Estimation of a Low-Intensity Filtered Poisson Process in Additive White Gaussian Noise

A Poisson–Gauss process is defined as the sum of a filtered Poisson process (a Poisson process passed through a linear filter) and a white Gaussian noise. This model describes the electrical signal at the output of an optical measuring unit consisted of a photodetector and its following amplifying circuits. The intensity of the Poisson process in this model is proportional to the received optical power. With the observations of a Poisson–Gauss process, three estimation problems are considered: minimum mean squared error estimation of the Poisson process at every fixed but arbitrary time, minimum mean squared error estimation of the Poisson intensity, and the maximum likelihood estimation of the intensity. The solutions to these problems are presented in terms of a complicated functional of the observed Poisson–Gauss process which is hard to compute for an arbitrary value of the Poisson intensity; however, under a low-intensity regime, nonlinear filtering schemes are developed to efficiently compute this functional. This special case provides a signal processing framework for single photon detection, a technology dedicated to measurement of low optical powers.