Window Method in the Frequency Domain

$$\begin{eqnarray*} (1)\quad{\hat H}(\omega) &=& (W\ast H)(\omega) \eqsp \int_{-\pi}^{\pi} W(\nu)H(\omega-\nu)\,d\nu\\ [5pt] &=& \zbox{\langle W, \hbox{\sc Shift}_\omega\{\hbox{\sc Flip}(\overline{H})\} \rangle} \end{eqnarray*}$$

For the ideal lowpass filter:

• Main-lobe is widened by convolution with rectangle to become the pass band
• Side-lobes are convolved with the rectangle to form the stop band (adjacent-sidelobe partial cancellation occurs, resulting SBA$<$ sidel-lobe-level, except for dc-pass where SBA=SLL)

$$\begin{eqnarray*} (2)\quad{\hat H}(\omega) &=& (H\ast W)(\omega) \eqsp \int_{-\pi}^{\pi} H(\nu)W(\omega-\nu)\,d\nu\\ [5pt] &=& \zbox{\langle H, \hbox{\sc Shift}_\omega\{\hbox{\sc Flip}(\overline{W})\} \rangle} \end{eqnarray*}$$

For the ideal lowpass filter:

• Passband is widened by convolution with main lobe to give a transition band
• Transition band is clearly one main-lobe-width wide
• Stop-band made from sliding width-$2\omega_c$ integral over the window side-lobes