Design of a Pure Digital IIR Notch Filter

$$H(z)=\frac{1-2cos({\omega }_0)z^{-1}+z^{-2}}{1-2r\cos ({\omega }_0)z^{-1}+r^2{z}^{-2}}$$

This gives us a notch filter where the width is tunable while $r<1$ The closer $r$ goes towards 1, the more narrow the frequency reponse.