Re: Fourier decomposition

[Eckard Blumschein <Eckard.Blumschein@xxxxxxxxxxxxxxxxxxxxxxxxxx>]

Hello Fred,

Maybe you did not get aware of my reply to Robert Gjerdingen: errors in harmonic
chord identification (last Thu 15 Sep 2005), or you did not consider it serious.

I anticipate: The majority of experts is either not able or at least not willing
to accept what I found out concerning Fourier decompositon of a contiuously
incoming signal:

Future events cannot be observed in advance. While this insight is beginning to
cause a lot of turmoil in quantum physics, it is self-evident in case of
hearing. Hearing is a real-time process. Accordingly, the signal of concern can
be completely expressed on a 'half-sided' time scale extending not from minus oo
to plus oo but from zero to oo 'elapsed time'. I am suggesting the shorter name
'E-time'.

Incidentally, if basics of physics is actually restricted to observables then
I do not see any basis for future spacetime, time travel, white holes, still
missing evidence for Higgs bosons and susy, superluminal propagation of signals
and similar highly questionable stuff.

The expression 'half-sided' implies what I consider an illusion: Future world is
believed to exists in advance. I argue: Even if this was correct, it would not
matter at all. Description in E-time is fully sufficient. Check it out:
Complex-valued Fourier transform does not at all contain more information than
real-valued Cosine transform except for an arbitrarily chosen zero of time or
linear phase, respectively. It is just twice padded with redundant copies of the
essential information.

Fortunately, natural evolution did not learn this human belief. The ear does not
have an arbitrary reference point ot time. I see time the other way round.
E-time is the basic physical quantity while ordinary time is just a human
abstraction and extrapolation, in short: Ordinary time is a human fabrication.
Our notorious trouble due to Fourier transform is selfmade.

You wrote:

> It is not possible to accurately recover the coefficients (amplitude and
> phase of the individual harmonics) of a function consisting of harmonic
> sinusoidal components,

I would like to clarify first what you are intending to deal with. You obviously
refer to a function of ordinary time, not of E-time. Notice, with Cosine
transform in IR+ there is no decomposition in terms of magnitude and phase but
simply the natural decomposition into a spectral profile with different positive
or negative amplitudes as it is to be seen at basilar membrane.

Moreover, you seem to tacitly assume a given theoretical function to be
analysed. This is not what the ear does. To the inner ear, any theory is
unknown. It has to decide itself what frequency content is in the signal. From
this point of view, it depends on its own frequency analyis what 'is' or is not
within the signal. With this respect, the ear is unique while STFT requires a
lot of arbitrary windowing.

> when the Period of the not necessarily present
> fundamental is not known, by using the normal computational procedure of
> either the Fourier Series or the Short term Fourier Transform.

In order to better understand how the ear's work works, ask for the natural
window of time. It is amazing.

> -----------------------
>
> I would appreciate any and all comments.

Hopefully you will express your appreciation by commenting on my comments.

For your convenience, I point you once again to

http://iesk.et.uni-magdeburg.de/~blumsche/M277.html

Yours sincerely,
Eckard