discrete-time Fourier transform

The transform X(e^{jw}) = sum_n x[n]e^{-jwn}, turning a discrete-time sequence into a continuous, 2*pi-periodic frequency-domain function.

The discrete-time Fourier transform (DTFT) of $x[n]$ is $X(e^{j\omega}) = \sum_n x[n]e^{-j\omega n}$ , a continuous function of $\omega$ built from a discrete sum. It is always $2\pi$ -periodic in $\omega$ , since the underlying discrete-time sinusoids repeat every $2\pi$ .