Discrete-time Fourier transform

Summary

The discrete-time Fourier transform turns a sequence $(x_n)_{n \in \Z}$ into a $2\pi$-periodic function $X(\omega) = \sum_n x_n e^{-i \omega n}$ of a continuous frequency $\omega$. It is the Fourier transform of the group $\Z$, whose dual group is the circle. Read backwards, it is a Fourier series: up to the sign of the index, the sequence consists of the Fourier coefficients of $X$. It is the basic tool for analysing digital filters and sampled signals. Sampling the frequency variable at $N$ points leads to the discrete Fourier transform. See the definition.