Re: [music-dsp] Sampling theorem extension
Now that I read up on it... Actually no. Every tempered distribution has a Fourier transform, and if that's compactly supported, the original distribution can be reconstructed via the usual Shannon-Whittaker sinc interpolation formula. That also goes for polynomials and sine modulated polynomials in the continuous domain. Whatever that means in general. Right, that makes intuitive sense. I guess what we lose is the model of sampling as multiplication by a stream of delta functions, but that is more of a pedagogical convenience than a basic requirement to begin with. But what does the convergence of the Shannon-Whittaker formula look like in the case of stuff like polynomials? In the usual setting we get nice results about uniform local convergence, but that requires the asymptotic behavior of the signal being sampled to behave nicely. In a case where it's blowing up at polynomial rate, it seems intuitively that there could be quite strong dependencies on samples far removed in time from any particular region. So the concern would be that it works fine for the ideal sinc interpolator, but could fall apart quite badly for realizable approximations to that. E On Fri, Jun 19, 2015 at 12:49 PM, Sampo Syreeni de...@iki.fi wrote: On 2015-06-12, Ethan Duni wrote: Thanks for expanding on that, this is quite interesting stuff. However, if I'm following this correctly, it seems to me that the problem of multiplication of distributions means that the whole basic set-up of the sampling theorem needs to be reworked to make sense in this context. Now that I read up on it... Actually no. Every tempered distribution has a Fourier transform, and if that's compactly supported, the original distribution can be reconstructed via the usual Shannon-Whittaker sinc interpolation formula. That also goes for polynomials and sine modulated polynomials in the continuous domain. Whatever that means in general. No? Yes. While the formalism apparently goes through, I don't have the slightest idea of how to interpret that wrt the usual L^2 theory. I can sort of get that the polynomial-to-series-of-delta-derivatives duality works as it should, and via the Schwartz Representation Theorem captures the asymptotic growth of tempered distributions. But how you'd utilize that in DSP or with its oversampling problems is thus far beyond me. -- 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 -- 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] Sampling theorem extension
On 2015-06-19, robert bristow-johnson wrote: i thought that, because of my misuse of the Dirac delta (from a mathematician's POV, but not from an EE's POV), i didn't think that the model of sampling as multiplication by a stream of delta functions was a living organism in the first place. i thought, from the mathematician's POV, we had to get around this by using the Poisson summation formula [...] In the framework of tempered distributions, all of that follows as well. You can actually do with Dirac deltas what you'd like to do, and what seems natural. Pretty much the only thing you can't do is freely multiply two distributions together, unless they're not just distributions, but functions as well. Convolve if one of the distributions has compact support, or you land within the conventional L_2 theory, or something like that... But otherwise, you can do the funkiest shit. Nota bene, this is not EE stuff per se. This is heady math stuff, used to formalize what you EEs wanted to do all along. It's the kind of collaboration where us math freaks provide the rubber...and then you EE folks can finally fuck your sister in peace and certainty. ;) (Sorry, can't help it, been looking at a lot of stand up comedy of late...) -- 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
[music-dsp] A brief history of Reverb
Sean Costello of Valhalla DSP recently presented at the Seattle chapter of AES with some interesting info on Reverberation. https://valhalladsp.wordpress.com/2015/06/19/slides-from-my-aes-reverb-presentation/ Doesn't tell you *how* to do it (there are many ways), but it's an interesting story. (via Sean's twitter feed) Eric -- 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] Sampling theorem extension
On 2015-06-19, Ethan Duni wrote: I guess what we lose is the model of sampling as multiplication by a stream of delta functions, but that is more of a pedagogical convenience than a basic requirement to begin with. In fact even that survives fully. In the local integration framework that the tempered distributions carry with them, you can convolve a polynomial by a delta function or any finite derivative of it, and you can also apply a Dirac comb to it so as to sample it. But what does the convergence of the Shannon-Whittaker formula look like in the case of stuff like polynomials? Precisely the same as it does in the case of any other function. You just have to take the convergence in the weak* sense, and then do some extra legwork to return that into a function, from the functional domain. What it returns to is precisely the unique polynomial (or whatnot) you're after. The reconstruction formula, using sinc functions, is exact in that circuitous sense. In the usual setting we get nice results about uniform local convergence, but that requires the asymptotic behavior of the signal being sampled to behave nicely. In a case where it's blowing up at polynomial rate, it seems intuitively that there could be quite strong dependencies on samples far removed in time from any particular region. So the concern would be that it works fine for the ideal sinc interpolator, but could fall apart quite badly for realizable approximations to that. All that is taken care of by the fact that the reconstruction is defined as a transposition of a functional wrt the Schwartz space to begin with. All the mechanics are local because of that. The asymptotics don't matter after that, and the Shannon-Whittaker formula is suddenly defined locally, so that growth rates upto polynomial don't matter at all. Of course some funky global, dual shit happens then: you actually need all of the samples from -inf to +inf in order to define any polynomial, and no finitely supported in time subset will suffice. -- 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] Sampling theorem extension
On 6/19/15 5:03 PM, Sampo Syreeni wrote: On 2015-06-19, Ethan Duni wrote: I guess what we lose is the model of sampling as multiplication by a stream of delta functions, but that is more of a pedagogical convenience than a basic requirement to begin with. pedagogical convenience, schmedagogical convenience... :-) In fact even that survives fully. In the local integration framework that the tempered distributions carry with them, you can convolve a polynomial by a delta function or any finite derivative of it, and you can also apply a Dirac comb to it so as to sample it. i thought that, because of my misuse of the Dirac delta (from a mathematician's POV, but not from an EE's POV), i didn't think that the model of sampling as multiplication by a stream of delta functions was a living organism in the first place. i thought, from the mathematician's POV, we had to get around this by using the Poisson summation formula ( https://en.wikipedia.org/wiki/Poisson_summation_formula ) to properly understand uniform sampling and that any manipulation of the naked Dirac delta (like adding up a string of them to make a Dirac comb) outside of the superficial representation with the integral (the sampling property of the Dirac impulse) was illegit. a naked Dirac delta unclothed by an integral is not legit (according to the priesthood of mathematicians) and, even when wearing the clothing of an integral, it's really just a functional that maps x(t) to x(0). personally, i have no problem with the pedantic manner that EEs are used to using the Dirac impulse (a.k.a. the local integration framework). works for me. and i can understand other issues (like the dimensional analysis of impulses and impulse responses) better, more directly, with the pedantic POV of the Dirac. -- r b-j r...@audioimagination.com Imagination is more important than knowledge. -- 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] Sampling theorem extension
On 2015-06-19, Ethan Duni wrote: We theoretically need all samples from -inf to +inf in the regular sampling theorem as well, [...] Not exactly. If you take the typical sampling formula, with equidistant samples, you need them all. But in theory pretty much any numerable number of samples from any compact interval will do. I'm not 100% certain, but with polynomials in the distributional setting, I think you'll actually need -inf to +inf in some sense (equidistant sampling being sufficient but probably not quite necessary), despite the bandlimitation which usually makes the function rigid enough to be analytic, whole, and so resamplable+continuabe from pretty much whatever you have at hand. This happens basically because the sinc function dies off linearly [...] Linearly? It dies off as 1/x. And that's part of the magic. You see: -1/x dominates any decaying exponential, being in a sense their limit -exp(x) dominates any monomial, being in a sense their limit -log(x) dominates any root, being in a sense their limit -there's a fourth one, plus some integral equalities, here This stuff basically delimits in real terms the Schwartz space used to construct tempered distributions. It also delimits the L_p spaces. The fact that the 1/x growth rate is the limit of decaying exponentials and that we go through the weak* topology of the dual space is somehow the reason why we can pass to the 1/x limit of the Shannon-Whittaker interpolation formula, both in the simpler L_2 theory and in the more general distributinal framework. And it's somehow clearly the reason why you can't have but polynomial growth in (tempered) distributions. I don't understand this stuff fully myself, yet, but it's evidently there. So the limiting growth rate of the sinc function cannot be an accident. I think it comes from the dominating real convergence rate of any polynomially bounded tempered distribution, when approximated via milder distributions in the weak* topology. [...] and we are dealing with signals with at most constant-ish asymptotic behavior - so the contribution of a given sample to a given reconstruction region is guaranteed to die off as you get farther away from the region in question. Not quite so. The proper way to say it is when probed locally by nice enough test functions, the reconstruction works the same. That's a bitch because some functions within the space of tempered distributions can be plenty weird. The main counter example I've found is f(x)=sin(x*e^x). That's bounded and continuous, so it induces a well behaved tempered distribution. Then we know that every derivative of a tempered distribution is also a tempered distribution. g(x)=f'(x)=cos(x*e^x)*D(x*e^x)=e^x*cos(x*e^x). That doesn't look polynomially growing at *all*, yet it's part of the space. (The reason is its fast oscillation while it grows.) So for any finite delay, we can get a finite error bound on the reconstruction. But in the case of a polynomial it seems to me that the reconstruction in a given region (around t=0 say) could depend very strongly on samples way off at t = +- 1,000,000,000, since the polynomial is eventually going to be growing faster than the sinc is shrinking. That's the problem: the local integration theory we use with the distributions doesn't work with your usual error metrics or notions of convergence. This sort of argument is meaningless there. What you need to do is bring in the whole set of test functions, in order to construct a nice functional, and then show it can be induced by a function which doesn't integrate in the normal sense against any L_2 function, say. So I'm not seeing how we can get any error bounds for causal, finite-delay approximations to the ideal reconstruction filter in the polynomial case. You'll have to go via the functional transposition operator. We also need the property that the reconstruction can be approximated with realizable filters in a useful way. The sinc convolution is just fine even in this setting. It's just that we just happened to prove its workability in a slightly more general setting. And yes, that blows my mind, too. :) -- 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] Sampling theorem extension
On 2015-06-19, Ethan Duni wrote: Not exactly. If you take the typical sampling formula, with equidistant samples, you need them all. Yeah, that's what we're discussing isn't it? Are we? You can approximate any L_2 bandlimited function arbitrarily closely with a finite number of samples. I don't think you can even approach a polynomial in the distributional sense absent the whole infinite set. But in theory pretty much any numerable number of samples from any compact interval will do. Sure, but that's not going to help us with figuring out what comes out of an audio DAC. Yes. And I'm sorry if I sound of like a know-it-all or show-off here. I really am anything *but*. Just interested in this stuff. :) Linearly? It dies off as 1/x. Yeah that's what I mean. Kind of informal, but die off was meant to imply this is what is in the denominator. Check. But 1/x is still pretty special in the denominator. Not quite so. The proper way to say it is when probed locally by nice enough test functions, the reconstruction works the same. I'm not sure we're on the same page here - the statement you were replying to was referring to the classical L2 sampling theorem stuff. If so, again sorry. I have tried to work as much in the distributional setting as I can. The sinc convolution is just fine even in this setting. ??? The sinc convolution is not implementable in any setting. It actually is in the distributional setting. When you go via the weak* topology of the relevant functional space, the functions they induce back implement pure sinc interpolation. The limit is exact. But yeah, in *reality* nothing of the sort can exist. You just have to approximate. It's just that there's nothing new there for any of us, I think. Delta-sigma, yadda-yadda, it's what them chips do all the time for us. Right? ;) -- 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
[music-dsp] [ot] math vs. EE
On 2015-06-19, robert bristow-johnson wrote: we EEs are fucking our sisters when we say that there *is* a function that is zero almost everywhere, yet has an integral of 1. (but when we take the rubber off, we find out that it's a distribution, not a function in the normal sense that one might recognize in anatomy class.) Us wannabe-math-freaks play with things like the by-now classical Cantor function. Continuous, monotonically increasing from 0 to 1, almost everywhere differentiable, with zero slope there. https://en.wikipedia.org/wiki/Cantor_function ...and then we *like* that shit. Cool sister bedamned. ;) -- 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] FPGA or SHARC Programmer wanted
OK, I will bite first. I should point out that I am a SHARC partisan with many, many designs behind me. That said, I have also worked with FPGAs. Modern DSPs such as the SHARC have become increasingly more powerful. Moore's law has applied to them as well as their more general purpose cousins. Assuming a 450M clock, you average about 9375 instructions per 48k sample. This is before you consider SIMD (x2) or multiple cores (for example ADSP-SC589). This can be a lot of instructions to create very powerful platforms. In a SHARC you have the opportunity to use either fixed point or floating point. Depending on your algorithm requirements, this may simplify your programming tasks considerably. You also have built in peripheral support for interfacing to external devices, for example UARTs for Midi or SPI/SPORTs to data converters. In general, I think you will find software development considerably easier in a SHARC (or any other DSP chip) In the FPGA case, you have the opportunity for fast parallelism. You could have many multipliers working at the same time. The catch is you have to create your algorithm in a much more down to the metal approach. FPGA suppliers have gotten better at DSP libraries and tool support but it is still much harder to program these devices. If you are building a product where all algorithms are going to be written in house by a dedicated staff, this may be acceptable. I don't think it will be very practical for anyone who wants to support a more open architecture. Sometimes a mixed approach is a better idea. You can interface a DSP FPGA as a pair. In this case the DSP is usually the master and the FPGA is a coprocessor. The FPGA can also be used as I/O expansion. The FPGA might just run a specific algorithm that is ultimately called by the DSP. There is also a trend to marry ARM cores to both FPGAs and DSPs. The Altera SoC and Xilinx Zynq are FPGA examples. The Analog Devices ADSP-SC589 is the SHARC example. I think the best use for the ARM in these cases is to expand and manage communications without burdening the signal processing processing. You could also do coefficient cooking and the like with the ARM core. Our company is working on modules that implement all of these ideas, so this is clearly how I vote on this issue. Al Clark www.danvillesignal.com On 6/18/2015 11:24 PM, Justin Tallent wrote: Agree completely. That could be a very useful discussion for those of us that are interested in exploring the possibility of going the FPGA route or staying in the more traditional DSP realm. - justin Sent from my iPhone On Jun 18, 2015, at 5:50 AM, padawa...@obiwannabe.co.uk padawa...@obiwannabe.co.uk wrote: Would love to see some of that discussion remain on-list as the landscape for embedded DSP is always in flux and a very interesting topic to hear practical personal experiences of development. cheers all, Andy On 18 June 2015 at 00:20 Al Clark acl...@danvillesignal.com wrote: Hi Ory, I have designed a lot of SHARC boards. I would be happy to discuss your project specifics via email or Skype. Al Clark www.danvillesignal.com On 6/17/2015 2:36 PM, Ory Hodis wrote: Hello, Starting a project and need consulting on which processor to go with (FPGA OR SHARC) in relation to future goals. Thank you -- 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 attachment: aclark.vcf-- 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] FPGA or SHARC Programmer wanted
I've contemplated several times trying out FPGA for physical modeling of instruments. I'd love to know how people are using FPGAs within the more traditional signal processing realm. They are become increasingly common in some hardware, specifically RME uses it for signal processing (powers their onboard mix and fx) but I'd love to know why that vs traditional DSP chips which most manufacturers still use. cheers, b On Fri, Jun 19, 2015 at 12:24 AM Justin Tallent j...@justintallent.com wrote: Agree completely. That could be a very useful discussion for those of us that are interested in exploring the possibility of going the FPGA route or staying in the more traditional DSP realm. - justin Sent from my iPhone On Jun 18, 2015, at 5:50 AM, padawa...@obiwannabe.co.uk padawa...@obiwannabe.co.uk wrote: Would love to see some of that discussion remain on-list as the landscape for embedded DSP is always in flux and a very interesting topic to hear practical personal experiences of development. cheers all, Andy On 18 June 2015 at 00:20 Al Clark acl...@danvillesignal.com wrote: Hi Ory, I have designed a lot of SHARC boards. I would be happy to discuss your project specifics via email or Skype. Al Clark www.danvillesignal.com On 6/17/2015 2:36 PM, Ory Hodis wrote: Hello, Starting a project and need consulting on which processor to go with (FPGA OR SHARC) in relation to future goals. Thank you -- 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 -- 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] FPGA or SHARC Programmer wanted
Modules are on our roadmap for fall release. Al Clark www.danvillesignal.com On 6/19/2015 9:16 AM, Niels Dettenbach (Syndicat.com) wrote: -BEGIN PGP SIGNED MESSAGE- Hash: SHA256 Am 19. Juni 2015 15:57:56 MESZ, schrieb Al Clark acl...@danvillesignal.com: Sometimes a mixed approach is a better idea. You can interface a DSP FPGA as a pair. In this case the DSP is usually the master and the FPGA is a coprocessor. The FPGA can also be used as I/O expansion. The FPGA might just run a specific algorithm that is ultimately called by the DSP. Wow, this sounds to be a very cool arch. Do you know of any audio equipment vor effect stuff in the market using such an approach today? Sorry for possibly being a bit green here in this topic, but are there any development kits / frameworks for SHARC-beginners easy to deploy/use for audio i/o (for audio effect experiments on a high perf level) to recommend - or still some SHARC/FPGA framework available? otherwise, sorry for the noise, Niels. - --- Niels Dettenbach Syndicat IT Internet http://www.syndicat.com -BEGIN PGP SIGNATURE- attachment: aclark.vcf-- 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] the original reference for Nyquist-Shannon theorem
Does anyone know what is the original published source for the Nyquist-Shannon theorem? Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 -- 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] the original reference for Nyquist-Shannon theorem
http://web.stanford.edu/class/ee104/shannonpaper.pdf is a reprint from 1949 2015-06-19 14:00 GMT+02:00 STEFFAN DIEDRICHSEN sdiedrich...@me.com: According to the german Wikipedia, Shannon published it here: Proc. IRE. Vol. 37, No. 1, 1949 And Nyqvist published his theorem here: Harry Nyquist https://de.wikipedia.org/wiki/Harry_Nyquist: Certain Topics in Telegraph Transmission Theory. In: Transactions of the American Institute of Electrical Engineers. Vol. 47, 1928, ISSN 0096-3860 http://dispatch.opac.dnb.de/DB=1.1/CMD?ACT=SRCHAIKT=8TRM=0096-3860 1928!! Steffan On 19.06.2015|KW25, at 13:53, Victor Lazzarini victor.lazzar...@nuim.ie wrote: Does anyone know what is the original published source for the Nyquist-Shannon theorem? Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 -- 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] the original reference for Nyquist-Shannon theorem
Fantastic, thanks Uli Steffan. Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 On 19 Jun 2015, at 13:03, Uli Brueggemann uli.brueggem...@gmail.com wrote: http://web.stanford.edu/class/ee104/shannonpaper.pdf is a reprint from 1949 2015-06-19 14:00 GMT+02:00 STEFFAN DIEDRICHSEN sdiedrich...@me.com: According to the german Wikipedia, Shannon published it here: Proc. IRE. Vol. 37, No. 1, 1949 And Nyqvist published his theorem here: Harry Nyquist https://de.wikipedia.org/wiki/Harry_Nyquist: Certain Topics in Telegraph Transmission Theory. In: Transactions of the American Institute of Electrical Engineers. Vol. 47, 1928, ISSN 0096-3860 http://dispatch.opac.dnb.de/DB=1.1/CMD?ACT=SRCHAIKT=8TRM=0096-3860 1928!! Steffan On 19.06.2015|KW25, at 13:53, Victor Lazzarini victor.lazzar...@nuim.ie wrote: Does anyone know what is the original published source for the Nyquist-Shannon theorem? Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 -- 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
Re: [music-dsp] the original reference for Nyquist-Shannon theorem
According to the german Wikipedia, Shannon published it here: Proc. IRE. Vol. 37, No. 1, 1949 And Nyqvist published his theorem here: Harry Nyquist https://de.wikipedia.org/wiki/Harry_Nyquist: Certain Topics in Telegraph Transmission Theory. In: Transactions of the American Institute of Electrical Engineers. Vol. 47, 1928, ISSN 0096-3860 http://dispatch.opac.dnb.de/DB=1.1/CMD?ACT=SRCHAIKT=8TRM=0096-3860 1928!! Steffan On 19.06.2015|KW25, at 13:53, Victor Lazzarini victor.lazzar...@nuim.ie wrote: Does anyone know what is the original published source for the Nyquist-Shannon theorem? Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 -- 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] Sampling theorem extension
Upon a little bit more thinking I came to the conclusion that the expressed in the earlier post (quoted below) idea should work. Indeed, the windowed signal y(t) can be represented as a series of windowed monomials, by simply windowing each of the terms of its Taylor series separately. If the statement can be shown for a monomial of an arbitrary order, then, given BLEP convergence, it will follow for their sum (windowed Taylor series) as well (since the series converges pointwise on a compact interval, the uniform convergence on that interval follows, if I'm not mistaken, and one can apply Fourier transform to each term separately). Now, a bandlimited version of a windowed monomial can be represented as a sum of time-shifted BLEPs of 0th and higher orders and therefore it is a bandlimited signal in the sense of both the classical definition and the proposed definition. Maybe I should write a short paper with a more detailed and more clear explanation? Regards, Vadim vadim.zavalishin писал 2015-06-13 14:50: Ethan Duni писал 2015-06-12 23:43: However, if I'm following this correctly, it seems to me that the problem of multiplication of distributions means that the whole basic set-up of the sampling theorem needs to be reworked to make sense in this context. I.e., not much point worrying about whether to call whatever exotic combination of derivatives of delta functions that result from polynomials as band limited or not, when we don't have a way to relate such a property back to sampling/reconstruction of well-tempered distributions in the first place. No? Kind of. Actually, I just had an idea of a much more clear definition of bandlimitedness, which doesn't rely on the sampling theorem (which is not applicable everywhere in the context of interest), or on the weird sequences of sinc convolution which converge only in the average (kind of Cesaro sense) at best. The definition applies only to real entire functions (that is entire functions giving real values for real argument). In the present context we are not interested in other functions. Particularly, any discontinuity of the function or its derivative will make the function non-bandlimited, so we don't need to cover those. Let x(t) be a real entire function (possibly not having a Fourier transform in any sense). Let's apply some arbitrary rectangular window to this signal: y(t)=w(t)*x(t). This creates the discontinuities of the function and its derivatives at the window edges. The signal y(t) is in L_2 and thus has Fourier transform. Let BL[y] be the bandlimited (using the Fourier transform, or equivalently, sinc convolution) version of that signal. Now instead of bandlimiting the signal y let's apply BLEP bandlimiting to the discontinuities of y and its derivatives, obtaining (if the ifninite sum of BLEPs converges) some other signal y'. The signal x is called bandlimited if for any rectangular window w(t), the signal y' exists (the BLEPs converge) and y'=BL[y]. This definition is well-specified and directly maps to the goals of the BLEP approach. The conjectures are - for the signals which are in L_2 the definition is equivalent to the usual definition of bandlimitedness. - if y' exists (BLEPs converge), then y'=BL[y] If the BLEP convergence is only given within some interval of the time axis (don't know if such cases can exist), then we can speak of signals bandlimited on an interval. -- Vadim Zavalishin Reaktor Application Architect | RD 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] Sampling theorem extension
On 2015-06-12, Ethan Duni wrote: Thanks for expanding on that, this is quite interesting stuff. However, if I'm following this correctly, it seems to me that the problem of multiplication of distributions means that the whole basic set-up of the sampling theorem needs to be reworked to make sense in this context. Now that I read up on it... Actually no. Every tempered distribution has a Fourier transform, and if that's compactly supported, the original distribution can be reconstructed via the usual Shannon-Whittaker sinc interpolation formula. That also goes for polynomials and sine modulated polynomials in the continuous domain. Whatever that means in general. No? Yes. While the formalism apparently goes through, I don't have the slightest idea of how to interpret that wrt the usual L^2 theory. I can sort of get that the polynomial-to-series-of-delta-derivatives duality works as it should, and via the Schwartz Representation Theorem captures the asymptotic growth of tempered distributions. But how you'd utilize that in DSP or with its oversampling problems is thus far beyond me. -- 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
