A discrete-time Fourier transform (or DTFT) is a Fourier transform of a function <math>f_n<math> 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 <math>f_n<math> that periodic. Thus the DTFT produces a continuous spectrum <math>F(\omega)<math>. The spectrum <math>F(\omega)<math> 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 <math>f_n<math> is given by:

<math>F(\omega) = \sum_{n=-\infty}^{\infty} f_n \,e^{-in\omega}.<math>

and its inverse transform recovers <math>f_n<math> by

<math>f_n =\frac{1}{2\pi}\int_{-\pi}^\pi F(\omega)\,e^{in\omega}\,d\omega.<math>