Re: Gaussian vs uniform noise audibility (Eckard Blumschein )

Subject: Re: Gaussian vs uniform noise audibility
From:    Eckard Blumschein  <Eckard.Blumschein(at)E-TECHNIK.UNI-MAGDEBURG.DE>
Date:    Fri, 23 Jan 2004 20:16:01 +0100

At 10:11 23.01.2004 -0600, beauchamp james w wrote:
>Eckard Blumschein writes:
>>Of course, complex representartion of a signal requires magnitude as well
>>as phase. (Magnitude is always positive while there are positive and
>>negative amplitudes.) This should once again persuade anybody that the
>>inner ear does not perform a complex Fourier transform. What about "any
>>operation" I agree on condition of just a single snapshot.
>>Hearing is, however, a continuous process where the complex Fourier
>>transform is doomed to hop from window to window in a clumsy manner.
>
>This is an artifact of using discrete computation. In a continuous (or
>"analog") formulation, the short-time transform can be absolutely
>continuous in time. And this can be approximated as closely as one wishes
>on a computer. However, it is easy to show that only a few samples per
>window are required due to the band-limited nature of the window
>function.

First of all, forget the wrong idea that the cochlea performs a complex
Fourier transform. The next step after (real-valued) frequency analysis is
rectification, and a magnitude cannot be rectified.

Complex Fourier transform requires a lot of arbitrariness. Short time FT
tries to abstains from the basic arbitrariness by close adaptation to the
natural current zero of time. You might be right in so far that this
hopping of the origin could be made quasi continuous in a similar manner as
I manage to do it more easily with the simpler FCT. Would you please be so
kind telling us some references if it is more than just your idea?

Admittedly I failed to overlook all variants of spectrograms and wavelets.
The vast variety indicates that none of the complex solutions is the
ultimate one or could at least be regarded similar to cochlear function.
Wouldn't continuous shift of the zero tacitly more or less change magnitude
into amplitude, not to tell the other trouble? I guess, equipping the
complex FT with a continuously shifted center of the window would end up at
an equivalent to FCT. Massively overlapping windows seem already to be a
preferred standard solution. The ear does not need such artificial
re-formulation. It performs what can be called action of a 'filter bank' or
in mathematical terms Fourier cosine transform.

You are correct: Equidistant sampling causes a lot of artifacts from which
hearing is not affected because it doesn't use regular sampling at all.
However, because digitalization is the best way to get rid of noise and
errors in transmission, I consider sampled data always the given input of
analysis.

In my opinion, non-causallity, clumsiness, arbitrary windows and other
shortcomings of or in connection with complex Fourier transform may neither
be denied nor be ascribed to discretization artifacts. Complex Fourier
transform has huge merits but it does not fit as natural to cochlear
frequency analysis as does Fourier cosine transform. It gave rise to a lot
of unnecessary confusion in particular concerning the role of phase and
exceptions from phase deafness.

Incidentally, my method to calculate the natural specrogram is not subject to
a limited bandwidth while sample rate sets a limit, of course. Seemingly
contradicting to the uncertainty principle, this is nonetheless plausible.

Sincerely and hard working,
Eckard Blumschein