Consider an M -band linear-phase filter bank with perfect reconstruction with all ana- lysis and

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

E 2 ( z ) = z − L DE 2 ( z − 1 ) D ,

where D is a diagonal matrix with entries 1 or −1, as described in Exercise 9.37, and

L is the order of E 2 ( z ) . Also show that

E 2, i = DE 2, L − i D .