- the signal being sampled has to be stationary
- you have an infinite number of samples
In that case, a sampling frequency of 2N+epsilon will perfectly reproduce the signal. Otherwise there can be issues.
I wonder, what are some examples of signals that fail to reproduce after sampling in a way that is "nearly Nyquist"?
There are many signals of practical interest that can be approximately reconstructed with a finite truncation of the series. Note, however, that any signal that has only a finite length, eg has a uniformly zero amplitude after some time t_final, does not have a finite bandwidth, and cannot be exactly reconstructed by any sampling scheme. This is the case whenever you stop sampling a signal, eg it is always the case whenever you step outside the mathematical abstraction and start running real code on a real computer. So any signal reconstructed from samples is always approximate, except for some relatively trivial special cases.
What if you take a function that has bounded support in the time domain, and then turn it into a periodic function? Might the resulting function have bounded support in the frequency domain even though the original function did not? I suppose doing this would force the Fourier transform to have discrete support? But under what conditions would it have bounded support?…
I guess technically a low-pass filter applied to a signal with finite support in the time domain, would result in a function which has infinite support in the time domain.
I suppose sinc(f t + c) doesn’t have bounded support, and it is unsurprising that a non-trivial linear combination of finitely many terms of this form would also not have finite support.
Still, such a linear combination could decay rather quickly, I imagine. (Idk if asymptotically faster than (1/t) , but (1/(f t)) is still pretty fast I think, for large f.)
Soon enough the decay should be enough that the amplitude should be smaller than the smallest that the speaker hardware is capable of producing, I suppose.
When you perform a finite sample reconstruction, this is essentially the unstated approximation you’re making.