Closed-form formulae for time-difference-of-arrival estimation

For a positioning system with L sensors, a maximum of L(L - 1)/2 distinct time-difference-of-arrival (TDOA) measurements, which are referred to as the full TDOA set, can be obtained. In this paper, closed-form expressions regarding optimum conversion of the full TDOA set to the nonredundant TDOA set, which corresponds to (L - 1) TDOA measurements with respect to a common reference receiver, in the case of white signal source and noise, are derived. The most interesting finding is that optimum conversion can be achieved via the standard least squares estimation procedure. Furthermore, the Cramér-Rao lower bound for TDOA-based positioning is produced in closed-form, which will be useful for optimum sensor array design. © 2008 IEEE.