dvsarwate wrote:> On Jan 7, 9:59 am, Andor <andor.bari...@gmail.com> wrote: > >> I think the main problem is to find the rough location of the zero- >> crossing in the sampled data. As my example shows, a zero-crossing of >> the underlying sampled function will not necessarily show up as a >> "zero-crossing" in the samples. I would opt for Ron's idea to use >> quick polynomial interpolation with order >= 3 (can be implemented >> with very small kernel FIR filters) to oversample the data by 4 or so >> to find "zero-crossings" in the samples and then use the method of >> sinc-interpolation outlined in your paper to find the exact location. >> >> This reminds me of the old comp.dsp debate on the maximum number of >> zero-crossings that a bandlimited function may have on any given >> interval. As it turned out (I think Matt Timmermans came up with the >> example of the prolate spheroidal wave functions), the number of zero- >> crossings is not limited ... > > But if there could be infinitely many zero-crossings, then no matter > how > much we oversample, we could never be sure that we have found them > all, could we? > > If a (low-pass) signal is truly band-limited to W Hz, meaning that > X(f) = 0 for |f| > W, then the magnitude of its derivative is bounded > by > 2\pi W max |x(t)| (cf. Temes, Barcilon, and Marshall, The > optimization > of bandlimited systems, Proc. IEEE, Feb. 1973). So, depending on > the sampling rate and the value of two successive positive samples, > we can be sure in *some* cases that there is no zero-crossing between > the two samples: the bound on how quickly x(t) can change does not > allow for the possibility that the signal could dip down below zero in > between the two positive samples.But the derivative grows with both frequency and amplitude. Can a bound be assigned to the derivative without knowing the pre-sampling amplitude? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
conventional wisdom how to upsample very large arrays, accurately?
Started by ●January 6, 2010
Reply by ●January 13, 20102010-01-13
Reply by ●January 13, 20102010-01-13
Andor wrote:> On 12 Jan., 19:48, Jerry Avins <j...@ieee.org> wrote: >> Andor wrote: >>> On 11 Jan., 16:16, Jerry Avins <j...@ieee.org> wrote: >>>> Andor wrote: >>>> ... >>>>> In the end I had to use 7th order polynomial interpolation FIR filters >>>>> on my test function to actually catch the zero-crossings using >>>>> upsampling by factor 4 (5th order was not enough). This can still be >>>>> implemented efficiently (8*3*1e9 = 24e9 MACS should easily be >>>>> computable in under a minute on a normal PC). >>>> I've been puzzling over this for a few days now. Set aside the search >>>> for actual locations of the zero crossings. If there are at least two >>>> samples in the period of the highest frequency present in the sampled >>>> signal, how can any zero crossing fail to be among those counted? >>> Like this: >>> http://www.zhaw.ch/~bara/files/zero-crossings_example.png >>> Plot of the function described above with f=0.98. The function has 80 >>> zero-crossings in the displayed time window, the samples of the >>> function have exactly one zero-crossing. >> I see now, thanks. Very nice! >> >> What procedures fail if we count zero touchings as zero crossings? Does >> it come down to the difference between open and closed intervals? >> >> Maybe I don't see after all. are the samples close to but not actually >> on the axis? I see only a grapg, no equation. How does .98 come in? > > Yup, the samples don't touch the axis. The equation for this function > can be found higher up in the thread. For convenience: > > x(t) = sin(pi f t) + sin(pi (1-f) t) + eps s((1-f)/2 t) > > where s(t) is the square-wave function with period 1 (replace with > truncated Fourier series to make x(t) bandlimited). x(t) is sampled > with sampling period T=1. The samples close to the axis have value eps > (in this example, eps = 0.02) and f=0.98.Aside from the possibility of Gibbs's phenomenon with the truncated s, it's clear now. Thanks again. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
