I was learning about the Nyquist theorem regards signal processing the area of interest which I will rephrase below:
Given a signal lasting infinitely long with a maximum frequency of f, then you can always replicate the signal perfectly by sampling by AT LEAST 2f.
However I quickly thought up what seemed to be a counter example which I cannot shake off:
Take a sine signal of frequency of 1Hz. This is the only frequency of such a wave so according to the theorem I can replicate the signal as long as I sample at 2Hz intervals. A valid set of samples would be to sample at every point the signal crosses the x axis which is sampling at 2f satisfying the condition. (At 0 amplitude)
This gives 0 for every sample. However this tells you nothing about the signal - it could be the flat signal of 0 amplitude, the same sine wave of double or half amplitude. Hence this seems to be a counter example to the theorem suggesting the same frequency needs to be greater than 2f.
Could someone please either explain the theorem (It may be I am misunderstanding the theorem) or confirm this counter example.
Thanks!