Show that the (L_{2}) norm of the transfer function [H(z)=frac{b_{1} z+b_{2}}{z^{2}+a_{1} z+a_{2}}] is [|H(z)|_{2}^{2}=frac{left(b_{1}^{2}+b_{2}^{2} ight)left(1+a_{2} ight)-2 b_{1}

Show that the \(L_{2}\) norm of the transfer function

\[H(z)=\frac{b_{1} z+b_{2}}{z^{2}+a_{1} z+a_{2}}\]

is

\[\|H(z)\|_{2}^{2}=\frac{\left(b_{1}^{2}+b_{2}^{2}\right)\left(1+a_{2}\right)-2 b_{1} b_{2} a_{1}}{\left(1-a_{1}^{2}+a_{2}^{2}+2 a_{2}\right)\left(1-a_{2}\right)} .\]