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<math> \begin{align} \\ | <math> \begin{align} \\ | ||
\mathcal{X}(\omega+2\pi) & = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega+2\pi)n}\\ | \mathcal{X}(\omega+2\pi) & = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega+2\pi)n}\\ | ||
| − | & =\sum_{n=-\infty}^{\infty}x[n]\\ | + | & =\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\\ |
\end{align} | \end{align} | ||
</math> | </math> | ||
Revision as of 18:47, 29 September 2014
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.
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}\\ \end{align} $
