## 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, *H*_{0}(*z*) = 1 − *z*^{−1}, and *H*_{1}(*z*) = 1 + *z*^{−1}

- Design the synthesis filters,
*G*_{0}(*z*) and*G*_{1}(*z*), in Figure 6.27 such that aliasing distortions are minimized. - Write the closed-form expression for
*υ*_{0}(*n*),*υ*_{1}(*n*),*y*_{0}(*n*),*y*_{1}(*n*),*w*_{0}(*n*),*w*_{1}(*n*), and the synthesized waveform, . In Figure 6.27, assume*y*(_{i}*n*) = , for*i*= 0, 1. - Assuming an alias-free scenario, show that , where
*α*is the QMF bank gain,*n*_{0}is a delay that depends on*H*(_{i}*z*) and*G*(_{i}*z*). Estimate the value of*n*_{0}. - Repeat steps (a) and (c) for
*H*_{0}(*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.