Optimal sensor configuration for passive position estimation

The goal of passive source localization is to acoustically detect objects producing noises by multiple sensors (e.g. microphones, hydrophones) and to estimate their position using only the sound information. While within the last four decades a lot of work was carried out on how to best measure the time delay of arrivals (TDOAs) and on finding an optimal location estimator, relatively little work can be found on how to best place the sensors. However, the performance of such estimators is strongly correlated to the sensor configuration. Therefore, we propose a procedure for an optimal sensor setup minimizing the condition numbers of an analytic linear least-squares (LLS) estimator and an iterative, linearized model (LM) estimator. An advantage of using the condition number as the cost function is that, unlike the Cramer Rao Lower Bound, it defines an upper bound for the estimation error. Further, no assumptions about the disturbance noise need to be made and a robust sensor configuration will be found, which is invariant to rotation and dilatation. The two condition numbers of the presented passive source localization algorithms are independent of the number of sensors. However, it will be shown, that the estimation error decreases proportionally to the inverse of the square-root of the number of sensors. Some analytical forms of optimal sensor configurations will be derived, which attain the global minimum of the condition number of the LLS estimator or which minimize the condition number of the LM estimator. Further, a sensor geometry using a minimum number of sensors is derived, which forces the condition numbers of both estimators equal to one. The interest of such a setup lies in a possible combination of both estimators. The LM estimator might then be initialized by the position estimate found by the LLS estimator. A variety of alternative estimators are closely related to the LLS estimator. Their performances will be compared, and it will be shown, that the optimal sensor geometry specially derived for the LLS estimator also increases their accuracies.