The effect of pulse pile-up on threshold crossing rates in a system with a known impulse response

The problem of counting rates in a counting detection system is viewed as a stochastic problem of mean threshold crossing rates for a dynamic system driven by a stationary random force. We present a general formula for calculating the rates of such a system at an arbitrary threshold for all values of event occurrence rates, given that the amplitude distribution of the incoming events and the impulse response of the detection system are known. From a single general formula we derive asymptotic expressions for counting rates at both limits of high and low incoming rates. We give a simple expression for the saturation counting rate of a detection system and show that for a high-order pile-up the average intensity and variation of the incoming signal can be determined by measuring the counting rates at two thresholds. For low incoming rates, we show how the unknown incoming distribution can be computed from the measured pulse-height spectrum. Based on the asymptotic results, we demonstrate how to construct an approximation to the impulse response function of the detection system, which facilitates numerical evaluation of the general formula. In each case, we present a comparison with numerical experiments. Throughout the paper, we illustrate how well-known experimental facts can be deduced from a single general formula.