Order

Derive an expression to calculate the order of Inverse Chebyshev low pass filter

The attenuation is given by

\[
\alpha = -20 \log_{10}\left(|T(j\omega)|\right)
\]

We know,

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

Hence,

\[
\alpha = -20 \log_{10}\left(|T(j\omega)|\right)
\]

\[
\alpha = -10 \log_{10}\left(\frac{\varepsilon^2 C_n^2(\omega)}{1 + \varepsilon^2 C_n^2(\omega)}\right)
\]

\[
\alpha = 10 \log_{10}\left(1 + \frac{1}{\varepsilon^2 C_n^2(\omega)}\right)
\]

We know that at

\[
\omega = \omega_p, \quad \alpha = \alpha_p = \alpha_{\max}
\]

\[
\alpha_{\max} = 10 \log_{10}\left(1 + \frac{1}{\varepsilon^2 C_n^2(\omega_p)}\right)
\]

\[
10^{\frac{\alpha_{\max}}{10}} = 1 + \frac{1}{\varepsilon^2 C_n^2(\omega_p)}
\]

\[
\frac{1}{\varepsilon^2 C_n^2(\omega_p)} = 10^{\frac{\alpha_{\max}}{10}} - 1
\]

\[
C_n^2(\omega_p) = \frac{1}{\varepsilon^2 \left(10^{\frac{\alpha_{\max}}{10}} - 1\right)}
\]

Where,

\[
\varepsilon = \sqrt{10^{\frac{\alpha_{\min}}{10}} - 1}
\]

Thus,

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

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

---

### Chebyshev Functions

For Chebyshev filters:

\[
C_n(\omega) = \cosh\left(n \cosh^{-1}(\omega)\right), \quad \omega > 1
\]

For **Inverse Chebyshev**:

\[
C_n\!\left(\frac{1}{\omega_p}\right) = \cosh\left(n \cosh^{-1}\!\left(\frac{1}{\omega_p}\right)\right), \quad \frac{1}{\omega_p} > 1 \; \text{or} \; \omega_p < 1
\]

Since \(\frac{1}{\omega_p} > 1\), we can write:

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

Taking inverse hyperbolic cosine:

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

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

If \(\omega_s\) is given, then:

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

This gives the **order of the Inverse Chebyshev filter**.