Ethan F wrote: >I see your nitpick and raise you. :o) Surely there are uncountably many such functions, >as the power at any apparent frequency can be distributed arbitrarily among the bands.
Ah, good point. Uncountable it is! Nigel R wrote: >But I think there are good reasons to understand the fact that samples represent a >modulated impulse train. I entirely agree, and this is exactly how sampling was introduced to me back in college (we used Oppenheim and Willsky's book "Signals and Systems"). I've always considered it the canonical EE approach to the subject, and am surprised to learn that anyone thinks otherwise. Nigel R wrote: >That sounds like a dumb observation, but I once had an argument on this board: >After I explained why we stuff zeros of integer SRC, a guy said my explanation was BS. I dunno, this can work the other way as well. There was a guy a while back who was arguing that the zero-stuffing used in integer upsampling is actually not a time-variant operation, on the basis that the zeros "are already there" in the impulse train representation (so it's a "null operation" basically). He could not explain how this putatively-LTI system was introducing aliasing into the output. Or was this the same guy? So that's one drawback to the impulse train representation - you need the sample rate metadata to do *any* meaningful processing on such a signal. Otherwise you don't know which locations are "real" zeros and which are just "filler." Of course knowledge of sample rate is always required to make final sense of a discrete-time audio signal, but in the usual sequence representation we don't need it just to do basic operations, only for converting back to analog or interpreting discrete time operations in analog terms (i.e., what physical frequency is the filter cut-off at, etc.). The other big pedagogical problem with impulse train representation is that it can't be graphed in a useful way. People will also complain that it is poorly defined mathematically (and indeed the usual treatments handwave these concerns), but my rejoinder would be that it can all be made rigorous by adopting non-standard analysis/hyperreal numbers. So, no harm no foul, as far as "correctness" is concerned, although it does hobble the subject as a gateway into "real math." Ethan D On Fri, Sep 1, 2017 at 2:38 PM, Ethan Fenn <et...@polyspectral.com> wrote: > This needs an additional qualifier, something about the bandlimited >> function with the lowest possible bandwidth, or containing DC, or >> "baseband," or such. > > > Yes, by bandlimited here I mean bandlimited to [-Nyquist, Nyquist]. > > Otherwise, there are a countably infinite number of bandlimited functions >> that interpolate any given set of samples. These get used in "bandpass >> sampling," which is uncommon in audio but commonplace in radio >> applications. > > > I see your nitpick and raise you. :o) Surely there are uncountably many > such functions, as the power at any apparent frequency can be distributed > arbitrarily among the bands. > > -Ethan F > > > On Fri, Sep 1, 2017 at 5:30 PM, Ethan Duni <ethan.d...@gmail.com> wrote: > >> >I'm one of those people who prefer to think of a discrete-time signal >> as >> >representing the unique bandlimited function interpolating its samples. >> >> This needs an additional qualifier, something about the bandlimited >> function with the lowest possible bandwidth, or containing DC, or >> "baseband," or such. >> >> Otherwise, there are a countably infinite number of bandlimited functions >> that interpolate any given set of samples. These get used in "bandpass >> sampling," which is uncommon in audio but commonplace in radio >> applications. >> >> Ethan D >> >> On Fri, Sep 1, 2017 at 1:31 PM, Ethan Fenn <et...@polyspectral.com> >> wrote: >> >>> Thanks for posting this! It's always interesting to get such a good >>> glimpse at someone else's mental model. >>> >>> I'm one of those people who prefer to think of a discrete-time signal as >>> representing the unique bandlimited function interpolating its samples. And >>> I don't think this point of view has crippled my understanding of >>> resampling or any other DSP techniques! >>> >>> I'm curious -- from the impulse train point of view, how do you >>> understand fractional delays? Or taking the derivative of a signal? Do you >>> have to pass into the frequency domain in order to understand these? >>> Thinking of a signal as a bandlimited function, I find it pretty easy to >>> understand both of these processes from first principles in the time >>> domain, which is one reason I like to think about things this way. >>> >>> -Ethan >>> >>> >>> >>> >>> On Mon, Aug 28, 2017 at 12:15 PM, Nigel Redmon <earle...@earlevel.com> >>> wrote: >>> >>>> Hi Remy, >>>> >>>> On Aug 28, 2017, at 2:16 AM, Remy Muller <muller.r...@gmail.com> wrote: >>>> >>>> I second Sampo about giving some more hints about Hilbert spaces, >>>> shift-invariance, Riesz representation theorem… etc >>>> >>>> >>>> I think you’ve hit upon precisely what my blog isn’t, and why it exists >>>> at all. ;-) >>>> >>>> Correct me if you said it somewhere and I didn't saw it, but an >>>> important *implicit* assumption in your explanation is that you are >>>> talking about "uniform bandlimited sampling”. >>>> >>>> >>>> Sure, like the tag line in the upper right says, it’s a blog about >>>> "practical digital audio signal processing". >>>> >>>> Personnally, my biggest enlighting moment regarding sampling where when >>>> I read these 2 articles: >>>> >>>> >>>> Nice, thanks for sharing. >>>> >>>> "Sampling—50 Years After Shannon" >>>> http://bigwww.epfl.ch/publications/unser0001.pdf >>>> >>>> and >>>> >>>> "Sampling Moments and Reconstructing Signals of Finite Rate of >>>> Innovation: Shannon Meets Strang–Fix" >>>> https://infoscience.epfl.ch/record/104246/files/DragottiVB07.pdf >>>> >>>> I wish I had discovered them much earlier during my signal processing >>>> classes. >>>> >>>> Talking about generalized sampling, may seem abstract and beyond what >>>> you are trying to explain. However, in my personal experience, sampling >>>> seen through the lense of approximation theory as 'just a projection' onto >>>> a signal subspace made everything clearer by giving more perspective: >>>> >>>> - The choice of basis functions and norms is wide. The sinc >>>> function being just one of them and not a causal realizable one >>>> (infinite >>>> temporal support). >>>> - Analysis and synthesis functions don't have to be the same (cf >>>> wavelets bi-orthogonal filterbanks) >>>> - Perfect reconstruction is possible without requiring >>>> bandlimitedness! >>>> - The key concept is 'consistent sampling': *one seeks a signal >>>> approximation that is such that it would yield exactly the same >>>> measurements if it was reinjected into the system*. >>>> - All that is required is a "finite rate of innovation" (in the >>>> statistical sense). >>>> - Finite support kernels are easier to deal with in real-life >>>> because they can be realized (FIR) (reminder: time-limited <=> >>>> non-bandlimited) >>>> - Using the L2 norm is convenient because we can reason about best >>>> approximations in the least-squares sense and solve the projection >>>> problem >>>> using Linear Algebra using the standard L2 inner product. >>>> - Shift-invariance is even nicer since it enables *efficient* >>>> signal processing. >>>> - Using sparser norms like the L1 norm enables sparse sampling and >>>> the whole field of compressed sensing. But it comes at a price: we have >>>> to >>>> use iterative projections to get there. >>>> >>>> All of this is beyond your original purpose, but from a pedagocial >>>> viewpoint, I wish these 2 articles were systematically cited in a "Further >>>> Reading" section at the end of any explanation regarding the sampling >>>> theorem(s). >>>> >>>> At least the wikipedia page cites the first article and has a section >>>> about non-uniform and sub-nyquist sampling but it's easy to miss the big >>>> picture for a newcomer. >>>> >>>> Here's a condensed presentation by Michael Unser for those who would >>>> like to have a quick historical overview: >>>> http://bigwww.epfl.ch/tutorials/unser0906.pdf >>>> >>>> >>>> On 27/08/17 08:20, Sampo Syreeni wrote: >>>> >>>> On 2017-08-25, Nigel Redmon wrote: >>>> >>>> http://www.earlevel.com/main/tag/sampling-theory-series/?order=asc >>>> >>>> >>>> Personally I'd make it much simpler at the top. Just tell them sampling >>>> is what it is: taking an instantaneous value of a signal at regular >>>> intervals. Then tell them that is all it takes to reconstruct the waveform >>>> under the assumption of bandlimitation -- a high-falutin term for "doesn't >>>> change too fast between your samples". >>>> >>>> Even a simpleton can grasp that idea. >>>> >>>> Then if somebody wants to go into the nitty-gritty of it, start talking >>>> about shift-invariant spaces, eigenfunctions, harmonical analysis, and the >>>> rest of the cool stuff. >>>> >>>> >>>> _______________________________________________ >>>> dupswapdrop: music-dsp mailing list >>>> music-dsp@music.columbia.edu >>>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>>> >>>> >>>> >>>> _______________________________________________ >>>> dupswapdrop: music-dsp mailing list >>>> music-dsp@music.columbia.edu >>>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>>> >>> >>> >>> _______________________________________________ >>> dupswapdrop: music-dsp mailing list >>> music-dsp@music.columbia.edu >>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>> >> >> >> _______________________________________________ >> dupswapdrop: music-dsp mailing list >> music-dsp@music.columbia.edu >> https://lists.columbia.edu/mailman/listinfo/music-dsp >> > > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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