Closed-form formula of Cramer–Rao lower bound for 3D TOA target localisation

• Published in: Electronics letters 2020-01, Vol.56 (1), p.43-45
• Main Authors: Li, Yin-Ya, Wang, Chang-Cheng, Qi, Guo-Qing, Sheng, An-Dong
• Format: Article
• Language: English
• Subjects: 3D TOA localisation; 3D TOA target localisation; Cramer–Rao lower bound; CRLB; equivalent closed‐form formulas; Fisher information matrix; geometry; matrix algebra; performance metric; Radar, sonar and navigation; time‐of‐arrival estimation; time‐of‐arrival target localisation; TOA sensors; worse localisation accuracy; worst sensor‐target geometries
• ISSN: 0013-5194; 1350-911X; 1350-911X
• DOI: 10.1049/el.2019.2669

This Letter focuses on the problem of analytically evaluating the performance metric using the Cramer–Rao lower bound (CRLB) for three-dimensional (3D) time-of-arrival (TOA) target localisation. The necessary and sufficient condition on the invertibility (non-singularity) of the Fisher information matrix (FIM) is presented, and then the existence of the CRLB is analysed. Two types of equivalent closed-form formulas of the CRLB are proposed, and any one of which can be adopted to analytically evaluate the best achievable accuracy of 3D TOA localisation. Moreover, three types of the worst sensor-target geometries described by the unit vectors of line-of-sight of TOA sensors are extensively illustrated, which will lead to a worse localisation accuracy or even result in singularities of the FIM and should be avoided in the deployment of 3D TOA sensors.