In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced
samples of a
function into a same-length sequence of equally-spaced samples of the
discrete-time Fourier transform (DTFT), which is a
complex-valued
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a
Fourier series, using the DTFT samples as coefficients of
complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a
frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.
The DFT is the most important
discrete transform, used to perform
Fourier analysis in many practical applications.
[ 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
Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a tr ...
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 it deals with a finite amount of data, it can be implemented in ]computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...
s 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 __NOTOC__
In mathematics the finite Fourier transform may refer to either
*another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the ''finite Fourier transform'' as a "co ...
".
Definition
The ''discrete Fourier transform'' transforms a sequence of ''N'' complex numbers into another sequence of complex numbers, which is defined by
where the last expression follows from the first one by Euler's formula.
The transform is sometimes denoted by the symbol , as in or or .
Motivation
can also be evaluated outside the domain