Re: real-time information

[Eckard Blumschein <Eckard.Blumschein@xxxxxxxxxxxxxxxxxxxxxxxxxx>]

Dear Paris Smaragdis,

At 16:54 05.03.2001 -0500, you wrote:
... The Fourier transform, as well as
>the cosine and sine families of transforms, are optimal 2nd order
>decompositions of markov processes.  That means that if you have time
>series with any temporal coherence (that would be any sound), and you
>wanted to find a set of basis functions that would decorrelate them, you
>would end up with a harmonic transform (and the virtues of decorrelation
>and independence are well understood even by stingy engineers).
>Harmonic transforms are not a convenient tool we came up with one day,
>they are formed by the nature of coherent time series.  Likewise,
>applying decompositions with other independence criteria you can derive
>constant-Q-ish, and even cochlea-lookalike transforms.

Incidentally, I do not understand why do you exclude non-Markovian
processes? Please excuse my ignorance, I imagine a Markov chain independent
of the way in which the present state arose from its predecessors. Doesn't
at least the output of cochlea contradict to the definition of a Markov
process?

I tried to draw attention to what I consider neglected functional
peculiarities of signal processing inside cochlea (and subsequently in the
brain too):

Presumably, the signal is not simply transformed at all, but more relevant
data, i.e. more recent ones, are gradually preselected. Also, the whole
signal is prepared for more robustness. In terms of information theory,
cochlea might even be interpreted as an adapter to a slower sampled
information channel with nonetheless larger capacity. The key functional
element is a continuously distributed memory with permanent partial renewal
over the whole partition and also permanent gradual forgetting. Notice, I
am using terms like partition and renewal in common, not in mathematical
sense.

Even if the virtue of decorrelation into mutually independent orthogonal
components may provide some similar benefits as does the actual function of
cochlea, I do not expect that any transform may fulfill the whole task
while avoiding detrimental sideeffects. That's why I am skeptical. Do we
really need a cochlea-lookalike transform in order to improve, for
instance, perceptual coding? I see three options.

Those who are convinced that the problems can be resolved by providing the
"optimal" criteria of decomposition will perhaps go on offering an
abundance of rather academic transforms which do not necessarily reflect
the reality. They are obviously not interested in function of audition,
being to them just an exchangeable object.

Those who studied shortcomings of the traditional approach are suggesting
corrections. I would be very much interested in your comments on some
intriguing ideas by Terhardt. Please find a link to his papers on FTT at my
home.

Hopefully, at least a few people will be able and willing to attack the
problems the other way round, that is, based on what is already
functionally understood, fairly regardless of the traditional approach and
the high level of sophistication in mathematics.

>...our ears ... could have strived for a form that resulted in a sparse
and informative decomposition.

I agree, with the addition that benefiting from decorrelation is only one
out of several selection criteria.

>There are very interesting 'coincidences' and very strong theories
>forming in the field of mathematical modelling of perception, and all of
>us in this field would appreciate less snubbing.  Don't blame math for
>being poor and unnatural.  It is only a tool, and when used correctly it
>is a lot more interesting than measuring cochlear responses.

Of course. However, I feel, we have to rehabilitate those who were more
than snubbed because they did not accept what meanwhile has proven
questionable or unjustified application of mathematics. I see this not just
a question of fairness but also a precondition of progress in practice.

Sincerely,
Eckard