Fourier transforms Continuous Fourier transform Fourier series
Discrete-time Fourier transform Discrete Fourier transform
Fourier analysis Related transforms
Relationship between the (continuous) Fourier transform and the discrete Fourier transform. Left column: A continuous function (top) and its Fourier transform (bottom).
Center-left column: Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right column: Original function is discretized (multiplied by
a Dirac comb) (top). Its Fourier transform (bottom) is a periodic summation (DTFT) of the original transform. Right column: The DFT (bottom) computes discrete samples of the continuous DTFT. The inverse DFT (top) is a periodic summation of the original samples.
The FFT algorithm computes one cycle of the DFT and its inverse is one cycle of the DFT inverse.
In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values. It can be said to convert the sampled function from its original domain (often time or position along a line) to the frequency domain.
The input samples are complex numbers (in practice, usually real numbers), and the output coefficients are complex as well. The frequencies
of the output sinusoids are integer multiples of a fundamental frequency, whose corresponding period is the length of the sampling interval. The combination of sinusoids obtained through the DFT is therefore periodic with that same period. The DFT differs from the discrete-time Fourier transform (DTFT) in that its input and output sequences are both finite; it is therefore said to be the Fourier analysis of finite-domain (or periodic) discrete-time functions.
Illustration of using Dirac comb functions and the convolution theorem to model the effects of sampling and/or periodic summation. At lower left is a DTFT, the spectral result of sampling s(t) at intervals of T. The spectral sequences at (a) upper right and (b)
lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. The respective formulas are (a) the Fourier series integral and (b) the DFT summation. Its similarities to
the original transform, S(f), and its relative computational ease are often the motivation for computing a DFT sequence.
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 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;[1] 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.
Definition
The sequence of N complex numbers is transformed into an N-periodic sequence of complex numbers:
(integers) [2] (Eq.1)
Each is a complex number that encodes both amplitude and phase of a sinusoidal component of function . The sinusoid's frequency is k/N cycles per sample. Its amplitude and phase are:
where atan2 is the two-argument form of the arctan function. Due to periodicity (see Periodicity), the customary domain of k actually computed is [0, N-1]. That is always the case when the DFT is implemented via the Fast Fourier transform algorithm. But other common domains are [-N/2, N/2-1] (N even) and [-(N-1)/2, (N-1)/2] (N odd), as when the left and right halves of an FFT output sequence are swapped.
The transform is sometimes denoted by the symbol , as in or or .[3]
Eq.1 can be interpreted or derived in various ways, for example:
• It completely describes the discrete-time Fourier transform (DTFT) of an N-periodic sequence, which comprises only discrete frequency components. (Discrete-time Fourier transform#Periodic data)
• It can also provide uniformly spaced samples of the continuous DTFT of a finite length sequence. (Sampling the DTFT)
• It is the cross correlation of the input sequence, xn, and a complex sinusoid at frequency k/N. Thus it acts like a matched filter for that frequency.
• It is the discrete analogy of the formula for the coefficients of a Fourier series:
(Eq.2)
which is also N-periodic. In the domain this is the inverse transform of Eq.1.
The normalization factor multiplying the DFT and IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1/N. A normalization of for both the DFT and IDFT, for instance, makes the transforms unitary.
In the following discussion the terms "sequence" and "vector" will be considered interchangeable.
Properties
Completeness
The discrete Fourier transform is an invertible, linear transformation
with denoting the set of complex numbers. In other words, for any N > 0, an N-dimensional complex vector has a DFT and an IDFT which are in turn N-dimensional complex vectors.
Orthogonality
The vectors form an orthogonal basis over the set of N-dimensional
complex vectors:
where is the Kronecker delta. (In the last step, the summation is trivial if , where it is 1+1+⋅⋅⋅=N, and otherwise is a geometric series that can be explicitly summed to obtain zero.) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.