摘要： In this paper, we study the time-of-arrival (TOA) based self-calibration problem of dual-microphone array for known and unknown rack distance, and also for different combinations of dimension for the af?ne spaces spanned by the receivers and by the senders. Particularly, we analyze the minimum cases and present minimum solvers for the case of microphones and speakers in 3-D/3-D, in 2-D/3-D, and in 3-D/2-D, with given or unknown rack length. We identify for each of these minimal problems the number of solutions in general and develop ef?cient and numerically stable, non-iterative solvers. Solving these problems are of both theoretical and practical interest. This includes understanding what the minimal problems are and how and when they can be solved. The solvers can be used to initialize local optimization algorithms for ?nding the maximum likelihood estimate of the parameters. The solvers can also be used for robust estimation of the parameters in the presence of outliers, using, e.g., RANSAC algorithms. We demonstrate that the proposed solvers are numerically stable in synthetic experiments. We also demonstrate how the solvers can be used with the RANSAC paradigm. We also apply our method for several real data experiments, using ultra-wide-band measurements and using acoustic data.