Discrete-time Fourier transform

A discrete-time Fourier transform (or DTFT) is a Fourier transform of a function $f_{n}$ of an integer (discrete) "time" variable n with an unbounded domain.

The DTFT differs from the discrete Fourier transform (DFT), however, in that the latter transforms a function function $f_{n}$ that periodic. Thus the DTFT produces a continuous spectrum $F(\omega )$. The spectrum $F(\omega )$ is periodic however, as a consequence of the discreteness of the input.

Essentially, the DTFT is the reverse of the Fourier series, in that the latter has a continous periodic input and a discrete unbounded spectrum. The applications of the two transforms, however, are quite different.

Definition

The DTFT of $f_{n}$ is given by:

$F(\omega )=\sum _{n=-\infty }^{\infty }f_{n}\,e^{-in\omega }.$

and its inverse transform recovers $f_{n}$ by

$f_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }F(\omega )\,e^{-in\omega }\,d\omega .$