Amplitude Distributions of the Poisson Process Transformed by Systems with Memory, with Special Reference to Shot Noise and Pile-Up

Many physical and biological phenomena can be described through processes obtained by a transformation of the Poisson process. Such processes have been simulated by the computer for various amplitude distributions and response functions which are common for processing of signals from nuclear detectors. The following amplitude distributions have been analyzed: delta function, gaussian, exponential, rectangular, and typical nuclear spectra. The system impulse response functions can have arbitrary shapes, such as exponential, RC integrated and/or differentiated, cusp, bipolar, etc. The following cases have been analyzed: continuous time series; pulsed time series (pulsed accelerator case); pulse shaping networks for minimal distortions at high rates; pile-up rejection systems; shot noise; systems with finite resolving time. A large number of cases have been simulated and systematically investigated, and probability density functions for each case are shown.