Discrete Fourier transform

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In mathematics, the discrete Fourier transform (DFT) is a discrete version of the Fourier transform that converts a finite sequence of numbers into another sequence of the same length, representing the amplitude and phase of different frequency components. In this way, it changes data from a description in terms of sampled values to a description in terms of oscillations. The inverse discrete Fourier transform reverses this process and recovers the original sequence. For data sampled at equally spaced points, the DFT can be understood more precisely as converting between sample values and the coefficients of a trigonometric polynomial that interpolates those values. It is therefore a basic tool for numerical work with smooth periodic functions, which can often be approximated well by trigonometric polynomials. In practice, the DFT is usually computed by efficient fast Fourier transform (FFT) algorithms. The DFT is used in many practical applications of Fourier analysis. In digital signal...

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