Analytical solutions in amplitude and time measurements from discrete sampling of pseudo-Gaussian signals

Analytical solutions for amplitude and time measurements from digitized signals of the pseudo-Gaussian shape are considered. The least squares method (l.s.m.) is chosen as the algorithm to determine amplitude $A$ and timestamp $t_0$. For a profile with an uncorrelated sampling the l.s.m. is reduced to analytical formulas consistent with the experiment. This permits to estimate the desired number of points $N_s$ on a profile. The obtained results for $N_s$ are illustrated with qualitative estimates in accordance with the Nyquist–Shannon–Kotelnikov sampling theorem. The optimality of the electronic filters forming the waveform is analyzed in terms of the excess noise factor. An analytical solution is found for the autocorrelation function from stochastic noise sources. It permits to define non-diagonal weight matrix elements in the l.s.m. with formulation of requirements for neglecting sampling correlations.