Theoretical facts on RSSI-based geolocation

We address the problem of locating a stationary emitter using the Received Signal Strength (RSS) at receivers with known locations. The Maximum Likelihood estimator for the emitter location requires the minimization of a non-convex cost function. Since this cost function exhibits numerous local minima, its global minimization is usually realized by means of a grid search and is therefore computationally expensive. In this document, we prove three novel theoretical properties of RSS-based cost functions for Maximum Likelihood localization. First, we show that local maxima of RSS-based cost functions occur at receivers locations. Thus, unlike local minima, the locations of local maxima are a-priori known since the receivers locations are known. Second, we show that the smallest local maximum is necessarily closer to the global minimum than any other local minimum. Third, we show that the global minimum of the non-convex cost function lies within a triangular area defined by the smallest local maxima. Combining these theoretical facts, we propose a procedure for delimiting a small geographical area that contains the global minimum of the cost function, with high probability. Therefore, localization can be achieved by grid search over this reduced area only, which significantly reduces computational costs.