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The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of () into a discrete-time Fourier transform (DTFT), which generally entails a type of distortion called aliasing. Choice of an appropriate sample-rate (see Nyquist rate) is the key to minimizing that distortion.
[note 3] Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or R n, notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S 1, the unit circle ≈ ...
Let X(f) be the Fourier transform of any function, x(t), whose samples at some interval, T, equal the x[n] sequence.Then the discrete-time Fourier transform (DTFT) is a Fourier series representation of a periodic summation of X(f): [d]
By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences u [ n ] {\displaystyle u[n]} and v [ n ] {\displaystyle v[n]} with transforms U {\displaystyle U} and V {\displaystyle V} :
Most of the important attributes of the complex DFT, including the inverse transform, the convolution theorem, and most fast Fourier transform (FFT) algorithms, depend only on the property that the kernel of the transform is a principal root of unity. These properties also hold, with identical proofs, over arbitrary rings.
The exponential form of Fourier series is given by: = = [],which is periodic with an arbitrary period denoted by . When continuous time is replaced by discrete time , for integer values of and time interval , the series becomes:
[citation needed] For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers.