# Band pass filter (digital resonance)

Q1:  Consider a discrete-time band-pass filter (BPF) that uses only two poles as

$$H(z) = G \frac{1}{\big(1-r \, e^{j \omega_0} z^{-1}\big)\big(1-r \, e^{-j \omega_0} z^{-1}\big)}$$

where this filter has a center frequency of $\omega_0$. If this BPF was designed to have a very narrow bandwidth by selecting $r$ to be very close to unity, show that the 3dB bandwidth of the BPF filter is approximately given by

$$BW_{3dB}=2(1-r)$$

(Hint:  Use a geometrical approach on the unit circle.)

• If this is homework, please add the home work tag – Hilmar Jul 29 '18 at 19:18
• this is not a band-pass filter (BPF), but is a low-pass filter (LPF). however if $$0 < 1-r \ll 1$$ then it's an LPF with a resonant peak at frequency $\omega_0$. – robert bristow-johnson Jul 30 '18 at 6:47