Apply the technique, known as balancing, described in Exercise 10.6, to the filters of Equations (10.204) and (10.205) and comment on the observed results.

Exercise 10.6,

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 \(z=-1\).

Transcribed Image Text:

Ho(z) H(z) -0.129 4095 -0.224 1439z- = +0.482 9629 +0.8365163z+0.224 1439z" -0.129 40952-3 (10.204) +0.8365163z' - 0.482 9629z (10.205)