I take it that you must have found the papers.
Yes T(x) is the nonlinear transfer characteristic of the system defined within the domain of applicability and x is the amplitude, which can be a voltage or displacement or whatever variable we are looking at. This domain is required because in nonlinear analysis one expands the nonlinear transfer characteristic in a Taylor series, which is only applicable from -1 to + 1, so one has to define what the max and min amplitudes will be before any analysis can be done and the problem normalized to those limits. This is the part that most people who look at nonlinear distortion fail to understand - that we need to always define this region and if the amplitudes go outside of this region then all bets are off.
In our study T(x) was frequency independent. In a general nonlinear system it does not have to be, but simplifies significantly if it is - best to walk before we run. So the frequency dependence was dropped later on in the paper. The general theory of nonlinear systems is so complex that only PhD level students tend to get into this. It's an incredibly intense theory involving functionals (functions of functions) and multi-dimensional spaces which get out of hand very quickly. Without some simplification one tends to not get anywhere. Most of this is discussed in the papers - there are two, by the way and you need to read both to understand the issues.
So why did this all die? That's because once we had a valid metric, we quickly learned that nonlinear distortion is not a significant problem and people (Lidia and I!) just started looking elsewhere for relevant problems. We are in the middle of a study of room reflections and the data is showing some very interesting results.
Yes T(x) is the nonlinear transfer characteristic of the system defined within the domain of applicability and x is the amplitude, which can be a voltage or displacement or whatever variable we are looking at. This domain is required because in nonlinear analysis one expands the nonlinear transfer characteristic in a Taylor series, which is only applicable from -1 to + 1, so one has to define what the max and min amplitudes will be before any analysis can be done and the problem normalized to those limits. This is the part that most people who look at nonlinear distortion fail to understand - that we need to always define this region and if the amplitudes go outside of this region then all bets are off.
In our study T(x) was frequency independent. In a general nonlinear system it does not have to be, but simplifies significantly if it is - best to walk before we run. So the frequency dependence was dropped later on in the paper. The general theory of nonlinear systems is so complex that only PhD level students tend to get into this. It's an incredibly intense theory involving functionals (functions of functions) and multi-dimensional spaces which get out of hand very quickly. Without some simplification one tends to not get anywhere. Most of this is discussed in the papers - there are two, by the way and you need to read both to understand the issues.
So why did this all die? That's because once we had a valid metric, we quickly learned that nonlinear distortion is not a significant problem and people (Lidia and I!) just started looking elsewhere for relevant problems. We are in the middle of a study of room reflections and the data is showing some very interesting results.
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I don't think it's necessarily insignificant to those people who hear differences between amplifiers, DACs, and even individual op-amps. I am referring to those who can do it blind, and repeatably. Which is to say, people who have never really been seriously studied because they fall out of the 95% of the population that has most interested researchers, and perhaps for other reasons.
What some manufacturers seem to do instead of trying to use a complicated metric is take whatever measurements they are interested in, then conduct listening tests.
Maybe one of the reasons for not using a hard-to-calculate metric could be because a poor score might not point to what physically in a design needs to be fixed, making it kind of useless for design feedback. Also, perhaps slow to evaluate design iterations.
Another reason might be that the metric hasn't been validated against the 5% for whom it may be important, so designers may still need to perform listening tests. In that case, how are complicated tests justified?
Finally, it might help if there was a demo app that would make it easy to run some tests and try it out. What you have is some research, not an easy to use product.
My first love is ... believe it or not ... the god-forsaken world of car audio
but i'm still struggling to fit an 'infinite line' in a car FWIW, I don't like the word "cylindrical" without the "dispersive" modifier (just to distinguish it from the "impulsive sphere" of a point source). Truth is, i think that well-executed line arrays have, potentially, some real advantages: less attenuation with distance offers a wider sweet-spot, management of floor & ceiling bounce to your advantage, and possibility of lower distortion and lower power compression from multiple drivers. BUT, i have not yet heard the best line-array executions, so my personal opinion wouldn't be worth very much.
As i mentioned earlier, i think ... i find this fascinating: some 90 years into audio reproduction, 60 years into stereo reproduction, 30 years into multi-channel reproduction ... and we still have healthy debates about whether our loudspeakers should mimic ideal 'points', or ideal 'lines'
it's all good ... competition improves the art!We haven't even begun to tackle the REAL difficult problem : Finite Line Sources. Topic for another thread
I told you what we did, what we found out and why its important. If you don't want to accept it because it isn't what you want to hear that's fine. I did the research, I believe its conclusions and I told you how that directed me to do other things.
Maybe I just don't care about that esoteric 5% that never seems to accept anything.
Car audio, I had enough of that to fill a lifetime.
But here I am going to Ford Audio next Friday to give a lecture.Actually, I find it interesting. Also, unfortunate that others haven't chosen to use it. I don't really understand why there isn't more interest in using it than you found and I was speculating as to some possible explanations for that.
Sorry to just throw this out there, but I see an opportunity so I might as well take it and ask away: Is there any truth to rectangular boxes favoring odd harmonics and cylinders favoring even harmonics? Where can one find more info on this?My first love is ... believe it or not ... the god-forsaken world of car audio
FWIW, I don't like the word "cylindrical" without the "dispersive" modifier (just to distinguish it from the "impulsive sphere" of a point source)...
A line array that extends from a (reflective) floor to a (reflective) ceiling WILL have first order reflections from the floor and ceiling ... but those reflections are EXACTLY what would be produced from an extended array. By USING reflections, we are EXTENDING the array.
Imagine holding a long pole, and pointing it at a mirror. Push that pole right up against a mirror. See how the reflection makes the pole "look" twice as long? We used a REFLECTION to EXTEND the length of the pole
Now imagine a pole placed, length-wise, between TWO mirrors ... so that the small ends of the pole are right up against the mirrors. The pole is suspended, length-wise, between the mirrors (if you will). If you can stick your head between those mirrors, how long does the pole look now? Did we somehow "avoid" reflections ... or "use" them to our advantage?
I think if I stick my head between the two mirrors I just see my face smiling back at me, quizzically
but I'm disappointed if this thread is going to end without at least touching on finite discrete line arrays. After all, there is just a limited number of them. Can't we just add naiively up their contributions at the LP?
I think if I stick my head between the two mirrors I just see my face smiling back at me, quizzically
but I'm disappointed if this thread is going to end without at least touching on finite discrete line arrays. After all, there is just a limited number of them. Can't we just add naiively up their contributions at the LP?
Disappointed? The thread title really does mention "infinite" something or other, right?
Tell ya what ... ask for a refund at the front desk But seriously ... one would think, all we gotta do, to analyze FINITE LINES, is put FINITE LIMITS on all those integrals that extend from z=0 to +infinity. Right? Turns out that's true
But the math gets ugly quick. There's some classic "far field" approximations (but not so far that anything and everything looks like a point source) ... but other than that, closed-form solutions are few and far between For now, i can offer a few things about finite lines:
- Once you limit the extension along the z-axis, you introduce a "z dependency" that does not exist with the infinite line (meaning, the response will now depend on the height of the listening position, which, as we learned, is not the case with the infinite line).
- If you think that the response from a truncated line will simply be a truncated version of what i've called the "dispersive cylinder" ... you are quite wrong. Doesn't work that way. "Simply" truncating the line does NOT "simply" truncate the dispersive cylindrical response
- The right way to employ a finite line is with some sophisticated "shading" ... meaning, each small "element" of the finite line is driven by a (possibly) different magnitude and phase function, that may even vary with frequency. ONLY by investigating this rather complex approach, can you hope to achieve something approaching a polar response that's somewhat independent of frequency and position.
YES ... truncating the infinite line is no SMALL challenge! But all hope is not lost
there's some VERY interesting work being done in this area, and i don't think the final chapter has yet been written ...
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Well I thought that to a first approximation they would turn into sums over N sources, or if taking "M" floor and ceiling reflections into account, sums over NxM sources
If you look at David Smith's classic paper that - sum over n sources - is essentially what he did for the far field response, with some additional complexities to address near field response, and then went on to discuss shading. The benefits of shading were well known even then and showed up in a program that was based on a sum over N sources.
I don't think it gets hard until you to take into account the directivity of the sources and then it gets very hard quickly, as you say and undoubtedly know better than me.
If you can provide links to some of current work in this area, I would be very interested to read it.
If it is so challenging to truncate a line, why is it that so many DIYers do that and find themselves happy with the result? Does it have something to do with the fact that in home we tend to listen over a very limited vertical window and within such a window the finite discrete array can be equalized quite well?
If you look at David Smith's classic paper that - sum over n sources - is essentially what he did for the far field response, with some additional complexities to address near field response, and then went on to discuss shading. The benefits of shading were well known even then and showed up in a program that was based on a sum over N sources.
I don't think it gets hard until you to take into account the directivity of the sources and then it gets very hard quickly, as you say and undoubtedly know better than me.
If you can provide links to some of current work in this area, I would be very interested to read it.
If it is so challenging to truncate a line, why is it that so many DIYers do that and find themselves happy with the result? Does it have something to do with the fact that in home we tend to listen over a very limited vertical window and within such a window the finite discrete array can be equalized quite well?
Yes, shading has been known for some time ... but typically refers to 'amplitude-only' shading, often frequency-independent. But what if we open up ALL the variables : allow for magnitude and phase shading, and allow those to be frequency-dependent ... which of course includes a phase function that's linear with frequency, aka time delay, that's potentially different for the different elements of the finite line. The optimization of the polar response can get pretty complex.
I'd look at the work Don Keele is doing with CBT ... it's quite complex, but he's got some simple approximations to the theory (and great videos on youtube)
I think the answer is yes. One of my favorite speakers, a couple decades ago, was the Martin Logan CLS. A certain amount of magic, to be sure
but nothing even close to resembling a postion- or frequency-independent polar response Not exactly a finite line source, i know, but a similar narrow listening "window"damn i thought this movie was going to have a better ending...foreign language films...maybe the sequel will be better!
when it comes to correction systems for an array am i correct in my lowball understanding that frequency and amplitude shading along with delay/time alignment can solve some of it's problems?
when it comes to correction systems for an array am i correct in my lowball understanding that frequency and amplitude shading along with delay/time alignment can solve some of it's problems?
You didn't really expect it to come to a satisfying conclusion did you, it's not like the good old days anymore where there is no doubt in your mind as to the difference between right and wrong? Directors these days like to leave things in limbo where you have to make up your own mind, after all, if they told you, then their job would be done.............
werewolf, a question if i may....
I get that perfect magnitude response = perfect impulse.
Doesn't that also equate to perfect phase response?
Isn't the full relationship of identities that of:
perfect magnitude = perfect impulse = perfect phase?
I mean we are talking infinities, ...line length, dynamic range, perfect reflectors, etc...
It just seems intuitive to me under those conditions that phase would have to be 0 degrees dead flat too ....
If not so, please shed some light , thanks!
Stepping away from the infinite....into real world line lengths..
I feel certain you've seen this stuff, but in case fellow DIYers haven't, the technology in line array rigs like
Martin's MLA YouTube,
or EAW's Anya YouTube
is simply fascinating....to me at least
Oh, I still have a pair of CLS...AcoustatX too...I agree with your assessment,
even if I've converted over to HiFi PA !
I get that perfect magnitude response = perfect impulse.
Doesn't that also equate to perfect phase response?
Isn't the full relationship of identities that of:
perfect magnitude = perfect impulse = perfect phase?
I mean we are talking infinities, ...line length, dynamic range, perfect reflectors, etc...
It just seems intuitive to me under those conditions that phase would have to be 0 degrees dead flat too ....
If not so, please shed some light , thanks!
Stepping away from the infinite....into real world line lengths..
I feel certain you've seen this stuff, but in case fellow DIYers haven't, the technology in line array rigs like
Martin's MLA YouTube,
or EAW's Anya YouTube
is simply fascinating....to me at least
Oh, I still have a pair of CLS...AcoustatX too...I agree with your assessment,
even if I've converted over to HiFi PA !
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