A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. 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. This article describes the algorithms, of which there are many; see discrete Fourier transform for properties and applications of the transform. Let x0, ...., xN-1 be complex numbers. The DFT is defined by the formula X_k = sum_^ x_n e^ nk } qquadk = 0,dots,N-1. Evaluating these sums directly would take O arithmetical operations. An FFT is an algorithm to compute the same result in only O(N log N) operations. In general, such algorithms depend upon the factorization of N, but (contrary to popular

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