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 image hk(lMm)gk(lMn), where M is the number of subbands.

6.3. For the down-sampling and up-sampling processes given in Figure 6.26, show that

image

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, image. In Figure 6.27, assume yi(n) = image, for i = 0, 1.
  3. Assuming an alias-free scenario, show that image, where α is the QMF bank gain, n0 is a delay that depends on Hi(z) and Gi(z). Estimate the value of n0.
  4. Repeat steps (a) and (c) for H0(z) = 1 − 0.75 ...

Get Audio Signal Processing and Coding now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.