Cheap non-linear zero-delay filters

[xhy3]

This also works with aliasing suppression for the tanh harmonics:

viewtopic.php?f=33&t=486252&p=6822981#p6822981

[andy-cytomic]

In emails with Mystran he has said that his method of using f(x)/x for non-linearities is already contained in the source code of certain circuit simulation packages, possibly as fallback methods for stability.

Well I haven't really found any source code utilizing the code, but in a book (and I regret forgetting which one it was, couldn't find it later) about circuit simulation methods there was a comparison between various iterative methods, including fixed-point, secant, this method and Newton. The author referred to the method mentioned here as the "BLAS-3" method and basically said that it's not been studied too carefully, but suspected the convergence (with iteration) should be somewhere between linear and that of secant method (which is about sqrt(2)).

Personally, from empirical observations, I suspect the convergence is damn near linear. As far as I'm concerned, the only advantage over Newton (or secant) is that a truncated iteration (most notably single evaluation only) tends to result in something that isn't totally useless (as could happen with Newton).

I actually followed this up and the book you are referring to that mentions the BIAS-3-A method is:
https://books.google.com.au/books?id=PZ ... navlinks_s
Fundamentals of Computer-Aided Circuit Simulation (2012)
William J. McCalla

Which reference this paper:
https://ieeexplore.ieee.org/abstract/document/1050153
BIAS-3-A program for the nonlinear D.C. analysis of bipolar transistor circuits (1972)
W.J. McCalla ; W.G. Howard

It is also called the "Line through the origin method" or "Conductance only" method etc etc.

Google got the image to text wrong, so you can only find this book by searching for "BLAS-3" but the actual method was called "BIAS-3-A"

[mystran]

Right... in any case the method isn't something original I came up, except as far as I rediscovered it before finding out that it was a rediscovery... which is what often happens when you try to figure out something that other people have also worked on in the past.

[andy-cytomic]

Right... in any case the method isn't something original I came up, except as far as I rediscovered it before finding out that it was a rediscovery... which is what often happens when you try to figure out something that other people have also worked on in the past.

Mystran: thanks so much for your multiple contributions to music-dsp, really appreciated.

I realise you didn't try and claim you came up with it, I just wanted to tie up the loose ends and show where it actually came from for everyone else here, since the names of the papers / books were never shown, and I think it's good for people to have references to read when it comes to different methods, so they can see them in a larger context.

I've made some variations based on this conductance only method which I use in production code for The Drop, so thanks again for making me aware of it

[itoa]

Regarding CEM3320 datasheet 91k vs 100k: if my limited EE understanding doesn't fail me, and we assume the stages are true low-pass stages, then one would expect voltage gain of 100k/91k ~ 1.1 per stage which results in total gain of 1.458 or so, eg (still assuming tanh()):

Code: Select all

 dV0/dt = f * tanh( 1.1 * in - r * limit(V3) - v0) 
 dV1/dt = f * tanh( 1.1 * v0 - v1 ) 
 dV2/dt = f * tanh( 1.1 * v1 - v2 ) 
 dV3/dt = f * tanh( 1.1 * v2 - v3 ) 

The resonance VCA is still a question mark. If it's safe to assume it's a simple OTA and the input resistor to ground (3.5k) is also what the other gain cells have (or had if they were OTAs), then we'd have roughly a gain of 1.96 ~ 2 from 51k resistor from audio out. Maybe one could calculate that from the values given in the datasheet but I'm too tired to figure out an obvious way right now. Anyway, the assumption would give:

Code: Select all

 r' = r / (1.458 * 1.96) ~ r / ( 2.86 ) 

 dV0/dt = f * tanh( 1.1 * in - r' * tanh(1.96 * V3) - v0) 
 dV1/dt = f * tanh( 1.1 * v0 - v1 ) 
 dV2/dt = f * tanh( 1.1 * v1 - v2 ) 
 dV3/dt = f * tanh( 1.1 * v2 - v3 )

That sounds quite reasonable actually (with resonance clipping before the stages start going foobar). Unfortunately I have no such filter to measure against..

edit: sound sample for the above (if I didn't make any mistakes) http://www.signaldust.com/files/cascade.mp3
(mp3 but high bitrate.. oh and 44.1kHz host rate with x4 oversampling)

I think that's not too bad, whether or not it models anything

What is "a typical" input level for such a filter model. I noticed -1 -> 1 sounds too saturated

[Urs]

If you read the original patent, a CEM 3320 model wouldn't have many tanh() terms. The distortion is asymmetric, but each stage is inverting, thus some distortion cancels out, but at low cutoffs you get considerable modulation bleed-through in a lowpass setting. At audio rate modulation, you can actually hear the modulation signal almost 1:1 on the output when Cutoff is all the way down.

(also, the levels within the stages are all over the place and vary from unit to unit, where "unit" for me are specimen in old Sequential synths - part of the differences we put into our models is simply gain staging based on measurements)

[mystran]

If you read the original patent, a CEM 3320 model wouldn't have many tanh() terms.

Yeah, that quoted old post of mine is really old and I wrote it before I'd looked into the patent for details. Such a tanh() filter is closed to the later (assuming the part numbers go roughly in order; not in the mood to start hunting for release dates) Curtis filters.

From memory, the short version of the patent (for CEM3320) is that ignoring the current mirrors (which are probably not negligible), there is an exp(vIn + vCV) terms for each stage where the signal itself is "small" such that the response is "approximately linear" where as the CV term is "large" and sets the bias such that the signal is (approximately) multiplied by the slope set by the CV term. There is one of these per pole and then one extra without a signal, where the latter is subtracted to (supposedly) cancel out the exp(vCV) part.

ps. Also before you try, I'll save you some trouble: the CEM3320 type circuit doesn't work well with the cheap method.

[Urs]

Doesn't work well with Newton Raphson either, because an overshoot in the exp() terms can have heavy consequence. Worse even when you try to reverse the computation (as described for the SEM-style filter in Vadim's book), because you'll easily get NaNs (when log() terms undershoot 0)

[mystran]

Doesn't work well with Newton Raphson either, because an overshoot in the exp() terms can have heavy consequence. Worse even when you try to reverse the computation (as described for the SEM-style filter in Vadim's book), because you'll easily get NaNs (when log() terms undershoot 0)

In my experience NR almost always works perfectly fine with exp() as long as you iterate log() whenever the voltage is large and then exp() again when it's small.

[itoa]

Sorry my electronics knowledge is quite basic, reading about bleeding and trying to imagine this, is each stage like?

dV1/dt = exp( (V0 - V1)*a + cv )

[mystran]

Sorry my electronics knowledge is quite basic, reading about bleeding and trying to imagine this, is each stage like?

dV1/dt = exp( (V0 - V1)*a + cv )

I'm not sure what "a" here is supposed to be, but you need to subtract the constant cv-term, something like:

dv1/dt = exp(v0 - v1 + cv) - exp(cv)

Also the current mirrors are probably not negligible, but I don't think I ever really looked into those too carefully, because personally I don't like the sound of this type of filter.

[itoa]

Thanks a lot.. "a" is an "unknown scaling factor"

I wonder how important is using exp here, while probably it operates in a small, "almost linear" exp curve range.