Design an analog Butterworth filter that has $-2$ passband attenuation at frequency of $20 \mathrm{rad} / \mathrm{sec} \&$ at least $-\operatorname{10dB}$ stop band attenuation at $30 \mathrm{rad} / \mathrm{sec}$.

Solution:

Given:

$ \begin{aligned}\\ &A_p=-2 d B \\\\ &A_s=-10 d B \\\\ &f\Omega_p=20 \mathrm{rad} / \mathrm{sec} . \\\\ &\Omega_s=30 \mathrm{rad} / \mathrm{sec} .\\ \end{aligned}\\ $

$ \begin{aligned}\\ A p &=20 \log \delta_q \\\\ -2 &=20 \log \delta_7 \\\\ \therefore \delta_7 &=0.7943\\ \end{aligned}\\ $

$ \begin{aligned}\\ A_s &=20 \log \delta_2 \\\\ -10 &=20 \log \delta_2\\ \end{aligned}\\ $

$ \Omega_c=\frac{\Omega_p}{\left(1 / \delta_1^2-1\right)^{1 / 2 N}}=\\ $

$ \left(\frac{(30}{\left.1 / 0.7943^2-1\right)^{1 / 8}})\right.\\ $

$ \Omega_c=32.079 \mathrm{rad} / \mathrm{sec} \\ $

$ \begin{aligned}\\ s_{i k} &=\Omega_c e^{j \pi / 2} e^{j(2 k+1) \pi / 2 N} \\\\ &=32.079 e^{j \pi / 2} e^{j(2 k+1) \pi / 8}\\ \end{aligned}\\ $

$ \begin{aligned}\\ S_0 &=32.079 e^{j \pi / 2} e^{j \pi / 8} \\\\ &=32.079 e^{j 5 \pi / 8} \\\\ &=32.079[\cos (5 \pi / 8)+j \sin (5 \pi / 8)] \\\\ &=-12.276+j 29.637\\ \end{aligned}\\ $

$ \begin{aligned}\\ S_1 &=32.079 e^{j \pi / 2} e^{j 3 \pi / 8} \\\\ &=32.079 e^{j 7 \pi / 8} \\\\ &=32.079[\cos (7 \pi / 8)+j \sin (7 \pi / 8)] \\\\ &=-29.637+j 12.276\\ \end{aligned}\\ $

$ \begin{aligned}\\ S_3 &=32.079 e^{j \pi / 2} e^{j\gt\pi / 8} \\\\ &=32.079 e^{j \pi \pi / 8} \\\\ &=32.079[\cos (11 \pi / 8)+j \sin (11 \pi / 8)] \\\\ &=-12.276-j 29.637\\ \end{aligned}\\ $

$ \begin{aligned}\\ H(S)=& \frac{S_0 S_1 S_2 S_3}{\left(S-S_0\right)\left(S-S_1\right)\left(S-S_2\right)\left(S-S_3\right)} \\\\ =& \frac{(-12.276+j 29.637)(-12.276-j 29.637)}{[S-(-12.276+j 29.637)][S-(-12.276-j 29.637)]} \\\\ & {[S-(-29.637-j 12.276)][S-(-29.637+j 12.276)] }\\ \end{aligned}\\ $

$ \begin{aligned}\\ &=\frac{\left[(2.276)^2+(29.637)^2\right]\left[(29.637)^2+(12.276)^2\right]}{\left[(s+12.276)^2+(29.637)^2\right]\left[(s+29.637)^2+(12.276)^2\right]} \\\\ &=\frac{1.0589 \times 10^6}{\left[(5+12.276)^2+(29.637)^2\right]\left[(s+29.637)^2+(2.276)^2\right]}\\ \end{aligned}\\ $