On the bounds of separability in sensor networks

A pair of target locations are separable if sensor observations can distinguish between the following choices: no targets are present, one target is present at either of the locations or a target is present at each location. The sensors of interest in this paper are binary proximity sensors, whose binary outputs are functions of the distance between the sensor and target. Sensors are deployed randomly according to a Poisson distribution. The probability that two target locations at a distance of r between them are separable is derived. This is extended to derive the probability of having at least Z among M uniformly distributed target locations to be non-separable from the origin. The bounds on this probability are expressed as a function of the sensor density λ.