>Which contains alias images of the original spectrum, which was my point.
There is no "original spectrum" pictured in that graph. Only the responses of the interpolators. There is no reference to any input signal at all. >No one claimed there was fractional delay involved. Fractional delay is a primary topic of this thread, and a major motivation for interest in polynomial interpolation in dsp in general. >Then how do you explain that taking noise sampled at 500 Hz, and >resampling it to 44.1 kHz gives an identical FFT graph? We've been over this already. It's because you're resampling the signal at such a large rate that the effects of the sampling are not visible. And you've chosen a signal with a flat spectrum, so there are no features of the signal spectrum visible - only the interpolator response. This goes exactly to the point that no resampling effects are present in the graphs. All we see are the interpolator spectra. The fact that there are various ways to generate a graph of an interpolator spectrum is entirely beside the point. >> If you resample to the original rate >> (in order to implement a fractional delay, say), then those weighted images >> will be folded back to the same place they came from. >That's exactly why they're called aliases. No, if you fold the images back to the same spots they originated, they are not aliases. All of the frequencies are mapped back to their original locations, none end up at other frequencies. Aliases are when signal images end up in new locations corresponding to different frequency bands. This distinction is crucial to understanding the operation of fractional delay interpolators: it's why they don't produce aliasing at their output. We just get a fractional delay filter with an imperfect spectrum. It's only the frequency response of the interpolator that gets aliased (introducing the zero at Nyquist for half-sample delay, for example), not the underlying signal content. That's why it's important to graph the frequency response of the interpolators directly, without worrying about signal spectra - to figure out what happens in the final digital interpolator, you take that continuous time interpolator spectrum, add a linear phase term for whatever delay you want, and then alias it according to your new sampling rate to get the final response of the digital interpolation filter. Signal aliasing only results if that involves a change in sampling rate. >Which is not the case on Olli's graph. Right, Ollie's graph shows only the intermediate stage, the spectrum of the polynomial interpolator in continuous time. This is an analytical convenience, we never actually produce any such signal. It's used as an input to figure out what the final response of a digital interpolator based on one of these polynomials will be. You can of course sample that at a very high rate and so neglect the aliasing of the interpolator response, but what is the point of that? You wouldn't use any of these interpolators if what you're trying to do is upsample a 500Hz sampled signal to 44.1kHz, the graphs show that they're crap for that. >I spent (wasted?) a considerate amount of time creating various >demonstrations and FFT graphs showing my point. Your time would be better spent figuring out a point that is relevant to what I'm saying in the first place. It is indeed a waste of your time to invent equivalent ways to generate graphs, since that is not the point. E On Fri, Aug 21, 2015 at 2:56 PM, Peter S <peter.schoffhau...@gmail.com> wrote: > On 21/08/2015, Ethan Duni <ethan.d...@gmail.com> wrote: > > The details of how the graphs were generated don't really matter. > > Then why do you keep insisting that they're generated by plotting > sinc^2(x) ? > > > The point > > is that the only effect shown is the spectrum of the continuous-time > > polynomial interpolator. > > Which contains alias images of the original spectrum, which was my point. > > > The additional spectral effects of delaying and > > resampling that continuous-time signal (to get fractional delay, for > > example) are not shown. > > No one claimed there was fractional delay involved. > > > There is no "resampling" to be seen in the graphs. > > I recreated the exact same graph via resampling a signal, proving that > is one method of generating that graph. > > >>I claim that they are aliases of the original spectrum. > > > > What we see in the graph is simply the spectra of the continuous-time > > interpolators. > > Then how do you explain that taking noise sampled at 500 Hz, and > resampling it to 44.1 kHz gives an identical FFT graph? > > How do you explain that an 50 Hz sine wave, resampled to 44.1 kHz, > contains alias frequencies at 450 Hz, 550 Hz, 950 Hz, 1050 Hz, 1450 > Hz, 1550 Hz, etc. ? What are those, if not "aliases" ? > > > Whether those are ultimately expressed as aliases depends on what you > then > > do with that continuous time signal. > > They're already "aliases"... You may filter them out, or do whatever > you want with them - that doesn't change the fact that they're aliases > of the original spectrum... > > > If you resample to the original rate > > (in order to implement a fractional delay, say), then those weighted > images > > will be folded back to the same place they came from. > > That's exactly why they're called aliases. > > > In that case, there > > is no aliasing, you just end up with a modified frequency response of > your > > fractional interpolator. > > Which is not the case on Olli's graph. > > > It is only if the interpolated continuous-time signal is resampled at a > > different rate, or just used directly, that those signal images end up > > expressed as aliases. > > Which was presented on Olli's graph, and that's what we're talking about. > > > The rest of your accusations are your usual misreadings and straw men. I > > won't be legitimating them by responding, and I hope you will accept that > > and give up on these childish tactics. It would be better for everyone if > > you could make a point of engaging in good faith and trying to stick to > the > > subject rather than attacking the intellects of others. > > I spent (wasted?) a considerate amount of time creating various > demonstrations and FFT graphs showing my point. And you accuse me of > "childish tactics". You are lame. > > -P > _______________________________________________ > music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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