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</center> | </center> | ||
---- | ---- | ||
− | == | + | == Outline == |
<font size = 3> | <font size = 3> | ||
*Definition | *Definition | ||
*Periodicity property | *Periodicity property | ||
*Example of computation of DTFT of a complex exponential | *Example of computation of DTFT of a complex exponential | ||
+ | *Conclusion | ||
+ | *References | ||
---- | ---- | ||
---- | ---- | ||
== 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 | + | 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 as the variable, <math>\omega</math>. The Fourier series is: |
<math> \begin{align} \\ | <math> \begin{align} \\ | ||
Line 86: | Line 88: | ||
If | If | ||
<math> \begin{align} \\ | <math> \begin{align} \\ | ||
− | \mathcal{X}(\omega) & =2\pi \delta(\omega - \omega_o)\\ | + | \mathcal{X}(\omega) & =2\pi \delta(\omega - \omega_o).\\ |
\end{align} | \end{align} | ||
</math> | </math> | ||
+ | The result works for <math>\omega</math> between <math>0</math> to <math>2\pi</math>. | ||
+ | |||
+ | But this is not the final answer. Because Fourier transform must be periodic but the answer we got above is not periodic. | ||
+ | |||
+ | Since <math>\mathcal{X}(\omega)</math> must be periodic with period <math>2\pi</math>. | ||
+ | |||
+ | The final answer is: | ||
+ | |||
+ | <math> \begin{align} \\ | ||
+ | \mathcal{X}(\omega) & = 2\pi rep_{2\pi}(\delta(\omega - \omega_o))\\ | ||
+ | \end{align} | ||
+ | </math> | ||
+ | ---- | ||
+ | ---- | ||
+ | == Conclusion == | ||
+ | Understand the basic calculation of Discrete-time Fourier Transform (DTFT) and inverse Fourier transform. | ||
+ | Notice an important property of DTFT: the periodicity property. | ||
+ | Master the basic DTFT computation of complex exponential. | ||
+ | ---- | ||
+ | ---- | ||
+ | == Reference == | ||
+ | [1]. Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26, 2009 | ||
+ | ---- | ||
+ | ---- | ||
+ | </font size> | ||
+ | ==[[Slecture_Fourier_transform_w_f_ECE438_review|Questions and comments]]== | ||
+ | |||
+ | If you have any questions, comments, etc. Please post them on | ||
+ | [[Slecture_Fourier_transform_w_f_ECE438_review|this page]]. | ||
+ | ---- | ||
+ | [[2014_Fall_ECE_438_Boutin|Back to ECE438, Fall 2014]] |
Latest revision as of 19:30, 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.
Contents
Outline
- Definition
- Periodicity property
- Example of computation of DTFT of a complex exponential
- Conclusion
- References
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 as the variable, $ \omega $. 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} $
Proof: $ \begin{align} \\ \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}\\ & = \infty, \omega = \omega_o; 0, else \\ \end{align} $
How to compute something that diverges:
$ \begin{align} \\ e^{j\omega_o n} & =\frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n}d\omega \\ \end{align} $
Assume $ \omega $ is between $ 0 $ to $ 2\pi $.
If $ \begin{align} \\ \mathcal{X}(\omega) & =2\pi \delta(\omega - \omega_o).\\ \end{align} $ The result works for $ \omega $ between $ 0 $ to $ 2\pi $.
But this is not the final answer. Because Fourier transform must be periodic but the answer we got above is not periodic.
Since $ \mathcal{X}(\omega) $ must be periodic with period $ 2\pi $.
The final answer is:
$ \begin{align} \\ \mathcal{X}(\omega) & = 2\pi rep_{2\pi}(\delta(\omega - \omega_o))\\ \end{align} $
Conclusion
Understand the basic calculation of Discrete-time Fourier Transform (DTFT) and inverse Fourier transform. Notice an important property of DTFT: the periodicity property. Master the basic DTFT computation of complex exponential.
Reference
[1]. Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26, 2009
Questions and comments
If you have any questions, comments, etc. Please post them on this page.