Yes, but I am afraid that they will be complex and hard to understand. First, you are correct in that the Gm is not well suited to nonlinearities like hysteresis. Fortunately we don't see that kind of nonlinearity very often. But your comment:
"For systems with dynamic nonlinearity, on the other hand, there is more than one possible instantaneous output value for a given instantaneous input value, and the necessary pattern of odd and even harmonics in quadrature does not exist. A system with hysteresis is an example of a system with dynamic nonlinearity."
is not quite accurate IMO.
In a full blown nonlinear analysis, the timing of a signal is important and two signals with different timings (or phasing) can yield two different results even if the spectra of the two signals is the same. For example two impulses will have different nonlinear outputs in a nonlinear system depending on the time difference between them. Or a phase difference between two sinusoids. This is what is meant by "dynamic nonlinearity" - the outputs depend on the temporal dynamics of the signals. But your claim that "there is more than one possible instantaneous output value for a given instantaneous input value" I don't think is true. To me this statement says that a single nonlinear system can have two different outputs for a single signal - that's not possible.
"Dynamic nonlinearity" is most easily thought of as a frequency dependent nonlinearity, but this too is not exactly correct, especially for highly nonlinear systems. I discuss this in the papers, how we pretty much have to simplify our analysis by assuming that the underlying systems are quasi-linear, as virtually any audio system is going to be. Otherwise, we do have consider things like spectrum/phase of the signal as the nonlinear aspects will depend on these. But, for a quasi-linear system the frequency aspects basically uncouple themselves from each other, meaning that I can look at the nonlinearity at a single frequency independent of other frequencies nonlinearities. For highly nonlinear systems, I cannot do that and this is an essential simplification of quasi-linear systems.
This means that the technique being used of measuring Gm at one frequency and then moving to the next is adequate as long as the nonlinearity remains small. The more linear the system, the better this approximation becomes.
I answered this above and also indicated that the technique you are using is adequate under the circumstances, but not so in the completely general case.
Your statement seems to be ambiguous, but here is what I believe is the case (and what I hope that you meant): Lower orders of nonlinearity are not very audible, but higher orders are, hence we must minimize the higher orders, while the lower orders are pretty benign.
The Gm is now almost two decades old and it was NOT common knowledge that higher orders of nonlinearity produce more audible distortion as you allude to. (Please don't confuse "higher Harmonics" with "higher orders of nonlinearity," they are not the same things, related yes, but not equivalent.) And if there are "many answers to that question" then you would have to show me where they are, because I don't know of any others.
Since our papers there has been a lot of talk about how one might also approach the problem, but no one, to my knowledge has actually proven that they are as good as or better than Gm. Howard discussed how simply weighting the harmonics produced "similar" results, but never showed what that actually meant in practice - "not nearly as good", "just as good", "better"? He just hypothesized about how these different techniques might be comparable, but never did such a comparison.
Assuming curiosity here... I take an appropriate device with a simple transfer characteristic and optimise (operating point etc), leave out non-local interactions (Voltage source, simple loads etc.), stabilise. One Voltage gain stage and one power gain stage. I'm less critical about bass amps.
Don't get me wrong! I am not at all saying that there are other metrics that have been tested in scientific terms to be "not nearly as good", "just as good" or "better". My point was simply that some guys did come up a long time ago with the concept of weighting the different harmonics in different ways in their approaches of assessing "sound quality". As far as I can tell, much of this was based on personal opinions or ideas, without proper controlled scientific experiments to assess the validity of these ideas. Here's one such example (yes, that's a bit of a mumbo-jumbo-story-telling thing, not strict science):
That said I do like the Gm metric! It is straight-forward to determine (assuming one can determine the levels and phase of harmonic distortion), it is just a plain number without too many ifs and buts, and the underlying concepts make sense to me.
Correct, lot's of people proposed things and recognized the problems with THD, but nobody did anything concrete about it, just hypothesized about the problems. That's why nothing ever took hold. I think that Gm, or something like it, may have taken hold had we not learned along the way how insignificant nonlinear distortion actually was. It happened to me. I was obsessed with linearity, developing accurate ways to quantify it, only to find out that subjectively nonlinearity (in loudspeakers) didn't matter very much! It was, perhaps, disappointing, but a good scientist takes his results and moves on from there. Had I not moved on I would never have found what I believe to be the keys to good sound quality. Good data doesn't always support ones in-grained biases.
These days, there is absolutely no problem to get, in a well engineered and designed amplifier, vanishing low values of non-linear distortions of all kinds, orders below threshold of audibility. CCIF IMD, DIN IMD, multitone IMD, HD, HD vs. freq, HD vs. amplitude, you name it. Thus, the discussion about distortion metrics and spectral components importance becomes pointless, at least for amplifiers, both pre and power amps. Only tube designs and some exotic, usually overpriced so-called high-end designs bring considerable distortion level, often higher than that of the speaker. However, such designs are colorizers, not the devices intended for true reproduction of the recorded signals.
Pavel
It's good to see someone who has the right perspective on this issue, or should I say non-issue.
Many people do not understand why a manufacturer would go out of there way to make their products "sound different". It seems obvious when its said that way - they want to differentiate them from in the market. To me, as audio products get better and better they should all start to sound the same with ever smaller variations in their perception. From a marketing perspective this would be a catastrophe.
It's good to see someone who has the right perspective on this issue, or should I say non-issue.
Many people do not understand why a manufacturer would go out of there way to make their products "sound different". It seems obvious when its said that way - they want to differentiate them from in the market. To me, as audio products get better and better they should all start to sound the same with ever smaller variations in their perception. From a marketing perspective this would be a catastrophe.
(Stress in bold added by myself.)
Earl, I hate to disagree with you (and it may only be a semantic matter), but for systems with a dynamic transfer function, it is possible for an instantaneous input value to have more than one possible instantaneous output value.
This is probably best illustrated with a simple example.
1) Static transfer function
Take a system with a sine input and the following output (expressed in sines):
The odd and even harmonics are, of course, in quadrature.
We can plot the instantaneous input values against the instantaneous output values to obtain a visual representation of the transfer function T(x). Normalised so that it extends from (-1, -1) to (1, 1) T(x) looks like this:
The plot of T(x) is a clear visual representation of the nonlinearity of this system, and it is clear that for every instantaneous value of the input, the output has only one possible value.
There is more than one way to represent T(x) as an expression. It needs to be an expression that is amenable to differentiation, and a simple polynomial would do the trick. As an alternative, a trigonometric expression of T(x) can be derived from the expression for the output:
Happily – and very conveniently – phase has vanished from this expression because the odd and even harmonics are in quadrature.
The coefficients Cn in this example are:
c0 = 0.181818182
c1 = 0.909090909
c2 = 0.181818182
c3 = -0.090909091
T(x) expressed in this way is amenable to differentiation, and it can now be plugged into the expression for Gm. (If anyone is interested to see how T(x) is derived, there is an exposition in the spreadsheet linked to in my first post.)
2) Dynamic transfer function
This is the same example, but with the harmonics no longer in quadrature. One possible realisation is:
The FFT spectrum and THD are exactly the same as before. However, the plot of the normalised transfer function T(x) now looks like this:
We now have a situation where the instantaneous value of the output depends on whether the input is ascending or descending. This means that, except for the extremes, there is more than one possible instantaneous output value for a given instantaneous input value (there are two in this instance).
I know of no way to represent this as an expression that is amenable to differentiation (twice) and which could therefore be plugged into the expression for Gm.
**********
Appendix: for the second example (dynamic nonlinearity), the normalised output y(t)
has the coefficients:
B0= 0.17784346
B1= 0.957217167
B2= 0.191443433
B3= 0.095721717
Let's define the time for a complete input cycle as 2.pi. Taking just one possible instantaneous input value, and normalising the output value, numerical solutions of y(t) show:
At t=pi/4, x(t)=0.707, y(t)=0.922
At t=3.pi/4, x(t)=0.707, y(t)=0.787
Thus we have two different instantaneous output values for the same instantaneous input value.
Stephen
I will have to look this over carefully, but right off the bat I would question if an output from a nonlinear system could ever have its harmonics not in quadrature (as your example requires.) This could be physically impossible. It seems to me that harmonic structures that do not yield single values transfer characteristics may not be physically possible since they do result in multi-valued functions and hence multi-valued outputs, which, it seems to me, must violate some physical principle.
Has any ever seen this happen in reality? Where one gets different outputs for the same input in different experiments on the same nonlinear system. Seems impossible to me.
I know that different initial conditions in a nonlinear system will react to the same stimulus in different ways, but then this is either not the same system or the same input.
Has any ever seen this happen in reality? Where one gets different outputs for the same input in different experiments on the same nonlinear system. Seems impossible to me.
I know that different initial conditions in a nonlinear system will react to the same stimulus in different ways, but then this is either not the same system or the same input.
It is not. In a real World DUTs, you may find all kinds of phase shifts of individual harmonic components of the non-linear distortion complex spectrum. The real DUT non-linearity is not only level dependent, but also frequency dependent, which makes the usual polynomial approximation not precise enough.
Well if Gm was inaudible below 3 as the Gm inventor suggests, why is there still some/same correlation at Gm=1? I cant ses how that scart answered your question about what level of Gm is inaudible.
//
For a static transfer function, the even harmonics (2f, 4f, 6f etc) need to have a phase of +/- 90 degrees relative to the fundamental, and the odd harmonics (3f, 5f, 7f etc) need to have a phase of 0 degrees or 180 degrees relative to the fundamental.
The odd harmonics (including the fundamental) can be said to be in quadrature to the even harmonics.
Since sine(wt + pi/2) = cos(wt), we could regard the output as being a series of alternating sines and cosines (as pi/2 = 90 degrees).
Yes, I agree, though I suspect that the deviation from the ideal static case is usually small.
Stephen
I do not think so. An example of a "real thing" is here
Digital Distortion Compensation for Measurement Setup
You can also find phase of distortion components when you work with LTSpice, which is able to calculate phases of distortion components.
The "nonlinearity" is not "level dependent" since it defines the level dependence. It can be "frequency dependent" that is discussed in the paper. In theory the harmonics cannot be "all kinds of phase shifts" due to the nonlinearity, but there can be phase shifts in the acoustic signal due to different frequency and phase dependent paths after the nonlinearity.
I think that this would be the point that I agree with. Dynamic nonlinearities, like hysteresis, are going to be smaller than the static nonlinearities, which are in themselves small quantities (loudspeakers are very nearly linear devices or they would be unlistenable.)
I am not speaking about acoustic signals, but electronic circuits. And you can both measure and simulate phase angles of the distortion components and they are definitely not only 90° or 180°. Maybe in theoretical oversimplification yes, but not in the real world. The theory is only that good as close it approaches the reality, no more.