Numerical related 3

Q.N What are the characteristics of Butterworth filter? Derive an expression to estimate the order ‘n’ of low pass Butterworth approximation. Use that formula to estimate that order of Butterwoth filter with following specification 

\[
\text{Given:} \quad \alpha_{\max} = 1 \, \text{dB}, \quad \alpha_{\min} = 20 \, \text{dB}, \quad \frac{\omega_s}{\omega_p} = 1.5
\]

\[
\text{We use the formula:} \quad
n = \frac{\log_{10} \left( \frac{10^{\alpha_{\min}/10} - 1}{10^{\alpha_{\max}/10} - 1} \right)}{2 \log_{10} \left( \frac{\omega_s}{\omega_p} \right)}
\]

\[
n = \frac{\log_{10} \left( \frac{10^{20/10} - 1}{10^{1/10} - 1} \right)}{2 \log_{10}(1.5)}
= \frac{\log_{10} \left( \frac{100 - 1}{1.2589 - 1} \right)}{2 \log_{10}(1.5)}
\]

\[
n = \frac{\log_{10}(99 / 0.2589)}{2 \cdot 0.1761}
= \frac{\log_{10}(382.43)}{0.3522}
= \frac{2.5825}{0.3522}
\approx 7.33 \approx 8
\]

\[
\therefore \text{Estimated order of the Butterworth filter: } \boxed{n = 8}
\]