Design Steps

Design Steps

Find the order and ripple factors

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

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

\[
\omega_{hp} = \frac{1}{\cosh\!\left(\tfrac{1}{n} \cosh^{-1}\!\left(\tfrac{1}{\varepsilon}\right)\right)}
\]

Find the component \(U_k\) and \(V\) for \(k = 0, 1, \dots, 2n+1\)

\[
u_k = \tfrac{1}{n} \cdot \tfrac{(2k+1)\pi}{2}, \quad k=1,2,\dots
\]

\[
v = \pm \tfrac{1}{n} \sinh^{-1}\!\left(\tfrac{1}{\varepsilon}\right)
\]

Compute the values of \(\sinh(v)\) and \(\cosh(v)\)

Determine Chebyshev Poles based on \(n\)

\[
S_k = \sigma(k) \pm j \omega(k)
\]

Where,

\[
\sigma(k) = \sin\!\left(\tfrac{(2k+1)\pi}{2n}\right)\sinh(v), 
\quad 
\omega(k) = \cos\!\left(\tfrac{(2k+1)\pi}{2n}\right)\cosh(v)
\]

for \(k = 0 \; \text{to} \; (2n+1)\)

Evaluate Inverse Chebyshev poles,

\[
S_k = \tfrac{1}{S(k)}
\]

Evaluate Chebyshev zeros Locations

\[
\omega(k) = \sec\!\left(\tfrac{k\pi}{2n}\right), \quad k=1,3,5,\dots \; \text{odd values}
\]