Uniform sampling of a periodic signal
You can, in a sense, if you can also sample your signal at an incommensurate sampling rate for the same period.

Any periodic signal can be decomposed into its constituent tones with a DFT and a proper algorithm.  (I have an algorithm I developed about 10 years ago that I have in the pipeline for my blog that I recently found out that McLeod described about 20 years ago.)

So, reduce your question down to a single pure tone.  The frequency for a pure tone, or an alias tone, can be found exactly (see my blog if you doubt this) for any frequency.

So, reduce your question down to a pure tone with a whole number of cycles in your sample frame.

The limitation with a DFT due to the discrete sampling is the aliasing effect.  If a tone is higher than the Nyquist frequency it will appear in the DFT as a frequency less that the Nyquist limit.  So, if a tone appears in bin $k$, where $k &lt; N/2$, the will be a conjugate value in bin $N-k$, which is over the Nyquist limit.  In actuality, you can't tell with the frequency is $k$ cycles per frame, or $N-k$.  The frequency could also be any multiple of N over those.

The possible frequency can be: $k$, $N-k$, $k+N$, $2N-k$ ...

By sampling at two different truly incommensurate rates, the sets of possible frequencies from each of the DFTs will only have one value in common, and that will be the actual frequency.  If the rates are nearly incommensurate, the next possible common value will be well out of range.

Here is a concrete example:


60 Hz Sampling  N = 60

49 Hz Sampling  N = 49

             49                   60 

 5 Hz   5, 44, 54, 93 ...    5, 55, 65, 115 ... 

47 Hz   2, 47, 51, 96 ...   13, 47, 63, 107 ...   

97 Hz   1, 48, 50, 97 ...   23, 37, 63, 97  ...



Now, suppose you thought you could be clever and use every other point of your 60Hz sampling, and every third point as your two different sampling rates.

One second sample frame



30 Hz Sampling  N = 30

20 Hz Sampling  N = 20

Possible Frequencies:

             20                                30             

 5 Hz   5, 15, 25, 35, 45, 55, 65 ...   5, 25, 35, 55, 65 ....  

47 Hz   7, 13, 27, 43, 47, 63, 67 ...  13, 17, 43, 47, 73 ....  



You can't tell whether the 5 Hz is 5, 25, 35, 55, etc.

You can't tell whether the 47 Hz is 13, 43, 47, etc.

So that doesn't work so well.

Hope this helps.

Ced



Followup:

This technique assumes you have "ideal" instantaneous sampling, so it is more theoretical than practical.