Design a two-band QMF filter bank satisfying the following specifications:

\[\begin{aligned}\delta_{\mathrm{p}} & =0.5 \mathrm{~dB} \\\delta_{\mathrm{r}} & =40 \mathrm{~dB} \\\Omega_{\mathrm{p}} & =0.45 \frac{\Omega_{\mathrm{s}}}{2} \\\Omega_{\mathrm{r}} & =0.55 \frac{\Omega_{\mathrm{s}}}{2} \\\Omega_{\mathrm{s}} & =2 \pi \mathrm{rad} / \mathrm{s} .\end{aligned}\]

(a) Design the filter using the standard direct-form parametrization.

(b) Design the filter by optimizing the parameters of the appropriate lattice FIR structure.