Apply the technique, known as balancing, described in Exercise 10.6, to the filters of Equations (10.204) and
Question:
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 \(z=-1\).
Step by Step Answer:
Digital Signal Processing System Analysis And Design
ISBN: 9780521887755
2nd Edition
Authors: Paulo S. R. Diniz, Eduardo A. B. Da Silva , Sergio L. Netto