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