## 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*becomes a random variable, and the true count probability must be obtained by subsequent averaging of the conditional Poisson count over the statistics of

_{v}*m*. Thus if

_{v}*m*has the probability density

_{v}*P*(

_{m}_{v}*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*, in general, depends on

_{v}**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 ...

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