Audio Signal Processing and Coding

PROBLEMS

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 h_k(lM − m)g_k(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

6.4. Using results from Problem 6.3, Prove Eq. (6.5) for the analysis-synthesis framework shown in Figure 6.1.

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

1. Design the synthesis filters, G₀(z) and G₁(z), in Figure 6.27 such that aliasing distortions are minimized.
2. 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 y_i(n) = , for i = 0, 1.
3. Assuming an alias-free scenario, show that , where α is the QMF bank gain, n₀ is a delay that depends on H_i(z) and G_i(z). Estimate the value of n₀.
4. Repeat steps (a) and (c) for H₀(z) = 1 − 0.75 ...