For the given specification design an analog Butterworth filter.

Solution:

$\begin{aligned} & \Omega_{\mathrm{p}}=0.2 \pi ; \quad \Omega_{\mathrm{s}}=0.4 \pi \\ & \frac{1}{\sqrt{1+\varepsilon^2}}=0.9 \Rightarrow \varepsilon=0.4843 \\ & \frac{1}{\sqrt{1+\lambda^2}}=0.2 \Rightarrow \lambda=4.899 \end{aligned}$

$\mathrm{N} \geq \frac{\log \left(\frac{\lambda}{\varepsilon}\right)}{\log \left(\frac{\Omega_{\mathrm{s}}}{\Omega_{\mathrm{p}}}\right)} \geq 3.34$

$\mathrm{N}=4$

$\mathrm{H}(\mathrm{s})=\frac{1}{\left(\mathrm{~s}^2+0.76537 \mathrm{~s}+1\right)\left(\mathrm{s}^2+1.8477 \mathrm{~s}+1\right)}$

Cut off frequency,

$\Omega_c=\frac{\Omega_p}{\varepsilon^{1 / N}}=0.24 \pi$

Obtain transfer function by …