What happens *just* under the Nyquist limit

[snatchingthepi]

TL;DR Summary

What does the reconstructed wave look like if we sample the input an infinitesimal amount under the Nyquist limit?

What does the reconstructed wave look like if we sample the input an infinitesimal amount under the Nyquist limit? I can intuitively picture how we can (ideally) reconstruct an input sampled at the Nyquist limit (and appropriate phase) because we are able able to get the extreme values of the input. But what if we sampled at the tiniest little itty bit under that limit?

My mind is showing me a reconstructed wave that eventually damps out, like an overdamped oscillator, but I can't justify that to myself. Does anyone know?

 

[tech99]

The idea of sampling is exactly the same the action of a modulator or mixer, when the local oscillator is the sampling frequency. The clock must be twice the highest frequency so that all the difference frequencies lie above the passband. If the LO is below this frequency, the higher input frequencies produce a difference frequency that lies within the passband. The spectrum is actually inverted, and we have severe distortion.

 

[eq1]

When you sample below Nyquist, during waveform reconstruction there will be a bunch of frequencies that all could've produced your set of samples, and you won't be able to tell which one, or ones, it was. [*] That set forms the aliases. The amount you sample below Nyquist will select which frequencies are in that set.

It's a similar idea to a beat. When two signals beat, as the two sinusoids get closer in frequency, the beat period gets longer. In your example, as the sample period gets closer to Nyquist, the frequencies in the alias set will get farther apart.

[*] I should mention this is from a pure math point of view. If one considers the physical circuit and real-life sampling system then some frequencies can likely be removed from the alias set. This is the idea behind an anti-alias filter, for example.

 

[tech99]

As far as I know, an anti-alliassing filter is an LPF which prevents frequencies higher than half the clock frequency from reaching the sampling device.

 

[eq1]

That's usually how an anti-aliasing filter is implemented but it doesn't explain why one needs to do it. I was trying to explain why one needs it.

See: https://en.wikipedia.org/wiki/Aliasing

Especially this line:
"When sampling a function at frequency fs (intervals 1/fs), the following functions yield identical sets of samples: {sin(2π( f+Nfs) t + φ), N = 0, ±1, ±2, ±3,...}"

So the idea of the anti-aliasing filter is to make the set of functions given above empty. In general, people don't worry about the negative Ns, because CW vs CCW rotation won't typically matter for most algorithms. But then there are the N=0,1,2,3,... terms to worry about. Since a LPF eliminates high F, if the knee of the filter is placed below f the set will be empty. Side note: This is why the basic RC LPF used as AAF with knee at Nyquist is usually an error. The 3db should be placed below because if you don't the N=0 term only gets 3db of attenuation which was probably not what was wanted.

 

[snatchingthepi]

Thank you all for the wonderful replies.

I will specify my question further. I am reading Easton's "Fourier Methods in Imaging" and on page 479 I have the following statement that I am having a hard time visualizing.

"...I'd the spatial frequency of the cosine is infinitesimally smaller than the Nyquist sampling frequency, then the amplitudes of alternate samples will not be identical: they will "march" down the sinusoid, eventually reaching the maximum amplitude at some sample index n_max and the minimum amplitude at a different (and possibly very distant) sample index n_min. The reconstructed function will have the correct extreme values at these samples, and the correct amplitude will be interpolated to the other coordinates by convolution with the ideal interpolator sinc[x/delta(x)]."

 

[tech99]

I think he is saying that the resulting beat has the correct peak values, which could be used for measurement purposes, but of course, it represents distortion and is of no obvious benefit.

 

[Baluncore]

Consider a sinewave, sample it at exactly twice that sinewave frequency. The record will depend on the initial phase. The two samples could be taken at 0 and π and show no output amplitude, or it could be sampled at π/2 and 3π/2 showing a full amplitude signal.

If the sinewave frequency was different by 1 Hz, the phase would drift through a full cycle over one second, generating a 1 Hz envelope, filled with a fundamental that is 1 Hz below half the sampling frequency.

If the frequency was different by an infinitesimal amount, the phase and so the envelope would drift through a full cycle over an infinite period starting at an unknown phase.

Depending on whether the sinewave frequency was higher or lower than half the sampling frequency, the sampled record would appear forward or backward in time, which is the conjugate in the frequency domain.

 

[snatchingthepi]

Exactly this! Thank you so much.

 

[sophiecentaur]

Consider a sinewave, sample it at exactly twice that sinewave frequency. The record will depend on the initial phase. The two samples could be taken at 0 and π and show no output amplitude, or it could be sampled at π/2 and 3π/2 showing a full amplitude signal.

If the sinewave frequency was different by 1 Hz, the phase would drift through a full cycle over one second, generating a 1 Hz envelope, filled with a fundamental that is 1 Hz below half the sampling frequency.

If the frequency was different by an infinitesimal amount, the phase and so the envelope would drift through a full cycle over an infinite period starting at an unknown phase.

Depending on whether the sinewave frequency was higher or lower than half the sampling frequency, the sampled record would appear forward or backward in time, which is the conjugate in the frequency domain.

Taking things to a limit in this is not always a fruitful way to understand a practical system. The basic Nyquist theorem only says that the original waveform 'can' be reconstructed but doesn't comment on the complexity - or the necessary time delay in a system operating very near f_N.
A practical Nyquist pre-filter is not a brick wall but designed to be 'gentle' enough to minimise this effect and the resulting 'beat' can be eliminated with only a short delay.