In 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 function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence.  An inverse DFT (IDFT) 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 used in the Fourier analysis of 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 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 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".

The discrete Fourier transform transforms a sequence of N complex numbers ${\displaystyle \left\{\mathbf {x} _{n}\right\}:=x_{0},x_{1},\ldots ,x_{N-1}}$ into another sequence of complex numbers, ${\displaystyle \left\{\mathbf {X} _{k}\right\}:=X_{0},X_{1},\ldots ,X_{N-1},}$ which is defined by:

Note: this definition omits the sampling interval, so the frequency axis becomes dimensionless and the amplitude is expressed only in relative scale. However, most software libraries, including FFT implementations, use this relative form.

The transform is sometimes denoted by the symbol ${\displaystyle {\mathcal {F}}}$, as in ${\displaystyle \mathbf {X} ={\mathcal {F}}\left\{\mathbf {x} \right\}}$ or ${\displaystyle {\mathcal {F}}\left(\mathbf {x} \right)}$ or ${\displaystyle {\mathcal {F}}\mathbf {x} }$.

Eq.1 can be interpreted or derived in various ways, for example:

Eq.1 can also be evaluated outside the domain ${\displaystyle k\in [0,N-1]}$, and that extended sequence is ${\displaystyle N}$-periodic. Accordingly, other sequences of ${\displaystyle N}$ indices are sometimes used, such as ${\textstyle \left[-{\frac {N}{2}},{\frac {N}{2}}-1\right]}$ (if ${\displaystyle N}$ is even) and ${\textstyle \left[-{\frac {N-1}{2}},{\frac {N-1}{2}}\right]}$ (if ${\displaystyle N}$ is odd), which amounts to swapping the left and right halves of the result of the transform.