$($a$)$ Consider an LSI system given by

$Q_1(z)=|z_1|\frac{1-z_1^{**-1}{z}^{-1}}{1-z_1{z}^{-1}}.$

Prove: $|Q_1(e^j)|=1.$

$($b$)$ Consider an LSI FIR $($non$-$recursive$)$ system given by

$H(z)=cpro{d}_k=1^q(1-z_k{z}^{-1}),$

where $z_k$ are distinct and $|z_k|1,$ for $k=1,2,$dots,$q.$ Determine the number $N$ of unique

LSI FIR $($non$-$recursive$)$ systems $H_k(z)$ such that $|H_k(z)|=|H(z)|,$ for for $k=1,2,$dots,$q.$