Exact and approximate maximum likelihood localization algorithms

Sensors at separate locations measuring either the time difference of arrival (TDOA) or time of arrival (TOA) of the signal from an emitter can determine its position as the intersection of hyperbolae for TDOA and of circles for TOA. Because of measurement noise, the nonlinear localization equations become inconsistent; and the hyperbolae or circles no longer intersect at a single point. It is now necessary to find an emitter position estimate that minimizes its deviations from the true position. Methods that first linearize the equations and then perform gradient searches for the minimum suffer from initial condition sensitivity and convergence difficulty. Starting from the maximum likelihood (ML) function, this paper derives a closed-form approximate solution to the ML equations. When there are three sensors on a straight line, it also gives an exact ML estimate. Simulation experiments have demonstrated that these algorithms are near optimal, attaining the theoretical lower bound for different geometries, and are superior to two other closed form linear estimators.