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I am designing a Chebyshev filter and I am required to calculate the poles of it. It requires $1\textrm{ dB}$ pass-band ripple, a ripple factor of $0.5088$ and an order of $3$.

The equations for calculating the real and imaginary poles are shown below:

\begin{align} s_k &= \sigma_k + j\omega_k\\ \sigma_k &= \sin\left(\frac{(2k-1)\pi}{2n}\right)\sinh(v)\\ \omega_k &= \cos\left(\frac{(2k-1)\pi}{2n}\right)\cosh(v) \end{align}

However, when I calculate the imaginary part I get the following values:

\begin{align} \sigma_1 &= -0.96\\ \sigma_2 &= 0\\ \sigma_3 &= +0.96 \end{align}

However, according to my solutions I should get $+0.96$ and $-0.96$. This checks out with the values from this link.

Is there some sort of rule to invert the values?

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  • $\begingroup$ I don't understand your question; I'd expect complex conjugate poles. $\endgroup$ – Marcus Müller Aug 16 '16 at 7:31
  • $\begingroup$ You get the correct poles, don't you? One real-valued one and a complex conjugate pair with imaginary parts $\pm 0.96$, so everything's OK, isn't it? You can number the poles in any way you like (if that was the misunderstanding). $\endgroup$ – Matt L. Aug 16 '16 at 7:42
  • $\begingroup$ Yes but I get -0.96 for n = 1 and 0.96 for n =3 ? When it should be the other way round according to my answers/online link? or is it because of the complex conjugate you invert the imaginary part? $\endgroup$ – Tricks Aug 16 '16 at 7:53
  • $\begingroup$ You mean $k$, the pole index, not $n$, the filter order, right? It doesn't matter at all. It's just about getting the correct poles. Depending on the exact form of the formula, the order of the poles can be interchanged, but this is completely irrelevant. You don't know which (version of the) formula is used by the website, do you? $\endgroup$ – Matt L. Aug 16 '16 at 9:10
  • $\begingroup$ I just checked the website, it does give the formula, but it doesn't say at all which pole in the table corresponds to which value of $k$ (and that's good, because that is totally irrelevant). $\endgroup$ – Matt L. Aug 16 '16 at 9:12
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As I've mentioned in a comment, the poles are not "ordered" in any way. You can come up with different versions of the formula for the pole locations, which will give you the same poles in different orders.

But anyway, looking at the formulae in your question, the first thing to notice is that there should be a negative sign in the formula for the real parts $\sigma_k$. Second, the values that you got must be the imaginary parts $\omega_k$ (and not $\sigma_k$ as written in your question). If I evaluate the formula for $\omega_k$ given in your question for $k=1,2,3$ I get the following values (for $n=3$ and $1$ dB pass band ripple): $$\begin{align}\omega_1&=0.966i\\ \omega_2&=0\\ \omega_3&=-0.966i\end{align}$$

So I get a positive sign for $k=1$ and a negative sign for $k=3$.

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