Two-Step mmWave Positioning Scheme with RIS-Part II: Position Estimation and Error Analysis

In this series of work, we propose a comprehensive two-step three-dimensional (3D) positioning scheme in a millimeter wave (mmWave) system, where the reconfigurable intelligent surface (RIS) is leveraged to enhance the positioning performance of mobile users (MUs). Specifically, the first step is the estimation error modeling and analysis, while the second step is the corresponding positioning algorithm design and bias analysis. The first step is introduced in Part I of this series of work, and the second step is investigated in this paper. Our aim in this series of work is to obtain the closed-form solution of the MU's position through a two-stage weight least square (TSWLS) algorithm. In the first stage, we construct the pseudolinear equations based on the angle of arrival (AOA) and the time difference of arrival (TDOA) estimation at the RISs, then we obtain a preliminary estimation by solving these equations using the weight least square (WLS) method. Based on the preliminary estimation in the first stage, a new set of pseudolinear equations are obtained, and a finer estimation is obtained by solving the equations using the WLS method in the second stage. By combining the estimation of both stages, the final estimation of the MU's position is obtained. Further, we study the theoretical bias of the proposed algorithm by considering the estimation error in both stages. Simulation results demonstrate the superiority of the proposed positioning scheme.