Cooley–Tukey algorithm
By far the most commonly used FFT is the Cooley–Tukey algorithm. This is a divide-and-conquer algorithm that recursively breaks down a DFT … See moreIn many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry
See moreAs defined in the multidimensional DFT article, the multidimensional DFT
transforms an array … See moreWikipedia text under CC-BY-SA license Interpret FFT results – obtaining magnitude and phase …
WebNov 19, 2015 · 3a. Extract amplitude of frequency components (amplitude spectrum) The FFT function computes the complex DFT and the hence …
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Fast Fourier Transform (FFT) — Python Numerical Methods
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Fast Fourier Transformation for polynomial multiplication
WebMar 15, 2023 · Fast Fourier Transform (FFT) can perform DFT and inverse DFT in time O (nlogn). DFT. DFT is evaluating values of polynomial at n complex nth roots of unity . So, for k = 0, 1, 2, …, n-1, y = (y0, y1, y2, …
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Plotting a fast Fourier transform in Python - Stack Overflow