Order of Chebyshev filter ‘n’

Derive an expression to calculate the order of low pass specification using Chebyshev approximations.

We know,  
At \( \omega = \omega_p \), attenuation is \( \alpha = \alpha_p = \alpha_{\text{max}} \)  
At \( \omega = \omega_s \), attenuation is \( \alpha = \alpha_s = \alpha_{\text{min}} \)

\[
|T_n(j\omega)|^2 = \frac{1}{1 + \varepsilon^2 C_n^2(\omega)}
\]

\[
|T_n(j\omega)| = \frac{1}{\sqrt{1 + \varepsilon^2 C_n^2(\omega)}}
\]

Then,

\[
\alpha = -20 \log \left( \frac{1}{\sqrt{1 + \varepsilon^2 C_n^2(\omega)}} \right)
\]

\[
\alpha = -20 \log \left( (1 + \varepsilon^2 C_n^2(\omega))^{-1/2} \right)
\]

\[
\alpha = -20 \cdot \left( -\frac{1}{2} \log (1 + \varepsilon^2 C_n^2(\omega)) \right)
\]

\[
\alpha = 10 \log (1 + \varepsilon^2 C_n^2(\omega))
\]

At \( \omega = \omega_s \), \( \alpha = \alpha_s = \alpha_{\text{min}} \), so:

\[
\alpha_{\text{min}} = 10 \log (1 + \varepsilon^2 C_n^2(\omega_s))
\]

\[
\frac{\alpha_{\text{min}}}{10} = \log (1 + \varepsilon^2 C_n^2(\omega_s))
\]

\[
10^{\frac{\alpha_{\text{min}}}{10}} = 1 + \varepsilon^2 C_n^2(\omega_s)
\]

\[
\varepsilon^2 C_n^2(\omega_s) = 10^{\frac{\alpha_{\text{min}}}{10}} - 1
\]

But from earlier, we also know:

\[
\varepsilon^2 = 10^{\frac{\alpha_{\text{max}}}{10}} - 1
\]

So,

\[
(10^{\frac{\alpha_{\text{max}}}{10}} - 1) \cdot C_n^2(\omega_s) = 10^{\frac{\alpha_{\text{min}}}{10}} - 1
\]

\[
C_n^2(\omega_s) = \frac{10^{\frac{\alpha_{\text{min}}}{10}} - 1}{10^{\frac{\alpha_{\text{max}}}{10}} - 1}
\]

\[
C_n(\omega_s) = \sqrt{ \frac{10^{\frac{\alpha_{\text{min}}}{10}} - 1}{10^{\frac{\alpha_{\text{max}}}{10}} - 1} }
\]

For \( \omega > 1 \), we use the Chebyshev polynomial definition:

\[
C_n(\omega_s) = \cosh \left( n \cosh^{-1}(\omega_s) \right)
\]

\[
\cosh \left( n \cosh^{-1}(\omega_s) \right) = \sqrt{ \frac{10^{\frac{\alpha_{\text{min}}}{10}} - 1}{10^{\frac{\alpha_{\text{max}}}{10}} - 1} }
\]

\[
n \cosh^{-1}(\omega_s) = \cosh^{-1} \left( \sqrt{ \frac{10^{\frac{\alpha_{\text{min}}}{10}} - 1}{10^{\frac{\alpha_{\text{max}}}{10}} - 1} } \right)
\]

\[
n = \frac{ \cosh^{-1} \left( \sqrt{ \frac{10^{\frac{\alpha_{\text{min}}}{10}} - 1}{10^{\frac{\alpha_{\text{max}}}{10}} - 1} } \right) }{ \cosh^{-1}(\omega_s) }
\]

If \( \omega_p \) is given instead:

\[
n = \frac{ \cosh^{-1} \left( \sqrt{ \frac{10^{\frac{\alpha_{\text{min}}}{10}} - 1}{10^{\frac{\alpha_{\text{max}}}{10}} - 1} } \right) }{ \cosh^{-1}\left( \frac{\omega_s}{\omega_p} \right) }
\]