Closed-form analytical solutions of the time difference of arrival source location problem for minimal element monitoring arrays.

Closed-form analytical solutions are found for the time difference of arrival (TDOA) source location problem. Solutions are found for both two-dimensional (2D) and three-dimensional (3D) source location by formulating the TDOA equations in, respectively, polar and spherical coordinate systems, with the radial direction coincident with the assumed geodesic path of signal propagation to a reference sensor. Quadratic equations for TDOA 2D and 3D source location based on the spherical intersection (SX) scheme, in some cases permitting dual physical solutions, are found for three and four sensor element monitoring arrays, respectively. A method of spherical intersection subarrays (SXSAs) is developed to derive from these quadratic equations globally unique closed-form analytical solutions for TDOA 2D and 3D source location, for four and five sensor element monitoring arrays, respectively. Errors in 2D source location for introduced bias in time differences of arrival are shown to have a strong geometrical dependence. The SXSA and SX methods perform well in terms of accuracy and precision at high levels of arrival time bias for both 2D and 3D source location and are much more efficient than nonlinear least-squares schemes. The SXSA scheme may have particular applicability to accurately solving source location problems in demanding real-time situations.