Re: [music-dsp] Instant frequency recognition
Haven't really been following the thread but I wonder if the sinusoid model is really that good. Don't we actually want to match something like SUM(k,1,N) e^jwkt and might not harmonics help us from falling down to the noise floor? On Mon, Aug 4, 2014 at 10:25 AM, Vadim Zavalishin vadim.zavalis...@native-instruments.de wrote: I think it can be done simpler. Just extend the inverse Fourier transform in the same way how the bilateral Laplace transform extends the direct Fourier transform. Any mistake in that reasoning? Regards, Vadim On 02-Aug-14 20:10, colonel_h...@yahoo.com wrote: On Fri, 1 Aug 2014, Vadim Zavalishin wrote: My quick guess is that bandlimited does imply analytic in the complex analysis sense. 1st off, I am fairly sure it is true that a BL signal cannot be zero over an interval, so two non-zero BL signals cannot differ by zero over an interval, so a function with cetain values over any interval is unique, so the rest of this may be cruft... However, an audio signal is most often a real valued function of a real value or a complex valued function of a real value whos imaginary part happens to be zero (often almost interchangably to little ill effect.) So to get an analytic complex function you'd have to extend the function. A non-zero analytic can't have a zero imaginary part, so we'd need a ``new'' imaginary part and to extent the real and imaginary parts to a neighborhood of the real line. Off the cuff I think you might use the real values of f on the real axis as boundary conditions for the Cauchy-Reimann equations in a neighborhood of the real axis to solve for a non-zero imaginary part for f(z) which would then be analytic. This is /if/ BL is enough to show such a solution exists tehn you're done (which I do not claim is false. I just can't see a way to get there.) Ron -- 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
Re: [music-dsp] Instant frequency recognition
SUM(k,1,N) a_k e^jwkt even On Wed, Aug 6, 2014 at 11:00 PM, Emanuel Landeholm emanuel.landeh...@gmail.com wrote: Haven't really been following the thread but I wonder if the sinusoid model is really that good. Don't we actually want to match something like SUM(k,1,N) e^jwkt and might not harmonics help us from falling down to the noise floor? On Mon, Aug 4, 2014 at 10:25 AM, Vadim Zavalishin vadim.zavalis...@native-instruments.de wrote: I think it can be done simpler. Just extend the inverse Fourier transform in the same way how the bilateral Laplace transform extends the direct Fourier transform. Any mistake in that reasoning? Regards, Vadim On 02-Aug-14 20:10, colonel_h...@yahoo.com wrote: On Fri, 1 Aug 2014, Vadim Zavalishin wrote: My quick guess is that bandlimited does imply analytic in the complex analysis sense. 1st off, I am fairly sure it is true that a BL signal cannot be zero over an interval, so two non-zero BL signals cannot differ by zero over an interval, so a function with cetain values over any interval is unique, so the rest of this may be cruft... However, an audio signal is most often a real valued function of a real value or a complex valued function of a real value whos imaginary part happens to be zero (often almost interchangably to little ill effect.) So to get an analytic complex function you'd have to extend the function. A non-zero analytic can't have a zero imaginary part, so we'd need a ``new'' imaginary part and to extent the real and imaginary parts to a neighborhood of the real line. Off the cuff I think you might use the real values of f on the real axis as boundary conditions for the Cauchy-Reimann equations in a neighborhood of the real axis to solve for a non-zero imaginary part for f(z) which would then be analytic. This is /if/ BL is enough to show such a solution exists tehn you're done (which I do not claim is false. I just can't see a way to get there.) Ron -- 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
Re: [music-dsp] Instant frequency recognition
Sorry, meant to say SUM(k,1,N) a_k e^jwk(t+p_k) It would seem that phase should be important, especially if instaneous frequency is desired . On Wed, Aug 6, 2014 at 11:01 PM, Emanuel Landeholm emanuel.landeh...@gmail.com wrote: SUM(k,1,N) a_k e^jwkt even On Wed, Aug 6, 2014 at 11:00 PM, Emanuel Landeholm emanuel.landeh...@gmail.com wrote: Haven't really been following the thread but I wonder if the sinusoid model is really that good. Don't we actually want to match something like SUM(k,1,N) e^jwkt and might not harmonics help us from falling down to the noise floor? On Mon, Aug 4, 2014 at 10:25 AM, Vadim Zavalishin vadim.zavalis...@native-instruments.de wrote: I think it can be done simpler. Just extend the inverse Fourier transform in the same way how the bilateral Laplace transform extends the direct Fourier transform. Any mistake in that reasoning? Regards, Vadim On 02-Aug-14 20:10, colonel_h...@yahoo.com wrote: On Fri, 1 Aug 2014, Vadim Zavalishin wrote: My quick guess is that bandlimited does imply analytic in the complex analysis sense. 1st off, I am fairly sure it is true that a BL signal cannot be zero over an interval, so two non-zero BL signals cannot differ by zero over an interval, so a function with cetain values over any interval is unique, so the rest of this may be cruft... However, an audio signal is most often a real valued function of a real value or a complex valued function of a real value whos imaginary part happens to be zero (often almost interchangably to little ill effect.) So to get an analytic complex function you'd have to extend the function. A non-zero analytic can't have a zero imaginary part, so we'd need a ``new'' imaginary part and to extent the real and imaginary parts to a neighborhood of the real line. Off the cuff I think you might use the real values of f on the real axis as boundary conditions for the Cauchy-Reimann equations in a neighborhood of the real axis to solve for a non-zero imaginary part for f(z) which would then be analytic. This is /if/ BL is enough to show such a solution exists tehn you're done (which I do not claim is false. I just can't see a way to get there.) Ron -- 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
Re: [music-dsp] Instant frequency recognition
I think it can be done simpler. Just extend the inverse Fourier transform in the same way how the bilateral Laplace transform extends the direct Fourier transform. Any mistake in that reasoning? Regards, Vadim On 02-Aug-14 20:10, colonel_h...@yahoo.com wrote: On Fri, 1 Aug 2014, Vadim Zavalishin wrote: My quick guess is that bandlimited does imply analytic in the complex analysis sense. 1st off, I am fairly sure it is true that a BL signal cannot be zero over an interval, so two non-zero BL signals cannot differ by zero over an interval, so a function with cetain values over any interval is unique, so the rest of this may be cruft... However, an audio signal is most often a real valued function of a real value or a complex valued function of a real value whos imaginary part happens to be zero (often almost interchangably to little ill effect.) So to get an analytic complex function you'd have to extend the function. A non-zero analytic can't have a zero imaginary part, so we'd need a ``new'' imaginary part and to extent the real and imaginary parts to a neighborhood of the real line. Off the cuff I think you might use the real values of f on the real axis as boundary conditions for the Cauchy-Reimann equations in a neighborhood of the real axis to solve for a non-zero imaginary part for f(z) which would then be analytic. This is /if/ BL is enough to show such a solution exists tehn you're done (which I do not claim is false. I just can't see a way to get there.) Ron -- 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
Re: [music-dsp] Instant frequency recognition
On Mon, 4 Aug 2014, Vadim Zavalishin wrote: I think it can be done simpler. Just extend the inverse Fourier transform in the same way how the bilateral Laplace transform extends the direct Fourier transform. Any mistake in that reasoning? Basically, allow t to be complex when you inverse transfrom? Hmmm. On 02-Aug-14 20:10, I wrote: A non-zero analytic can't have a zero imaginary part, so we'd need a ``new'' imaginary part This (or what I incompletely expressed) is wrong. Clearly f(z)=z has a zero imaginary part all along the real axis. For a non zero f(z), im(f(z)) can't be zero on an open set in z which would have to have extent in both the real imaginary direction, so a ``normal'' f(z)=f(x)+j0 is fine. Sorry 'bout that. Ron -- 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
Re: [music-dsp] Instant frequency recognition
On Fri, 1 Aug 2014, Vadim Zavalishin wrote: My quick guess is that bandlimited does imply analytic in the complex analysis sense. 1st off, I am fairly sure it is true that a BL signal cannot be zero over an interval, so two non-zero BL signals cannot differ by zero over an interval, so a function with cetain values over any interval is unique, so the rest of this may be cruft... However, an audio signal is most often a real valued function of a real value or a complex valued function of a real value whos imaginary part happens to be zero (often almost interchangably to little ill effect.) So to get an analytic complex function you'd have to extend the function. A non-zero analytic can't have a zero imaginary part, so we'd need a ``new'' imaginary part and to extent the real and imaginary parts to a neighborhood of the real line. Off the cuff I think you might use the real values of f on the real axis as boundary conditions for the Cauchy-Reimann equations in a neighborhood of the real axis to solve for a non-zero imaginary part for f(z) which would then be analytic. This is /if/ BL is enough to show such a solution exists tehn you're done (which I do not claim is false. I just can't see a way to get there.) Ron -- 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
Re: [music-dsp] Instant frequency recognition
On 01-Aug-14 05:22, colonel_h...@yahoo.com wrote: On Fri, 18 Jul 2014, Sampo Syreeni wrote: Well, theoretically, all you have to know is that the signal is bandlimited. When that is the case, it's also analytic, which means that an arbitrarily short piece of it (the analog signal) will be enough to reconstruct all of it as a simple power series. I believe it is true than band limited implies C^infinity, but the function is not complex, so it's a different use of the term analytic than in complex analysis, My quick guess is that bandlimited does imply analytic in the complex analysis sense. This must be a dual of Laplace transform of a bandlimited signal being analytic (entire). Although I could be missing something here. 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
Re: [music-dsp] Instant frequency recognition
Sorry, I meant Laplace transform of a timelimited signal. On 01-Aug-14 10:06, Vadim Zavalishin wrote: On 01-Aug-14 05:22, colonel_h...@yahoo.com wrote: On Fri, 18 Jul 2014, Sampo Syreeni wrote: Well, theoretically, all you have to know is that the signal is bandlimited. When that is the case, it's also analytic, which means that an arbitrarily short piece of it (the analog signal) will be enough to reconstruct all of it as a simple power series. I believe it is true than band limited implies C^infinity, but the function is not complex, so it's a different use of the term analytic than in complex analysis, My quick guess is that bandlimited does imply analytic in the complex analysis sense. This must be a dual of Laplace transform of a bandlimited signal being analytic (entire). Although I could be missing something here. 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
Re: [music-dsp] Instant frequency recognition
On Fri, 18 Jul 2014, Sampo Syreeni wrote: Well, theoretically, all you have to know is that the signal is bandlimited. When that is the case, it's also analytic, which means that an arbitrarily short piece of it (the analog signal) will be enough to reconstruct all of it as a simple power series. I believe it is true than band limited implies C^infinity, but the function is not complex, so it's a different use of the term analytic than in complex analysis, so you don't get the analytic-an arbitrarily short piece tells you everything theorem (for a complex function od a complex variable having one derivite means it has derivaties of all orders at that point. For a real function of a real variable this isn't true. --even if you think of a signal as complex, time is akmost always still real) A real C^infinity function can be zero over an interval (e.g. f(x)=0 x=0, =exp(-1/x) x0 ) so the difference between two C^infinty can be zero over an interval. However, BL is stronger than C^infinity (a gaussian is C^infinity but not BL) so it is possble it could still be true, just your sketch of a proof is incomplte. Ron -- 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
Re: [music-dsp] Instant frequency recognition
On 2014-07-17, Ethan Duni wrote: 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. Well, theoretically, all you have to know is that the signal is bandlimited. When that is the case, it's also analytic, which means that an arbitrarily short piece of it (the analog signal) will be enough to reconstruct all of it as a simple power series. But of course that's the sort of thing you don't actually get to apply in practice. 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. Yes. 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. A typical example would be FM demodulation in software, or e.g. the detector part of a phased locked loop. There you know that you'll be dealing with a signal that is short term sinusoidal at full amplitude, and want to derive an instantaneous differential frequency estimate with as little delay as possible. That's the kind of problem you can solve optimally in closed form under a wide variety of conditions, e.g. using just three successive samples as input, and those kinds of solutions are actually in use. -- Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front +358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2 -- 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
Re: [music-dsp] Instant frequency recognition
Ethan Duni wrote: .. The thing about this approach is that it requires very strong prior knowledge of the signal structure - That reminds me of theories for which the applicability in the end appears to be such that the domain for the solution is the empty set... If you know there's some number of harmonic components, in principle you can play a matrix solution game to find them out, but if the signal is random, even if it is frequency limited, you'll have to factor in all kinds of practical cases which include delays, beating patterns, transients measuring as harmonics, general signal estimation is quite a b*tch unless you go about all the relevant theories and practicalities in a qualitative and quantitative reasonable way, and understand all the connections. T. -- 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
Re: [music-dsp] Instant frequency recognition
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
Re: [music-dsp] Instant frequency recognition
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
Re: [music-dsp] Instant frequency recognition
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
Re: [music-dsp] Instant frequency recognition
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. One such application would be a CV to controls(Midi,OSC,whatever) converter. As virtually all the soundcard inputs are AC coupled, it is often impossible to plug the CV from your ancient analog modular into your soundcard's input. What you can do is connect the output of your analog oscillator (which is a periodic waveform of given amplitude and shape) to the input of the soundcard, track its frequency and use it to generate controls for ... whatever: other oscillators, filters ... Giulio Da: Ethan Duni ethan.d...@gmail.com A: A discussion list for music-related DSP music-dsp@music.columbia.edu Inviato: Venerdì 18 Luglio 2014 2:35 Oggetto: Re: [music-dsp] Instant frequency recognition 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
Re: [music-dsp] Instant frequency recognition
On 16-Jul-14 12:31, Olli Niemitalo wrote: What does O(B^N) mean? -olli This is the so called big O notation. f^(N)(t)=O(B^N) means (for a fixed t) that there is K such that |f^(N)(t)|K*B^N where f^(N) is the Nth derivative. Intuitively, f^(N)(t) doesn't grow faster than B^N Regards, Vadim On Thu, Jul 10, 2014 at 4:02 PM, Vadim Zavalishin vadim.zavalis...@native-instruments.de wrote: Hi all, a recent question to the list regarding the frequency analysis and my recent posts concerning the BLEP led me to an idea, concerning the theoretical possibility of instant recognition of the signal spectrum. The idea is very raw, and possibly not new (if so, I'd appreciate any pointers). Just publishing it here for the sake of discussion/brainstorming/etc. For simplicity I'm considering only continuous time signals. Even here the idea is far from being ripe. In discrete time further complications will arise. According to the Fourier theory we need to know the entire signal from t=-inf to t=+inf in order to reconstruct its spectrum (even if we talk Fourier series rather than Fourier transform, by stating the periodicity of the signal we make it known at any t). OTOH, intuitively thinking, if I'm having just a windowed sine tone, the intuitive idea of its spectrum would be just the frequency of the underlying sine rather than the smeared peak arising from the Fourier transform of the windowed sine. This has been commonly the source of beginner's misconception in the frequency analysis, but I hope you can agree, that that misconception has reasonable foundations. Now, recall that in the recent BLEP discussion I conjectured the following alternative definition of bandlimited signals: an entire complex function is bandlimited (as a function of purely real argument t) if its derivatives at any chosen point are O(B^N) for some B, where B is the band limit. Thinking along the same lines, an entire function is fully defined by its derivatives at any given point and (therefore) so is its spectrum. So, we could reconstruct the signal just from its derivatives at one chosen point and apply Fourier transform to the reconstructed signal. In a more practical setting of a realtime input (the time is still continuous, though), we could work under an assumption of the signal being entire *until* proven otherwise. 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. Now suppose that instead of sinusoidal signals we get a sawtooth. In the beginning we detect just a linear segment. This is an entire function, but of a special class: its derivatives do not fall off smoothly as O(B^N), but stop immediately at the 2nd derivative. From the BLEP discussion we know, that so far this signal is just a generalized version of the DC offset, thus containing only a zero frequency partial. As the sawtooth transition comes we can detect the discontinuity in the signal, therefore dropping the assumption of an entire signal and use some other (yet undeveloped) approach for the short-time frequency detection. Any further thoughts? 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 -- 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
Re: [music-dsp] Instant frequency recognition
I see, so the limiting case that turns the inequality to an equality is a sinusoid (or a corresponding complex exponential). If the signal is band-limited, it must be a bounded sum of those, and the derivatives must thus also be bounded sums of derivatives of those, and your criterion will be satisfied. At least that part of your theory seems consistent. -olli On Wed, Jul 16, 2014 at 1:39 PM, Vadim Zavalishin vadim.zavalis...@native-instruments.de wrote: On 16-Jul-14 12:31, Olli Niemitalo wrote: What does O(B^N) mean? -olli This is the so called big O notation. f^(N)(t)=O(B^N) means (for a fixed t) that there is K such that |f^(N)(t)|K*B^N where f^(N) is the Nth derivative. Intuitively, f^(N)(t) doesn't grow faster than B^N Regards, Vadim On Thu, Jul 10, 2014 at 4:02 PM, Vadim Zavalishin vadim.zavalis...@native-instruments.de wrote: Hi all, a recent question to the list regarding the frequency analysis and my recent posts concerning the BLEP led me to an idea, concerning the theoretical possibility of instant recognition of the signal spectrum. The idea is very raw, and possibly not new (if so, I'd appreciate any pointers). Just publishing it here for the sake of discussion/brainstorming/etc. For simplicity I'm considering only continuous time signals. Even here the idea is far from being ripe. In discrete time further complications will arise. According to the Fourier theory we need to know the entire signal from t=-inf to t=+inf in order to reconstruct its spectrum (even if we talk Fourier series rather than Fourier transform, by stating the periodicity of the signal we make it known at any t). OTOH, intuitively thinking, if I'm having just a windowed sine tone, the intuitive idea of its spectrum would be just the frequency of the underlying sine rather than the smeared peak arising from the Fourier transform of the windowed sine. This has been commonly the source of beginner's misconception in the frequency analysis, but I hope you can agree, that that misconception has reasonable foundations. Now, recall that in the recent BLEP discussion I conjectured the following alternative definition of bandlimited signals: an entire complex function is bandlimited (as a function of purely real argument t) if its derivatives at any chosen point are O(B^N) for some B, where B is the band limit. Thinking along the same lines, an entire function is fully defined by its derivatives at any given point and (therefore) so is its spectrum. So, we could reconstruct the signal just from its derivatives at one chosen point and apply Fourier transform to the reconstructed signal. In a more practical setting of a realtime input (the time is still continuous, though), we could work under an assumption of the signal being entire *until* proven otherwise. 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. Now suppose that instead of sinusoidal signals we get a sawtooth. In the beginning we detect just a linear segment. This is an entire function, but of a special class: its derivatives do not fall off smoothly as O(B^N), but stop immediately at the 2nd derivative. From the BLEP discussion we know, that so far this signal is just a generalized version of the DC offset, thus containing only a zero frequency partial. As the sawtooth transition comes we can detect the discontinuity in the signal, therefore dropping the assumption of an entire signal and use some other (yet undeveloped) approach for the short-time frequency detection. Any further thoughts? 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 -- 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
Re: [music-dsp] Instant frequency recognition
On Thu, 10 Jul 2014, Vadim Zavalishin wrote: From the BLEP discussion we know, that so far this signal is just a generalized version of the DC offset, thus containing only a zero frequency partial. The ``no new partials'' rule comes from integral exp(kx) = exp(kx)/k so integration just scales the exponentials but the formula isn't true for k=0 so the rule doesn't work for constants, which are naughty. Ron -- 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
[music-dsp] Instant frequency recognition
Hi all, a recent question to the list regarding the frequency analysis and my recent posts concerning the BLEP led me to an idea, concerning the theoretical possibility of instant recognition of the signal spectrum. The idea is very raw, and possibly not new (if so, I'd appreciate any pointers). Just publishing it here for the sake of discussion/brainstorming/etc. For simplicity I'm considering only continuous time signals. Even here the idea is far from being ripe. In discrete time further complications will arise. According to the Fourier theory we need to know the entire signal from t=-inf to t=+inf in order to reconstruct its spectrum (even if we talk Fourier series rather than Fourier transform, by stating the periodicity of the signal we make it known at any t). OTOH, intuitively thinking, if I'm having just a windowed sine tone, the intuitive idea of its spectrum would be just the frequency of the underlying sine rather than the smeared peak arising from the Fourier transform of the windowed sine. This has been commonly the source of beginner's misconception in the frequency analysis, but I hope you can agree, that that misconception has reasonable foundations. Now, recall that in the recent BLEP discussion I conjectured the following alternative definition of bandlimited signals: an entire complex function is bandlimited (as a function of purely real argument t) if its derivatives at any chosen point are O(B^N) for some B, where B is the band limit. Thinking along the same lines, an entire function is fully defined by its derivatives at any given point and (therefore) so is its spectrum. So, we could reconstruct the signal just from its derivatives at one chosen point and apply Fourier transform to the reconstructed signal. In a more practical setting of a realtime input (the time is still continuous, though), we could work under an assumption of the signal being entire *until* proven otherwise. 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. Now suppose that instead of sinusoidal signals we get a sawtooth. In the beginning we detect just a linear segment. This is an entire function, but of a special class: its derivatives do not fall off smoothly as O(B^N), but stop immediately at the 2nd derivative. From the BLEP discussion we know, that so far this signal is just a generalized version of the DC offset, thus containing only a zero frequency partial. As the sawtooth transition comes we can detect the discontinuity in the signal, therefore dropping the assumption of an entire signal and use some other (yet undeveloped) approach for the short-time frequency detection. Any further thoughts? 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
