I'm trying to derive the transfer function for Chebyshev filter.

$$|H(\Omega)|^2=\frac{1}{\sqrt{(1+\epsilon^2T_n^2(\frac{\Omega}{\Omega_c})}}$$

where $$T_n(x)=cos(Ncos^{-1}(x)) \forall x \le 1$$

$$T_n(x)=cosh(Ncosh^{-1}(x)) \forall x \ge 1$$

$$H(s)=\frac{1}{\sqrt{(1+\epsilon^2T_n^2(\frac{s}{j\Omega_c}))}}$$

To calculate the poles I made the following substituion which looks like this:

$$cos^{-1}(\frac{s}{j\Omega_c})=\alpha + j \beta$$

So

$$s=j\Omega_c(\cos\alpha \cosh\beta - j \sin\alpha \sinh\beta)$$ $$s=\Omega_c \sin\alpha \sinh\beta + j \Omega_c \cos\alpha \cosh\beta$$

where $\alpha=\frac{(2k-1)\pi}{2N}$ and $\beta = \frac{1}{N} \sinh^{-1}{\frac{1}{\epsilon}}$

But with this substitution my poles yield to be stability or produce an ambiguity on the stability end.

But if I make the substituion as $$cos^{-1}(\frac{s}{j\Omega_c})=\alpha - j \beta$$

poles seem stable.

Can someone help me with this. is my substitution wrong?

I'm trying to derive the transfer function for Chebyshev filter.

$$|H(\Omega)|^2=\frac{1}{\sqrt{(1+\epsilon^2T_n^2(\frac{\Omega}{\Omega_c})}}$$

where $$T_n(x)=\cos(N\cos^{-1}(x)) \forall x \le 1$$

$$T_n(x)=\cosh(N\cosh^{-1}(x)) \forall x \ge 1$$

$$H(s)=\frac{1}{\sqrt{(1+\epsilon^2T_n^2(\frac{s}{j\Omega_c}))}}$$

To calculate the poles I made the following substituion which looks like this:

$$\cos^{-1}(\frac{s}{j\Omega_c})=\alpha + j \beta$$

So

$$s=j\Omega_c(\cos\alpha \cosh\beta - j \sin\alpha \sinh\beta)$$ $$s=\Omega_c \sin\alpha \sinh\beta + j \Omega_c \cos\alpha \cosh\beta$$

where $\alpha=\frac{(2k-1)\pi}{2N}$ and $\beta = \frac{1}{N} \sinh^{-1}{\frac{1}{\epsilon}}$

But with this substitution my poles yield to be stability or produce an ambiguity on the stability end.

But if I make the substitution as $$\cos^{-1}(\frac{s}{j\Omega_c})=\alpha - j \beta$$

poles seem stable.

Can someone help me with this. is my substitution wrong?