| Line 25: | Line 25: | ||
<math> \begin{align} \\ | <math> \begin{align} \\ | ||
| − | \mathcal{X}_1(\omega) & = \sum_{n=-\infty}^{\infty}x[n] | + | \mathcal{X}_1(\omega) & = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} \\ |
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | Inverse DTFT is : | ||
| + | |||
| + | x[n] & =\frac{1}{2pi} | ||
| + | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
Revision as of 19:00, 28 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 :
x[n] & =\frac{1}{2pi}
\end{align} </math>
