Numerical formulae for TOA-based microphone and source localization

This paper presents numerical formulae for the Time-of-Arrival (TOA)-based microphone and source localization problem, which determines the positions of microphones and sources based on distances between each microphone and each source respectively. This is a purely geometrical problem in mathematics. Concretely, we show when the number of microphones or the number of sources is at least nine, the formulae of the microphone positions and source positions are given simply from the distance-matrix. A similar statement is given in the cases of at least eight microphones or sources if we know an extra information about the distance between any two microphones or two sources. The accuracy of these formulae are proven shortly by ten thousand independent experiments of which coordinates of points have an independent Uniform distribution.