Automatic position calibration of multiple microphones

We describe a method to determine automatically the relative three dimensional positions of multiple microphones using at least five loudspeakers in unknown positions. The only assumption we make is that there is a microphone which is very close to a loudspeaker. In our experimental setup, we attach one microphone to each loudspeaker. We derive the maximum likelihood estimator and the solution turns out to be a non-linear least squares problem. A closed form solution which can be used as the initial guess for the minimization routine is derived. We also derive an approximate expression for the covariance of the estimator using the implicit function theorem. Using this, we analyze the performance of the estimator with respect to the positions of the loudspeakers. The algorithm is validated using both Monte-Carlo simulations and a real-time experimental setup.