RIS- and Multi-Snapshot-Enabled SISO 3D Position and Velocity Estimation With Single Base Station

Reconfigurable intelligent surface (RIS) panels can act as cost-effective anchors for radio localization, complementing conventional base station (BS) anchors. This paper investigates joint three-dimensional position and velocity estimation (3D-JPVE) in single-input single-output (SISO) systems with only one BS available. We first theoretically show that 3D-JPVE is infeasible when relying solely on a single RIS or on multiple snapshots alone. To address this, we propose combining RIS deployment with multi-snapshot utilization to enable realizable 3D-JPVE. A two-stage method is developed for multi-snapshot channel parameter estimation, comprising a tensor-based coarse estimation step followed by a maximum likelihood refinement step. In particular, we introduce a third-order tensor formulation to decompose the challenging 3D joint angle-of-departure and Doppler shift estimation (3D-JADE) into two tractable subproblems, which are jointly solved via a low-complexity alternating optimization approach. Building on the channel parameter estimates, we further design a two-stage low-complexity method for optimal 3D-JPVE: coarse estimation is obtained from differential measurements through linear equations, and the preliminary results are refined iteratively using the original measurements. Moreover, we derive the closed-form Cramer-Rao lower bound (CRLB) and show that the proposed 3D-JPVE method approaches CRLB-level accuracy. Simulation results confirm the statistical efficiency of the proposed estimators and demonstrate substantial 3D-JPVE performance gains when deploying active RIS compared to passive RIS.