Design Butterworth filter using the impulse invariant method for the following specification,

$$ \begin{array}{r} 0.8 \leq\left|H\left(e^{j \omega}\right)\right| \leq 1, \quad 0 \leq \omega \leq 0.2 \pi \\ \left|H\left(e^{j \omega}\right)\right| \leq 0.2, \quad 0.6 \pi \leq \omega \leq \pi \end{array} $$

Solution:

$ \begin{aligned} & \frac{1}{\sqrt{1+\varepsilon^2}}=0.8 \Rightarrow \varepsilon=0.75 \\\\ & \frac{1}{\sqrt{1+\lambda^2}}=0.2 \Rightarrow \lambda=4.899\\ \end{aligned} $

Ws $=0.6$ pi rad: $w_p=0.2$ pi rad . then assume $\mathrm{T}=1 \mathrm{sec}$

$ \frac{\omega_{\mathrm{s}}}{\omega_{\mathrm{p}}}=\frac{\Omega_{\mathrm{s}} \mathrm{T}}\\{\Omega_{\mathrm{p}} \mathrm{T}}=\frac{\Omega_{\mathrm{s}}}{\Omega_{\mathrm{p}}}=\frac{0.6 \pi}{0.2 \pi}=3\\ $

$ \mathrm{N} \geq \frac{\log \left(\frac{\lambda}{\varepsilon}\right)}{\log \left(\frac{\Omega_{\mathrm{s}}}{\Omega_{\mathrm{p}}}\right)} \geq \frac{\log \left(\frac{4.899}{0.75}\right)}{\log 3} \geq 1.71\\ $

For $\mathrm{N}=2$ the transfer …