# 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 $(M-1)/2$ units.

$\bar{h}[n]=h[n-\frac{M-1}{2}]$

Then we multiply the result by the window function w[n] where

$w[n]=\left\{ \begin{array}{ll} 1, & 0 \leq n < M,\\ 0, & else. \end{array} \right.$

Below is 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 a zoom on of of the the little "spiky corners" of the graph of the magniture of $W(\omega)$

for $N=1000$ $\omega_c=\frac{\pi}{2}$..