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

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 (7121.11 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 FDA and should be avoided in the deployment of 3D TOA sensors.