Information, error, and imaging in deadtime-perturbed doubly stochastic Poisson counting systems

The detection of a fluctuating signal in the presence of noise is considered for a doubly stochastic Poisson counting system that is subject to fixed nonparalyzable detector deadtime. Explicit expressions are obtained for the likelihood-ratio detection of a modulated source of arbitrary statistics in the presence of Poisson noise counts. Receiver operating characteristics (ROC curves) are presented for an unmodulated (amplitude-stabilized) source with detector dead-time as a parameter; increasing deadtime causes a decrease in the probability of detection for a fixed false-alarm rate. Probability of error curves are presented for an amplitude-stabilized source, both in the absence of modulation and in the presence of triangular modulation, illustrating the deleterious effects of modulation, noise, and deadtime on receiver performance. Expressions for the average mutual information and channel capacity of the system are obtained and graphically presented for the simple counting receiver and for the maximum-likelihood counting receiver; the channel capacity decreases with decreasing signal level and with increasing deadtime and modulation depth. Representative examples of the appropriate counting distributions are provided. Finally, a maximum-likelihood estimate of the mean signal level is obtained for a simple image detection system with a deadtime-perturbed counting array. An expression for the statistical confidence level of the estimate is also obtained. The results are valid for an arbitrary deadtime-perturbed doubly stochastic Poisson counting system and as such are expected to find application in a broad variety of disciplines including photon counting and lightwave communications, operations research, nuclear particle counting, and neural counting and psychophysics.