"Andor" <andor.bariska@gmail.com> wrote in message news:1186154018.682574.177520@22g2000hsm.googlegroups.com...> glen herrmannsfeldt wrote: >> Fred Marshall wrote: >> >> (snip of discussion on basis function for non-uniform sampling) >> >> > Here's a guess: >> > The basis set has to be 1.0 at the intended sample time *and* zero at >> > all >> > the other known sample times (?). >> > So, knowing the sample times, one might think one could construct a >> > polynomial that fits all those points I suppose. > > In general, there are an infinite number of sample points, and the > polynomial becomes a power series. But the idea is correct. > >> > I'd expect to see some >> > weird functions if the sample times are bunched and sparse. In fact, I >> > expect it can be proven that zeros can't be bunched too closely >> > together >> > without the functions "blowing up" outside. Bandwidth limitations >> > limit the >> > regular spacing of zeros .. as with the sinc. >> >> If by "blowing up" you mean infinite, no, it shouldn't do that. > > If by "blowing up" Fred means growing without bound, then that seems > very good intuition to me. You can see it in your functions below if > you let a->0 or 1. > >> >> As the spacing gets less uniform, though, the basis functions can >> get large, which is the cause for the sensitivity of the reconstruction >> to the accuracy of the samples, including any noise. >> >> Consider sampling at points ..., -3, -3+a, -2, -2+a, -1, -1+a, 0, a, 1, >> 1+a, 2, 2+a, ... That is, all integers and integers plus a constant a. >> What do the functions look like when a=0.5? They must be sinc(2x). >> But sinc(2x)=sin(2pi x)/(2 pi x) = cos(pi x)sin(pi x)/(pi x) where >> the cos(pi x) adds the extra zeros where they are needed. >> >> If you put two samples between each integer, at a and b, and >> have a=1/3 and b=2/3, then >> 3 sinc(3x)=sin(3 pi x)/(pi x)=(4 cos(pi x)**2 -1) sin(pi x)/(3 pi x) >> or (2 cos(pi x)-1) (2 cos(pi x)+1) sin(pi x) / (3 pi x) >> >> where the cos terms supply the new zeros. >> >> This gives some hint as to what the functions will look like >> for other a and b in factored form. >> >> For samples at integers and integers+a, the function >> for x=0 (and shifted, for other integers) >> >> f(x)=sin(pi x)sin(pi (x-a))/sin(-pi a)/(pi x) > > I'm not quite sure how you arrived at that function. Does that follow > from what you wrote above? > >> >> the function for the sample at a will be the mirror image around a, >> g(x)=f(a-x) >> >> g(x)=sin(pi (a-x)) sin(-x)/sin(-pi a)/(pi (a-x)) > > If you compare g and f with the kernels given in [1], these certainly > seem to be the correct interpolation functions in the case where every > second sample shifted. > >> >> as a gets close to 0 or 1 the peak gets larger as >> abs(sin(pi a)) gets smaller. > > Yup. > > Regards, > Andor > > [1] J. L. Yen, "On Nonuniform Sampling of Bandwidth-Limited Signals," > IRE Trans. Circuit Theory, vol. 3, pp. 251-257, Dec. 1956.Andor and Glen, Well, it wasn't *entirely* intuition but close enough. I was thinking of supergained functions which "blow up" in the sense that the approximated function gets very large - perhaps not infinite - outside the region of interest (and sometimes in the "invisible" region to use antenna pattern language). This occurs when you push the approximant between zero and 2pi without bounding what happens beyond 2pi. A limiting case for supergaining is the vanDerMaas antenna pattern function that has perfectly flat sidelobes / sinc-like functions with non-decaying tails / extending beyond 2pi and to infinity (this is seen if the independent variable is taken as the angle and not the cosine of the angle), and has a window transform that *is* necessarily infinite at the edges to produce the never-edning sinusoidal component. Andor interprets what I called "bunched up" as the separation between samples approaches zero. That's a good way of looking at it. You might ask: What happens to the implied bandwidth when that happens? I think it must go up. There's another way of looking at this: if we assume a strictly bandlimited function is sampled irregularly. Reconstruction can happen by convolving the samples with a sinc / i.e. passing them through a perfect lowpass filter. Looking at it that way, the basis set doesn't change from the most familiar one - but the construction expressions are more complicated is all. Glen's construction is interesting nonetheless! Fred