On Jan 31, 2022, at 19:21, Sampo Syreeni <de...@iki.fi> wrote: > On 2022-01-30, Ethan Duni wrote: >> IMO the physics part is the realization that (wavefunctions of) certain >> physical quantities (position/momentum, time/energy) can be related as >> Fourier transform pairs (and the associated scaling with Planck's constant). >> Once you make that assumption, uncertainty relationships follow immediately >> from the properties of the FT, as you say. > > Yes, and the whole idea of position/momentum pairs being related via Fourier > machinery (furthermore the theory of noncommutative operators) kind of begs > the question. Where's the pairing? > > > Well, it actually kind of suggests itself from the empirical evidence, from > the start.
More than merely empirical evidence - the pairs are specifically related by their units. Momentum is the change in position divided by the change in time. Position is necessarily related to this ratio because it's one of the units. The uncertainty principle always involves complementary or conjugate variables. You can't refine the measurement of one without getting sloppier with the other. >> This is a very helpful perspective, thanks for this. > > I neglected to show you what happens here, by the way. So... > > What happens when you sample a continuous time signal by projecting it onto a > basis consisting of equispaced polynomial splines is that you get a > well-invertible problem, true, but only at the discretely spaced sampling > instants. What happens in-between the points is rather different from the > usual framework. Bandlimitation no longer automatically applies, even if you > *can* easily shift your sampling basis via linear operations. (E.g. shifting > the first order term f(x)=x some time t forward really *is* just about adding > a constant term and a phase shift, so that f becomes y(x+a)-b. The same > analysis carries over to the higher order terms, via a centered Taylor > series, which always converges for any series of compact support in either > time or frequency.) > > So, you *can* invert, and as long as your analysis basis is nice enough, your > biorthogonal reconstruction assumes a not-too-nasty-form as well. It might > even be orthonormal itself, and repetitive. But in general it won't be > bandlimited, and it won't lead to perfect reconstruction of bandlimited > signals. That's because your basis is now composed of things which have > infinite bandwidth, so that they can *only* cancel to finite bandwidth via > periodic negative interference; only at sampling instants, and not between. > The the whole sampling apparatus necessarily becomes time-invariant, and of > uncompact spectral support. While you can fully invert what happens at the > sampling instants, you cannot control what happens *between*, and what > happens there, does not happen linearly, so that intermodulation products and > the like could be controlled. There's nothing about Nyquist-Shannon that need involve linear interpolation. You can invert by running the sample impulse train through the same filter that was used to band-limit before sampling. This gives you the exact intermediate values, assuming that the band limiting was done sufficiently to meet the requirements. Brian Willoughby
