An algebraic solution to the time difference of arrival equations

A closed form, algebraic solution to the time difference of arrival equations is presented. Prior derivations assumed that the receiving antennas are coplanar. Our derivation allows the receiving antennas to be located anywhere in three dimensional space. Previous techniques address the over-determined problem by considering a weighted average of the set of minimally determined solutions. Our technique employs the generalized inverse as a means of forming an optimal solution based on the geometry of the receiving sites. Our solution is additionally compared with a gradient search method to demonstrate its ability to avoid local minima. Finally, analysis is presented to demonstrate the effect of noisy measurements on the solution.