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.)