Yeah, that's basically the chirp decomposition I was referring to earlier. I.e., if you can write the signal as (for example) A*cos(f(t)), then you can take the derivative of f(t) and call the resulting function the "instantaneous frequency," in a well-defined, meaningful way. But that only works for a certain constrained class of signals. It runs into several fundamental problems if you try to apply it to a generic signal. And even if you solve those, you're in all cases using data segments much longer than any of the fundamental periods in question, in order to do the estimation.
But that's not what the OP in this thread was suggesting. The idea was to do frequency estimation on arbitrarily short time segments: > Particularly, if we get a mixture of > several static sinusoidal signals, they all will be properly restored from > an arbitrarily short fragment of the signal. That is not the same thing as instantaneous frequency in the chirp sense. The idea is to estimate a *fixed* frequency from a very short time fragment. The thing about this approach is that it requires very strong prior knowledge of the signal structure - to the point of saying quite a lot about how it behaves over all time - in order to work. I.e., if you have a signal that you know is a sine wave with given amplitude and phase, you can work out its frequency from a very short length of time. But that's only because you have very strong prior knowledge that relates the behavior of the signal in any short time period to the behavior of the signal over all time. I guess my point is that I'm struggling to think of an application where such strong prior knowledge exists, and where we'd still need to estimate frequencies from data. E On Thu, Jul 17, 2014 at 4:19 PM, zhiguang e zhang <ericzh...@gmail.com> wrote: > This post explains the concept instantaneous frequency well: (It is > basically used to distinguish amplitude from phase) > > > http://math.stackexchange.com/questions/85388/does-the-phrase-instantaneous-frequency-make-sense > > EZ > On Jul 17, 2014, at 6:40 PM, Ethan Duni <ethan.d...@gmail.com> wrote: > > >> Sinc interpolation would be theoretically correct, but, remember, > >> that this thread is not about "strictily theoretically correct" > frequency > >> recognition, but rather about some "more intuitive" version with the > >> concept of "instant frequency". > > > > What is "instant frequency?" I have to say that I find this concept to be > > highly counter-intuitive on its face. How can we speak meaningfully about > > frequency on time scales shorter than one period? > > > >> Maybe we could attempt exactly fitting a set of samples into a sum > >> of sines of different frequencies? Each sine corresponding to 3 degrees > >> of freedom. > > > > Yeah, this is called sinusoidal modeling. But I don't see how it give you > > any handle on "instantaneous frequency." If you're operating on time > scales > > shorter than the periods of the frequencies in question, then the basis > > functions you're using in sinusoidal modeling do not exhibit any > meaningful > > periodicity, but instead look something like low-order polynomials. The > > frequency parameters you'd estimate would be meaningless as such - they'd > > jump around all over the place from frame to frame, depending on exactly > > how the frame being analyzed lined up with the basis functions. I.e., > > they'd just be abstract parameters specifying some > low-order-polynomial-ish > > shapes, and not indicating anything meaningful about periodicity. > > > > The only way I can see to speak meaningfully about instantaneous > frequency > > is if you were to decompose a signal with some kind of chirp basis - on a > > time scale much longer than any particular period seen in the chirp > basis. > > Then you could turn around and say that the frequency is evolving > according > > to the chirp parameter, and talk about an instantaneous frequency at any > > particular time. But not that this requires doing the analysis on a > rather > > *long* time scale, so you can be confident that the chirp structure > you're > > finding actually corresponds to some real signal content. > > > > E > > > > > > On Thu, Jul 17, 2014 at 1:30 AM, Vadim Zavalishin < > > vadim.zavalis...@native-instruments.de> wrote: > > > >> On 16-Jul-14 15:29, Olli Niemitalo wrote: > >> > >>> Not sure if this is related, but there appears to be something called > >>> "chromatic derivatives": > >>> > >>> http://www.cse.unsw.edu.au/~ignjat/diff/ > >>> > >> > >> Seems pretty much related and going further in the same direction > >> (alright, I just briefly glanced at chromatic derivatives). Anyway, it > >> seems that for the discrete time signals the situation is somewhat > >> different from what I described for continuous time in that there are no > >> derivative discontinuities for discrete time signals. At the same time > it's > >> not possible to locally compute the derivatives of the discrete time > >> signals, so the local Taylor expansion idea is not applicable anyway > (the > >> same applies to the the chromatic derivatives, I'd guess). However, > >> instead, we could simply apply the inter-/extra-polation to the obtained > >> sample points. The most intuitive would be applying Lagrange > interpolation, > >> which as we know, converges to the sinc interpolation. However (again, > >> remember the BLEP discussion), any finite order polynomial contains only > >> the generalized DC component. Not very useful for the frequency > estimation. > >> So, the question is, what kind of interpolation should we use? Sinc > >> interpolation would be theoretically correct, but, remember, that this > >> thread is not about "strictily theoretically correct" frequency > >> recognition, but rather about some "more intuitive" version with the > >> concept of "instant frequency". Maybe we could attempt exactly fitting a > >> set of samples into a sum of sines of different frequencies? Each sine > >> corresponding to 3 degrees of freedom. > >> > >> > >> Regards, > >> Vadim > >> > >> -- > >> Vadim Zavalishin > >> Reaktor Application Architect > >> Native Instruments GmbH > >> +49-30-611035-0 > >> > >> www.native-instruments.com > >> -- > >> dupswapdrop -- the music-dsp mailing list and website: > >> subscription info, FAQ, source code archive, list archive, book reviews, > >> dsp links > >> http://music.columbia.edu/cmc/music-dsp > >> http://music.columbia.edu/mailman/listinfo/music-dsp > >> > > -- > > dupswapdrop -- the music-dsp mailing list and website: > > subscription info, FAQ, source code archive, list archive, book reviews, > dsp links > > http://music.columbia.edu/cmc/music-dsp > > http://music.columbia.edu/mailman/listinfo/music-dsp > > -- > dupswapdrop -- the music-dsp mailing list and website: > subscription info, FAQ, source code archive, list archive, book reviews, > dsp links > http://music.columbia.edu/cmc/music-dsp > http://music.columbia.edu/mailman/listinfo/music-dsp > -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
