Position calibration of microphones and loudspeakers in distributed computing platforms

We present a novel algorithm to automatically determine the relative three-dimensional (3-D) positions of audio sensors and actuators in an ad-hoc distributed network of heterogeneous general purpose computing platforms such as laptops, PDAs, and tablets. A closed form approximate solution is derived, which is further refined by minimizing a nonlinear error function. Our formulation and solution accounts for the lack of temporal synchronization among different platforms. We compare two different estimators, one based on the time of flight and the other based on time difference of flight. We also derive an approximate expression for the mean and covariance of the implicitly defined estimator using the implicit function theorem and approximate Taylors´ series expansion. The theoretical performance limits for estimating the sensor 3-D positions are derived via the Crame´r-Rao bound (CRB) and analyzed, with respect to the number of sensors and actuators, as well as their geometry. We report extensive simulation results and discuss the practical details of implementing our algorithms in a real-life system.