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

Gordonjcp gordonjcp at gjcp.net
Wed May 22 23:51:13 CEST 2019

Hi folks,

I was giving a bit of thought to how ladder filters tune and of course
you need to warp Fc a bit for the frequency response to "track" properly
at higher cutoff settings.

Looking at various bits and pieces it looks like the trick is to use
some function of tan() to calculate how much the filter cutoff needs
"bent down" by.  I can't figure this out, because I suck at maths.

However if I use the "quick tanh approximation" x/(1+x) to warp my
normalised cutoff (ie. 6.28 * Fc / Fs) - which is at best a very very
roughly tanh() shaped function and a very loose approximation - my
actual filter resonant frequency comes out absolutely bob on.

Can anyone explain the maths to me?


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