tl;dr version: The justification for DSP (equi-distant samples) is the Whittaker-Shannon interpolation formula, which follows from the Poisson summation formula plus some hand-waving about distributions (dirac delta theory). Am I right?
On Fri, Mar 28, 2014 at 4:50 AM, Ethan Duni <ethan.d...@gmail.com> wrote: > Hi Robert- > > > i dunno what "non-standard analysis" you mean. > > I'm referring to the stuff based on hyperreal numbers: > > http://en.wikipedia.org/wiki/Hyperreal_number > > These are an extension of the extended real numbers, where each hyperreal > number has a "standard part" (which is an extended real) and an > "infintesimal part" (which corresponds to a convergence rate). The basic > idea is that each hyperreal number represents an equivalence class of > functions which converge (in the extended reals, so "converging" to > infinity is allowed) to the same limit at the same rate. The limit is given > by the standard part of the number, and the convergence rate by the > infintesimal part. So you can make sense of statements like "0/0" or > "infinity - infinity" in this context, by comparing the infintesimal parts. > I.e., the usual epsilon-delta limit approach from standard analysis is > embedded into the arithmetic of the hyperreals. So using this approach you > can rigorously do the kinds of "sloppy" algebraic manipulations of "dx" > type terms that we often see in introductory calculus classes, for one > example. > > > the only truly rigorous usage of the Dirac delta is to keep it clothed > with a surrounding integral. > > That's true, but a Dirac delta in the context of non-standard analysis > isn't "naked" - it comes clothed with an associated limiting process given > by the infintesimal part. I.e., consider a sequence of functions that > "converges" to a Dirac delta, as is used in the standard approach (there's > the boxcar example you've already given, or you can use a two-sided > exponential decay, or a Gaussian distribution with variance shrinking to > zero, or any number of other things). For any such sequence, there is an > associated hyperreal Dirac delta, which expresses all of the relevant > analytic properties of that class of sequences - the fact that it tends to > zero everywhere except the origin and blows up there, and also the rates at > which each point converges. Using that, we should be able to do the usual > "non-rigorous" algebraic manipulations used in undergrad engineering > "proofs," but make them rigorous (with a bit of care - you have to work out > what effects the non-standard versions of various operations have, take the > "standard part" at appropriate places to get back to the final answer, > etc.). > > Anyway the whole thing is a bit of a curiosity. It's generally easier to > just do the proofs the standard way if you're really interested, and just > use the regular sloppy approach if you aren't. But still kind of neat I > think, that the "fake" way can actually be made rigorous by embedding the > relevant analytic framework into an extended number system. > > E > > > > On Thu, Mar 27, 2014 at 7:17 PM, robert bristow-johnson < > r...@audioimagination.com> wrote: > > > On 3/27/14 5:27 PM, Ethan Duni wrote: > > > >> it is, at least, if you accept the EE notion of the Dirac delta function > >>> > >> and not worry so much about it "not really being a function", which is > >> literally what the math folks tell us. > >> > >> I may be misremembering, but can't non-standard analysis be used to make > >> that whole Dirac delta function approach rigorous? > >> > > > > i dunno what "non-standard analysis" you mean. the only truly rigorous > > usage of the Dirac delta is to keep it clothed with a surrounding > integral. > > so naked Dirac deltas are a no-no. then we can't really have a notion > of > > a Dirac comb function either. > > > > > > I know that it works for > >> the whole "algebraic manipulation of delta-x terms" that we also like to > >> do > >> in engineering classes, intuitively seems like we could play the same > >> trick > >> with Dirac delta's and associated stuff. But I don't recall whether it > >> actually works out entirely... although Wikipedia suggests that maybe it > >> does ( > >> http://en.wikipedia.org/wiki/Dirac_delta_function# > >> Infinitesimal_delta_functions). > >> > >> Not that it's worth the trouble to really work out - we already know > what > >> the correct answers are from measure theory/distributions - but it's > nice > >> to keep in mind that these pedantic math complaints are actually kind of > >> baseless, at least if some care is taken to adhere to the rules of > >> non-standard analysis and so avoid various pitfalls. > >> > > > > i just treat the Dirac delta in time as if it has a Planck Time (10^(-43) > > second) width. then it's a true function and it still has, to within an > > immeasureable degree of accuracy, the same properties that i want. > > > > L8r, > > > > > > -- > > > > 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 > > > -- > 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
