Optimum sensor placement for localization in three dimensional under log normal shadowing

In scenarios of wireless source localization using sensor networks, the geometry of the sensor nodes in the network heavily influences the accuracy of the source location estimate. This paper thus considers the optimal sensor placement in three dimensions so that a source can be localized optimally from the Received Signal Strength (RSS) at various non-coplanar sensors, under Lognormal Shadowing. We assume that the source is uniformly distributed in a sphere and the sensors must be placed outside a larger concentric sphere to avoid proximity to the hazards posed by the source. The mathematical problem becomes one of maximizing the smallest eigenvalue or the determinant of the expectation of an underlying Fisher Information Matrix (FIM), or minimizing the trace of the inverse of this matrix, subject to the above constraint. We show that the optimality is achieved if and only if the expectation of the underlying FIM is a scaled diagonal, and provide methods for achieving this condition.