6.1. Prove the identities shown in Figure 6.10.
6.2. Consider the analysis-synthesis filter bank shown in Figure 6.1 with input signal, s(n). Show that hk(lM − m)gk(l − Mn), where M is the number of subbands.
6.3. For the down-sampling and up-sampling processes given in Figure 6.26, show that
6.5. Consider Figure 6.27,
Given s(n) = 0.75 sin(πn/3) + 0.5cos(πn/6), n = 0, 1, … , 6, H0(z) = 1 − z−1, and H1(z) = 1 + z−1
- Design the synthesis filters, G0(z) and G1(z), in Figure 6.27 such that aliasing distortions are minimized.
- Write the closed-form expression for υ0(n), υ1(n), y0(n), y1(n), w0(n), w1(n), and the synthesized waveform, . In Figure 6.27, assume yi(n) = , for i = 0, 1.
- Assuming an alias-free scenario, show that , where α is the QMF bank gain, n0 is a delay that depends on Hi(z) and Gi(z). Estimate the value of n0.
- Repeat steps (a) and (c) for H0(z) = 1 − 0.75 ...