Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) is a discrete version of the Fourier transform that converts one finite sequence of function values into another of the same length. The DFT converts back and forth between two different representations of a trigonometric polynomial: a representation in terms of the function values at equispaced sample points, and a representation in terms of the polynomial coefficients. The DFT (especially in its fast form) is a useful tool when considering any sufficiently smooth periodic function, because by taking sufficiently many sample points, a trigonometric polynomial will approximate it to any desired degree of accuracy, and trigonometric polynomials represented in terms of their coefficients are computationally convenient objects to work with.

The DFT is used in many practical applications of Fourier analysis. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function). In image processing, the samples can be the values of pixels along a row or column of a raster image. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.

Since the DFT deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms; so much so that the terms "FFT" and "DFT" are often used interchangeably. Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term "finite Fourier transform".