Moog Ladder Filter - Research Paper

[Richard_Synapse]

Go ahead, great paper! And btw, this method works fine with oversampling, too. We have been using NR on virtual-analog filters almost a decade ago in realtime and with oversampling. It is very efficient once optimized, and also stable.

In a sense it can even work better with oversampling, because the input and output tends to be smoother, changing less from one time-step to another, so the probability that you need to iterate NR a lot is actually lower.

Exactly!

Richard

[andy-cytomic]

@pscorbett : I had a quick read of your paper, and all looks good, just a few little things:

- A ladder filter is specifically one with two lines of devices with capacitors to join them (to make each rung), so what you are doing is modelling in the paper is a "cascade filter", not a ladder filter, but you are using the specific approximate non-linearities from a transistor ladder filter. Cascade just means a bunch of one poles in series.
- There is no such thing as a "zero delay feedback filter", filtering requires phase delay (IIR) or sampled delay (FIR) to work. What you are doing is numerically solving a non-linear implicit filter using trapezoidal integration.
- You repeated the Antti Moog paper in the references, better twice than not at all in my book.
- I saw "the the" somewhere, so just do a quick spellcheck on the latex input text

I recommend the title be more like:

Non-linear Implicit Modelling of a Transistor Ladder Filter using OTA Abstraction

The implicit means you solve a continuous system of equations where the current output depends on the current output itself in some way.

https://en.wikipedia.org/wiki/Trapezoid ... equations)
"The trapezoidal rule is an implicit second-order method..."
and Tanh is a non-linear function, hence Non-linear Implicit Modelling.

This is what wikipedia has to say about "zero delay feedback filter":
"The page "Zero delay feedback filter" does not exist. You can create a draft and submit it for review or request that a redirect be created, but consider checking the search results below to see whether the topic is already covered."

PS: a lot of old DSP used explicit numerical integration, or a mix of implicit and explicit. These methods are best termed what they are and are usually defined by their Butcher tables:
https://en.wikipedia.org/wiki/List_of_R ... ta_methods

Explicit methods are those where the current output does not depend on the current output itself (in possibly a non-linear way) but only on the previous outputs (and current input / previous inputs), which are all known so no zero finding is required.

[mystran]

The discussion of the PHONES and EXT amplifier is not very clear. Labelling the parts in figure 5 would already help and without building this thing (eg. in Spice) it is not clear to me what the biased tanh() is supposed to be modelling exactly.. except as far as Figures 6 and 7 suggest it's not a terribly accurate model.

I'm sure things would be more clear if I had the original schematics open to cross-reference, but I think it is somewhat lazy writing to expect the reader to cross-reference random schematics just to try to guess what is being discussed. Now, when I look at the actual schematics (Minimoog right?) what I see is also a complete VCA between the filter and the headphone jack and even if we assume this whole thing is totally linear, then at least we need to scale the feedback by the VCA envelope.

If the LTP formed by Q27 and Q33 would maintain more or less constant combined emitter current, then this would indeed be a tanh() and whatever asymmetry would be from different base offsets, but I'm not sure if R58 qualifies as a constant current source here. The fact that there's a 200k resistor (which is not an E12 value like 99% of the other resistors in the thing) also suggests that the intention could have been to get "close enough to symmetric" and I'd probably suggest measuring a couple of units (or varying parameters in Spice randomly by 5% or so) before concluding too much about the details.

[mystran]

Now, on the topic of E12 values, those are nominally specified for 10% tolerance. The 200k resistor is found in E24 that is designed for 5% tolerances. While there's no point even trying to find 10% resistors, everyone stocks a ton of E12 'cos usually they are close enough and when you try to draw your circuits using mostly E12 you automatically simplify your bill of materials without having to think about whether you already used the same value somewhere else. In fact the 27k R65 is the only one that isn't in E6, but you kinda tend to need enough "in between" values that E12 is still reasonable (even I have a few of each, so I can breadboard random schematics at ease).

Now, when you say 200k or 220k in Spice you get 200k or 220k. When you buy 1% today just because you can't be bothered with anything less 'cos a 5000 reel (ie. small quantity) of them costs €0.008 a piece (checking Mouser for the 1/2W specified by Moog .. though we could probably mostly get away with 1/4W at €0.003 or even 1/10W at €0.001 .. these are SMD but the 1/10W is still 0603 package, so within the realm of what you can do even with a soldering iron) .. we're looking at 198k-202k and 218-222k or so...

...but back in the days Moog specced 5% 'cos the modern 1% would have been expensive. There are a few resistors specced at 1% in the parts list so it's not like they couldn't get them, they'd just been more expensive... and the 200k R61 is specced at 5% so the worst case is 190-210k and a 220k 5% would be 210-230k or so.

So why would Moog say R61 should be 200k if in the worst-case it might be the same as 220k (which btw is E3 ... 1, 2.2, 4.7, 10, 22 .. )? The resistors probably cost the same, but having to add another value into your BoM adds to the logistics cost.

This strongly suggests that Moog was aware that 220k would be "less than ideal" and probably wanted "less rather than more" .. so it might be interesting to simulate the thing with say 195k or something and see what happens with the symmetry. Perhaps Moog had excess of 200k left over from another project? That's also possible, but I find it somewhat less likely.

[Z1202]

Some argument about the terminology.

- A ladder filter is specifically one with two lines of devices with capacitors to join them (to make each rung), so what you are doing is modelling in the paper is a "cascade filter", not a ladder filter, but you are using the specific approximate non-linearities from a transistor ladder filter. Cascade just means a bunch of one poles in series.

I think some people here (including myself in my book) have been consistently using the term "ladder" filter to describe any filter made by creating feedback around a set of other serially connected filter. Typically 1-poles, but that's not essential, what is essential is that those are practically useable filters and not bare integrators (in the latter case we'd be getting a generalized SVF). In a generalized sense, one can even have multiple feedback tap points, or multiple feedback injection points in such "ladder filter". This is not "ladder filter" in a strict electrical engineering sense, but it has a very similar topological idea.

I agree to the terminology that the cascade filter OTOH is just a serial connection of other filters, not necessarily 1-poles though.

- There is no such thing as a "zero delay feedback filter", filtering requires phase delay (IIR) or sampled delay (FIR) to work. What you are doing is numerically solving a non-linear implicit filter using trapezoidal integration.

I assume this is a typo on Andy's side and he meant "zero delay filter", which is what some people sometimes mistakenly call ZDF filters? Formally even the former term makes sense (although it's usually simply called zero-phase filter), that filter doesn't exist "in reality", since it's not causal, but it's a widely employed theoretical concept.

At any rate the term "Zero-delay feedback filter" is firmly and commonly established in our community here, I'm not sure, what's the problem with that, and the term is correct in the sense that there is no delay in the feedback path. "Implicit trapezoidal integration" is 99% mathematically equivalent to what "ZDF" usually means here, but implies and focuses on somewhat different aspects methodologically. I would be cautious using the term "ZDF filter" in a conference or journal paper though, it's better to refer to such filters via "trapezoidal integration" or "delayless feedback" (depending on which of the aspects you want to highlight), which are more accepted terms in the scientific community.

- You repeated the Antti Moog paper in the references, better twice than not at all in my book.

I'm not sure which book Andy refers to, but, as the argument goes, I believe I've seen somewhere in the paper text (which I just skimmed over) mentioning that the g/s notation was introduced by Urs and Andy. Somehow I was under the impression that the idea of instantaneous linearization of block diagram components for the purpose of solving zero-delay feedback was introduced by me in the paper "Preserving the LTI system topology in s- to z-plane transforms" in 2008, which is the first paper on the TPT formulation of the ZDF apporoach. Although I didn't really check the history, I think it's somewhat unlikely (although, of course possible), that someone else randomly picked up exactly the same notation. Maybe that's what happened, though, just saying.

[mystran]

Also I'd guess C22, C21 and even C26 are probably all meaningless at audio frequencies, probably (ie. just there to improve stability) and C23 into base of Q27 is just another buffered hipass pole, likely C20 is negligible too, so fitting a static waveshaper is probably reasonable... but I think you might want to provide some more convincing arguments as to why this should be (biased) tanh() in particular (which might not be entirely wrong, it's an LTP.. but is the asymmetry due to differential offset or is it due to not quite maintaining the fixed combined emitter currents.. my experience with these circuits is not sufficient to answer without simulation in Spice).

[mystran]

Some argument about the terminology.

- A ladder filter is specifically one with two lines of devices with capacitors to join them (to make each rung), so what you are doing is modelling in the paper is a "cascade filter", not a ladder filter, but you are using the specific approximate non-linearities from a transistor ladder filter. Cascade just means a bunch of one poles in series.

I think some people here (including myself in my book) have been consistently using the term "ladder" filter to describe any filter made by creating feedback around a set of other serially connected filter.

I sort of agree with Andy in that one should call it "4-pole cascade" unless it's modelling an actual transistor ladder. The ladder is called a ladder because it literally looks like a ladder, it's that specific filter (unrelated to RC and LC and perhaps digital waveguide ladders found elsewhere, but those are also called ladders 'cos they look like ladders when drawn in a certain way). I don't think anyone is seriously going to confuse the term "4-pole cascade" with any other type of filter, at least not in the context of music-dsp. On the other hand, I disagree with the idea that you necessarily need to simulate it differentially, because if we take the "ideal and forward biased" assumption then the ground-referenced tanh-based formulation is mathematically equivalent optimization of the model.

I'm still divided about how I think about the "OTA with ladder non-linearities" idea though. I know I have derived this same equivalence when I was first trying to understand these things and it did help me gain a deeper understanding of the topic, but at the same time in retrospective I'm not entirely sure if that is actually the most straight-forward way to reach the correct solution.

The ladder filter is bit of a mind bender. Even after all these years I'm not sure if I fully understand why it works the way it works, even though I've done the usual derivations myself many times over. The part that makes me uneasy in particular is the idea that we can assume the emitter current across the transistor pairs other than the first stays constant. It must certainly stay constant on average because there is only one current supply in the bottom, but with the capacitor in there, is this truly so (to a good approximation anyway) in the short-term when we acknowledge imperfect real-world components?

At any rate the term "Zero-delay feedback filter" is firmly and commonly established in our community here, I'm not sure, what's the problem with that, and the term is correct in the sense that there is no delay in the feedback path.

I don't like the ZDF term either, I much prefer "trapezoidal" because that is the term that actually describes what is going on when we use trapezoidal integration to simulate a continuous time system... but I agree the term ZDF is something we're stuck with and in the interest of trying to avoid any confusion I've come to refer to these filters as ZDF/TPT/trapezoidal most of the time.

[Z1202]

The part that makes me uneasy in particular is the idea that we can assume the emitter current across the transistor pairs other than the first stays constant. It must certainly stay constant on average because there is only one current supply in the bottom, but with the capacitor in there, is this truly so (to a good approximation anyway) in the short-term when we acknowledge imperfect real-world components?

There are two parallel current "rails" (going through emitter-collector pairs of transistors). If we ignore the current leakage through the bases, then the current can only flow from one side to the other via a capacitor, thereby the current escaped through a capacitor from one side appears on the other side in the same amount, leaving the sum unchanged.

I don't like the ZDF term either, I much prefer "trapezoidal" because that is the term that actually describes what is going on when we use trapezoidal integration to simulate a continuous time system...

Let me venture a guess. You prefer trapezoidal because, as you admitted earlier, you prefer thinking in equations. In that case, I'd agree that trapezoidal is a much more adequate description of what we're doing. OTOH I'm coming more from the block-diagram side, in which case ZDF is a more precise description of what is happening.

There is even a difference between trapezoidal and ZDF, but it shows up only under extreme conditions. Somewhat remarkably, the difference in this thinking can have real impact. E.g. I remember reading somewhere about people puzzling over why differentiator-based filters (real electronic circuits) often have inferior behavior. Thinking in block diagram terms gives a relatively easy answer

I sort of agree with Andy in that one should call it "4-pole cascade" unless it's modelling an actual transistor ladder. I don't think anyone is seriously going to confuse the term "4-pole cascade" with any other type of filter, at least not in the context of music-dsp.

TBH when one says "4-pole cascade", I'd just assume a series of filters making up a 4-pole in total, but without any feedback. Would be rather confusing to me.

[mystran]

The part that makes me uneasy in particular is the idea that we can assume the emitter current across the transistor pairs other than the first stays constant. It must certainly stay constant on average because there is only one current supply in the bottom, but with the capacitor in there, is this truly so (to a good approximation anyway) in the short-term when we acknowledge imperfect real-world components?

There are two parallel current "rails" (going through emitter-collector pairs of transistors). If we ignore the current leakage through the bases, then the current can only flow from one side to the other via a capacitor, thereby the current escaped through a capacitor from one side appears on the other side in the same amount, leaving the sum unchanged.

Right.. and with the bases held at the same potential by the resistor divide, the Vbe voltages differ by the capacitor voltage. If I recall correctly, the TIS97 has beta around 500 (modern datasheet from onsemi says 250-700, but I remember seeing "typical" figures in some older datasheet around 500) so these are higher gain than a basic 2n3904 (or TIS92 with typical beta 160 used elsewhere in the schematic). This suggests that ideally (by Moog's design) the base current is negligible, but in practice with real-world parts it might not be entirely the case. The schematic does use TIS97 for a bunch of other stuff too, but I wonder if that's partially "well, it won't hurt" when they probably had a bunch of them left over after having to match the pairs in the filter.. because frankly I don't see why they'd need very high beta for the current source of the LTP in VCA when the filter uses a TIS92/2n3904 for the same purpose (probably because they want to match with complementary TIS93/2n3906, I don't think there's one for TIS97) .. but I suspect unless they already had a large stock of TIS97 lying around, the transistor was probably chosen for the ladder and precisely because of the large beta? Or perhaps they just wanted to use beta as high as possible everywhere and had to compromise where they needed complementary pairs? No idea.

I don't know. I have not found the energy to actually try to figure out what finite beta does here. Perhaps it just means the capacitors higher up see less current overall so the poles aren't quite on top of each other before resonance.

There is even a difference between trapezoidal and ZDF, but it shows up only under extreme conditions.

In that case I have no idea what ZDF might mean. The way I've always approached "ZDF" is that we use trapezoidal integration and then we (typically) prewarp the "control voltage" going into the model so that we can get the cutoff where we want (even if technically we are simulating a filter with a higher cutoff and errors from the trapezoidal integration just pull it down to where we want).

[Z1202]

I don't know. I have not found the energy to actually try to figure out what finite beta does here. Perhaps it just means the capacitors higher up see less current overall so the poles aren't quite on top of each other before resonance.

I'd guess that the variation in capacitor values will have a much larger effect on the poles than the base current leakage. OTOH I imagine the base currents need to be somehow matched against the resistor values in the voltage divider, so that the transistors are kept in the active mode no matter what the (practical) current values are.

The transistor mismatch within one pair is a different story, as we strictly speaking don't get a difference of tanh's anymore. The equations do not look analytically solvable in this case. I pursued this a bit once, but didn't come to anything conclusive, except maybe some hints at cutoff signal leakage, not unlike in the feedback amplifier.

In that case I have no idea what ZDF might mean. The way I've always approached "ZDF" is that we use trapezoidal integration and then we (typically) prewarp the "control voltage" going into the model so that we can get the cutoff where we want (even if technically we are simulating a filter with a higher cutoff and errors from the trapezoidal integration just pull it down to where we want).

The way I've always approached "ZDF" is basically described by the TPT (replacing all analog integrators with their digital models, usually TDF2 BLT, and then resolving the arising zero-delay feedback). Practically, I treat ZDF as the de-facto established "umbrella" term for both trapezoidal and TPT.

The main difference between equation- and block diagram-based approaches is that the latter implies certain causal relationships, which equations are ignorant about. As mentioned, this can lead to different predictions (or different level of detail) in extreme condition. The linked paper, as well as Section 3.13 of VAFilterDesign go into some details. The paper is written much later and contains new insights.

[mystran]

OTOH I imagine the base currents need to be somehow matched against the resistor values in the voltage divider, so that the transistors are kept in the active mode no matter what the (practical) current values are.

I don't see how this would be very sensitive to the resistor values though. In the mini we have a total of 1020 ohms from +10V to ground (=10mA through resistor chain? so about 100mW converted to heat?) with 150ohm or 1.5V between each transistor pair and the datasheet gives the usual 0.65V for Vbe-on.. so basically these will dump as much as current to their emitters as they possibly can and the only thing limiting the whole thing is the control current through Q28.

The main difference between equation- and block diagram-based approaches is that the latter implies certain causal relationships, which equations are ignorant about. As mentioned, this can lead to different predictions (or different level of detail) in extreme condition. The linked paper, as well as Section 3.13 of VAFilterDesign go into some details. The paper is written much later and contains new insights.

Ok, I read sections 2.2 and 2.3 to sample the paper... and came to the conclusion that you might be losing it. The discussion about the causality (or lack of) for derivatives in particular is horrible non-sense. That's the same level of non-sense as trying to argue that 0.9999999... is smaller than 1 and only really demonstrates fundamental misunderstanding of what limits actually mean.

The whole idea that there are integrator or differentiator based filters (or circuits, or systems, or whatever) is fundamentally flawed, because at the end of the day it's only really a matter of how you choose to write your equations and the whole point of equational reasoning is that you can convert back and forth as you see fit... because they are equivalent.

ps. I much more reasonable argument to prefer integrators is that in a sense only integration carries state in the form of the integration constants as set by initial conditions. That has nothing to do with causality, you can integrate forwards or backwards or sideways or up and down.. but while you can take the derivative as the inverse of integration, this in a sense will lose some information as strictly speaking the integral as an inverse of a derivative is multi-valued until you fix the initial conditions.

[Z1202]

I can bring a whole lot of counterarguments to what you're saying about differentiators/integrators and causality. Actually so much, that I find it hard to structure, so I just write a random list.

• Claiming that integrators and differentiators are flawed is kinda like claiming current and voltage sources are flawed. The truth is, all of them is idealized models and idealized models is pretty much the only thing that we're dealing with in engineering or science. We just need to become aware of the limitations of those models, and that's what the article is dedicated to.
• Your belief in the equations being an unflawed approach (or so it would seem to me) misses that equations represent a particular idealized model as well. Even a simple equation such as Ohm's law is an idealized model (and the article also goes into some detail on that). So equations have the same kind of limitations, just at different places. In a way, equations are worse, because it's not immediately clear, where those limitations are.
• Causality is implicitly all-penetrating in the circuit design (whenever we use buffered connections, or things like current/voltage sources). Trying to ignore it is strange to say the least.
• Your argument about 0.99999 and 1 being a "similar kind of nonsense", is pretty much what the paper is talking about, it just doesn't call it nonsense.
• Integrators and differentiators exist as real components (IIUC) and are occasionally used as they are.
• Your claim about equivalence of integration and differentiation misses the main point of the article (they are not equivalent in a causal setting, and that's what must be responsible for the respective circuits, employing physical differentiatiors, failing).

[urosh]

If the LTP formed by Q27 and Q33 would maintain more or less constant combined emitter current, then this would indeed be a tanh() and whatever asymmetry would be from different base offsets, but I'm not sure if R58 qualifies as a constant current source here. The fact that there's a 200k resistor (which is not an E12 value like 99% of the other resistors in the thing) also suggests that the intention could have been to get "close enough to symmetric" and I'd probably suggest measuring a couple of units (or varying parameters in Spice randomly by 5% or so) before concluding too much about the details.

We're talking about external preamp here, right?
Q27, Q32 and Q33 from very primitive opamp, Q27 and Q32 being differential input stage and Q33 both VAS and output.
Input LTP is suboptimal design from symmetry POV. Problems are R65 and R58 (which define bias point, not impedances at bases). When all transistors are biased properly, voltage accross R65 is 0.6V -> current is 0.6/27K=22uA, voltage across R58 is 9.4V -> LTP tail current is 9.4/150K=62uA. So at DC bias point Q27 current is 22uA, Q32 current is 40uA. So, LTP is biased inherently asymetrical.
However, I don't think this matters much for distortion mentioned in paper. Question is, what saturates first LTP or VAS? VAS by the looks of it, because it's single transistor without even current recourse as load. On negative swing Q33 gets starved, no current source to pull the weight, bottom of waveform starts to clip first. Top of the waveform clips when Q33 runs into saturation. Is biased tanh good model for this? Not really.

[mystran]

• Claiming that integrators and differentiators are flawed is kinda like claiming current and voltage sources are flawed.

Hold on. I think I phrased really poorly. I wasn't trying to say that integrators and differentiators are flawed, but rather I was trying to say that one shouldn't classify something as one or the other.

[*]Your belief in the equations being an unflawed approach (or so it would seem to me) misses that equations represent a particular idealized model as well.

I don't understand how these two concepts here are anything other than orthogonal to each other.

[*]Causality is implicitly all-penetrating in the circuit design (whenever we use buffered connections, or things like current/voltage sources). Trying to ignore it is strange to say the least.

But do you actually have a buffered connection in real life? Or do you have a certain set of constraints that can only be satisfied by drawing minimal current? Do you actually have a voltage source.. or do you have something that mostly appears to be a voltage source if you don't look at the voltage too carefully, your bypass capacitor is close enough and you don't draw too much current? When three things are all connected to the ground, are they actually at the same voltage?

All this stuff is idealization on top of idealization... yet mostly it still works pretty well as long as you remember to leave some tolerances, make sure to stay within SOA and .. don't forget your bypass caps.

[*]Your argument about 0.99999 and 1 being a "similar kind of nonsense", is pretty much what the paper is talking about, it just doesn't call it nonsense.

You dropped the ellipsis, which suggests you did not appreciate what I was referring to:

https://en.wikipedia.org/wiki/0.999...

[*]Your claim about equivalence of integration and differentiation misses the main point of the article (they are not equivalent in a causal setting, and that's what must be responsible for the respective circuits, employing physical differentiatiors, failing).

See the remark I added in the post scriptum of my post about this. Causality is orthogonal to the discussion, rather we need to set the integration constant with an "initial condition" somewhere in time or space and a derivative loses that piece of information: the state.. but that "state" need not be temporal in nature.

[Z1202]

Hold on. I think I phrased really poorly. I wasn't trying to say that integrators and differentiators are flawed, but rather I was trying to say that one shouldn't classify something as one or the other.

This is where we disagree. I believe integration should be the default classification, but that's strongly related to causality, where we don't seem to be on the same page. At least your argument about integrators having a state is kind of related.

[*]Your belief in the equations being an unflawed approach (or so it would seem to me) misses that equations represent a particular idealized model as well.

I don't understand how these two concepts here are anything other than orthogonal to each other.

Which two concepts? Equations and differentiators/integrators? I don't see how they are orthogonal. They are identical in the absence of causality, and almost identical in its presence.

But do you actually have a buffered connection in real life?

That's what I said: those are idealizations. But the ones explicitly and implicitly applied in lots of our thinking about circuits and filters. E.g. when we're talking about a differential amplifier at the base of Moog ladder, we assume that the transistor below it acts as a current source. Don't tell me, you don't apply this kind of simplification

All this stuff is idealization on top of idealization... yet mostly it still works pretty well as long as you remember to leave some tolerances, make sure to stay within SOA and .. don't forget your bypass caps.

Same holds for differentiators and integrators as idealization. The paper just highlights their edge-case differences (thanks to causality) and effectively gives you a list (possibly incomplete) of "things to remember".

You dropped the ellipsis, which suggests you did not appreciate what I was referring to:

https://en.wikipedia.org/wiki/0.999...

That was just quick typing, I did mean the ellipsis. Actually this question is pretty insightful, I didn't check the Wiki page, but I do have an opinion on this topic covered elsewhere.

See the remark I added in the post scriptum of my post about this. Causality is orthogonal to the discussion, rather we need to set the integration constant with an "initial condition" somewhere in time or space and a derivative loses that piece of information: the state.. but that "state" need not be temporal in nature.

I'm not sure how this remark relates to what I was saying, and I'm not sure to what specific discussion causality is orthogonal to. As long as the question of equivalence of TPT and trapezoidal goes, it's anything but orthogonal.