[SOLVED] Bireciprocal lattice wave digital filter

[bubulescu]

Hello

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

γ=(re(s)−1)/(re(s)+1)

where re(s) is the realpart of the pole, gives this result:

**broken link removed**

Among the few references I have, 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 seemed as if the poles need to be "normalized" to the single real pole, s[(N+1)/2]. That's what I did:

γ=(re(s)/s[(N+1)/2]−1)/(re(s)/s[(N+1)2]+1)

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

**broken link removed**

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 are 3 results using the same settings as in the thesis (As=68=>Ap,ωs=2/3=>ωp,f0=2), between a normal Cauer IIR filter (V(o3)), Barbara Dai's non-quantized coefficients (V(o1)) and my coefficients used with the "normalizing" described above (V(y1), γ1=−0.0912405,γ2=−0.3412645,γ3=−0.729655):

**broken link removed**

While not there (too may traces makes comparison difficult), if the 11 bit quantized values are used for V(o1), the response actually gets closer to the IIR.


Anticipated thanks,
Vlad

 

[bubulescu]

In the meantime someone gave me the answer: the first formula, γi=(σi−1)/(σi+1) is correct, but the determination of the poles in the s-domain is wrong, in that all the four parameters, As, Ap, ωs and ωp, 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.