Interesting Papers re temporal resolution Options

[ncdrawl]

http://www.physics.sc.edu/kunchur/Acoustics-papers.htm

http://www.physics.sc.edu/kunchur/papers/FAQs.pdf

[Canar]

If I may grossly oversimplify your argument Axon, you're partially arguing that he's choosing an extreme edge case to test. However, if his intent is to map the boundaries of audibility, wouldn't an edge case be acceptable? I find the conclusion that ultrasonics are perceptible fascinating, and if he's found a case in which they actually are audible, should we not hear it out? Even though it does not represent most cases, if they are truly audible in this case, isn't that worth considering?

As an archivist, I want transparency in all cases, so I don't have to worry about the edge cases. That's why I use FLAC and not MP3. If there is any case where 44.1kHz is not sufficient, isn't that worth devising solutions for?

[Axon]

If I may grossly oversimplify your argument Axon, you're partially arguing that he's choosing an extreme edge case to test. However, if his intent is to map the boundaries of audibility, wouldn't an edge case be acceptable? I find the conclusion that ultrasonics are perceptible fascinating, and if he's found a case in which they actually are audible, should we not hear it out? Even though it does not represent most cases, if they are truly audible in this case, isn't that worth considering?

As an archivist, I want transparency in all cases, so I don't have to worry about the edge cases. That's why I use FLAC and not MP3. If there is any case where 44.1kHz is not sufficient, isn't that worth devising solutions for?

The use of a 7khz square wave as an input here, in this context, seems particularly unrepresentative to me, as a -10db ultrasonic third harmonic, with a signal completely absent of energy at 14khz from other sources, is a profoundly special case. It's not merely that it's an edge case - it is way, way over the edge to begin with. It's like arguing that 16 bits is insufficient because you can hear the noise with the gain raised ~20db above normal (as even shown by Meyer/Moran). Of course you can - but that situation never actually happens in the real world, where music is normalized near 0dbFS and released for an audience that actually wishes to listen to it. More generally, Kunchur never really justifies that input signal very well, and without careful delineation, nothing's stopping anybody from boosting 21khz levels arbitrarily high to get arbitrarily low measured thresholds (like with, for instance, a bipolar pulse train).

In the final reduction ad absurdum, it's hard to tell apart his conclusions apart from a claim that (say) 200khz bandwidth is necessary for audio, because if you play extremely powerful 200khz and 202khz tones, the inevitable intermodulation is audible. The existence of any form of intermodulation, combined with the existence of an ultrasonic bandwidth, necessarily implies that some classes of signals will show audible differences when filtered before distortion. Morevoer, this audibility will exist at any amount of filtering greater than zero, because I'll always be able to hand you a signal that will break threshold at the intermodulation frequency.

A test with ultrasonic content at ranges more representative of real situations would restore validity, but I think that is not going to save his conclusions. In that case, if audibility is shown in the first place, it will almost certainly be above 22us. And at that point it no longer has anything to do with time resolution. But it would be a convincing proof of CD's insufficiency - but before that point is reached, the question is, how would that be possible when every prior attempt has failed?

This post has been edited by Axon: Jul 28 2009, 20:18

[andy_c]

I'd like to add a few comments regarding the "Temporal resolution of hearing probed by bandwidth restriction" paper.

One potential issue is the second-harmonic contribution of the test setup. Since this contribution is within the audible band (14 kHz), special attention needs to be paid to it. Of course, an ideal square wave has no even-order harmonics by virtue of its half-wave symmetry, i.e. f(t +/- T/2) = -f(t). But there are ways that the second harmonic can creep back in. Two ways I can think of are the duty cycle of the square wave not being exactly 50 percent, and second-harmonic distortion of the test setup's transducers and electronics. Either of these situations will eliminate the half-wave symmetry and introduce even-order harmonics. The text below figure 1 states "The acoustic output from the transducer is devoid of even numbered harmonics because of the square-wave signal fed to it". Of course, in the real world, that signal can't be entirely devoid of second harmonic, and indeed figure 4 shows its presence. Oddly, figure 4 shows the frequency components in terms of power, where dB would have been better if clarity were the intent. In any case, the power ratio of second harmonic to fundamental is shown as 1e-6, giving a voltage ratio of 1e-3 (-60 dB) at the mic preamp output where this measurement was presumably taken.

One interesting thing that can be done is to make the generous assumption that the electronics and transducers have zero second-harmonic distortion, and assume this second-harmonic component is due entirely to the duty cycle of the square wave not being exactly 50 percent. One could then derive the Fourier series coefficients of a rectangular wave having duty cycle d, where 0 < d < 1. Then one could figure out what values of d correspond to a second-harmonic component 60 dB down from the fundamental. This would correspond to a "best case" scenario, because with electronics and transducers having non-zero second-harmonic distortion, the tolerance on the duty cycle of the square wave would have to be even tighter to take into account those additional second-harmonic components. Suppose we have a rectangular wave with a symmetrical voltage swing of +/- Vp and a duty cycle d. It's not too hard to show that the Fourier series coefficients vn of this function are:

vn = (4Vp/(pi*n)) sin(pi*d*n) for n >= 1 (i.e. this excludes the DC term)

The ratio r of the second harmonic to the fundamental is:

r = (1/2) sin(2*pi*d)/sin(pi*d)

Now we can solve numerically for the value of d that makes r = +/- 1e-3. This will give two answers - one slightly less than 0.5 and one slightly greater. Plugging this into MathCad, we get r1 = 0.49968 and r2=0.50032. So the allowable range of the duty cycle is 50 +/- 0.032 percent. This is an extremely stringent requirement, and yet this number is optimistic for two reasons. The first is the previously mentioned assumption that the electronics and transducers have no second-harmonic distortion. The second is that real-world square wave generators use nonlinear circuits which in general will have slightly asymmetric rise and fall characteristics. The Fourier series coefficients assume zero rise and fall times. I'm finding it hard to believe that he actually achieved this number, especially with two transducers involved (the headphones and microphone). So let's look at table 1, where he lists the harmonic components of the transducer output to see what he actually measured for the second harmonic in each condition. Well, those measurements aren't there, only the fundamental and the third and fifth harmonics. The entire issue is papered over with the statement "The acoustic output from the transducer is devoid of even numbered harmonics because of the square-wave signal fed to it". Since changes in this second harmonic value could explain the experimental results without needing some hypothesis regarding the alleged ability of the ear to detect signals above the frequency limit of human hearing, it's essential to include these data. Yet he fails to do so.

[Axon]

I'd like to add a few comments regarding the "Temporal resolution of hearing probed by bandwidth restriction" paper.

One potential issue is the second-harmonic contribution of the test setup. Since this contribution is within the audible band (14 kHz), special attention needs to be paid to it. Of course, an ideal square wave has no even-order harmonics by virtue of its half-wave symmetry, i.e. f(t +/- T/2) = -f(t). But there are ways that the second harmonic can creep back in. Two ways I can think of are the duty cycle of the square wave not being exactly 50 percent, and second-harmonic distortion of the test setup's transducers and electronics. Either of these situations will eliminate the half-wave symmetry and introduce even-order harmonics. The text below figure 1 states "The acoustic output from the transducer is devoid of even numbered harmonics because of the square-wave signal fed to it". Of course, in the real world, that signal can't be entirely devoid of second harmonic, and indeed figure 4 shows its presence.

Well, given that he actually measured the acoustic field at the listening position and no even harmonics were observed at that point, I figure that is reasonable enough evidence that they do not exist... but read on.

Oddly, figure 4 shows the frequency components in terms of power, where dB would have been better if clarity were the intent. In any case, the power ratio of second harmonic to fundamental is shown as 1e-6, giving a voltage ratio of 1e-3 (-60 dB) at the mic preamp output where this measurement was presumably taken.

In his first paper (to Proceedings of Meetings on Acoustics) Kunchur explicitly rejects the use of FFTs in the power spectrum analysis:

An enormous time (of the order of two years) and effort were spent to develop the instrumentation and the methods for checking for artifacts. For example, for just the Fourier spectrum shown in Fig. 4, it took a few months to develop the instrumentation setup and to write the C code (FFT was not used). To measure one such spectrum takes over a week.

IIRC, when questioned about this he said the implementation was an "elementary" one derived from textbook sources. I assert that anything using a power spectrum that can't use an FFT is unquestionably not elementary.