For the given specification design an analog Butterworth filter.

$$ \begin{array}{r} 0.9 \leq|\mathrm{H}(\mathrm{j} \Omega)| \leq 1, \quad \text { for } 0 \leq \Omega \leq 0.2 \pi \\ |\mathrm{H}(\mathrm{j} \Omega)| \leq 0.2, \text { for } 0.4 \pi \leq \Omega \leq \pi \end{array} $$

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 substituting s by,

$ \frac{\mathrm{s}}{\Omega_{\mathrm{c}}}=\frac{\mathrm{s}}{0.24 \pi} $

$ \mathrm{H}(\mathrm{s})=\frac{0.323}{\left(\mathrm{~s}^2+0.577 \mathrm{~s}+0.0576 \pi^2\right)\left(\mathrm{s}^2+1.393 \mathrm{~s}+0.0576 \pi^2\right)} $