(New page: Category:ECE438 (BoutinFall2009) =ECE 438 Fall_2009 Prof. Boutin: graph of the magnitude of the DFT of a windowed filter= This is an approximation of the graph of the magniture of <m...) |
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=ECE 438 Fall_2009 Prof. Boutin: graph of the magnitude of the DFT of a windowed filter= | =ECE 438 Fall_2009 Prof. Boutin: graph of the magnitude of the DFT of a windowed filter= | ||
| − | + | Consider the ideal low-pass filter | |
| + | |||
| + | <math>H(\omega)=\left\{ \begin{array}{ll}1, & \text{ when } |\omega|< \omega_c\\ | ||
| + | 0, & \text{ when } \omega_c< |\omega|< \pi \end{array}\right.,</math> | ||
| + | |||
| + | which corresponds to the DT signal | ||
| + | |||
| + | <math>h[n]=\left\{ | ||
| + | \begin{array}{ll} | ||
| + | \frac{\omega_c}{\pi}, & \text{ if }n=0,\\ | ||
| + | \frac{\sin (\omega_c n)}{\pi n}, & else. | ||
| + | \end{array} | ||
| + | \right.</math> | ||
| + | First we shift <math>h[n]</math> by <math>\frac{M-1}{2}</math> units. | ||
| + | |||
| + | <math>\bar{h}[n]=h[n-\frac{M-1}{2]}</math> | ||
| + | |||
| + | |||
| + | Belowis an approximation of the graph of the magniture of <math>H_{FIR}(\omega)=W(\omega)*\bar{H}(\omega)</math> for <math>M=1,000</math> and <math>\omega_c=\frac{\pi}{2}</math>. | ||
[[Image:hFIR.png|500px]] | [[Image:hFIR.png|500px]] | ||
Revision as of 10:21, 14 October 2009
ECE 438 Fall_2009 Prof. Boutin: graph of the magnitude of the DFT of a windowed filter
Consider the ideal low-pass filter
$ H(\omega)=\left\{ \begin{array}{ll}1, & \text{ when } |\omega|< \omega_c\\ 0, & \text{ when } \omega_c< |\omega|< \pi \end{array}\right., $
which corresponds to the DT signal
$ h[n]=\left\{ \begin{array}{ll} \frac{\omega_c}{\pi}, & \text{ if }n=0,\\ \frac{\sin (\omega_c n)}{\pi n}, & else. \end{array} \right. $ First we shift $ h[n] $ by $ \frac{M-1}{2} $ units.
$ \bar{h}[n]=h[n-\frac{M-1}{2]} $
Belowis an approximation of the graph of the magniture of $ H_{FIR}(\omega)=W(\omega)*\bar{H}(\omega) $ for $ M=1,000 $ and $ \omega_c=\frac{\pi}{2} $.
This is the graph of the magniture of $ W(\omega) $ for $ N=100 $.
This is the graph of the magniture of $ W(\omega) $ for $ N=10000 $.
