Okay, no big deal. It's easy to come off the wrong way in complicated, fast moving email threads.
Ethan D On Wed, Sep 6, 2017 at 6:37 PM, Nigel Redmon <earle...@earlevel.com> wrote: > Ethan, I wasn't taking a swipe at you, by any stretch. In fact, I wasn't > even addressing your ADC comment. It was actually about things like the > idea of making DACs with impulses. As I noted, we don't because there are > ways that are easier and accomplish the same goal, but it is feasible. I've > had people say in the past to me it's absurd, and I've assured them that a > reasonable and practical approximation of it would indeed produce a > reasonable approximation of a decent DAC. That's a pretty relative > statement because the quality depends on how hard you want try, but I > subsequently saw that Julius Smith make the same assertion on his website. > > Sorry you misinterpreted it. > > On Sep 7, 2017, at 5:34 AM, Ethan Duni <ethan.d...@gmail.com> wrote: > > Nigel Redmon wrote: > >As an electrical engineer, we find great humor when people say we can't > do impulses. > > I'm the electrical engineer who pointed out that impulses don't exist and > are not found in actual ADCs. If you have some issue with anything I've > posted, I'll thank you to address it to me directly and respectfully. > > Taking oblique swipes at fellow list members, impugning their standing as > engineers, etc. is poisonous to the list community. > > >What constitutes an impulse depends on the context—nano seconds, > milliseconds... > > If it has non-zero pulse width, it isn't an impulse in the relevant sense: > multiplying by such a function would not model the sampling process. You > would need to introduce additional operations to describe how this finite > region of non-zero signal around each sample time is translated into a > unique sample value. > > >For ADC, we effectively measure an instantaneous voltage and store it as > an impulse. > > I don't know of any ADC design that stores voltages as "impulse" signals, > even approximately. The measured voltage is represented through modulation > schemes such as PDM, PWM, PCM, etc. > > Impulse trains are a convenient pedagogical model for understanding > aliasing, reconstruction filters, etc., but there is a considerable gap > between that model and what actually goes on in a real ADC. > > >If you can make a downsampler that has no audible aliasing (and you > can), I think the process has to be called linear, even if you can make a > poor quality one that isn't. > > I'm not sure how you got onto linearity, but the subject is > time-invariance. > > I have no objection to calling resamplers "approximately time-invariant" > or "asymptotically time-invariant" or somesuch, in the sense that you can > get as close to time-invariant behavior as you like by throwing resources > at the bandlimiting filter. This is qualitatively different from other > archetypical examples of time-variant systems (modulation, envelopes, etc.) > where explicitly time-variant behavior is the goal, even in the ideal case. > Moreover, I agree that this distinction is important and worth > highlighting. > > However, there needs to be *some* qualifier - the bare statement > "(re)sampling is LTI" is incorrect and misleading. It obscures that fact > that addressing the aliasing caused by the system's time-variance is the > principle concern in the design of resamplers. The fact that a given design > does a good job is great and all - but that only happens because the > designer recognizes that the system is time-invariant, and dedicates > resources to mitigating the impact of aliasing. > > >If you get too picky and call something non-linear, when for practical > decision-making purposes it clearly is, it seem you've defeated the purpose. > > If you insist on labelling all resamplers as "time-invariant," without any > further qualification, then it will mess up practical decision making. > There will be no reason to consider the effects of aliasing - LTI systems > cannot produce aliasing - when making practical system design decisions. > You only end up with approximately-LTI behavior because you recognize at > the outset that the system is *not* LTI, and make appropriate design > decisions to limit the impact of aliasing. So this is putting the cart > before the horse. > > The appropriate way to deal with this is not to get hung up on the label > "LTI" (or any specialized variations thereof), but to simply quote the > actual performance of the system (SNR, spurious-free dynamic range, etc.). > In that way, everything is clear to the designers and clients: the system > is fundamentally non-LTI, and deviation from LTI behavior is bounded by the > performance figures. Then the client can look at that, and make an > informed, practical decision about whether they need to worry about > aliasing in their specific context. If not, they are free to say to > themselves "close enough to LTI for me!" If so, they can dig into the > non-LTI behavior and figure out how to deal with it. Insisting that > everyone mislabel time-variant systems as LTI short-circuits that whole > process and so undermines practical decision-making. > > Ethan D > > On Tue, Sep 5, 2017 at 1:05 AM, Nigel Redmon <earle...@earlevel.com> > wrote: > >> As an electrical engineer, we find great humor when people say we can't >> do impulses. What constitutes an impulse depends on the context—nano >> seconds, milliseconds... >> >> For ADC, we effectively measure an instantaneous voltage and store it as >> an impulse. Arguing that we don't really do that...well, Amazon didn't >> really ship that Chinese garlic press to me, because they really relayed an >> order to some warehouse, the shipper did some crazy thing like send it in >> the wrong direction to a hub, to be more efficient...and it was on my >> doorstep when I checked the mail. What's the diff... >> >> Well, that's the most important detail (ADC), because that defined what >> we're dealing with when we do "music-dsp". But as far as DAC not using >> impulses, it's only because the shortcut is trivial. Like I said, audio >> sample rates are slow, not that hard to do a good enough job for >> demonstration with "close enough" impulses. >> >> Don't anyone get mad at me, please. Just sitting on a plane at LAX at >> 1AM, waiting to fly 14 hours...on the first leg...amusing myself before >> going offline for a while >> >> ;-) >> >> >> On Sep 4, 2017, at 10:07 PM, Ethan Duni <ethan.d...@gmail.com> wrote: >> >> rbj wrote: >> >> >1. resampling is LTI **if**, for the TI portion, one appropriately >> scales time. >> >> Have we established that this holds for non-ideal resampling? It doesn't >> seem like it does, in general. >> >> If not, then the phrase "resampling is LTI" - without some kind of >> "ideal" qualifier - seems misleading. If it's LTI then what are all these >> aliases doing in my outputs? >> >> >no one *really* zero-stuffs samples into the stream >> >> Nobody does it *explicitly* but it seems misleading to say we don't >> *really* do it. We employ optimizations to handle this part implicitly, but >> the starting point for that is exactly to *really* stuff zeroes into the >> stream. This is true in the same sense that the FFT *really* computes the >> DFT. >> >> Contrast that with pedagogical abstractions like the impulse train model >> of sampling. Nobody has ever *really* sampled a signal this way, because >> impulses do not exist in reality. >> >> >7. and i disagree with the statement: "The other big pedagogical problem >> with impulse train representation is that it can't be graphed in a >useful >> way." graphing functions is an abstract representation to begin with, so >> we can use these abstract vertical arrows to represent >impulses. >> >> That is my statement, so I'll clarify: you can graph an impulse train >> with a particular period. But how do you graph the product of the impulse >> train with a continuous-time function (i.e., the sampling operation)? Draw >> a graph of a generic impulse train, with the scaling of each impulse >> written out next to it? That's not useful. That's just a generic impulse >> train graph and a print-out of the sequence values. The useful graph here >> is of the sample sequence itself. >> >> >if linear interpolation is done between the subsamples, i have found >> that upsampling by a factor of 512 followed by linear interpolation >> >between those teeny-little upsampled samples, that this will result in 120 >> dB S/N >> >> What is the audio use case wherein 512x upsampling is not already >> sufficient time resolution? I'm curious why you'd need additional >> interpolation at that point. >> >> Ethan D >> >> On Mon, Sep 4, 2017 at 1:49 PM, Nigel Redmon <earle...@earlevel.com> >> wrote: >> >>> 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. >>> >>> >>> I agree that the zero-stuff-then-lowpass technique is much more obvious >>> when we you consider the impulse train corresponding to the signal. But I >>> find it peculiar to assert that these two sequences are "identical." If >>> they're identical in any meaningful sense, why don't we just stop there and >>> call it a resampler? The reason is that what we actually care about in the >>> end is what the corresponding bandlimited functions look like, and >>> zero-stuffing is far from being an identity operation in this domain. We're >>> instead done constructing a resampler when we end up with an operation that >>> preserves the bandlimited function -- or preserves as much of it as >>> possible in the case of downsampling. >>> >>> >>> Well, when I say they are identical, the spectrum is identical. In other >>> words, they represent the same signal. The fact that it doesn’t make it >>> a resampler is a different thing—an additional constraint. We only have >>> changed the data rate (not the signal) when we insert zeros. Most of the >>> time, we want to also change the signal (by getting rid of the aliases, >>> that were above half the sample rate and now below). That’s why my article >>> made a big deal (point #3) of pointing out that the digital samples >>> represent not the original analog signal, but a modulated version of it. >>> >>> Of course, we differ only in semantics, just making mine clear. When I >>> say they represent the same signal, I don’t just mean the portion of the >>> spectrum in the audio band or below half the sample rate—I mean the whole >>> thing. >>> >>> >>> On Sep 4, 2017, at 12:14 PM, Ethan Fenn <et...@polyspectral.com> wrote: >>> >>> 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. >>> >>> >>> In that case I'd suggest some more editing is in order, since the >>> article stated this pretty overtly at least a couple times. >>> >>> 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). >>> >>> >>> Here's the way I see it. There are three classes of interesting objects >>> here: >>> >>> 1) Discrete time signals, which are sequences of numbers. >>> 2) Scaled, equally-spaced ideal impulse trains, which are a sort of >>> generalized function of a real number. >>> 3) Appropriately bandlimited functions of a real number. >>> >>> None of these are exactly identical, as sequences of numbers are not the >>> same sort of beast as functions of a real number. But obviously there is a >>> one-to-one correspondence between objects in classes 1 and 2. Less >>> obviously -- but more interestingly and importantly! -- there is a >>> one-to-one correspondence between objects in classes 1 and 3. So any >>> operation on any of these three classes will have a corresponding operation >>> in the other two. >>> >>> This is what the math tells us. It does not tell us that any of these >>> classes are identical to each other or that thinking of one correspondence >>> is more correct than the other. >>> >>> 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. >>> >>> >>> I agree that the zero-stuff-then-lowpass technique is much more obvious >>> when we you consider the impulse train corresponding to the signal. But I >>> find it peculiar to assert that these two sequences are "identical." If >>> they're identical in any meaningful sense, why don't we just stop there and >>> call it a resampler? The reason is that what we actually care about in the >>> end is what the corresponding bandlimited functions look like, and >>> zero-stuffing is far from being an identity operation in this domain. We're >>> instead done constructing a resampler when we end up with an operation that >>> preserves the bandlimited function -- or preserves as much of it as >>> possible in the case of downsampling. >>> >>> This is why it is more natural for me to think of the discrete signal >>> and the bandlimited function as being more closely identified. The impulse >>> train is a related mathematical entity which is useful to pull out of the >>> toolbox on some occasions. >>> >>> I'm not really interested in arguing that the way I think about things >>> is superior -- as I've stated above I think the math is neutral on this >>> point, and what mental model works best is different from person to person. >>> It can be a bit like arguing what shoe size is best. But I do think it's >>> counterproductive to discourage people from thinking about the discrete >>> signal <-> bandlimited function correspondence. I think real insight and >>> intuition in DSP is built up by comparing what basic operations look like >>> in each of these different universes (as well as in their frequency domain >>> equivalents). >>> >>> -Ethan >>> >>> >>> >>> On Mon, Sep 4, 2017 at 2:14 PM, Ethan Fenn <et...@polyspectral.com> >>> wrote: >>> >>>> Time variance is a bit subtle in the multi-rate context. For integer >>>>> downsampling, as you point out, it might make more sense to replace the >>>>> classic n-shift-in/n-shift-out definition of time invariance with one that >>>>> works in terms of the common real time represented by the different >>>>> sampling rates. So an integer shift into a 2x downsampler should be a >>>>> half-sample shift in the output. In ideal terms (brickwall filters/sinc >>>>> functions) this all clearly works out. >>>> >>>> >>>> I think the thing to say about integer downsampling with respect to >>>>> time variance is that it's that partitions the space of input shifts, >>>>> where >>>>> if you restrict yourself to shifts from a given partition you will see >>>>> time >>>>> invariance (in a certain sense). >>>> >>>> >>>> So this to me is a good example of how thinking of discrete time >>>> signals as representing bandlimited functions is useful. Because if we're >>>> thinking of things this way, we can simply define an operation in the space >>>> of discrete signals as being LTI iff the corresponding operation in the >>>> space of bandlimited functions is LTI. This generalizes the usual >>>> definition, and your partitioned-shift concept, in exactly the way we want, >>>> and we find that ideal resamplers (of any ratio, >>>> integer/rational/irrational) are in fact LTI as our intuition suggests they >>>> should be. >>>> >>>> -Ethan F >>>> >>>> >>>> >>>> On Mon, Sep 4, 2017 at 1:00 AM, Ethan Duni <ethan.d...@gmail.com> >>>> wrote: >>>> >>>>> Hmm this is quite a few discussions of LTI with respect to resampling >>>>> that have gone badly on the list over the years... >>>>> >>>>> Time variance is a bit subtle in the multi-rate context. For integer >>>>> downsampling, as you point out, it might make more sense to replace the >>>>> classic n-shift-in/n-shift-out definition of time invariance with one that >>>>> works in terms of the common real time represented by the different >>>>> sampling rates. So an integer shift into a 2x downsampler should be a >>>>> half-sample shift in the output. In ideal terms (brickwall filters/sinc >>>>> functions) this all clearly works out. >>>>> >>>>> On the other hand, I hesitate to say "resampling is LTI" because that >>>>> seems to imply that resampling doesn't produce aliasing. And of course >>>>> aliasing is a central concern in the design of resamplers. So I can see >>>>> how >>>>> this rubs people the wrong way. >>>>> . >>>>> It's not clear to me that a realizable downsampler (i.e., with >>>>> non-zero aliasing) passes the "real time" definition of LTI? >>>>> >>>>> I think the thing to say about integer downsampling with respect to >>>>> time variance is that it's that partitions the space of input shifts, >>>>> where >>>>> if you restrict yourself to shifts from a given partition you will see >>>>> time >>>>> invariance (in a certain sense). >>>>> >>>>> More generally, resampling is kind of an edge case with respect to >>>>> time invariance, in the sense that resamplers are time-variant systems >>>>> that >>>>> are trying as hard as they can to act like time invariant systems. As >>>>> opposed to, say, modulators or envelopes or such, >>>>> >>>>> Ethan D >>>>> >>>>> >>>>> On Fri, Sep 1, 2017 at 10:09 PM, Nigel Redmon <earle...@earlevel.com> >>>>> wrote: >>>>> >>>>>> Interesting comments, Ethan. >>>>>> >>>>>> Somewhat related to your points, I also had a situation on this board >>>>>> years ago where I said that sample rate conversion was LTI. It was a >>>>>> specific context, regarding downsampling, so a number of people, one by >>>>>> one, basically quoted back the reason I was wrong. That is, basically >>>>>> that >>>>>> for downsampling 2:1, you’d get a different result depending on which set >>>>>> of points you discard (decimation), and that alone meant it isn’t LTI. Of >>>>>> course, the fact that the sample values are different doesn’t mean what >>>>>> they represent is different—one is just a half-sample delay of the >>>>>> other. I >>>>>> was surprised a bit that they accepted so easily that SRC couldn’t be >>>>>> used >>>>>> in a system that required LTI, just because it seemed to violate the >>>>>> definition of LTI they were taught. >>>>>> >>>>>> On Sep 1, 2017, at 3:46 PM, Ethan Duni <ethan.d...@gmail.com> wrote: >>>>>> >>>>>> 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/?ord >>>>>>>>>> er=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 >>>>>>> >>>>>> >>>>>> _______________________________________________ >>>>>> 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 >>> >> >> _______________________________________________ >> 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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