3.5 COUNTING WITH RANDOM FIELDS
It has been shown that, with deterministic receiver fields, the count variable is Poisson with parameter mv; the latter is dependent on the field energy over the observation volume. When the received field is random, the parameter mv becomes a random variable, and the true count probability must be obtained by subsequent averaging of the conditional Poisson count over the statistics of mv. Thus if mv has the probability density Pmv(m), (0 < m < ∞), the count probability over V is obtained by
This counting probability for k is no longer a Poisson probability, and obviously depends on the probability density of mv induced by the stochastic field. Since the conditional probability in the integand is Poisson, we call the class of probabilities P(k) generated from Eq. (3.5.1) conditional Poisson (CP) probabilities. In the literature they have also been called doubly stochastic Poisson probabilities. Thus, the count probability resulting from the photo-detection of a random field always belongs to the class of CP probabilities. We call the count k a CP random variable. We emphasize again that the density of mv, in general, depends on V, and, therefore, CP densities are generally functions of V and again are nonstationary in time and space.
Although higher moments of the CP count can be also related to the higher moments of mv (see Problem 3.11), the integrations ...