Theo wrote: >I get there are certain statistical ideas involved. I wonder >however where those ideas in practice lead to, because >of a number of assumptions, like the "statistical variance" >of a signal. I get that a self correlation of a signal in some >normal definition gives an idea of the power, and that you >could take it that you compute power per frequency band. >But what does it mean when you talk about variance ?
>Of course to determine a statistical measure about a spectrum, >either based on sampled signals or (where the analysis comes >from and is only generally correct for signal from - to + inf) on a >continuous signal, and based either on a Fourier integral/summation >or a Fast Fourier analysis (with certain analysis length and frequency >bin accuracy), you could use the general big numbers theorem and >presume there's a mean and a variance. It would be nice to at least > make credible why this is an ok analysis, because a lot of signals are >far from Gaussian distributed in the sense of the frequency spectrum. So we are employing prob/stat terms like mean and variance, and normalizing the power spectrum so that it looks like a probability density. However, this is only a matter of semantics, we are not required to actually treat the signals in question as random. The whole thing works the same way regardless of whether we apply it to the power spectral density of a random signal, or the power spectrum of a deterministic signal (which is what we've been doing so far here). The goal is to find some simple features of the spectrum that capture something about how "bright" it is - so the center of mass of the spectrum, and maybe also its spread. Then we can compare these features to make estimates of whether one signal is "brighter" than another, for example. This is not required to be a complete characterization of the spectrum in question - as you note, absent some other assumption like Gaussianity, the first two moments will not be sufficient to completely characterize it. It's only supposed to give us some (hopefully) meaningful indication of certain broad properties of the spectrum. The hope would be that two (different) spectra with the same first moment will have similar "brightness," and so that statistic is sufficient to capture the property in question. These are simply features of a power spectrum, much like familiar quantities of bandwidth, peak level, transition width, etc. They do admit a prob/stat interpretation, which is interesting but secondary to the primary motivation here. E On Thu, Feb 25, 2016 at 11:04 AM, Theo Verelst <theo...@theover.org> wrote: > Evan Balster wrote: > >> ... >> >> To that end: A handy, cheap algorithm for approximating the >> power-weighted spectral >> centroid -- a signal's "mean frequency" -- which is a good heuristic for >> perceived sound >> brightness <https://en.wikipedia.org/wiki/Brightness#Brightness_of_sounds>. >> In spite of >> its simplicity, ... >> > Hi, > > Always interesting to learn a few more tricks, and thanks to Ethan's > explanation I get there are certain statistical ideas involved. I wonder > however where those ideas in practice lead to, because of a number of > assumptions, like the "statistical variance" of a signal. I get that a self > correlation of a signal in some normal definition gives an idea of the > power, and that you could take it that you compute power per frequency > band. But what does it mean when you talk about variance ? Mind you I know > the general theoretics up to the quantum mechanics that worked on these > subjects long ago fine, but I wonder what the understanding here is? > > Some have remarked about the analysis of a signal into ground frequency > and harmonics that it might be hard to summarize and make an ordinal > measure for "brightness" as a one dimensional quantity, I mean of you look > at a number of peaks in a frequency graph, how do you sum up the frequency > of the signal, if there is one, and the meaning of the various harmonics in > the spectrum, if they are to be taken as a measure of the brightness? So a > trick is fine, though I do not completely understand the meaning of a > brightness measure for frequency analysis. > > Of course to determine a statistical measure about a spectrum, either > based on sampled signals or (where the analysis comes from and is only > generally correct for signal from - to + inf) on a continuous signal, and > based either on a Fourier integral/summation or a Fast Fourier analysis > (with certain analysis length and frequency bin accuracy), you could use > the general big numbers theorem and presume there's a mean and a variance. > It would be nice to at least make credible why this is an ok analysis, > because a lot of signals are far from Gaussian distributed in the sense of > the frequency spectrum. > > T. > > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp > >
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