[sdiy] tanh() and approximations and warping, why is my maths wrong?

Bernard Arthur Hutchins, Jr bah13 at cornell.edu
Wed Jun 12 05:19:29 CEST 2019

Digital filters are designed by either a direct method (e.g., least squares, Remez) or through the indirect use of an “analog prototype” (Impulse-Invariant, Bilinear-Z-Transform).  In the case of an analog prototype, some desirable analog performance characteristic is preserved in the process.  With Bilinear-Z, it is the SHAPE of the frequency response that is preserved.  It is classic (50 years) and terms such as “prototype”, “cutoff warping”, and “pre-warping” (Andy above) are always seen with BZT:


[See pages 27-33.]  The warping is tangential, not hyperbolic tangential.    It is easy to see (absent a clear statement of the application involved), how confusion comes about.

Neil above is also likely correct that the original questioner was not thinking digital filter but some sort of analog ladder filter.  (When we hear “ladder” we tend to think Moog Ladder or cascaded 4-pole, although digital ladder filters, “lattice” filters, and leap-frog-ladders are found digitally.

But why would anyone propose to correct a high-end tracking error of a FILTER with a method (tanh) that is not even used for a VCO?  I’m missing something.

- Bernie

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