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Re: Gaussian vs uniform noise audibility
At 12:06 23.01.2004 -0800, Julius Smith wrote:
>At 11:16 AM 1/23/2004, Eckard Blumschein wrote:
>>First of all, forget the wrong idea that the cochlea performs a complex
>>Fourier transform.
>
>This implies phase is discarded.
No! Do not consider me a moron. You and largely the rest of the world grew
up with the erroneous believe that there is no equivalent alternative to
complex spectral analysis. Complex calculus is indeed tremendously useful.
No matter whether one prefers magnitude and phase or real and imaginary
part, one always has to consider both constituents except for the case one
of them equals zero. Given, a function of time like 2A cos(omega t) does
not have any imaginary part at all. Entrance into complex plane is payed by
mandatory arbitrary omission of A exp(- i omega t) or A exp(i omega t).
Neither the magnitude A nor the phase omega t can be discarded.
At that point, you will object: Aren't anti-symmetrical functions, i.e.
functions of time with odd symmetry like sinus, also needed in frequency
analysis?
No again, on condition, causality has been taken into account. In brief:
Future signals cannot be analyzed yet. Even sin(omega t) can be continued
as its mirror into fictive future time like an even function. Of course,
this wouldn't hold for its derivative or antiderivative. However, our topic
is just frequency analysis within cochlea.
>However, phase information does exist as
>the phase of the basilar membrane vibration,...
I don't take amiss this fallacy. It has to do with the missing natural
justification for fixing any reference point on the time scale. Our ears
are not synchronized with anything. When Descartes introduced Cartesian
coordinates, he imagined a spatially infinite world. Time is
correspondingly believed to also expand from minus infinite to plus
infinite. However, elapsed time definitely ends at the 'NOW' being the only
clever choice for a natural time scale. Take subsequent snapshots of a
sinusoid at NOW each. Try the same with any cochlear pattern. By chance,
you might observe sin or cos. In other words, so called linear phase is
arbitrary as is time. I don't deny that delay or according phase difference
is reasonable with respect to a second signal or a different reference.
Without such reference, a sinusoidal function cannot be a identified as
sin, cos or something complex in between, and the reference is lacking in
nature. The only natural reference is the NOW, which is steadily on the
move. This causes the trouble of permanently lagging window position in
case of arbitrarily centered complex Fourier transform.
>Since basilar membrane filtering is generally
>modeled as linear, any corresponding short-time-Fourier-transform would
>have to be complex to model basilar membrane filtering. Subsequent
>half-wave rectification does not eliminate all phase information,
An old specialist of power electronics like me cannot retrace how you
imagine rectification of a complex-valued function of time.
My wife is a teacher for adults. Perhaps she would more heedfully
anticipate what you and many others are feeling rather than thinking. I
will try and elucidate how engineers handle a similar case: Consider an
ideal sinusoidal voltage as a real input into a circuit that may also
contain a first (small) resistor and a reactance in series. Parallel to the
first resistor there are a diod and a much larger second impedance in
series. The voltage across the first resistor is a complex quantity with
respect to the source but pretty independent of the diod. However,
piecewise linear calculation requires to refer to the current through the
diod as a real one. In case of hearing, phase of the stimulus does not
matter since it anyway relates to an arbitrary reference.
As a rule, recognized experts like you tend to be cautious against
radically uncommon views. Therefore I would like to ask you: Look at
pattern of BM motion (e.g. T. Ren's) or of firing in the auditory nerve.
They do not resemble magnitude, nothing to say about phase. As far as I can
judge, they resemble the pattern of the natural (real-valued) spectrogram.
More in detail: Magnitude cannot account for the different patterns with
rarefaction vs. condensation clicks while positve and negative amplitudes
of the natural spectrogram clearly differ from each other.
In all, I didn't find any tenable argument in favor of complex cochlear
function. On the other hand, Fourier cosine transform, the natural
spectrogram and joint autocorrelation already resolved a lot of so far
poorly understood questions.
Incidentally, I recall a textbook denying any difference between time
domain and frequency domain. I do not fully share this opinion. In
particular, I consider it necessary to clearly distinguish between real
world and fictitious complex domain.
Eckard