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Re: Gaussian vs uniform noise audibility
At 07:21 23.01.2004 +0200, Israel Nelken wrote:
>... whether the ear is a fourier analyser, ... I think
>everyone's answer (certainly mine) is strictly speaking no - although
>spectrograms are reasonable 0th order approximations to what we hear...
Don't consider me a knowing-all if I try to clarify:
- Usual (complex) Fourier analysis transforms the real signal into a
magnitude/phase representation with respect to an arbitrarily chosen
temporal reference, missing within the ear.
- Function of the inner ear does definitely not resemble the complex
Fourier transform but the slightly different real-valued Fourier cosine
transform plus one-way rectification.
- The usual spectrogram is a rather imperfect freak inbetween. Nonetheless,
I agree, it provides a first but partially misleading picture of what we
hear. I argue that e.g. pitch perhaps reflects autocorrelation resulting
from joint cochlear and subsequent neural signal processing.
>...spectral representation is mathematically 'true' (within the
>obvious limits of the application of a mathematical theory to real
Oliver Heaviside's trick of creating Hermitian symmetry has proven very
clever on the expense of physical adequacy. Integration over time from
minus infinite to plus infinite is only reasonable if one replaces the
unknown future by mutually compensating mirrors of the past. Corresponding
results often exhibit non-causality.
>Furthermore, for real-life signals the inverse fourier transform is
>equal to the original function almost everywhere,
The same is true for the Fourier cosine transform.
>so that any operation on the signal that is formulated in terms of
>its temporal waveform can be also formulated in terms of the fourier
>spectrum (amplitdue AND phase).
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.
>...there's a full equivalence between time and spectral processing.
Complex spectral analysis is not only restricted to linearity and to
snapshots of band-limited signals. It also demands a return into time
domain because transition into complexity was based on neglect of either
exp(i omega t) or exp(-i omega t).
Those who thoughtlessly claim full equivalence between a function of time
and its complex spectrum tend to also high-handed deny the actual
equivalences between R and R+ in case of (apparent) symmetry and between
complex-valued magnitude/phase representation and the underestimated
real-valued time/frequency representation. The latter merely lacks the
information about the arbitrary reference point of complex analysis, i.e. a
single value that does not belong to the original signal.
>Finally, there's the question of the usefullness of the spectral
>description of random processes, which is yet a somewhat different
I consider the adequate model of cochlea (FCT + rectification) a good
precondition for correct conclusions concerning random stimuli while
non-linearity of rectification further invalidates the pipe dream of
equivalence between the signal and its complex matrices.
>The independence of the spectral components of a gaussian
>process is a mathematical result, independent of the physiology of
>hearing, but it has consequences for hearing.
Given, adequacy of the mathematical model is questionable. How trustworthy
are the results?
>Since we are sensitive to spectral correlations, non-gaussianity can >be
detectable by such sensitivity. This doesn't assume anything about >the use
of Fourier transforms in the ear!
I googled for 'non-gaussianity': 3750 results, mostly cosmic microwave