Difficulties acquiring sample period of a signal
I think you are mixing up two things here. By "best sampling rate", I suppose your concern is how to select the $\Delta f$ in the spectrum such that the time-domain signal could be fully reconstructed from the sampled spectrum. This can be answered by the dual of the sampling theorem.

Consider the Fourier transform pair $s(t)\xrightarrow{\mathcal{F}}S(f).$ Let me first review the sampling theorem:


  Sampling in time with sampling rate $F_s=\frac{1}{\Delta t}$ causes replication in the frequency domain at every $F_s$ (i.e. frequency aliasing). To prevent frequency aliasing, $s(t)$ should be bandlimited to $B$, and the sampling interval should satisfy $$\Delta t=\frac{1}{F_s}&lt;\frac{1}{2B}$$
  the signal's spectrum is recovered by filtering the replicated spectrum.


The dual theorem states that If the sampling interval in frequency domain is not short enough, it will result in time aliasing: 


  Sampling in frequency with rate $\frac{1}{\Delta f}$ causes replication in the time domain at every $\frac{1}{\Delta f}$ (i.e. time aliasing). To prevent time aliasing, $s(t)$ should be time-limited to $T$ and the sampling interval should satisfy $$\Delta f&lt;\frac{1}{2T}$$ The signal at the time domain can be recovered by time-windowing the replicated signal.


Getting back to your signal $s_1(t)$, since it is not time-limited (is not theoretically zero outside a time horizon), you should truncate it somewhere (e.g. at $T$). Then, it will satisfy the criteria and you can choose $\Delta f=\frac{1}{2T}$.