For a two-band perfect reconstruction filter bank, assume the analysis filter (H_{0}(z)) and the synthesis filter (G_{0}(z))

Question:

For a two-band perfect reconstruction filter bank, assume the analysis filter \(H_{0}(z)\) and the synthesis filter \(G_{0}(z)\) satisfy the condition of Equation (10.142). Assume also that these filters were designed to generate a wavelet so that they have enough zeros placed at \(z=1\).

(a) Show that a filter bank consisting of an analysis lowpass filter whose impulse response is

\[\hat{h}_{0}(n)=\frac{1}{2}\left[h_{0}(n)+h_{0}(n-1)\right]\]

and the synthesis lowpass filter impulse response is

\[\frac{1}{2} \hat{g}_{0}(n)=\left[g_{0}(n)-\frac{1}{2} \hat{g}_{0}(n-1)\right]\]

still represents a perfect reconstruction filter bank.

(b) Show this procedure affects the number of zeros of \(\hat{H}_{0}(z)\) and \(\hat{H}_{1}(z)\) at \(z=-1\).