Consider an \(M\)-band linear-phase filter bank with perfect reconstruction with all analysis and synthesis filters having the same length \(N=L M\). Show that, for a polyphase matrix \(\mathbf{E}_{1}(z)\) corresponding to a linear-phase filter bank, then \(\mathbf{E}(z)=\mathbf{E}_{2}(z) \mathbf{E}_{1}(z)\) will also lead to a linear-phase perfect reconstruction filter bank if

\[\mathbf{E}_{2}(z)=z^{-L} \mathbf{D} \mathbf{E}_{2}\left(z^{-1}\right) \mathbf{D}\]

where \(\mathbf{D}\) is a diagonal matrix with entries 1 or -1, as described in Exercise 9.37, and \(L\) is the order of \(\mathbf{E}_{2}(z)\). Also show that

\[\mathbf{E}_{2, i}=\mathbf{D E}_{2, L-i} \mathbf{D}\]

Exercise 9.37,

Show that if a filter bank has linear-phase analysis and synthesis filters with the same lengths \(N=L M\), then the following relations for the polyphase matrices are valid:

\[\begin{aligned}& \mathbf{E}(z)=z^{-L+1} \mathbf{D E}\left(z^{-1}\right) \mathbf{J} \\& \mathbf{R}(z)=z^{-L+1} \mathbf{J R}\left(z^{-1}\right) \mathbf{D}\end{aligned}\]

where \(\mathbf{D}\) is a diagonal matrix whose entries are 1 if the corresponding filter is symmetric and -1 otherwise.