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Re: Non-linear additions to linear models. (was On pitch and periodicity (was "correction to post"))

Hi Dick,
My last observation is on your suggestion of adding non-linearity to some linear model to cover what some people may call illusions. As an aside, I believe Helmholtz was forced to add in the quadratic function only because experimentalists (Seebeck I believe) were breathing down his neck proving the existence of the missing fundamental problem. I would have to strongly disagree with some of the conclusions reached from such quadratic and cubic expansions. In my opinion, I think when people say that a new paradigm was needed I assumed that it meant a totally new approach to signal analysis that did not necessarily adhere to any assumptions of linearity. Take for example a system based on rate of change of signal energy, it could right away explain some minor psycho-acoustic phenomenon associated with changes in static pressure or that tricky bias term that comes up when one is analyzing sounds like speech. But as I am sure you would point out, much more would be needed before such a statement could have any validity as the basis for a system theory. I agree. On that note, I believe that till some such system is offered for review, non-linear additions to linear models will have to do for rest of us who are appalled by the associated mathematics. Regards,
Randy Randhawa

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:
Dear Dick,
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.
Randy Randhawa

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"?