(New page: Category:slecture Category:ECE438Fall2014Boutin Category:ECE Category:ECE438 Category:signal processing <center><font size= 4> Downsampling in the Frequency Domain ...) |
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| − | A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student | + | A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student John S. |
Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. | Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. | ||
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==Introduction== | ==Introduction== | ||
| + | Remember for time domain, Downsampling is defined as:<br><br> | ||
| + | Image1<br><br> | ||
| + | Now let's describe this process in the frequency domain. | ||
==Derivation== | ==Derivation== | ||
| + | First we'll take the Discrete Time Fourier Transform of the original signal and the downsampled version of it.<br> | ||
| + | <math>\begin{align} | ||
| + | \mathcal{X}(\omega) &= \mathcal{F }\left \{ x_2[n] \right \} = \mathcal{F }\left \{ x_1[Dn] \right \}\\ | ||
| + | &= \sum_{n=-\infty}^\infty x_1[Dn]e^{-j2\omega f} | ||
| + | \end{align}</math> | ||
| + | <br>make the substitution of <math>n=\frac{m}{\D}</math> | ||
==Example== | ==Example== | ||
==Conclusion== | ==Conclusion== | ||
---- | ---- | ||
Revision as of 13:41, 9 October 2014
Downsampling in the Frequency Domain
A slecture by ECE student John S.
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Contents
Introduction
Remember for time domain, Downsampling is defined as:
Image1
Now let's describe this process in the frequency domain.
Derivation
First we'll take the Discrete Time Fourier Transform of the original signal and the downsampled version of it.
$ \begin{align} \mathcal{X}(\omega) &= \mathcal{F }\left \{ x_2[n] \right \} = \mathcal{F }\left \{ x_1[Dn] \right \}\\ &= \sum_{n=-\infty}^\infty x_1[Dn]e^{-j2\omega f} \end{align} $
make the substitution of $ n=\frac{m}{\D} $
