Consider the notch filter given in

$H(z)=b_0[1-(2rcos)z^{-1}+r^2{z}^{-2}].$

$($a$)$ Determine the magnitude of its frequency response $H(e^j)$ for $$~~$$

and determine $b_0$ to normalize this response.

$($b$)$ Compute and plot $|H(e^j)|$ and $\frac{?}{{?}_H(e^j)}$ for $r=0\mathrm{.}9$ and $=2\frac{}{5}.$

$($c$)$ Determine an analytic expression for the $3-dB$ bandwidth of the above notch filter and verify it using your plot in $($b$)$ above.