From: "Ethan Duni" <ethan.d...@gmail.com> Date: Thu, February 18, 2016 4:48 pm -------------------------------------------------------------------------- > I've noticed > in my (cursory) searches that some people use amplitude spectra and others > use power spectra, but the only thing I've found in the way of comparison > tests was to do with whether it gets normalized by fundamental frequency or > not. � i haven't even found that in the lit. �which is why i was interested when Evan brought this topic up. � > Let's start in continuous time, with some real signal x(t) with FT X(w). > Recall the differentiation property, d/dt x(t) <=> jwX(w). Next, let's use > Parseval's theorem (ignoring the normalization constants because they'll > cancel out later): > > integral( |x(t)|^2 dt) = integral( |X(w)|^2 dw), and likewise integral( > |d/dt x(t)|^2 dt) = integral( |w|^2 |X(w)|^2 dw). > > Thus, the ratio of the time-domain integrals gives: > > integral( |d/dt x(t)|^2 dt)/integral( |x(t)|^2 dt) = integral( |w|^2 > |X(w)|^2 dw)/integral( |X(w)|^2 dw) > > I.e., if we run a differentiator, then compute the ratio of the power in > that to the power in the original signal, the result is the second moment > of the (normalized) power spectrum. � it's "second moment" because both positive and negative frequencies are used. � > This corresponds to the system Evan > proposed in the OP, without the later square root modification. So that's > something, but presumably we want to get the *first* moment of the > normalized power spectrum. � the first moment is 0. due to the symmetry of what we're looking at. � but i think that we were supposed to be integrating only positive values of w. �and then this centroid becomes more like a mean, not so much a variance. > One option is to replace the differentiator with an inverse pinking filter, > as rbj suggested. Are there any good references on design of inverse > pinking filters? > same as the old standby:�http://www.firstpr.com.au/dsp/pink-noise/� � but swap the poles and the zeros. � > Another option is to stick some square roots on these quantities, as Evan > suggested in a subsequent post. But moving those through the integrals > means, according to Jensen's inequality, that we get an over-estimate of > the first moment of the normalized power spectrum. How big the > overestimation is depends on the shape of the spectrum, but this may well > be quite usable regardless and should be substantially cheaper than the > inverse pinking filter approach. > > Next let's consider how this would work in discrete time. Naively, we might > simply replace the differentiator with a first difference. Recall the > relevant DTFT property: x[n] - x[n-1] <=> (1-e^(-jw))X(w). This gets us the > graph and explanation that Evan provided in the OP: for sufficiently small > values of w, it is approximately linear, so we can simulate the > continuous-time case via oversampling. We could also add a high frequency > compensation filter, or again, just replace the difference/sqrt() approach > with an inverse pinking filter designed according to whatever criteria. > > Are we all on the same page with this analysis so far? > > I notice that various sources define spectral centroid in terms of > amplitude spectrum, rather than power spectrum. This makes the analysis > more difficult, since we can't rely on Parseval's theorem directly. But > this is part of why I asked what the consensus is on definitions - is it > worth analyzing, or is it just something people do when using FFT based > methods, without much further thought on the alternatives? � i like power or magnitude-square more than just magnitude. �we can do Parseval on it. �or lot'sa other calculus. � -- r b-j � � � � � � � � �r...@audioimagination.com � � � "Imagination is more important than knowledge." �
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