>Could you give a little bit more of a clarification here? So the >finite-order polynomials are not bandlimited, except the DC? Any hints >to what their spectra look like? How a bandlimited polynomial would look >like?
>Any hints how the spectrum of an exponential function looks like? How >does a bandlimited exponential look like? I hope we are talking about >one and the same real exponential exp(at) on (-infty,+infty) and not >about exp(-at) on [0,+infty) or exp(|a|t). The Fourier transform does not exist for functions that blow up to +- infinity like that. To do frequency domain analysis of those kinds of signals, you need to use the Laplace and/or Z transforms. Equivalently, you can think of doing a regular Fourier transform after applying a suitable exponential damping to the signal of interest. This will handle signals that blow up in one direction (like the exponential), but signals that blow up in both directions (like polynomials) remain problematic. That said, I'm not sure why this is relevant? Seems like you aren't so much interested in complete exponential/polynomial functions over their entire domain, but rather windowed versions that are restricted to some small time region? Isn't that what is used in the band limited impulse stuff? E On Tue, Jun 9, 2015 at 1:32 AM, Vadim Zavalishin < vadim.zavalis...@native-instruments.de> wrote: > Creating a new thread, to avoid completely hijacking Theo's thread. > Previous message here: > http://music.columbia.edu/pipermail/music-dsp/2015-June/073769.html > > On 08-Jun-15 18:29, Sampo Syreeni wrote: > >> On 2015-06-08,vadim.zavalishin wrote: >> >> The sampling theorem applies only to the class signals which do >>> have Fourier transform, separating this class further into >>> bandlimited and non-bandlimited. However, it doesn't say anything >>> about the signals which do not have Fourier transform. >>> >> >> Correct. But the class of signals which do have a Fourier theory is >> much, *much* larger than you appear to think. Usually continuous time >> Fourier theory is set in the class of tempered distributions, which >> does include such things as Dirac impulses, trains of them with >> suitably regular supports, and of course shifts, sums, scalings and >> derivatives of all finite orders, and so on. Not everything you could >> possibly ever want -- products are iffy, for instance -- but >> everything you need in order to bandlimit the things and get the >> most usual forms of converence in norm we're interested in. >> > > Oh, I didn't know there's a rigorous framework behind Dirac deltas. > > This can be further extrapolated to the polynomial signal case >>> (which are implicitly assumed bandlimited in the BLEP, BLAMP and >>> higher order derivative discontinuity approaches). >>> >> >> They are not implicitly assumed anything, but instead a) the part of >> them which lands outside of your bandlimit are carefully bounded to >> where aliasing is perceptually harmless, or b) they're used so that >> they're fed bandlimited signals, which via the theory of Chebyshev >> polynomials and the usual properties of the full Fourier transform >> lead to exactly as many fold band expansion as the order of the >> polynomial used has. That stuff then works perfectly fine with >> oversampling. >> > > Could you give a little bit more of a clarification here? So the > finite-order polynomials are not bandlimited, except the DC? Any hints > to what their spectra look like? How a bandlimited polynomial would look > like? > > >> Other signals (such as exponential function, FM'ed sine etc) are >>> also interesting. >>> >> >> Sure. And all that is well understood. The exponential function is >> easy because of its connection via the Euler equation to sinusoids, >> and via the usual energy norm, to the second order theory of the >> Gaussian. >> > > I'm failing to see how Euler equation can relate exponentials of a real > argument to sinusoids of a real argument? Any hints here? > > > FM, that's usually expanded in the terms of Bessel functions -- >> nasty, but in the narrow band, small modulation index limit, again >> well understood from decade of radio work. >> > > The question of FM rather popped in connection with my conjecture, which > I will write once again below in a more detailed form. > > >> So, the sampling rate theorem doesn't work for this kind of >>> signals, in the sense of giving no answer whether they are >>> bandlimited or not. >>> >> >> The latter two aren't. But they do possess Fourier transforms and >> once we have that, we can study which portion of their spectrum goes >> outside of any given bandlimit. >> > > Any hints how the spectrum of an exponential function looks like? How > does a bandlimited exponential look like? I hope we are talking about > one and the same real exponential exp(at) on (-infty,+infty) and not > about exp(-at) on [0,+infty) or exp(|a|t). > > >> Therefore we would like to have a generalization of the sampling >>> theorem, which includes these signals. >>> >> >> You can't really have any general version of such, because the class >> of general signals is just too big to admit such a discrete basis. >> But certain special cases can in fact be handled. >> > > Yes, this is what I was referring to. Currently I'm interested in the > class of functions which are representable as a sum of a real function, > which, if analytically extended to the complex plane, is entire and > isolated derivative discontinuity functions (non-bandlimited versions of > BLEPs BLAMPs etc). > > > Let f(t) be a signal such that the statement A(f) holds, let F[n] >>> be its naively sampled counterpart and and let {D_n} be a sequence >>> of DACs such that the statement B({D_n}) holds. Then the sequence >>> D_n(F) converges to f. >>> >> >> There is nor there can be such a general theory. In fact the idea >> that you can just naïvely sample pretty much anything is *highly* >> > > Well, I hope that I can naively sample no matter what. The question is > what are the consequences of that sampling. Particularly, whether I can > restore the signal back, right? And this is exactly what the sampling > theorem is about. > > I made some initial conjectures in this regard in the mentioned >>> thread, where A had to do with the rolloff speed of the function's >>> derivatives and D_n was just a sequence of time-windowed sincs >>> with increasing window size. >>> >> >> Won't work. Not even close. If you have no upper limit to the order >> of continuity, I can plug in one of the vast class of continuous, >> nowhere differentiable functions, and integrate down to something >> which fulfils your condition yet will fail any normal sampling >> theory under the assumption of shift-invariance. If you don't have >> such a limit, then you're essentially already back to the theory of >> Schwartz spaces, which are used in the construction of the topology >> of the space of tempered distributions I mentioned above. >> > > See the above statement about the entire function. I just didn't mention > this in the context of the other thread, trying to be brief. > > Anyway, here is my conjecture. > > The signals in our class of interest are a sum of two parts: > - the "continuous part" a real function of real argument which is > simultaneously entire in the complex plane. > - the "discontinuous part" which is a sum of "non-bandlimited BLEPS (0th > order), BLAMPS (1st order bleps) and higher-order bleps", where the > discontinuity points are isolated but are allowed to have infinite > "multiplicity" (that is at a single isolated point there may be > discontinuities in all the derivatives). > > We are interested in: > - whether the continuous part of our signal is bandlimited? > - what is the rolloff speed of the derivative discontinuity amplitudes > (as the order of the discontinuity grows) at each multiple-discontinuity > point? > > Because, if the continuous part is bandlimited, then we have "just" to > replace the discontinuities by their bandlimited versions (the essence > of the BLEP approach) and the remaining question is only: if there are > infinitely many discontinuities at a given point, whether the sum of > their bandlimited versions will converge. > > The convergence of BLEPs is defined by the rolloff speed of the > derivatives. My conjecture is that so is the bandlimitedness of the > entire function, and that the rolloff speed requirement is exactly the > same for both conditions! > > If this conjecture is true, this should automatically imply that all > polynomials are bandlimited, since all their higher-order derivatives > are zeros. > > Here is the older thread where I was asking the same question: > http://music.columbia.edu/pipermail/music-dsp/2014-June/072679.html > > -- > Vadim Zavalishin > Reaktor Application Architect | R&D > 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
