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 $.
