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In digital signal processing, the decimator performs decimation, which is downsampling a signal. In other words, when a digital signal is downsampled, the signal's sampling rate would be reduced. | In digital signal processing, the decimator performs decimation, which is downsampling a signal. In other words, when a digital signal is downsampled, the signal's sampling rate would be reduced. | ||
| + | where <br> | ||
| + | <math>s_D [m]=\left\{ \begin{array}{ll} | ||
| + | 1,& \text{ if } n \text{ is a multiple of } 4,\\ | ||
| + | 0, & \text{ else}. | ||
| + | \end{array}\right. = {\frac{1}{D}} \sum_{k = -\infty}^{D-1} e^{jk {\frac{2 \pi}{D} m}}</math> | ||
== Derivation == | == Derivation == | ||
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<font size="3"></font> | <font size="3"></font> | ||
| + | |||
| + | <font size="3">To summarize, the Nyquist theorem states that any bandlimited signal can be perfectly reconstructed from its sampling if sampled at a rate greater than twice its bandwidth <span class="texhtml">(''f''<sub>''s''</sub> > 2''f''<sub>''M''</sub>)</span>. However, the Nyquist condition is not necessary for perfect reconstruction as shown in the example above. </font> | ||
<font size="3"></font> | <font size="3"></font> | ||
Revision as of 19:35, 9 October 2014
Downsampling
A slecture by ECE student Yerkebulan Yeshmukhanbetov
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Outline
- Introduction
- Derivation
- Example
- Conclusion
- References
Introduction
In digital signal processing, the decimator performs decimation, which is downsampling a signal. In other words, when a digital signal is downsampled, the signal's sampling rate would be reduced.
where
$ s_D [m]=\left\{ \begin{array}{ll} 1,& \text{ if } n \text{ is a multiple of } 4,\\ 0, & \text{ else}. \end{array}\right. = {\frac{1}{D}} \sum_{k = -\infty}^{D-1} e^{jk {\frac{2 \pi}{D} m}} $
Derivation
Example
Example
Conclusion
To summarize, the Nyquist theorem states that any bandlimited signal can be perfectly reconstructed from its sampling if sampled at a rate greater than twice its bandwidth (fs > 2fM). However, the Nyquist condition is not necessary for perfect reconstruction as shown in the example above.
