Volterra serries nonlinear modelling

[Aleksey Vaneev]

Many people can tell the difference between an early Line6 POD and a real tube amp easily, for example, but there are many differences between the two, I am not sure what they hear is due to the nonlinearity differences involved.

There's always a difference between everything. What's important is what works and what's not. At the same time, a lot of reviewers and users are semantically biased: e.g. tubes are good, transistors are bad, etc. I think this is a historic injustice, maybe just maybe due to the fact tube devices were historically designed by audiophile nerds while transistor equipment was designed by schooled engineers without much audiophile experience and understanding of harmonic coloration. Audiophile nerds would never go transistor route out of perfectionism and experience they have: also learning something completely new is a hard task. But to me, tube or transistor - they are just "functions". To sound good, transistor device requires as much attention to detail as any good tube device requires it.

[stratum]

maybe just maybe due to the fact tube devices were historically designed by audiophile nerds while transistor equipment was designed by schooled engineers without much audiophile experience and understanding of harmonic coloration.

Actually many simply weren't designed at all. Somebody just turned up the volume a bit too much, and the amp was overdriven. It wasn't intended to be used in that way at all. Luckily it was connected to a bandwidth limited speaker and the result was fine instead of being totally trashy (the latter result would be obtained with today's full range speakers). The rest is history. There are other differences that helped to make this accident happen: Tubes draw much less current from coupling capacitors compared to BJT's, so luckily the accident did not include blocking distortion.

[stratum]

There are even other instances of pure luck: Of example since distortion obtained in this way was obtained only at an high volume, the amount of energy contained in the air could be used to enhance the sustain obtained from guitar strings, if the player were to stand at the right spot. I guess it feels like pure magic to the player when that happens. One of the criticisms of modelling amps was that this behavior was not obtained easily, perhaps due to input/output processing latency.

[Zaphod (giancarlo)]

All these components are rather simple - triodes are simple. What makes them special is the phase/transient. Otherwise a waveshaper is boring. All those eq have really different phase... Also harmonic distortion has a phase... And it is frequency dependant. All those things mixed up are the secret of analog world. If you stack minimum phase eq or waveshapers you have a big mess... It is the reason why people working with real analog gear tells you the mix is less busy

It is also the reason why analog studios don't buy all the same outboard and- apart the main console - don't use the same things for all tracks

About the tube vs transistor: it is just a little story created more from magazines than real engineers. I can say many ones like api as much as pultec, depending on the task.
We sold as many solid state pultec emulatios as normal original tube ones. There is a market on both of them

[Miles1981]

There is a visual difference between the simple waveshaper and the ODE, the cited paper says also clearly that the spectrum is not respected. Which means that users would notice the difference.
For a triode, there are indeed two waveshapers, one for the positive side, another for the negative side, but there are also clearly shifts that cannot be reproduced with shappers and as such differneces in the spectrum. I would assume that listeners would notice the difference?
In some ways, static vs ODE is related to the old UAD plugins and their new ones (I'm pretty much confident this is one of the first differences, if there are even additional differences!). I would assume that the race to better modeling is because simple/static modeling is not enough to get these "even harmonies that add warmth" to the signal.

[Zaphod (giancarlo)]

Exactly

[Aleksey Vaneev]

For a triode, there are indeed two waveshapers, one for the positive side, another for the negative side.

You didn't quite understand me. Triode's signal follows the so called "load lines". Each load line is a waveshaper and it already has "positive" and "negative" ends. Then triode's signal oscillates between 2 such load lines. Such topology as I've mentioned recreates quite complex state transitions. For example, waveshaper1 can be a biased tanh() function while waveshaper2 can be a symmetric clip() function. Then you create a derivative oscillation signal (can be a high-passed input signal, or even output signal if feedback is needed) that oscillates (crossfades) between "tube" waveshaper1 and "tape" waveshaper2. The result is quite complex, and I believe this is the essense of the triode's function.

By the way, Koren's triode equations can be used to "extract" load line equations (e.g. expressed as a polynomial). The load line (which is literally drawn as a line on triode's usual 3D voltage graph) depends on the circuitry's parameters.

[Miles1981]

Sorry, I only played with triodes and their ODE equations, so I don't really understand the complexity of the two waveshapers (seems hard to understand without schema! do you have a reference paper on this?).

[Aleksey Vaneev]

Sorry, I only played with triodes and their ODE equations, so I don't really understand the complexity of the two waveshapers (seems hard to understand without schema! do you have a reference paper on this?).

Sorry, I do not have a paper. What I talk about is the result of my own study of triodes, I've analyzed a lot of information on the internet. However, topology I've mentioned is probably not usable for "scientific" or exact SPICE-like modeling nor verifiable. But it's good and versatile for Audio DSP. You can embed EQ into this topology. The only problem is that waveshapers and the oscillation signal should be carefully tuned so that the resulting frequency response is flat enough. You have to finetune gains and biases of waveshapers and oscillation signal, apply filters to them if needed.

[stratum]

Sorry, I only played with triodes and their ODE equations, so I don't really understand the complexity of the two waveshapers (seems hard to understand without schema! do you have a reference paper on this?).

At a glance that claim sounded reasonable for someone who was building a plugin intended to capture harmonic distortion associated with EQ or tube saturation circuits which only have mild nonlinearities so I didn't pay much attention when Aleksey said this. I guess it wasn't intended for overdrive circuits, but it's an interesting claim nevertheless, because I couldn't see from where these two waveshapers could come from physically or why they should be added to approximate a triode.

[Aleksey Vaneev]

I guess it wasn't intended for overdrive circuits, but it's an interesting claim nevertheless, because I couldn't see from where these two waveshapers could come from physically or why they should be added to approximate a triode.

This is a good site to get you started on load lines: http://www.trioda.com/tools/triode.html

When you map a load line as a 1-dimensional non-linear function you get a waveshaper.

When triode operates it in fact oscillates between load lines (amperage is oscillation signal while plate "power supply" voltage is fixed e.g. at 400V). In my opinion, it is perfectly enough to just have 2 extreme load lines and oscillate between them linearly.

But if you can afford e.g. calculating Koren equation in real-time you can evaluate load line precisely, without resorting to linear interpolation between 2 load lines. But anyway the general topology remains the same: a waveshaper which represents the load line. Waveshaper is applied to the input (grid) voltage. Then oscillation signal (or amperage) chooses the load line, or waveshaper.

[stratum]

it is perfectly enough to just have 2 extreme load lines and oscillate between them linearly.

What would be the source of the low frequency oscillator? The common wisdom has it that the dynamic aspect of the nonlinearity of a tube is due to the voltage across some capacitor, perhaps the coupling capacitor or the bias bypass capacitor, or both. It may be oscillatory in nature such that a triode might be approximated by a summation of two wave shapers if we knew the exact instantaneous value of that LFO signal. If you are saying that we do not need hard math to know it as it is simply observable with a scope, maybe it is true, as I haven't examined it. It is probably a value correlated to the amount of low frequency (bass) signal in a mix, or perhaps just overall signal amplitude.

[Aleksey Vaneev]

it is perfectly enough to just have 2 extreme load lines and oscillate between them linearly.

What would be the source of the low frequency oscillator? The common wisdom has it that the dynamic aspect of a nonlinearity of a tube is due to the voltage across some capacitor, perhaps the coupling capacitor or the bias bypass capacitor, or both. It may be oscillatory in nature such that a triode might be approximated by a summation of two wave shapers if we knew the exact instantaneous value of that LFO signal. If you are saying that we do not need hard math to know it as it is simply observable with a scope, maybe it is true, as I haven't examined it. It is probably a value correlated to the amount of low frequency (bass) signal in a mix.

Oscillator signal I'm talking about is not necessarily a low-frequency oscillator. It is triode's current in fact. But I do not treat it as such, I treat it as signal that oscillates between load lines. This is how it works physically. You just have to "unlearn" certain things and learn triode load lines - how in fact grid voltage transforms into triode's output signal. All other circuitry around triode are just various filters and possibly some feedback elements.

[aMUSEd]

Why is it called Volterra?

[Aleksey Vaneev]

Why is it called Volterra?

https://en.wikipedia.org/wiki/Volterra_series
First proposed by Vito Volterra.