>It shows *exactly* the aliasing.... It shows the aliasing left by linear interpolation into the continuous time domain. It doesn't show the additional aliasing produced by then delaying and sampling that signal. I.e., the images that would get folded back onto the new baseband, disturbing the sinc^2 curve. This is how we end up with a zero at Nyquist when we do half-sample delay, for example. And also how we end up with a perfectly flat response if we do the trivial resampling (original rate, no delay).
Those differences would be quite small for resampling to 44.1kHz with no delay, since the oversampling ratio is considerable, so you'd have to look carefully to see them. This is a big hint that they are not portrayed: Ollie knows what he is doing, so if he wanted to illustrate the effects of the resampling, he would have constructed a scenario where they are easily visible. And probably mentioned a second sample rate, explicitly shown both the sinc^2 and its aliased counterpart, etc. The effect would be shown in a visible, explicit manner, if that was what the graph was supposed to show. But all of those things depend on parameters like oversampling ratio and delay, so it would be a much more complicated picture. What we're shown here is just the effects of polynomial interpolation to get to the continuous time domain. The additional effects of delaying and then sampling that signal back into the discrete time domain are not visible. It seems that you have assumed that some resampling must be happening because the graph only goes up to 22kHz. But that's just the range of the graph, you don't need to do any resampling of anything to graph sinc^2 over any particular range of frequencies. >Oh, it's the *exact* same graph! (Minus some >difference above 20 kHz, due to my soundcard's anti-alias filter.) >You get the same graph if you sample that continuous time signal >at a 44.1 kHz sampling rate (with some further aliasing from the >sampling). But that's not quite the exact same graph. And why are you putting a sound card in the loop? This is all just digital processing in question here. You don't even need to process any signals, there are analytic expressions for all of the quantities involved. That's how Ollie generated graphs of them without reference to any particular signals. Again, the differences in question are small due to the high oversampling ratio, so it's going to be quite difficult to see them in macroscopic graphs like this. If you want to see the differences, just make a plot of both sinc^2 and its aliased versions (for whatever oversampling ratios and/or delays), and look at the differences. It won't be interesting for high oversampling ratios and zero delay - which is exactly why that scenario is a poor choice for illustrating the effects in question. The fact that sampling a continuous time signal at a very high rate results in a spectrum that closely resembles the continuous time spectrum (over the sampled bandwidth) is beside the point. It just means that you're operating in a regime where the effects are very hard to spot. It doesn't follow from that resemblance that resampling must be occurring to get a plot of the spectrum of the continuous time signal. E On Fri, Aug 21, 2015 at 10:51 AM, Peter S <peter.schoffhau...@gmail.com> wrote: > On 21/08/2015, Ethan Duni <ethan.d...@gmail.com> wrote: > >>Creating a 22000 Hz signal from a 250 Hz signal by interpolation, is > >>*exactly* upsampling > > > > That is not what is shown in that graph. The graph simply shows the > > continuous-time frequency response of the interpolation polynomials, > > graphed up to 22kHz. No resampling is depicted, or the frequency > responses > > would show the aliasing associated with that. > > It shows *exactly* the aliasing.... > http://morpheus.spectralhead.com/img/interpolation_aliasing.png > > There are about 88 alias images visible on the graph. > The linear interpolation curve is not "smooth", so it contains aliasing. > > > It's just showing the sinc^2 > > response of the linear interpolator, and similar for the other > polynomials. > > If the signal you interpolate is white noise, and the spectrum of the > signal is a flat spectrum rectangle like the one displayed, then after > resampling, you get *exactly* the spectrum you see on the graph, > showing 88 alias images. > > Proof: > I created 60 seconds of white noise sampled at 500 Hz, then resampled > it to 44.1 kHz using linear interpolation. After the upsampling, it > sounds like this: > > http://morpheus.spectralhead.com/wav/noise_resampled.wav > > Its spectrum looks like this: > http://morpheus.spectralhead.com/img/noise_resampled.png > > Looks familiar? Oh, it's the *exact* same graph! (Minus some > difference above 20 kHz, due to my soundcard's anti-alias filter.) It > is an FFT graph of the upsampled white noise, and it shows *exactly* > the aliasing. Good morning! > > > This is what you'd get if you used those interpolation polynomials to > > convert a 250Hz sampled signal into a continuous time signal, not a > > discrete time signal of whatever sampling rate. > > Nope. You get the same graph if you sample that continuous time signal > at a 44.1 kHz sampling rate (with some further aliasing from the > sampling). Just as I've shown. > > Besides, I think the graph was created via numerical means using FFT, > because it has noise at the low ampliutes (marked on the image). > Therefore, it doesn't show a continuous time sinc^2 graph, because > that wouldn't be noisy. > > -P > _______________________________________________ > music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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