Hi Ethan, Good comments and questions…I’m going to have to skip the questions for now (I’m in a race against time the next few days, then will been off the grid, relatively speaking, for a couple of weeks—but I didn’t want to seem like I was ignoring your reply; I think any quick answers to your questions will require some back and forth, and I won’t be here for the rest).
First, I want to be clear that I don’t think people are crippled by certain viewpoint—I’ve said this elsewhere before, maybe not it this thread or the article so much. It’s more than some things that come up as questions become trivially obvious when you understand that samples represent impulses (this is not so much a viewpoint as the basis of sampling). The fact that 5,17,-12,2 at sample rate 1X and 5,0,0,0,17,0,0,0,-12,0,0,0,2,0,0,0 at sample rate 4X are identical is obvious only for samples representing impulses. 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 said, OK, then why does inserting zeros work? He gave a one-word answer: “Serendipity.” So, he clearly knew how to get the job of SRC done—he wasn’t crippled—but he didn’t know why, was just following a formula (that’s OK—there are great cooks that only follow recipes). But I think there are good reasons to understand the fact that samples represent a modulated impulse train. We all learn early on that we need the sample rate to be more than double the highest signal frequency. This usually is accompanies by diagram showing a sine wave under sampled, and a dotted line drawing of an alias at a lower frequency, maybe chat about wagon wheels going backwards in movies. But if you think about the frequency spectrum of a PAM signal, it’s apparent that the aliased image “sidebands” (radio term) will stretch down into the audio band (the band below half SR) if the signal stretches above it. So, you’d better filter it so that doesn’t happen. The best part is this doesn’t apply to just initial sampling, but is equally apparent for any interim upsampling and processing in the digital domain. Cheers, Nigel > On 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 > <mailto:earle...@earlevel.com>> wrote: > Hi Remy, > >> On Aug 28, 2017, at 2:16 AM, Remy Muller <muller.r...@gmail.com >> <mailto: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 >> <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 >> <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 >> <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 >>>> <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 <mailto:music-dsp@music.columbia.edu> >> https://lists.columbia.edu/mailman/listinfo/music-dsp >> <https://lists.columbia.edu/mailman/listinfo/music-dsp> > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> > https://lists.columbia.edu/mailman/listinfo/music-dsp > <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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