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.