🚀 go-pugleaf

In article <1i7r3rINNm0q@network.ucsd.edu> mbk@lyapunov.ucsd.edu (Matt Kennel) writes:
>Dick Pierce is naturally correct.  This somewhat unsatisfying answer points
>out some of the disadvantages of classical Fourier analysis, in that
>spectral analysis does not easily distinguish transients with broadband
>frequency content from long-term broadband noise signals.
>{it also doesn't distinguish deterministic chaotic signals from broadband
>noise but that's totally irrelvant here}

I'd hardly call it a disadvantage.  The analysis is correct.  If you
want a given frequency resolution, you must have at LEAST a given
time span to determine that resolution, regardless of the
kind or type of analysis.  It's not a question of
"disadvantages of classical Fourier analysis", it's a question
of misapplication.  One is entirely able to take short-term
spectra that will differentiate, one is able to differentiate on
the phase spectrum, and so on.  Nothing magical, nothing lost,
nothing gained.  Rabiner and Crochiere say some words
about the subject, but pick up a good book on short term spectral
analysis for a better treatment. (None come to mind right now,
but I know I saw a few at the vendors at ICASSP last spring.)

>It may be that the
>physical realization of some systems may be described more faithfully
>with a mode of analysis that can exploit this difference.  The mathematical
>foundations of this field is called wavelet analysis, but it remains to
>be seen how much success it has in solving physically relevant problems.

Wavelet analysis is nothing new mathmatically. There have
been non-perfect reconstruction wavelet-like systems in the field
since about 1980, and the basic mathmatics has been around for
a century or so.  What IS new is the constructions for perfect
reconstruction, smoothness criteria, scaling, translation,
and so on. Don't get the idea I don't LIKE wavelets, but
I'm a bit convinced that they have become a "buzzword"
even inside the research community. A quick look
at the spectrogram machines of the 1950's shows right then
and there that a pseudo-fourier <I say pseudo because phase
is discarded in the only visible output> analysis can localize
an event. The window-switching used in PAC and ASPEC (see
AES and ICASSP papers on such) also localizes events within a 
different, perceptual, frame of reference.

The question of the "physical realization" of such systems is only
relevant to the issue if one can relate it to the perceptual process.
In the ear, the "frequency analysers" are both variable-bandwidth
(with frequency, see Scharf and others) and most incredibly
non-linear, so determining this question is not particularly
easy, unless given a particular signal.  In any case, the point
is that the "physical realization" doesn't matter, what matters
is the effect on perception, something that is the ultimate
"reality" but that is often fabulously unrelated to the "physical
realization" of the source.

>Quoting from Numerical Recipes,

>"What makes the wavelet basis interesting is that, unlike sines and
>cosines, individual wavelet functions are quite localized in space;
>simultaneously, like sines and cosines, individual wavelet functions are
>quite localized in frequency or characteristic scale.  As we
>will see below, the particular kind of dual localization achieved by
>wavelets renders large classes of functions and operators sparse, or sparse
>to some high accuracy, when transformed into the wavelet domain."
This, of course, leaves out MDCT, OBT, LOT, and all sorts of other
systems that can provide a system that is localized in time
GIVEN A PARTICULAR FREQUENCY RESOLUTION.  A wavelet with a given
frequency resolution has the same minimum length as a short-term fourier
spectra providing the same resolution; wavelet properties, like
scaling and translation come in to play in other (quite useful
indeed) ways.  A wavelet does indeed model a simple signal very
well, and in a different way than either fourier mathmatics or
LPC-like mathmatics.   One can indeed generate a wavelet spectrum
but it provides no more convenient interpretation than a multi-resolution
fourier analysis. <about which there is nothing magical>

In any case, as RDP points out, a "transient low frequency"
is an onxymoron. Transients by definition are fast, and
fast events have a wide spectrum, again by definition.

Relating this to the effects of absolute phase are yet ANOTHER
story.  The ear is, after a bunch of filtering, a leading-edge
detector, so there is SOME suggestion that one might be able
to distinguish assymetric waveforms based on when the
signal triggers the inner hair cells, but determining this
is an individual by individual and ear by ear process
at best.

What's also certain is that the best of transducers are going
to do SO much to a strong transient signal that I have great
faith in being able to observe (objectively) the differences in the
spectrum of the positive and negative-going spikes coming
out of most any transducer.  In that sense, systems are phase
sensitive, but you're asking "which distortion do you
prefer" rather than "which is right". In such cases,
taste is all one can rely on.
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