The discrete-time Fourier transform, or DTFT, represents a discrete-time sequence by a continuous periodic frequency spectrum.

For a suitable sequence $x[n]$,

$$X(e^{i\omega})=\sum_{n=-\infty}^{\infty}x[n]e^{-i\omega n}.$$

The spectrum is periodic in angular frequency:

$$X(e^{i(\omega+2\pi)})=X(e^{i\omega}).$$

The inverse relation reconstructs the sequence:

$$x[n]=\frac1{2\pi}\int_{-\pi}^{\pi}X(e^{i\omega})e^{i\omega n},d\omega.$$

The DTFT is the discrete-time counterpart of the continuous-time Fourier transform, but discreteness in time makes the frequency representation repeat every $2\pi$ radians per sample.

Discrete-time convolution becomes multiplication in frequency:

$$x*h\quad\longleftrightarrow\quad X(e^{i\omega})H(e^{i\omega}).$$

This makes the DTFT a natural language for sampled signals and digital filters.