Design an elliptic filter satisfying the specifications of Exercise 13.10, using parallel connection of second-order direct-form structures, with \(L_{2}\) and \(L_{\infty}\) norms for scaling, employing the closed-form equations of Section 13.2.4.

Exercise 13.10

Design an elliptic filter satisfying the specifications below:

\[\begin{aligned}A_{\mathrm{p}} & =0.4 \mathrm{~dB} \\A_{\mathrm{r}} & =50 \mathrm{~dB} \\\Omega_{\mathrm{r}_{1}} & =1000 \mathrm{rad} / \mathrm{s} \\\Omega_{\mathrm{p}_{1}} & =1150 \mathrm{rad} / \mathrm{s} \\\Omega_{\mathrm{p}_{2}} & =1250 \mathrm{rad} / \mathrm{s} \\\Omega_{\mathrm{r}_{2}} & =1400 \mathrm{rad} / \mathrm{s} \\\Omega_{\mathrm{s}} & =10000 \mathrm{rad} / \mathrm{s} .\end{aligned}\]

Realize the filter using:

(a) parallel direct-form sections;

(b) cascade of direct-form sections;

(c) cascade of optimal state-space sections;

(d) cascade of state-space sections without limit cycles.
The specifications must be satisfied when the coefficients are quantized in the range of 18 to 20 bits, including the sign bit.