Separating event points by Binary Proximity Sensors: An asymptotic analysis

Let n points be chosen in a sensing area and let identical events of interest occur only in these n chosen points. Binary Proximity Sensors are used to estimate which of these n points had events occurring in them. We restrict to at most one event per event point. Assume that the sensors are identical. The number of sensors dropped and the sensing radius are the two design parameters. We analytically derive the necessary and sufficient conditions on the two parameters to ensure that any of the 2ⁿ event configurations are decodable from sensor observations. The necessary and sufficient conditions are derived for various settings of the event-points and sensor deployments. These results have been derived as scaling laws, i.e., these laws are initially derived for n; and then conditions required if n → ∞ are calculated. We have also proposed the extension to higher dimensions from the 1-D case and we also pose a problem similar to the information theoretic Rate-Distortion problem.