That's interesting. One of the best "tonality estimators" ever written (in order to make good masking models for coders you need some estimate of the tone vs. noise character per critical band, well, actually 1/3 band,but anyhow) was a magnitude-phase predictor (polynomial forward predictor 1 -3 3 ) that predicted magnitude and phase separately, then took the actual data for the current block from an FFT and measured the error divided by the two magnitudes of the current and predicted.
This worked remarkably well for estimating masking thresholds that were for neither pure tone nor pure noise.
It was very, very ad-hoc. I invented it after trying about a gazillion things.
But, now, the comments from Ranjit seem, to me, to have something in common here.
Which is interesting. I am not going to assert that either is right, but there is a convergence of 'things that work'.
I commend you on your bravery.
At 12:53 PM -0400 9/6/11, Ranjit Randhawa wrote:
I felt that it would not only be negligent on my part but also cowardice, to end this thread without at least offering a possible approach to a new "paradigm", even if I am not capable of describing it in excruciating detail at the moment. So, with the greatest trepidation, here goes.
If one were to consider a pure sinusoid in the phase domain (one where the axis are x(t) and dx(t)/dt), the locus would be a circle. The area of this circle would give us the magnitude, though how to determine this requires a different approach as the integration over 2pi would be zero.
If we consider the product x(t)*dx(t)/dt as the rate of change of energy it would have a sign associated with it, then it is possible to determine this area, though the resulting algorithm would be too simple and fall apart for more complex signals since we don't know the period. To get a more general approach, it would be better to consider the circle in sectors of harmonically increasing sizes, thereby converting the sinusoid signal to a harmonic series, the area of each sector becomes the magnitude of a related tonal harmonic with the smaller sectors associated with magnitudes of higher frequencies.
We see then that we start with a single sinusoid being considered not as a single valued entity but as a harmonic series and this therefore immediately answers two questions, the first being the reported psycho-acoustical behavior whereby some people have indicated an ability to recognize harmonics of a pure tone and shown by others to be possible by using beats. The second being that the extent of the traveling wave can have an explanation in that the stiffness of the BM would limit activity for higher frequencies as these frequencies would have smaller areas for lower strength signals and the TW would grow as this strength increases.
More importantly, since we are directly using energy to determine magnitudes, the missing fundamental would show a magnitude and this magnitude would vary depending upon the relative phases of the components. This approach also provides for a much reduced computational method to determine the period, an alternative form of auto-correlation. These two assertions are based on the method chosen for determining the harmonic series and the one chosen by me was picked from the field of psychology and called "evaluative bivalence", whereby one makes use of sign associated with the rate of change of energy in the summation process.
The modified auto-correlation does work rather well for quasi-periodic signals and would welcome any suggestions for practical use, since I don't believe it is used by the auditory system. I believe it would fail in the "party room" environment. Other options are possible based on the summation process.
There are many consequences of this approach as it now becomes possible to provide a more exact method to explain source location capabilities, pitch explanations, and I would like to say cochlear functions but have to admit my knowledge at that level is focused only on what has been reported on the behavior of the Traveling Wave. The rest of it is a mystery to me.
I would like to apologize if this blurb causes some kind of angst among some in this LIST. It was not the intent. I simply wanted to show that sticking with existing mathematics has not made much progress in being to explain our original discussion and that was "The Case of the MIssing Fundamental". Thanks for your understanding and kindness and sorry for this delay,
On 8/4/2011 1:42 PM, Richard F. Lyon wrote:
I'll be the first to agree that linear systems theory is sometimes stretched beyond where it makes sense, and that you need to use nonlinear descriptions to describe pitch perception and most other aspects of hearing, and more so when you get up to cognitive levels.
I'm sorry to hear that you "gave up on linear systems", because I don't think it's possible to do much sensible with nonlinear systems when you don't have linear systems as a solid base to build on. Certainly at the level of HRFTs, cochlear function, and pitch perception models, a solid understanding of linear systems theory is in indispensible prerequisite. Then, the nonlinear modifications needed to make better models will seem less "tortured".
At 10:33 AM -0400 8/4/11, Ranjit Randhawa wrote:
While linear system theories seem to work reasonably well with mechanical systems, I believe they fail when applied to Biological systems. Consider that even Helmoholtz had to appeal to non-linear processes (never really described) in the auditory system to account for the "missing fundamental" and "combination tones". Both of these psycho-acoustical phenomenon have been well established and explanations for pitch perception are either spectral based or time based with some throwing in learning and cognition to avoid having to make the harder decision that maybe this field needs a new paradigm. This new paradigm should be able to provide a better model that explains frequency (sound!) analysis in a fashion such that the nothing is missing and parameter values can be calculated to explain pitch salience, a subject that seems to be never discussed in pitch perception models.
Furthermore, such a new approach should also be able to explain why the cochlear is the shape it is, which as far as I can see has never been touched upon by existing signal processing methods. Finally, are these missing components "illusions" that are filled in so to speak by our higher level cognitive capabilities? It is remarkable that this so called filling in process is as robust as it is, to be more or less common to everyone, and therefore one wonders if all the other illusions are really not illusions but may have a perfectly good basis for their existence. If they were "illusions" one would expect a fair amount of variation in the psycho-acoustic experimental results I would think.
I myself gave up on linear systems early in my study of this field and have felt that other systems, e.g. switching, may offer a better future explanatory capability, especially when it comes to showing some commonality of signal processing between the visual and the auditory system. To this end, I am quite happy to accept that I do not consider myself an expert in linear system theory.
On 8/2/2011 1:49 PM, Richard F. Lyon wrote:
At 5:55 PM +0300 8/2/11, ita katz wrote:
The periodicity is determined by the least-common-multiple of the periodicities of the present harmonics, so if (for example) a sound is composed of sines of frequencies 200Hz, 300Hz, and 400Hz, the periods are 5msec, 3 1/3msec, and 2.5msec, so the least-common-multiple is 10msec (2 periods of 5msec, 3 periods of 3.33msec, and 4 periods of 2.5msec), which is of course the periodicity of the sum of the sines, or in other words 100Hz. (actually it is the same as the greatest-common-divisor of the frequencies).
Ita, that explanation is sort of OK, but as written implies that the auditory system has the ability to do number-theory operations on periods (or frequencies), and depends on there being harmonics present and separately measureable.
It would be much more robust to say that "The pitch is determined based on an approximately common periodicity of outputs of the cochlea," which I believe is consistent with your intent.
Why is this better? First, it doesn't say the periodicity is determined; what is determined is the pitch (even that is a bit of stretch, but let's go with it). Second, it doesn't depend on whether the signal is periodic, that is, whether harmonics exist. Third, it doesn't depend on being able to isolate and separately characterize components, harmonic or otherwise. Fourth, it doesn't need "multiples" (or divisors), but relies on the property of periodicity that a signal with a given period is also periodic at multiples of that period, so it only needs to look for "common" periodicities--which doesn't require any arithmetic, just simple neural circuits. Fifth, it admits approximation, so that things like "the strike note of a chime" and noise-based pitch can be accommodated. Sixth, it recognizes that the cochlea has a role in pitch perception. It's still not complete or perfect, but I think presents a better picture of how it actually works, in a form that can be realistically modeled.
Is this "tortured use of existing signal processing techniques" as Randy puts it? I don't think so. Is it "a unique way to do frequency analysis and to meet the dictum in biology that 'form follows function'"? Sure, why not? But why call it "frequency analysis"? How about "a unique way to do sound analysis" (if by "unique" we mean common to many animals)?
I do have some sympathy for Randy's concern that we are far from a complete understanding, and that hearing aids are not as good as they would be if we understood better, but yes, he sounds way too harsh in overblowing it so. I'm wondering what's behind that, and whether it's just confusion about all the confusing literature on pitch perception, which I agree is a complicated mess -- or is the problem, indicated by Randy's previous posts, just that he doesn't understand basic linear systems and signal processing, and that's why it all seems "tortured"?