Optical Communications, 2nd Edition

3.5 COUNTING WITH RANDOM FIELDS

It has been shown that, with deterministic receiver fields, the count variable is Poisson with parameter m_v; the latter is dependent on the field energy over the observation volume. When the received field is random, the parameter m_v becomes a random variable, and the true count probability must be obtained by subsequent averaging of the conditional Poisson count over the statistics of m_v. Thus if m_v has the probability density P_mv(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 m_v 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 m_v, 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 m_v (see Problem 3.11), the integrations ...