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The literature on this method seems scarce, but I know that it has been used on radar systems to 'get away with' not having to sample nearly as fast as Nyquist would otherwise dictate, (at the cost of additional time). Or so I think.

So, how does "Equivalent Time Sampling" work, and where can it be used? How does it square with the Nyquist Criterion?

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Note that the Nyquist criterion applies to bandlimited signals; but the so-limited band does not need to be baseband, or even contiguous, just not overlapping (aliased) with any other sub-bands after folding.

Periodic sampling fan-folds an infinite spectrum in to "fans" with a width of half the sampling rate. Any frequency bands, after this fan-folding, that do not overlap with any other frequency bands in the spectrum when so folded can be reconstructed, according to one form of Nyquist theorem.

In practice, any random phase jitter in undersampling will cause the bands too far above Fs/2 to become more phase noise than signal. If the phase noise is mostly white and uncorrelated with the signal, then sampling for a much longer time frame may allow averaging out this phase noise sufficient to reconstruct a repetitive narrow-band signal, with some probability of accuracy for a given extended sampling time window.

So, some oscilloscopes have a mode where they undersample a bandpass-filtered narrowband input, then store and composite a much longer sampling time by folding the samples down to baseband.

In addition, signal reconstruction of a bandlimited signal may be possible with non-periodic sampling of known sample times, as long as the local density of samples is high enough. This is numerically much less stable, as estimating a reconstruction of the spectrum or waveform from a minimum number randomly spaced samples becomes more of a sensitive linear optimization problem, rather than a closed form transform or convolution. However some oscilloscopes might use a pseudo-random non-periodic undersampling, but averaged over a long time period, and let the user determine when the density of samples "looks good enough" when reconstructing an image of an undersampled repetitive waveform.

Randomized sampling can avoid certain aliasing problems, such as perfectly periodic sampling not allowing reconstruction of an arbitrary phase sinusoid with a frequency exactly at Fs/2 (it's smooshed at the creases in the fan-fold).

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    $\begingroup$ Thanks hotpaw2. I understand your post, however isn't what you described is also referred to as 'under sampling'? (As in, sampling at least double a pass band signal's bandwidth and reconstructing it at baseband by using its aliased version). If so, do 'under sampling' and 'equivalent time sampling' essentially mean the same thing? Is this just a terminology mash up? $\endgroup$ – Spacey May 2 '12 at 19:38
  • $\begingroup$ Not quite. The terminology seems to not apply to undersampling alone, but to the combination of undersampling with a form of non-periodic sampling. $\endgroup$ – hotpaw2 May 3 '12 at 12:42
  • $\begingroup$ To get the effect you describe, the sampling itself doesn't need to be non-periodic. When observing a periodic signal, you can get the desired effect while still sampling uniformly as long as the signal period is not an integer multiple of the sampling period. $\endgroup$ – Jason R May 3 '12 at 13:45

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