Discrete-time Fourier transform

The discrete-time Fourier transform is the DT analogue of the continuous-time Fourier transform.

The DTFT of a DT signal $x$ is the function $X:\mathbb{R}\to \mathbb{C}$ defined by:

$$X(e^{j\omega })=\sum _{n=-\infty }^{\infty }x[n]e^{-j\omega n}$$

where $X$ is the Fourier transform or spectrum of $x$. The inverse DTFT of a DT spectrum $X$ is the DT signal $x:\mathbb{Z}\to \mathbb{C}$ defined by:

$$x[n]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X(e^{j\omega })e^{j\omega n}d\omega$$

Properties

The spectrum $X(e^{j\omega })$ of a DT signal $x$ is a periodic function of $\omega$ with period $N=2\pi$.

Its relationship with the DTFS is given below:¹

Footnotes

1. From Prof Simpson-Porco’s lecture notes. ↩