Solution:

$0 \cdot 8 \leq\left|H\left(e^{j \omega}\right)\right| \leq 1 \quad 0 \leq \omega \leq 0 \cdot 2 \pi$

$\left|H\left(e^{j \omega)} | \leq 0.2 \quad 0.6 \pi \leq \omega \leq \pi\right.\right.$

Given:

$\delta p=0.8 \quad \omega_{p}=0.2 \pi$

$\delta_{S}=0.2 \quad \omega_{S}=0.6 \pi$

Step 1: Obtain analog specifications.

$\Omega p=\frac{2}{T} \tan \left(\frac{\omega P}{2}\right)$

$=\frac{2}{0.1} …

Step 3: Cutoff frequency $\Omega c$

$\Omega c=\frac{\Omega p}{\left[\frac{1}{\delta^2 p}-1\right]^{1 / 2 N}}$

$\Omega_{C}=\frac{6.4983}{\left[\frac{1}{0.8^{2}}-1\right]^{1 / 4}}$

$\Omega_{C}= \ 7.5035 \ rad/sec$

Step 4: Transfer Function H(S)

N is even.

$H(s)=\frac{\Omega_{p}^{N}}{\prod_{k=1}^{N / 2} s^{2}+\Omega_{e} b_{k}S+\Omega_{C}^{2}}$

$H(s)=\frac{(7.5035)^{2}}{\pi_{k = 1} \ s^{2}+7.5035 b_{R} S+(5.5035)^{2}}$

$=\frac{56.3025}{S^{2}+7.5035 b_{1} S+56.3025}$

$b_{1}=2 \sin \left[\left(\frac{2 k-1}{2 N}\right) \pi\right]$ …