Estimates of the maximum likelihood of parameters of weak optical signals during photodetecting by means of recording photoelectron emission moments

A likelihood function is obtained for estimating parameters of weak optical signals during detecting by means of recording emission moments of each photoelectron. Likelihood equations are derived and solved by an example of optimal estimates of a Gaussian pulse. Expressions for compatible and incompatible estimates of all three unknown parameters of the pulse are obtained. Analytical expressions for the Kramer-Rao bounds determining the quality of the estimates obtained are also derived. The variances of compatible estimates of the signal amplitude are demonstrated to be three times those of incompatible estimates, while the variances of the estimates of all parameters are inversely proportional to the product of the signal amplitude and its duration.