Actually its you thats trying to separate them, by specifically claiming in your paper that its the group delay that we hear rather than the resulting amplitude response variation. Correlation with increasing group delay does not mean that is what we are hearing.
Just because group delay and frequency response are inseparable in a minimum phase system doesn't mean its the group delay aspect of that which we can hear, and we have every reason to believe that its not.
Its well known that the ear is very sensitive to amplitude response variations but extremely insensitive to phase response variations, and group delay is essentially just showing the slope of the phase curve. Given variations in the phase and variations in the amplitude in a minimum phase system, which are we more likely to hear ? The obvious answer is the amplitude variations.
It would be a steep road indeed to prove that it was the phase / group delay variations that we were perceiving negatively rather than the amplitude variations that go hand in hand with them in a minimum phase system.
If you really wanted to isolate group delay as a single variable and see if we are sensitive to group delay variations on their own you should have done so with a non-minimum phase test where you had two systems with the same amplitude response, one minimum phase, and one non-minimum phase with large narrowband variations in excess phase (and therefore excess group delay) which are comparable to the group delay variations you see on a real speakers response. Something that should be easy to do with DSP these days.
My guess is that the degree of group delay we're talking about stripped of the amplitude changes will NOT be audible, even at high SPL's, but since as far as I can remember from your paper you didn't test this, we will never know.
I have no problem with the central precept of your paper - that these response variations become more audible and more obnoxious at higher SPL's and that this SPL dependent aspect has not be acknowledged or studied in any significant way before, but I take issue with pinning the blame on group delay audibility when group delay has not been tested in isolation.
Your results could just as easily be interpreted as "narrow band amplitude response variations become more audible and objectionable at higher SPL's".
So why did you not test with non minimum phase signals to see if group delay variations on their own with no amplitude response variations were more audible at high SPL's ? If this was not done you can't prove whether its the amplitude response variations or the group delay variations that we hear.
How is linear systems theory relevant to the perception of the ear ? I think we all DO know that the ear is not linear, especially at one SPL vs another. As average SPL levels increase the muscle in the ear (I forget its name) that acts as a crude AGC system comes into play, altering the response of the ear.
As for harshness, we all know that a high frequency tone above 3Khz or so can quickly become painful and obnoxious as SPL increases. As playback SPL increases a point will be reached where those high frequencies become obnoxious if they are present in excess relative to the rest of the spectrum, which ties into non flat frequency response.
Yes it is conjecture on my part, but so is your position that its only the group delay that matters. In your study the amplitude variations went hand in hand with the group delay variations - thus you have not isolated them and cannot say whether its the group delay or amplitude response that was being perceived. It's not my job to isolate those for you...
Sure, you see a correlation with increased group delay and SPL and listener ratings, but a similar correlation can be seen between increased non-flatness of amplitude response (in fact a seemingly better correlation if Feyz's graphs are correct) and without any effort to isolate the two we can't be sure which we're hearing, although given the ears sensitivity to amplitude response and insensitivity to phase it seems reasonable to assume its the amplitude variations that matter.
Just for fun I'll give some more conjecture on what it is we hear on a system with narrow band non flatness at high frequencies and why it might cause perceived harshness at higher SPL's.
1) A non flat response with narrow band variations will have an increased crest factor between the average perceived SPL level and particular tones/notes within the music. For example +/- 3dB peak variations in response will result in individual notes which fall on a peak in the response being 3dB louder than they should be relative to overall playback level.
If this is at a critical frequency like 3Khz or a sensitive frequency above 3Khz, at high SPL levels this may be enough to push it into the "discomfort" region at a significantly lower overall SPL level than the flatter response speaker. This ties in with my observation that a very flat response can be listened to at a higher SPL without discomfort.
2) Narrowband amplitude response variations will cause amplitude modulation of any frequency modulated sounds as they ride up and down the slopes. I gave the example before of vibrato on the high pitched components of a vocalist. I find it exceptionally irritating if the vibrato of a signers voice happens to fall on the side of a resonance or other sharp frequency response discontinuity - the amplitude modulation is clearly audible, and at higher SPL can be very uncomfortable. This amplitude modulation effect is usually the first thing I notice on a speaker with a high Q resonance or sharp response discontinuity at high frequencies.
Those are just two reasons I can think of that narrow band non-flatness of amplitude response at high frequencies can sound objectionable, and more so at higher SPL's.
As I said, I think your basic findings that the response variations resulting from group delay can be far more audible at higher SPL's is very interesting and valid, however I don't at all see that it has been proven that its the actual group delay we're hearing, and not simply the resulting amplitude response variations that go with it in a minimum phase system.
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Because we did not have to, it had already been proven. Brian Moore had shown in his AES paper that pure group delay (no FR variations) was auidible and he alluded to the fact (claiming in his paper that this was known fact) that this would vary with SPL level. Hence we were not interested in that aspect of the problem. We were interested in the audibility of diffraction-like signals which did modify the FR.
What you suggest would make an interesting test, not what I was interested in, but interesting none-the-less. Why don't you do it and let us know how it comes out?
I think that everyone is reading into the study more than is there. We were interested in knowing - since we already knew that GD alone was audible - how the audibility of diffraction-like signals (which while they involve a pure delay their level is frequency dependent) was affected by variations in the key parameters that define them. These parameters were 1) delay, 2) diffracted signal level and 3) overall SPL level. The paper proved that all three were statistically significant factors and that the audibility of each of them increased with increasing values of the parameters. The most interesting result was the fact that the audibility increased with increasing SPL. This menas that perception of a linear effect is nonlinear shedding doubt on any claims of a subject hearing "nonlinear distortion". Say what you want, this was not widely recognized at the time, if it was recognized at all (its still not widely recognized, although some seem to believe that it is obvious - I reject that claim however.)
So test your hypothesis and let us know the result. Better yet, publish the result in a refereed journal, because otherwise no one is going to accept it.
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This is what I wrote:
Note I was referring to loudness curves (plural), meaning since hearing sensitivity is different to different levels of SPL, as documented by those curves, the hearing is nonlinear.
I don't know others much, but I do tend to increase the volume to get detail better, which is inline with the equal loudness curves. As SPL goes higher, the loudness curves tend to become flatter, which I interpret as the hearing can catch more information from wider frequency range with higher SPL.
Below I have eye balled and taken an IS0 2003 equal loudness curves from web into Excel, then converted it into what can be called ear's transfer function with SPL. The last curve magnifies on the most sensitive region of the ear. I think it is noteable that 3Khz is the most sensitive region, and that the curves do not monotonously become flatter as increased SPL, the overall trend is that they get flatter but it is not monotonous. For instance 80 phons gets good high frequency but then 100 phons loses it back, which I will come back to this.
An externally hosted image should be here but it was not working when we last tested it.
The below is the FR curves for the filters used in the paper that I came up with which I tried as much as possible to match the two published in the paper. Since I don't know the details how of they were generated I tried my best.
They are for +4db shelve cases, and yellow is 0.2msec delay, red is 0.4 msec, light blue 0.6 msec, blue 0.8 msec and orange 1 msec delay.
I made them 10db offset from each other for display purposes.
What I find interesting here, combined with the equal loudness curves, the area of 3Khz and how these scored.
In overall average 0.8 msec delay ones had lower score than 0.6 msec and 1msec delay ones as I displayed earlier. And here it can be seen that 0.8 msec one has a dip at 3Khz, where as 0.6msec has a peak at 3Khz. Considering the ear's high sensitivity at 3Khz, this may be explained by that.
But if this is true, it also means, where exactly the peaks fall on the ear's sensitive region may be at least as important as how often and how many peaks and dips are on the other regions. The latter is controlled by the amount of delay, the frequency of the dips. So my point is, the relationship to delay is not very strong, and effected by where exactly the delay causes peaks in the FR. These are not given consideration in the paper, as I wrote before.
They are for +4db shelve cases, and yellow is 0.2msec delay, red is 0.4 msec, light blue 0.6 msec, blue 0.8 msec and orange 1 msec delay.
I made them 10db offset from each other for display purposes.
What I find interesting here, combined with the equal loudness curves, the area of 3Khz and how these scored.
In overall average 0.8 msec delay ones had lower score than 0.6 msec and 1msec delay ones as I displayed earlier. And here it can be seen that 0.8 msec one has a dip at 3Khz, where as 0.6msec has a peak at 3Khz. Considering the ear's high sensitivity at 3Khz, this may be explained by that.
But if this is true, it also means, where exactly the peaks fall on the ear's sensitive region may be at least as important as how often and how many peaks and dips are on the other regions. The latter is controlled by the amount of delay, the frequency of the dips. So my point is, the relationship to delay is not very strong, and effected by where exactly the delay causes peaks in the FR. These are not given consideration in the paper, as I wrote before.
There is actually very strong correlation with the non-flatness of amplitude response (in this case the height and depth of the peaks and notches).
These will demonstrate it better. The below are the graphs for amplitude (2db, 4db, 6db) with each delay per each SPL,as can be seen there is not a single case the score does not increase with increased amplitude of non-flatness, it is monotonous:
Continuing from above, these are the graphs of amplitude with SPL for each delay, again not a single case of decrease with increase amplitude of non-flatness:
But same is not true for delay and SPL. They have several cases where increased delay or increased SPL doesn't correspond with increased score. I have taken below some examples of them (overall average tendency is there, but there are exceptions, as below)
Examples of cases containing not always increasing score with increase in delay:
Examples of cases containing not always increasing score with increase in SPL:
My theory is that, these cases above are a result of the equal-loudness curves. The cases where increased delay not scoring better could be explained by it is placing peaks or dips to different parts of ear's sensitive region(s). The cases where increased SPL not scoring better could be explained by the not monotonous nature of the equal-loudness curves. As SPL is incresing not only its peak sensitive section moves a bit, but also it doesn't always get flatter in its frequency response in the whole frequency region, as I had depicted earlier.
And again, my overall point is, there is more to the scores obtained by just delay of reflection, diffraction. And the most important contributer to the scores is the amplitude non-flatness. Which means make the amplitude flat, it will be less detactable by ear.
Examples of cases containing not always increasing score with increase in delay:
Examples of cases containing not always increasing score with increase in SPL:
My theory is that, these cases above are a result of the equal-loudness curves. The cases where increased delay not scoring better could be explained by it is placing peaks or dips to different parts of ear's sensitive region(s). The cases where increased SPL not scoring better could be explained by the not monotonous nature of the equal-loudness curves. As SPL is incresing not only its peak sensitive section moves a bit, but also it doesn't always get flatter in its frequency response in the whole frequency region, as I had depicted earlier.
And again, my overall point is, there is more to the scores obtained by just delay of reflection, diffraction. And the most important contributer to the scores is the amplitude non-flatness. Which means make the amplitude flat, it will be less detactable by ear.
Interesting theory, now you need to test your hypothesis. Personally, I don't think that it will work out to be correct.
To me this is all beside the point (although I am not sure that I can deciper exactly what you are trying to say).
The goal was to study the detectability of diffraction in a waveguide and to find those parameters that were significant. Some of what was found was kind of obvious - that the greater the diffraction the more audible, and the longer the delay, the more audible (the dip at 6 kHz is not statistically significant as I said before, so its not worth addressing that issue, but your "coincidence" of 3 kHz peak and dip is interesting). By far the most important result was that the higher the SPL the more audible a given diffraction signal is. That's a key point (previously unknown to my knowledge) and nothing that you have said or shown refutes it. Your explaination of "why" is different than mine but in the region between 1 kHz and 10 kHz where our study was concentrated, the loudness differences (and using Fletcher-Munson curves is not going to impress anybody because those have been discredited) are not significant.
It is impressive to see all the work that you have done, but as far as I am concerned it does not change my point of view at all. I will admit that it would be very interesting to compare a delayed signals set of peaks and dips with the same FR but minimum phase - that would add a lot to the discussion, but unfortunately nobody has done that. (And I won't accept a "simple" test either. Too much of the bogus audio dogma is based on such uncontrolled tests.)
So because one paper by Brian Moore "alluded" to the fact that pure group delay audibility depended on SPL, you didn't think it was necessary to try to substantiate this properly independent of amplitude variations ? You consider his allusion alone to be sufficient proof ?
That doesn't make any sense to me when the whole point of your paper seems to be that the discovery group delay audibility varies with SPL was not widely accepted before that, if at all.
Yet you haven't shown that because you haven't isolated the effects of group delay from the effects of non flat frequency response in your own tests. So what does your paper prove then ? Merely that the non flat response may be more audible at higher SPL's but you haven't identified whether its group delay or frequency response, as Feyz points out. A seemingly huge omission.
It's more than interesting, its a key point - do we hear the group delay or the frequency response variations ? A pretty major point I would have thought. If you want to prove conclusively that its group delay causing the effects your subjects heard you really needed to isolate it from frequency response variations. Again its not my job to perform research to validate yours. 🙄
Yes but where are the prior studies that show conclusively that pure group delay was more audible at higher SPL's, other than an allusion made by Brian Moore ?
But nothing in your study separates group delay from frequency response variations. So all we know is that a non flat response (group delay and/or frequency response) is more audible at high SPL's. A useful piece of information to be sure, but why try to correlate it with delay time and not frequency response ? It seems almost as if you were looking for results that agreed with your hypothesis without considering other possible explanations.
Now that Feyz is questioning the assumption that the group delay is what matters you're very quick to dismiss his objections and uncooperative when it comes to describing certain conditions of the original test, so that the methodology and results can be independently verified or analysed. If the research is sold it will withstand the scrutiny of others, yet you don't even seem to want to reveal the basic parameters of the test signals, forcing Feyz to try to reverse engineer them.
Sorry but this comment just comes across as a case of sour grapes. I never made any claims that I am right, nor do I have the time or resources to do such research, I merely offered it into the discussion as one possible explanation of why certain types of non-flat frequency response could be deemed to be audible and/or objectionable to help stimulate the conversation.
Your comment and general tone is what I would refer to as the "movie critic fallacy". The idea that the only one who can criticize or point out the flaws in something is someone who can do it better, which is nonsense. A movie critic doesn't have to know how to make a better movie than the director (or even any movie at all) to be able to point out plot holes or other inconsistencies in a movie. It's an easy retort to say "well you do it better then".
Likewise I have no interest whatsoever in performing pure research of this type, nor do I have the time or resources, but that doesn't mean I can't read a research paper and point out a flaw or inconsistency in it. If the paper is solid it will stand up to any scrutiny.
Regarding 3Khz - an interesting point is its not just that we're more sensitive at 3Khz in terms of amplitude, but that the 2-4Khz region, 3Khz in particular elicits a certain negative psychological response, and that high levels of this frequency are very uncomfortable.
I posted a link to an interesting article about this in another thread, here's the original article:
Cover Your Ears! - ScienceNOW
And here's the thread I posted it in:
http://www.diyaudio.com/forums/multi-way/132777-flux-modulation-2.html#post2765428
Many people find scratching of blackboards particularly uncomfortable (me among them) and it turns out that its the frequencies around 3Khz that elicit this negative response, which researchers discovered by filtering out various frequencies of a recorded scratching sound and playing it to test subjects.
Basically for most people 3Khz is more than just the most sensitive frequency of our hearing, but also the most uncomfortable. Anything that puts a resonance or peak in the response at 3Khz is going to be very uncomfortable for most listeners, and much more so at high SPL's.
Haven't we just established further back in the thread that the effects of diffraction are always minimum phase, because the delayed signal is always lower in amplitude than the direct signal ?
So how would the minimum phase peaks and dips you propose be any different to the minimum phase results of diffraction ?
It wasn't "alluded to", it was stated as fact and since I consider Brian Moore to be the most authoritive researcher in this regard, yes, I do consider that sufficient for me.
Its easy to nit-pick someone else's work after the fact, so yes, step up to the plate and do your own work or chill-out.
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I rather like empirical, phenomenological approach:
Subjective impressions reported of the sound of an exponential horn as compared to that of a modern waveguide such as OS, are that the horn has a "dark" or "rough" sound that some folk find objectionable. Geddes has shown theoretically and experimentally that refraction or phase related products created in the horn are the source of the "dark", "rough" sound.
I verified that diminution of these refractive products with foam as Geddes describes does eliminate much of the "dark," "rough" characteristics from a large exponential horn. It also sounds quieter. Subjectively, a lot quieter, although, using pink noise, the Ratshack meter tells me it's just a couple of dB. Now I can play those horns 10dB louder before they sound too loud, than I could before I diminished the refractive products.
It appears then that my hearing is non linear with respect to phase delay or non linear distortion. 10 dB is a big deal.
It has been reported by instrument makers that generally instruments which have strong resonances in the higher part of their ranges tend to sound "bright". (See, for instance Schleske, here: Master Studio for Violinmaking - Martin Schleske Munich, Germany Tonal color and the resonance profile Take time to play with demo.)
Intuitively then, it's not unreasonable for me to think after looking at Schleske's demonstration that phase anomalies rather than only amplitude of frequency distribution are having effect on subjective experience of the horn since, if it were only greater energy distribution in the more sensitive area of our hearing, the sound ought to get brighter, rather than darker as it does.
If I think that, then I ought to have a "theory," right?😉
I think our hearing discerns the phase delayed sounds in the horn as noise which masks some upper mid range and high frequency sounds which (1) makes the sound seem "darker" and (2) since it's noise it makes the sound seem loud and "rough."😀
Hey, it's not a bad theory. It fits the phenomenology.
A gazillion micro echoes gotta be noise.
Subjective impressions reported of the sound of an exponential horn as compared to that of a modern waveguide such as OS, are that the horn has a "dark" or "rough" sound that some folk find objectionable. Geddes has shown theoretically and experimentally that refraction or phase related products created in the horn are the source of the "dark", "rough" sound.
I verified that diminution of these refractive products with foam as Geddes describes does eliminate much of the "dark," "rough" characteristics from a large exponential horn. It also sounds quieter. Subjectively, a lot quieter, although, using pink noise, the Ratshack meter tells me it's just a couple of dB. Now I can play those horns 10dB louder before they sound too loud, than I could before I diminished the refractive products.
It appears then that my hearing is non linear with respect to phase delay or non linear distortion. 10 dB is a big deal.
It has been reported by instrument makers that generally instruments which have strong resonances in the higher part of their ranges tend to sound "bright". (See, for instance Schleske, here: Master Studio for Violinmaking - Martin Schleske Munich, Germany Tonal color and the resonance profile Take time to play with demo.)
Intuitively then, it's not unreasonable for me to think after looking at Schleske's demonstration that phase anomalies rather than only amplitude of frequency distribution are having effect on subjective experience of the horn since, if it were only greater energy distribution in the more sensitive area of our hearing, the sound ought to get brighter, rather than darker as it does.
If I think that, then I ought to have a "theory," right?😉
I think our hearing discerns the phase delayed sounds in the horn as noise which masks some upper mid range and high frequency sounds which (1) makes the sound seem "darker" and (2) since it's noise it makes the sound seem loud and "rough."😀
Hey, it's not a bad theory. It fits the phenomenology.
A gazillion micro echoes gotta be noise.
interesting thread.
apart from all the theory exposed here, i'm only wondering how is possible to get the harmonics right when you stuff a pile of foam into the horn. i've never heard a solution based on foam damping of the horn when the sound was not severely harmonically challenged and the tone washed out.
apart from all the theory exposed here, i'm only wondering how is possible to get the harmonics right when you stuff a pile of foam into the horn. i've never heard a solution based on foam damping of the horn when the sound was not severely harmonically challenged and the tone washed out.
As I wrote in my post, the curves I replicated are taken from ISO 226:2003:
Equal-loudness contour - Wikipedia, the free encyclopedia
It is straight forward math that shows if a signal is amplitude reduced and delayed and than added (or substracted) back into the original, this results in minimum phase behaviour. You may not accept it, but as I wrote it is result of straight forward math. I gave the math in previous posts. Same is used on an older published JAES paper loudspeaker cabinet reflections to show they are minimum phase, it is even included in the Loudspeaker Anthologies, don't remember the exact name and date or the author but I can find it.
I don't think it is beside point, because if you start with wrong assumptions, the conclusions will be tainted at best. Wrong assumption here is the acceptance that a delayed echo will make the system non-minimum phase always, which is incorrect. In reality it makes it non-minimum phase only when its amplitude is higher than the original it is being added to.
This is the paper:
JAES Loudspeaker Cabinet Reflection Effects, James M Kates, 1977
AES E-Library Loudspeaker Cabinet Reflection Effects
In my case I have shown particularly for baffle diffraction it is always minimum phase, even including higher order diffractions.
I don't know what goes on in a horn/waveguide, on many posts you made here, you have been saying that a delayed signal when added back to the original makes the whole non-minimum phase. This is the simplest case and so is the simplest to show in math that such is not correct. Even though the study you made doesn't say explicitly so, it takes this this wrong assumption as a base
JAES Loudspeaker Cabinet Reflection Effects, James M Kates, 1977
AES E-Library Loudspeaker Cabinet Reflection Effects
In my case I have shown particularly for baffle diffraction it is always minimum phase, even including higher order diffractions.
I don't know what goes on in a horn/waveguide, on many posts you made here, you have been saying that a delayed signal when added back to the original makes the whole non-minimum phase. This is the simplest case and so is the simplest to show in math that such is not correct. Even though the study you made doesn't say explicitly so, it takes this this wrong assumption as a base
You see this is exactly why the point about minimum phase is "beside the point" because nothing that we did relies on a MP assumption being the case or not. I only mentioned the MP aspect in discussing what an additional experiment might want to do. Weather the situation is MP or not has nothing to do with what we did and thats why its not mentioned anywhere.
As for the math that proves that what you say is true, I might have to review that because I don't see how it can be the case, but I am not going to argue the point because it is, as I said, "beside the point".
Feyz
I read the paper, quite interesting. Since I am a physicist and not a EE I have no background in Minimum Phase (MP) (never use it myself - the concept is not used in Physics), and I had a misconception that you made clear in this paper. A system can have Group Delay (GD) and still be minimum phase - I had misunderstood that point. All that I am interested in is GD. I don't care if the system is MP or not, its doesn't matter. It was GD that Brian Moore proved was audible, MP had nothing to do with it. So it was GD that I was interested in.
Since GD is audible and the signals that I had studied had both GD and amplitude effects, there could be an issue about which is the more audible. I would not expect amplitude variations to change in audibility with level (this is where I disagree with your loudness argument), but I fully expected the GD effects to change in audibility with amplitude because this is precisely what Brian Moore said was the case. I can see where it is likely that both effects are factors and sorting out one from the other would not be a trivial task.
That said, in the context of what I was interested in it is not necessary to sort out this complexity because the effects of HOMs are audible and they do change with absolute SPL. The why is not important, what is important to to realize that when a horn has HOMs, that turning up the SPL will eventually make an inaudible HOM audible and that these effects will start to be annoying. This is precisely what does happen in reality and minimizing the HOMs does allow for a much higher SPL before this annoying sound becomes a problem.
So you see that none of this argument has any bearing on what I was interested in. I proved the point - reducing the HOM will make the device sound better, especiually at higher SPLs. But further it is important to couple this to our other studies of nonlinear distortion where we showed that it was, for the most part, not audible. People had for decades associated the poor sound quality of a horn to its "nonlinear distortion", but with what we have shown this is not the case. The audible distortion is linear, with a nonlinear perception - quite a different thing.
I read the paper, quite interesting. Since I am a physicist and not a EE I have no background in Minimum Phase (MP) (never use it myself - the concept is not used in Physics), and I had a misconception that you made clear in this paper. A system can have Group Delay (GD) and still be minimum phase - I had misunderstood that point. All that I am interested in is GD. I don't care if the system is MP or not, its doesn't matter. It was GD that Brian Moore proved was audible, MP had nothing to do with it. So it was GD that I was interested in.
Since GD is audible and the signals that I had studied had both GD and amplitude effects, there could be an issue about which is the more audible. I would not expect amplitude variations to change in audibility with level (this is where I disagree with your loudness argument), but I fully expected the GD effects to change in audibility with amplitude because this is precisely what Brian Moore said was the case. I can see where it is likely that both effects are factors and sorting out one from the other would not be a trivial task.
That said, in the context of what I was interested in it is not necessary to sort out this complexity because the effects of HOMs are audible and they do change with absolute SPL. The why is not important, what is important to to realize that when a horn has HOMs, that turning up the SPL will eventually make an inaudible HOM audible and that these effects will start to be annoying. This is precisely what does happen in reality and minimizing the HOMs does allow for a much higher SPL before this annoying sound becomes a problem.
So you see that none of this argument has any bearing on what I was interested in. I proved the point - reducing the HOM will make the device sound better, especiually at higher SPLs. But further it is important to couple this to our other studies of nonlinear distortion where we showed that it was, for the most part, not audible. People had for decades associated the poor sound quality of a horn to its "nonlinear distortion", but with what we have shown this is not the case. The audible distortion is linear, with a nonlinear perception - quite a different thing.
One more thing.
Your use of Cepstrum in this context was quite interesting. It may be just what I need to resolve out the HOMs from the main mode. I am going to give that a try.
Your use of Cepstrum in this context was quite interesting. It may be just what I need to resolve out the HOMs from the main mode. I am going to give that a try.
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