Chebyshev Approximations for Magnitude Response

General Approach to achieve a function of form is

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

We search a function \[F_n(\omega)\] so that the ripple is limited to a certain magnitude.

From Lissajous Function

\[
y = \cos^n(\cos^{-1} x) = C_n(x)
\]

Replacing \( x \) by \(\omega\) and scaling by a constant \(\varepsilon\) (\(\varepsilon < 1\)) gives the Chebyshev Polynomial:

\[
y = \varepsilon \cdot \cos^n(\cos^{-1} \omega) = C_n(\omega)
\]

This fulfills the requirement for \( 0 \leq \omega \leq 1 \).

For \(-1 < \omega < 1\),

\[
C_n(\omega) = \varepsilon \cdot \cos^n(\cos^{-1} \omega)
\]

But when \(\omega > 1\), \(\cos^{-1}(\omega)\) becomes imaginary and the inverse cosine function becomes a hyperbolic function.

Let

\[
\cos^{-1}(\omega) = j z
\]

Then,

\[
\omega = \cos(j z)
\]

We have

\[
\omega = \frac{e^{j(jz)} + e^{-j(jz)}}{2} = \frac{e^{-z} + e^{z}}{2}
\]

which is the definition of the hyperbolic cosine function:

\[
\omega = \cosh(z)
\]

Thus,

\[
z = \cosh^{-1}(\omega) \quad \text{for} \quad \omega > 1
\]

For \(\omega > 1\):

\[
C_n(\omega) = \cos^n(\cos^{-1} \omega) = \cos(n j z)
\]

Since \(\cos(n j z) = \cosh(n z)\),

\[
C_n(\omega) = \cosh \bigl( n \cosh^{-1}(\omega) \bigr)
\]

We have,

\[
C_n(\omega) = 
\begin{cases}
\cosh \left( n \cosh^{-1}(\omega) \right), & \omega > 1 \\
\cos^n \left( \cos^{-1}(\omega) \right), & 0 \leq \omega \leq 1
\end{cases}
\]

Then, the Chebyshev magnitude response takes the form:

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

where,

\[
C_n(\omega) = 
\begin{cases}
\cosh \left( n \cosh^{-1}(\omega) \right), & \omega > 1 \\
\cos^n \left( \cos^{-1}(\omega) \right), & 0 < \omega < 1
\end{cases}
\]

Here, \(|T_n(j\omega)|\) is the magnitude response of the low-pass Chebyshev filter, and \(C_n(\omega)\) is the Chebyshev polynomial of degree \(n\).

Also, the Chebyshev polynomial can be obtained by the recursive relation:

\[
C_{n+1}(\omega) = 2 \omega C_n(\omega) - C_{n-1}(\omega)
\]

with initial conditions:

\[
C_0(\omega) = 1, \quad C_1(\omega) = \omega
\]