>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.
Right, this is what I was getting at with the convergence line of thinking. We theoretically need all samples from -inf to +inf in the regular sampling theorem as well, but we end up with reasonable error behavior when using realizable reconstruction filters instead. This happens basically because the sinc function dies off linearly 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. 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. 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. Which suggests that what we need here is actually something stronger than a sampling theorem. That a particular class of signals can be reconstructed from its (infinite) samples is necessary but not sufficient. We also need the property that the reconstruction can be approximated with realizable filters in a useful way. E On Fri, Jun 19, 2015 at 2:03 PM, Sampo Syreeni <de...@iki.fi> 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. >> > > 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 > -- 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
