From Theoretical to Practical Source Placement for Active Geometry Calibration of Distributed Microphone Arrays

Distributed microphone arrays (DMAs) are widely used in various audio/speech processing applications, such as target localization, beamforming, and event monitoring. These applications typically require the geometric structure of DMAs as a priori knowledge, thus geometry calibration (also known as self-localization) is often employed to estimate microphone positions (MPs). Recent advances in geometry calibration adopt a set of randomly distributed calibration sources, which may result in a relatively high Cramér-Rao lower bound (CRLB). Alternatively, in this work, we consider an active geometry calibration framework, where a mobile robot acts as a movable calibration source. A lower CRLB of geometry calibration can be achieved by strategically guiding the robot to emit calibration signals at potentially optimal positions. To establish a theoretical foundation, we first derive the optimal CRLB (oCRLB) for synchronous and asynchronous DMAs using time-of-arrival (ToA) measurements in both 2D and 3D settings. Accordingly, the closed-form solutions for the optimal source placement are obtained in some cases, while the numerical methods are employed otherwise. However, the oCRLB hinges on perfectly known MPs, an assumption that is fundamentally incompatible with geometry calibration. To remove this assumption, we formulate the CRLB under MP uncertainty and develop a practical source placement approach. This study fills the gap in both theoretical analysis and practical design of source placement for DMA geometry calibration. Simulation results confirm the theoretical findings and demonstrate the effectiveness of the proposed methods.