The Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) and its inverse efficiently. The DFT is a mathematical transformation used to analyze the frequency content of discrete signals, transforming a sequence of complex numbers into another sequence of complex numbers. The basic idea is to express a discrete signal as a sum of sinusoids, which can provide insights into the signal's frequency characteristics.
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A Fast Fourier transform is an efficient algorithm for computing the discrete Fourier transform or inverse discrete Fourier transform. The discrete Fourier transform specifies the mathematical result; a Fast Fourier transform specifies how to compute it. Different Fast Fourier transform algorithms reorganize the same finite sums, with the selected sign and normalization. The Cooley-Tukey FFT algorithm is one important family of Fast Fourier transforms, using factorization of the record length.[ref]
https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/06d7972059805d20e6a7db3a52ec23f2_MIT6_02F12_chap13.pdf. Balakrishnan, Terman, and Verghese, MIT 6.02 notes, §13.2.6, p. 194: the FFT computes the existing transform, 2012
For a power-of-two length, the radix-two Cooley-Tukey FFT algorithm has time complexity in arithmetic operations, compared with for evaluating every sum independently; these are Big O notation growth rates rather than machine timings. It returns the same eight coefficients in the example. In exact arithmetic this reorganization introduces no approximation; actual implementations still incur roundoff error.
An efficient algorithm to calculate the discrete Fourier transform.