Editing Discrete-time Fourier transform (section)

{{distinguish|text=the [[discrete Fourier transform]]}}
{{short description|Fourier analysis technique applied to sequences}}
{{Fourier transforms}}

In [[mathematics]], the '''discrete-time Fourier transform''' ('''DTFT''') is a form of [[Fourier analysis]] that is applicable to a sequence of discrete values.

The DTFT is often used to analyze samples of a continuous function. 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. In simpler terms, when you take the DTFT of regularly-spaced samples of a continuous signal, you get repeating (and possibly overlapping) copies of the signal's frequency spectrum, spaced at intervals corresponding to the sampling frequency. Under certain theoretical conditions, described by the [[sampling theorem]], the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the [[discrete Fourier transform]] (DFT) (see {{slink|#Sampling the DTFT}}), which is by far the most common method of modern Fourier analysis.

Both transforms are invertible. The inverse DTFT reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence.  The [[fast Fourier transform]] (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.