Cramér-Rao Lower Bound Analysis of Positioning with Planar Large Intelligent Surfaces Under Rician Channel

In this paper we derive the Fisher information matrix (FIM) and Cramér-Rao lower bound (CRLB) for positioning a terminal with a planar large intelligent surface (LIS), under Rician channel. For a disk-shaped continuous LIS and a terminal located on the central perpendicular line (CPL) of the LIS, we obtain expressions for the CRLBs in the form of a single integration. For the situation that the terminal is far from the LIS compared to the LIS radius and the situation with an asymptotically large LIS, closed-form approximations of the CRLBs are derived, based on which scalings and properties of the positioning precision with respect to different system parameters are obtained. For positioning a terminal with arbitrary location, we derive closed-form expressions of the CRLBs when the terminal is far from the LIS and the wavelength is small. When the surface area is small and the path-loss exponent is not larger than 5, the CRLBs of a CPL-terminal are smaller than those of a non-CPL terminal for all three dimensions, while the reverse may occur when the surface area grows large enough. We also carry out a comparative study of the continuous and discrete models of the LIS. Numerical results are presented to validate the precision of our theoretical analysis and approximated performance.