Bandpass Filter from Lowpass

A bandpass can be derived by designing the appropriate lowpass filter (with cut-off frequency equal to half the width of the bandpass), and then modulating it to the desired center frequency $\omega_c$ . In the frequency domain, this means shifting two copies of the lowpass prototype, one to the left by $-\omega_c$ , and the other to the right by $\omega_c$ . The two shifted copies are added together to give a bandpass filter having a real impulse response.

To design a bandpass filter:

• Design a lowpass prototype filter of the desired width
  (cut-off $\omega_c = \omega_u - \omega_0$ = $(\omega_u-\omega_l)/2$ )
• Shift to the left and right by $\omega_0$ and sum:
  $h_{\hbox{bp}}(n)=2\cos(n\omega_0)h_{\hbox{lp}}(n)$