Thanks for an interesting read!

Regarding the article ... there actually is one small issue with digital sampling at twice the frequency of interest (i.e. the Nyquist frequency). While the Sharon-Nyquist Theorem correctly states that the frequency information is maintained, the phase and/or amplitude information is obscured. This was a lesson that one of my advanced controls professors taught me ... I hate to admit how long I had been ignorant of this ... Here is the simple way to convince yourself of this

1) Draw a simple cosine starting at time = 0

2) Draw the result of sampling that cosine starting at time zero at twice the cosine's frequency ... great! you just produced a string of 1, -1, 1, -1 ....

3) Now perform the same task as "2" but offset your initial sample point by just a little bit past t=0 ... now you have (for example) 0.9, -0.9, 0.9, -0.9 ... if you only have this information, how can you distinguish between a phase and an amplitude offset?

4) Of course the degenerate case is if you offset your sample by 90 degrees, where you get all zeros

The problem is that you have enough information to reproduce the frequency but not the phase & amplitude of the signal ... I am purposely avoiding the time-frequency analysis discussion for now because I need to get back to my thesis :)

In practice, one needs at least 4 times the sampling rate relative to the frequency of interest (2 for frequency, 1 for phase, 1 for amplitude ... in controls the rule of thumb is 10 to 20 times but these are noisier systems) to reproduce the signal's frequency, phase & amplitude ... and yes we're typically pretty crumby at absolute phase information, but the phase of one frequency relative to the other can be considered here as well.

I hope that this helps!

Thanks again for a great read!