Fast Fourier transform

The fast Fourier transform (FFT) is a family of algorithms for computing the discrete Fourier transform with substantially fewer operations than direct evaluation.

Discrete Fourier transform: A linear transform that expresses a finite sequence as a sum of complex sinusoidal components. The FFT computes this transform without evaluating every input-output pair directly.
Complex number: A number with a real part and an imaginary part, often represented as a point in the complex plane. Fourier coefficients and phase factors are naturally expressed as complex values.
Spectral analysis: The study of a signal's frequency components and their amplitudes or phases. The FFT makes discrete frequency spectra practical to calculate from sampled data.
Direct discrete Fourier transform: The direct evaluation of each output of the discrete Fourier transform from all input samples. It requires quadratic work, unlike the FFT's typical near-linearithmic operation count.
Cooley–Tukey algorithm: A family of FFT algorithms that recursively factors a transform into smaller transforms. Its factorization of transform length is the best-known route to FFT speedups.
Root of unity: A complex number whose positive integer power equals one. The DFT's periodic sampling points are roots of unity.
Digital signal processing: The computational analysis and transformation of signals represented as discrete samples. FFT algorithms are standard tools for frequency-domain processing of sampled signals.
Bluestein's algorithm: An FFT method that rewrites a transform of arbitrary length as a convolution. It can handle awkward transform lengths by reducing them to a convolution computation.
Butterfly diagram: A diagram of paired data combinations used to organize stages of an FFT computation. Its crossing connections show how small transforms combine into the full result.
Euler's formula: The identity relating complex exponentials to sine and cosine: e^(ix) = cos(x) + i sin(x). It connects the FFT's complex multiplications to sinusoidal oscillations.

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