I am trying to model a bireciprocal Cauer filter in LTspice but I don't get the expected results. More precisely, using

\$\gamma = \frac{re(p_i)-1}{re(p_i)+1} \$

where \$re(p_i)\$ is the realpart of the pole, gives this result:

"un-normalized" coefficients

Among the few references, one that gives a numerical example is a thesis, "Design and Realization Methods for IIR Multiple Notch Filters and High Speed Narrow-band and Wide-band Filters, L. Barbara Dai" and, simply by looking at the numbers and comparing them with what I had, it seemd as if the poles need to be "normalized" to the single real pole, \$p_{\frac{N+1}{2}}\$. That's what I did:

\$\gamma = \frac{\frac{re(p_i)}{p_{\frac{N+1}{2}}}-1}{\frac{re(p_i)}{p_{\frac{N+1}{2}}}+1}\$

so, even if the numerical values still differed, but a not as before, I got this result:

"normalized" coefficients

The example used here is not the one used in the thesis, but I seem to get good results (I cannot verify them) with either stop-band, or transition-band optimizations and for any (odd) order.

So, my question is: is this the way to do it, "normalize" poles by dividing each to the single, real pole?


Just for the sake of comparison, here is a comparison using the same settings as in the thesis (\$As=68 => Ap, \omega_s=\frac{2}{3} => \omega_p, f0=2\$), between a normal Cauer IIR filter (V(o3)), Barbara Dai's non-quantized coefficients (V(o1)) and the my coefficients used with the "normalizing" described above (V(y1), \$ \gamma_1=-0.0912405, \gamma_2=-0.3412645, \gamma_3=-0.729655 \$):



In the meantime someone gave me the answer: the first formula, \$\gamma_i=\frac{\sigma_i-1}{\sigma_i+1}\$ is correct, but the determination of the poles in the s-domain is wrong, in that all the four parameters, \$A_s\$, \$A_p\$, \$\omega_s\$ and \$\omega_p\$, need to be specified such that the resulting \$N\$ is closest to integer, without using ceil(). In other words, the best approach is to impose \$N\$ and deduce one of the four parameters from the other three.

With the same example from the thesis (p.27) and a quick change in my formulas, I get this result:


where the blue trace is from the thesis and the black one - mine. Even if there still are differences, they are minor, closer to the truth and probably due to one of the other 2 possible optimizations (which are too difficult to achieve with LTspice). Either way, this is the right path.

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