>What's causing you to be unable to reconstruct the waveform? There are an infinite number of different nyquist-frequency sinusoids that, when sampled, will all give the same ...,1, -1, 1, -1, ... sequence of samples. The sampling is a many-to-one mapping in that case, and so cannot be inverted.
See here: https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Critical_frequency Or consider what happens if you shift a nyquist-frequency sinusoid by half a period before sampling it. You get ..., 0, 0, 0, 0, ... - which is quite obviously the zero signal. It is not going to reproduce a nyquist frequency sinusoid when you run it through a DAC. E On Tue, Aug 18, 2015 at 1:28 PM, Peter S <peter.schoffhau...@gmail.com> wrote: > On 18/08/2015, Ethan Duni <ethan.d...@gmail.com> wrote: > >>Assume you have a Nyquist frequency square wave: 1, -1, 1, -1, 1, -1, 1, > > -1... > > > > The sampling theorem requires that all frequencies be *below* the Nyquist > > frequency. Sampling signals at exactly the Nyquist frequency is an edge > > case that sort-of works in some limited special cases, but there is no > > expectation that digital processing of such a signal is going to work > > properly in general. > > Not necessarily, at least in theory. > > In practice, an anti-alias filter will filter out a signal exactly at > Nyquist freq, both when sampling it (A/D conversion), and both when > reconstructing it (D/A conversion). But that doesn't mean that a > half-sample delay doesn't have -Inf dB gain at Nyquist frequency. It's > another thing that the anti-alias filter of a converter will typically > filter it out anyways when reconstructing - but we weren't talking > about reconstruction, so that is irrelevant here. > > A Nyquist frequency signal (1, -1, 1, -1, ...) is a perfectly valid > bandlimited signal. > > > But even given that, the interpolator outputting the zero signal in that > > case is exactly correct. That's what you would have gotten if you'd > sampled > > the same sine wave (*not* square wave - that would imply frequencies > above > > Nyquist) with a half-sample offset from the 1, -1, 1, -1, ... case. > > More precisely: a bandlimited Nyquist frequency square wave *equals* a > Nyquist frequency sine wave. Or any other harmonic waveform for that > matter (triangle, saw, etc.) In all cases, only the fundamental > partial is there (1, -1, 1, -1, ... = Nyquist frequency sine), all the > other partials are filtered out from the bandlimiting. > > So the signal 1, -1, 1, -1, *is* a Nyquist frequency bandlimited > square wave, and also a sine-wave as well. They're identical. It *is* > a bandlimited square wave - that's what you get when you take a > Nyquist frequency square wave, and bandlimit it by removing all > partials above Nyquist freq (say, via DFT). You may call it a square, > a sine, saw, doesn't matter - when bandlimited, they're identical. > > > The > > incorrect behavior arises when you try to go in the other direction > (i.e., > > apply a second half-sample delay), and you still get only DC. > > What would be "incorrect" about it? I'm not sure what is your > assumption. Of course if you apply any kind of filtering to a zero DC > signal, you'll still have a zero DC signal. -Inf + -Inf = -Inf... Not > sure what you're trying to achieve by "applying a second half-sample > delay"... That also has -Inf dB gain at Nyquist, so you'll still have > a zero DC signal after that. Since a half-sample delay has -Inf gain > at Nyquist, you cannot "undo" it by applying another half-sample > delay... > > > But, again, that doesn't really say anything about interpolation.It just > > says that you sampled the signal improperly in the first place, and so > > digital processing can't be relied upon to work appropriately. > > That's false. 1, -1, 1, -1, 1, -1 ... is a proper bandlimited signal, > and contains no aliasing. That's the maximal allowed frequency without > any aliasing. It is a bandlimited Nyquist frequency square wave (which > is equivalent to a Nyquist frequency sine wave). From that, you can > reconstruct a perfect alias-free sinusoid of frequency SR/2. > > What's causing you to be unable to reconstruct the waveform? > > -P > _______________________________________________ > music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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