Show that if a filter bank has linear-phase analysis and synthesis filters with the same lengths (N=L

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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.

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Digital Signal Processing System Analysis And Design

ISBN: 9780521887755

2nd Edition

Authors: Paulo S. R. Diniz, Eduardo A. B. Da Silva , Sergio L. Netto

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