Consider Figure 6.27,
Given $s(n)=0.75 \sin (\pi n / 3)+0.5 \cos (\pi n / 6), n=0,1, \ldots, 6, H_0(z)=1-$ $z^{-1}$, and $H_1(z)=1+z^{-1}$
a. Design the synthesis filters, $G_0(z)$ and $G_1(z)$, in Figure 6.27 such that aliasing distortions are minimized.
b. Write the closed-form expression for $v_0(n), v_1(n), y_0(n), y_1(n)$, $w_0(n), w_1(n)$, and the synthesized waveform, $\hat{s}(n)$. In Figure 6.27, assume $y_i(n)=\hat{y}_i(n)$, for $i=0,1$.
c. Assuming an alias-free scenario, show that $\hat{s}(n)=\alpha s\left(n-n_0\right)$, where $\alpha$ is the QMF bank gain, $n_0$ is a delay that depends on $H_i(z)$ and $G_i(z)$. Estimate the value of $n_0$.
d. Repeat steps (a) and (c) for $H_0(z)=1-0.75 z^{-1}$ and $H_1(z)=1+$ $0.75 z^{-1}$.
FIGURE CANT COPY
Figure 6.27. A two-band maximally decimated analysis-synthesis filter bank.