wavelet effect: clarifying ("R.Rossi.A.Faria" )

Subject: wavelet effect: clarifying
From:    "R.Rossi.A.Faria"  <regis(at)LSI.USP.BR>
Date:    Wed, 19 Mar 1997 21:24:40 -0300

I'd like to apologize for the gargage and bad formatting on my earlier message. Yes it was written on a word processor. Hope it's better now. I am not properly an wavelet expert (neither a Fourier's one) but let's go.. Gregory J. Sandell wrote: > > I'd like to ask Mr. Faria to clarify a few details of his interesting > query. > > The wavelet approach has several advantages over normal Fourier filtering > > since its filters have local support both in time and frequency, making it > > easy to locate transients on some frequency bands. > > When you say "normal Fourier filtering" I assume you mean (1) a method such as > the Phase Vocoder, where filters are typically allocated one-per-harmonic > of an estimated fundamental, but not (2) a method such as McCauley&Quatieri > where an arbitrary number of filters are allocated dynamically according to > the signal. Is that true? When I mentioned normal Fourier filtering I just meant to include under that those methods developed from and based upon the Fourier's spectral representations of signals, where frequencies are accessed individually. Normal Fourier filtering or transforming methods are then those methods whose mathematics treat frequency by frequency, where there is a frequency variable. By the other hand, behind wavelet transform is the notion of "band of frequencies" (or scale instead of frequency), that is, methods based on wavelet filtering or so would process bands of frequencies, opposed to the notion of processing frequencies one by one, individually assessing each one (as is implied in the Fourier series or transform formulae). Wavelet trasformed domains have coeficcients which do not simply represent a frequency component, as with Fourier coefficients on the transformed domain. As far as I know from the basics, Phase Vocoder functioning may be interpreted both as a filter bank analysis representation (where each filter can be allocated to an harmonic) or as a short-time Fourier analysis of successive intervals. However, while addressing individual frequencies (as short time Fourier transform does) it goes under my "normal Fourier filtering" terminology. I am not acquainted with McCauley&Quatieri method to comment about it. > Transients consist of non-harmonic information, and method (1) smears > non-harmonic information over several bands. Then is what you are > saying that wavelets more accurately assigns them to appropriate > frequency bands? Yes. > > Another advantage > > of wavelet filtering is its property of separating bands with quality > > factor (Q) constant over the frequency axis, in a way the basilar membrane > > in > > the inner ear also resolves frequency bands. This property makes wavelets > > closer to ear's acoustic pre-processing, on stages before neural > > processing. > > Is the nature of the advantage that (1) a person examining wavelet > analysis output is observing "how the ear experiences the sound", or (2) > the method is more economical by ignoring detail that goes beyond > the resolving properties of the ear, or (3) both? I believe the right answer is (3) both, but it demands a little more practical experimentation to fully affirm that. One important aspect, however, is to limit the action range of the expression "how the ear experiences the sound", because "wavelet analysis" is performed at the early stages of the auditory system, say, the "hardware" which transmits and tranduces the acoustic stimuli to the hair cells. > > (1) the result of mixing the original tone with a reconstruction of the sound > > from its wavelet coefficients (obtained in the forward transform) taking only > > the coefficients in level n and "clamping" other coefficients (from all other > > levels) to zero value (this is reconstructing only the level n and mixing it > > to the original sound). > > Can you clarify what "in level n" means? Do you mean "only those > coefficients whose value is n or greater? Wavelet time-frequency analysis is sometimes referred to as time-scale analysis. This is because it extracts information from a signal over different scales of resolution, adding details as you step to a finer resolution scale. Many researchers often explain this property using the geographical maps : if you want to examine details you need a map with higher resolution, i.e, lower scale, while higher scales give you a visual of the whole, withou details. Level n in wavelet terminology would be in scale n, which covers an specific band of frequencies. When I mean "level n" I mean only level n coefficients. Hope having clarified a bit. Regis Rossi A. Faria Computer Music Group Laboratorio de Sistemas Integraveis - LSI University of Sao Paulo Brazil http://www.lsi.usp.br/~regis/regis.html

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