Determine the order and the poles of a type I lowpass Chebyshev filter that has a $1 \mathrm{~dB}$ ripple in the passband and passband frequency $\Omega_p=1000 \pi$, a stopband frequency of $2000 \pi$ and an attenuation of $40 \mathrm{~dB}$ or more.

Solution:

Given data $\alpha_p=1 \mathrm{~dB} ; \Omega_p=1000 \pi \mathrm{rad} / \mathrm{sec} ; \alpha_s=40 \mathrm{~dB}$ $\Omega_s=2000 \pi \mathrm{rad} / \mathrm{sec}$

$ N \geq \frac{\cos h^{-1} \sqrt{\frac{10^{0.1 \alpha_s}-1}{10^{0.1 \alpha_p}-1}}}{\cos h^{-1} \frac{\Omega_s}{\Omega_p}} \geq \frac{\cos h^{-1} \sqrt{\frac{10^4-1}{10^{0.1}-1}}}{\cos h^{-1} \frac{2000 \pi}{1000 \pi}}=4.536 $

$ \begin{aligned} & then N=5 \\ & \varepsilon=\sqrt{10^{0.1 \alpha_p}-1}=0.508 ; \quad \mu=\varepsilon^{-1}+\sqrt{1+\varepsilon^{-2}}=4.17 …