Maximum likelihood detection of PPM signals governed by arbitrary point-process plus additive Gaussian noise

The maximum likelihood decision statistic for pulse-position modulated (PPM) signals governed by an arbitrary discrete point process in the presence of additive Gaussian noise is derived. Sufficient conditions are given for determining when the optimum PPM symbol detection strategy is to choose the PPM symbol corresponding to the maximum slot statistic. In particular, it is shown that for the important case of Webb distributed avalanche photodiode output electrons in the presence of Gaussian noise, the optimum decision rule is to choose the largest slot observable