Re: Gaussian vs uniform noise audibility (Julius Smith )

Subject: Re: Gaussian vs uniform noise audibility
From:    Julius Smith  <jos(at)CCRMA.STANFORD.EDU>
Date:    Tue, 27 Jan 2004 09:13:38 -0800

Yes, a "sliding cosine transform" can be used in place of the usual
"hopping short-time Fourier transform", and in that case, phase information
is contained in the time variation of the sliding transform
coefficients.  I didn't realize you were doing something like that, so my
argument was based on different assumptions.  Even the short-time Fourier
transform hopping by half its window length each frame can be stripped of
all phase information and still be used as the basis of a convincing sound
synthesis, at least for smoothly changing sounds. -- jos

At 03:08 AM 1/26/2004, Eckard Blumschein wrote:
>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