Discrete-time Fourier Transform (DTFT)
A slecture by ECE student Xian Zhang
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Contents
outline
- Definition
- Periodicity property
- Example of computation of DTFT of a complex exponential
Definition
The discrete-time Fourier transform (DTFT) of a discrete set of real or complex numbers x[n] with n=all integers, is a Fourier series, which produces a periodic function of a frequency variable. With w has units of radians/sample, the Fourier series is:
$ \begin{align} \\ \mathcal{X}_1(\omega) & = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} \\ \end{align} $
Inverse DTFT is :
$ \begin{align} \\ \quad x[n] & =\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathcal{X}(\omega)e^{j\omega n}d\omega \\ \end{align} $
Periodicity property
$ \mathcal{X}(\omega) $ is periodic with period $ 2\pi $. Because,
$ \begin{align} \\ \mathcal{X}(\omega+2\pi) & = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega+2\pi)n}\\ & =\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}e^{-j2\pi n}\\ & =\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\\ & =\mathcal{X}(\omega)\\ \end{align} $
Example of computation of DTFT of a complex exponential
Given: $ \begin{align} \\ x[n] =e^{j\omega_o n}\\ \end{align} $
$ \begin{align} \\ Proof: \mathcal{X}(\omega) & = 2\pi rep_{2\pi}(\delta(\omega - \omega_o))\\ \end{align} $
Firstly, let's try:
$ \begin{align} \\ \mathcal{X}(\omega) & = \sum_{n=-\infty}^{\infty}e^{j\omega_o n}e^{-j\omega n}\\ & = \sum_{n=-\infty}^{\infty}e^{j(\omega-\omega_o) n}\\ \end{align} $
