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/?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 >>>>>> >>>>> >>>>> _______________________________________________ >>>>> 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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