$3-$ Consider a system with two complex conjugate poles at $p_{1,2}=e^{+-j}$ :

$H(z)=\frac{b_0}{(1-r{e}^j{z}^{-1})(1-r{e}^{-j}{z}^{-1})}=\frac{b_0}{1-(2rcos)z^{-1}+r^2{z}^{-2}},$

a$-$ Find the input$-$output relationship

b$-$ Find the impulse response

c$-$Find geometrical pole$-$zero pattern, and magnitude response,

d$-$ Change the $H(z)$ transfer function to compensate for a high quality BPF