Consider a two-band linear-phase filter bank whose product filter \(P(z)=\) \(H_{0}(z) H_{1}(-z)\) of order \((4 M-2)\) can be expressed as

\[P(z)=z^{-2 M+1}+\sum_{k=0}^{M-1} a_{2 k}\left(z^{-2 k}+z^{-4 M+2+2 k}\right)\]

Show that such a \(P(z)\) :

(a) Can have at most \(2 M\) zeros at \(z=-1\).

(b) Has exactly \(2 M\) zeros at \(z=-1\) provided that its coefficients satisfy the following set of \(M\) equations:

\[\sum_{k=0}^{M-1} a_{2 k}(2 M-1-2 k)^{2 n}=\frac{1}{2} \delta(n), \quad n=0,1, \ldots,(M-1) .\]

Use the fact that a polynomial root has multiplicity \(p\) if it is also the root of the first \(p-1\) derivatives of the polynomial.