A lattice-like realization with second-order section allows the design of linear-phase two-band filter banks with even order where both \(H_{0}(z)\) and \(H_{1}(z)\) are symmetric. In this case

\[\mathbf{E}(z)=\left(\begin{array}{cc}\alpha_{1} & 0 \\0 & \alpha_{2}\end{array}\right)\left[\prod_{i=I}^{1}\left(\begin{array}{cc}1+z^{-1} & 1 \\1+\beta_{i} z^{-1}+z^{-2} & 1+z^{-1}\end{array}\right)\left(\begin{array}{cc}\gamma_{i} & 0 \\0 & 1\end{array}\right)\right]\]

(a) Design an analysis filter bank such that \(H_{1}(z)=-1+z^{-1}+2 z^{-2}+z^{-3}-z^{-4}\).

(b) Determine the corresponding \(H_{0}(z)\).

(c) Derive the general expression for the polyphase matrix of the synthesis filter.