Cram\'er-Rao Bounds of Near-Field Positioning Based on Electromagnetic Propagation Model

The adoption of large-scale antenna arrays at high-frequency bands is widely envisioned in the beyond 5G wireless networks. This leads to the near-field regime where the wavefront is no longer planar but spherical, bringing new opportunities and challenges for communications and positioning. In this paper, we improve the near-field positioning technology from the classical spherical wavefront model (SWM) to the more accurate and true electromagnetic propagation model (EPM). A generic near-field positioning model with different observation capabilities for three electric field types (vector, scalar, and overall scalar electric field) is developed based on the complete EPM. For these three observed electric field types, the Cram\'er-Rao bound (CRB) is adopted to evaluate the achievable estimation accuracy. The expressions of the CRBs for different electric field observations are derived by combining electromagnetic propagation concepts with estimation theory. Closed-form expressions can be further obtained as the terminal is assumed to be on the central perpendicular line (CPL) of the receiving antenna surface. Moreover, the above discussions are extended to the system with multiple receiving antennas. In this case, the CRBs using various electric field types are derived, and the effect of different numbers of receiving antennas is deeply investigated. Numerical results are provided to quantify the CRBs and validate the analytical results. Also, the impact of different system parameters, including electric field type, wavelength, size of the receiving antenna, and number of antennas, is evaluated.