Linear Detection in a Poisson Regime with Random Pulse Heights

The detection of light signals using photomultipliers is presented in terms of an idealized model in which the received signal consists of a series of pulses of random height with arrival times that are Poisson distributed with an instantaneous rate proportional to transmitted signal intensity. Linear detection, effected through multiplication by locally generated functions followed by integration, is considered. An optimization algorithm based on Chernoff bounds for the error probability is developed. For the standard Poisson regime with constant pulse heights this algorithm derives correctly the logarithmic local functions predicted rigorously for maximum likelihood detection. For random pulse heights the functions resemble smoothly truncated logarithmic functions of the transmitted signal intensities, and the logarithmic singularity for zero values of the signal is either removed or greatly weakened. Calculations for several classes of pulse height distributions show that the degradation in performance caused by pulse height fluctuations can be predicted with fair accuracy from an "excess noise factor" defined in terms of the ratio of pulse height variance to the square of its mean.