Consider tree-structured orthogonal filter banks as discussed in Example 3.10, and in particular…

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Consider tree-structured orthogonal filter banks as discussed in Example 3.10, and in particular the full tree of depth 2.

(a) Assume ideal sinc filters, and give the frequency response magnitude of G(2) i0 (ejω), i = 0,…, 3. Note that this is not the natural ordering one would expect.

(b) Now take the Haar filters, and give g(2) i [n], i = 0,…, 3. These are the discrete-time Walsh-Hadamard functions of length 4.

(c) Given that {g0[n], g1[n]} is an orthogonal pair, prove orthogonality for any of the equivalent filters with respect to shifts by 4.