Prove the three statements on the structure of linear phase solutions given in Proposition 3.11….

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Prove the three statements on the structure of linear phase solutions given in Proposition 3.11.

PROPOSITION 3.11

In a two-channel, perfect reconstruction filter bank, where all filters are linear phase, the analysis filters have one of the following forms:

(a) Both filters are symmetric and of odd lengths, differing by an odd multiple of 2.

(b) One filter is symmetric and the other is antisymmetric; both lengths are even, and are equal or differ by an even multiple of 2.

(c) One filter is of odd length, the other one of even length; both have all zeros on the unit circle. Either both filters are symmetric, or one is symmetric and the other one is antisymmetric (this is a degenerate case)