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== Definition == | == 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: | 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: | ||
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| + | <math> \begin{align} \\ | ||
\mathcal{X}_1(\omega) & = \sum_{n=-\infty}^{\infty}x[n]*e^{-j\omega n} </math> | \mathcal{X}_1(\omega) & = \sum_{n=-\infty}^{\infty}x[n]*e^{-j\omega n} </math> | ||
Revision as of 18:55, 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} $
