What did people do with time-varying filters prior to 2014?

[soundmodel]

To answer the question:

search.php?keywords=Filter+synthedit&te ... mit=Search

https://web.archive.org/web/20240000000 ... sicdsp.org

No the main problem is not about lack of resources, but having experience in the prior time to know what some time-varying techniques (SVF, ZDF/TPT) changed in that domain.

[BertKoor]

I've thought about building a small aeroplane. The options are to buy a kit, or study aerodynamics myself which I find too much investment. But I cannot present a kit as my own design.
Most kits are from fibreglass and have bent wing tips. What were plane builders using before fiberglass? How did they fly properly without bent tips?

Have your cake and eat it? If you manage that, then don't leave that cake out in the rain. For it took so long to bake it, and you'll never have that recipe again...

[earlevel]

This small paper claims that SVF is not known because of poor documentation:

Digital State-Variable Filters
Fons ADRIAENSEN
http://kokkinizita.linuxaudio.org/papers/digsvfilt.pdf

It references a 2020 paper though, so maybe the writer is not informed.

---

To answer viewtopic.php?p=8872347#p8872347 below:

Based on the superior advantages of the SVF presented in the first post's paper, people should be driven to use the SVF, if they'd be aware of the results from that paper or elsewhere.

Amusing...well, he didn't say "not known", he said, "So one may wonder why it is not used more often". Anyway, that paper is post 2020, but it's detailed in Hal Chamberlin's Musical Applications of Micro Processors in 1980. That's why it's sometime referred to as the Chamberlin state variable (Chamberlin moved one of the delay elements outside the loop, compared to your paper).

The filter has widespread use over decades. If your criteria is why it isn't used more in synths, well, I suppose it is, but for a four-pole solution people often gravitate to other forms, often the Moog-style loop of four first order sections.

Hal Chamberlin popularized the filter 44 years ago, and it's one of the most widely known forms. Here's my short article on it in 2003: The digital state variable filter

[mystran]

So you're saying that ZDF/TPT is the theoretical first principle for non-LTI?

No idea what you mean by "first principle" but basically traditional BLT transform takes the transfer function of a continuous-time LTI system and substitutes s<-2/T*(z-1)/(z+1), then simplifies resulting discrete-time transfer function to obtain a rational that we can implement as a direct form.

If the system is not LTI, then "transfer function" is ill-defined concept, because "transfer functions" in the conventional sense only apply to LTI systems. However, what we can do is observe that the subsitution s<-2/T*(z-1)/(z+1) or it's reciprocal 1/s<-T/2*(z+1)/(z-1) is in fact the trapezoidal rule of numerical integration and we can use the trapezoidal method to integrate (basically) any continuous-time system, LTI or not.

So rather than evaluating the continuous-time transfer function and substituting the trapezoidal rule there, we can take the (time-domain) differential equations and use the same trapezoidal method to integrate those instead. If the system "happens to be" LTI then we end up with the exact same discrete-time transfer function either way... but when the system is not LTI, doing the subsitution in time-domain allows us to preserve the original state variables: each integrator remains an integrator after the transform. Ofcourse the resulting system drawn as a block diagram often has "delay-free loops" so we can't just treat the difference equations as "code" but rather must actually solve it as "math" .. but one of the insights of this whole "ZDF/TPT business" is that such a "solver" need not be terribly expensive computationally.

Finally, since in both cases we're really using the same approximation (the trapezoidal rule) for Laplace s, we can also take the "prewarping" commonly applied to conventional bilinear transform and apply it the same way to our time-domain trapezoidal rule, which gives us full equivalence in the LTI case and in fact if we evaluate the digital transfer function (for an LTI system) then simplify we end up with the same direct form coefficients we would have obtained by substituting directly into the continuous-time transfer function.

So.. in a sense, the bilinear transform as applied to transfer functions to obtain direct form coefficients is a mathematical shortcut to trapezoidal integration that applies when the system is LTI. If the system is not LTI, you can still do trapezoidal "the hard way" and this what is known as ZDF/TPT.

[soundmodel]

So you're saying that ZDF/TPT is the theoretical first principle for non-LTI?

No idea what you mean by "first principle"

A "first principle" is a foundational assumption or proposition - it is foundational in that it cannot be deduced from other assumptions or propositions.

Or, in this context, a foundational way to design non-LTI filters. Or a very general or "pure" method for non-LTI filters. Or a kind of a "ground truth".

But, in fact, since it bases on LTI theory, then I am not sure. I always thought time-varying should be either reparametrization of time-invariant or then functions that give such properties naturally. Interpolation does not sound like it affects the filters except by trying to fit them together afterwards.

I am not versed with the methods enough to answer this, but I had this naive idea that zero delay filters could be the ground truth of non-LTI filters, if "zero delays" implies that the outputs of such filters cannot be phase-shifted (which would cause instabilities).

[soundmodel]

This paper:

Fast Modulation of Filter Parameters
A practical guide
Ammar Muqaddas
http://www.solostuff.net/wp-content/upl ... v1.1.1.pdf

does seem to treat the ZDF/TPT as a "ground truth" for filters with fast parameter modulation.

[whyterabbyt]

This paper:

Fast Modulation of Filter Parameters
A practical guide
Ammar Muqaddas
http://www.solostuff.net/wp-content/upl ... v1.1.1.pdf

does seem to treat the ZDF/TPT as a "ground truth" for filters with fast parameter modulation.

The author is a poster at KVR, named as s0lo. You can see the conversations he was having here at KVR about it, at viewtopic.php?t=523958

[soundmodel]

I wonder how much is it true that information regarding such filters is a bit scattered?

In viewtopic.php?p=7378192#p7378192 it suggests that at least the existence of filters with nice time-varying properties does not seem to be theoretically simple and may not even be related to particular structures. Rather, it seems that it's possible that the filters with nice time-varying properties can exist in many forms that also produce filters with non-nice properties. In this sense the thread's question is flawed, because it presumed that SVFs guarantee such properties (based on the Wishnick paper).

OTOH the A. Wishnick paper refers to:

J. Laroche, “On the stability of time-varying recursive
filters,” J. Audio Eng. Soc., vol. 55, no. 6, pp. 460–471, June
2007.

Which is supposed to have some criteria for stability in the time-varying case.

However, I also fail to understand how it seems so difficult to define time-varying stable filters.

Here viewtopic.php?p=7382444#p7382444 the poster suggests that TPT structures still have a high probability of having such properties. So in this sense we could, possibly, just erase what's prior to 2014 (or some year) and study the TPT now. Right? Because in any case, we would always want the time-varying behavior in musical filters.

[mystran]

I am not versed with the methods enough to answer this, but I had this naive idea that zero delay filters could be the ground truth of non-LTI filters, if "zero delays" implies that the outputs of such filters cannot be phase-shifted (which would cause instabilities).

The "zero delay" is just a term people used to describe filters where if you draw the signal flow as a graph, there are loops with "zero delays" around the loop. Such a filter requires you to solve simultaneous equations rather than just order the computations. That's all there is to it.

[soundmodel]

I am not versed with the methods enough to answer this, but I had this naive idea that zero delay filters could be the ground truth of non-LTI filters, if "zero delays" implies that the outputs of such filters cannot be phase-shifted (which would cause instabilities).

The "zero delay" is just a term people used to describe filters where if you draw the signal flow as a graph, there are loops with "zero delays" around the loop. Such a filter requires you to solve simultaneous equations rather than just order the computations. That's all there is to it.

Yes, but what does this imply about time-varying behavior and/or stability?

Although based on what was recently written here, it should not imply robustly, because TPT structures can fail to have desirable properties. Yet they have a high probability of having such properties.

Also, since TPT operates in the time-domain, then it seems like the stability issue is mostly just a matter of parametrization, and there should not, by default, exist a need to interpolate between finite frequency-domain information and continuous-time.

It does, after all, make naive sense that if a circuit is delayless, then it will not produce discontinuities in phase responses, because everything must then be continuous, as long as the components are (and they are in analog).

[S0lo]

Hi soundmodel,

It's been a while since I wrote that article so I'm not in a fresh mindset as I was. Besides, although the main point has been made there, I still don't consider that article complete, as my understanding of these concepts is still half backed.

In short, frequency domain analysis (i.e using transfer functions) is only valid when the system is LTI. For none LTI, you have to do time domain analysis. And the moment, you change the cutoff, resonance or whatever parameter. During the change the system is none LTI.

Now you might ask, how then is RBJ and many other filters that are derived completely in frequency domain (as in direct forms) do still work while modulating cutoff and resonance with an LFO or manually. Well it's because that modulation is slow. that is, not fast enough to induce problems. Yes, mathematically it's no longer the filter you designed, it's something else. But practically it works!!

Once you go beyond say 100Hz or so of speed in modulation, you run into the probability of the filter exploding, specially if resonance is high. When I say explode, I mean a numerical overflow of the filter output and possibly it's memory variables. In theory, the system is divergent (ie. output goes to infinity).

This "overflow" explosion is the only thing that concerns me in the article I wrote. Any other artifacts, good or bad, I consider subjective and is a matter of flavor or design. It's a valid musical filter/system that may or may not be put to use in some musical context as long as it doesn't explode. Other research papers might have other (or more) criterion than this, as in "nice time-varying properties", "bad behaviour" or "artifact-free". I haven't studied these criterion, and frankly don't care too much as long as the filter keeps working. What concerns me in that article is what makes the filter literally stop working.

[S0lo]

I am not versed with the methods enough to answer this, but I had this naive idea that zero delay filters could be the ground truth of non-LTI filters, if "zero delays" implies that the outputs of such filters cannot be phase-shifted (which would cause instabilities).

The "zero delay" is just a term people used to describe filters where if you draw the signal flow as a graph, there are loops with "zero delays" around the loop. Such a filter requires you to solve simultaneous equations rather than just order the computations. That's all there is to it.

Yes, but what does this imply about time-varying behavior and/or stability?

AFAIK, it doesn't contribute to stability (or instability) unless it contributes to it in the LTI case.

I have no proof for that. But I've demonstrated 3 cases in the article:

1. The zero delay SVF when done with TPT (time domain) doesn't explode at fast modulation.
2. The zero delay SVF, When done in frequency domain does explode at fast modulation.
3. The SVF, When done in time domain but with delay in feedback, doesn't explode at fast modulation.

Mind you that all these three cases are of the exact same SVF filter. No changes at all in the LTI case. From these 3 cases, I'm confident that the culprit for instability at fast modulation is frequency domain analysis. (Unless you can prove otherwise in a case by case basis).

[soundmodel]

I am not versed with the methods enough to answer this, but I had this naive idea that zero delay filters could be the ground truth of non-LTI filters, if "zero delays" implies that the outputs of such filters cannot be phase-shifted (which would cause instabilities).

The "zero delay" is just a term people used to describe filters where if you draw the signal flow as a graph, there are loops with "zero delays" around the loop. Such a filter requires you to solve simultaneous equations rather than just order the computations. That's all there is to it.

Yes, but what does this imply about time-varying behavior and/or stability?

AFAIK, it doesn't contribute to stability (or instability) unless it contributes to it in the LTI case.

I have no proof for that. But I've demonstrated 3 cases in the article:

1. The zero delay SVF when done with TPT (time domain) doesn't explode at fast modulation.
2. The zero delay SVF, When done in frequency domain does explode at fast modulation.
3. The SVF, When done in time domain but with delay in feedback, doesn't explode at fast modulation.

Mind you that all these three cases are of the exact same SVF filter. No changes at all in the LTI case. From these 3 cases, I'm confident that the culprit for instability at fast modulation is frequency domain analysis. (Unless you can prove otherwise in a case by case basis).

But this is what I was trying to suggest several times, but as I am inexperienced, then I cannot use the proper vocabulary. But the main naive idea I had is just that frequency-domain things necessarily need windows or "finite snapshots", and windowed-processing leads to discontinuities or at least difficulties in managing continuities.

Time-domain methods are continuous, if the functions are. Therefore a time-domain continuous filter that's also BIBO is always stable, right?

Trivially, a varying system cannot be described by any transfer function that's the same for all times, while a LTI system can.

I cannot find it now, but some sources have suggested that if given the choice one should almost always do filters in time-domain. While using frequency-domain only for analysis purposes.

I sort of get the feeling that frequency-domain was thrown in, because it's not possible to infer from time-domain equations what some analog system does.

However, viewtopic.php?p=7378192#p7378192 suggested that TPT as a structure does not guarantee stability. But it has a high probability of having stability.

[mystran]

Time-domain methods are continuous, if the functions are. Therefore a time-domain continuous filter that's also BIBO is always stable, right?

If we are talking about BIBO stability, then this is sort of a tautology, because .. well a filter is "BIBO stable" if bounded input leads to bounded output (duh). For LTI filters this implies it eventually decays to zero (or by extension some constant term if you add one). I believe such a system is called "asymptotically stable" (all initial conditions converge to the same final state), but BIBO stability need not imply asymptotic stability if the system is non-linear. For example, a self-oscillating filter is BIBO stable (hopefully), but need not ever stop oscillating.

That said, it can actually sometimes be quite difficult to formally define what we mean by "stable" in any given context... and you obviously can't prove that something is "stable" unless you've formally defined the stability conditions.

[S0lo]

But this is what I was trying to suggest several times, but as I am inexperienced, then I cannot use the proper vocabulary. But the main naive idea I had is just that frequency-domain things necessarily need windows or "finite snapshots", and windowed-processing leads to discontinuities or at least difficulties in managing continuities.

Sorry my terminology might have caused a confusion. By analysis I don’t mean processing. I meant mathematical derivation. Totally different things. There is no windowing involved or anything like that.

One of the reasons to use frequency domain derivation is because it’s much easier in many cases. For example, the equivalent of a transfer function (Z domain) in discrete time domain is a recurrence relation which is not always the easiest thing to manipulate, understand and design for.