A linear closed-form algorithm for source localization from time-differences of arrival

Microphone arrays often operate in the near field, which complicates the problem of determining a source location from time-difference-of-arrival (TDOA) measurements typically derived from generalized cross-correlation functions. Each TDOA satisfies the equation of a hyperboloid in space and methods have been developed to either solve for intersecting hyperboloids or make some approximation to them, keeping source-location determination a nonlinear, somewhat complex problem. We introduce a closed-form, analytic solution for the problem (the GS algorithm). It is so simple that we were surprised that, until very recently, there have been no other solutions similar to ours. The method uses a minimum of five microphones in three dimensions, one more than other solutions, but, for nonsingular layouts of the microphones, it is very fast and accurate. First, the new method is compared to other closed-form methods for accuracy and sensitivity to noise using simulated data. Then, several variants of GS are compared to two other real-time algorithms, LEMSalg and SRP-PHAT, using real, human-talker data from a large array in a noisy room.