Derivation of CRLB for linear least square estimator in wireless location systems

Abstract

Linear estimators, including the well-adopted linear least squares (LLS) estimator, have been extensively utilized for wireless location estimation for their simplicity and closed-form property. However, there exists information lost from the linearization of the location estimator to the nonlinear location estimation, which prevents the linear estimator from approaching the Cramer-Rao lower bound (CRLB). In this paper, the linearized location estimation problem based CRLB (L-CRLB) is derived to provide a portrayal in order to fully characterize the behavior of the linearized location estimator. The relationships between the L-CRLB and the CRLB are obtained and theoretically proven in this paper. Furthermore, the geometric layout between the mobile station (MS) and the base stations (BSs) that can achieve the minimum L-CRLB is also acquired. As can be suggested by the L-CRLB, an LLS location estimator can achieve higher accuracy if the MS is located inside the geometry confined by the BSs compared to the case that the MS is situated outside of the geometric layout. This result will be beneficial to the deployment of BSs or the signal selection schemes targeting for location estimation. Simulation results utilizing the LLS estimator as one of the implementation of the linearized location estimators further validate the theoretical proofs and the effectiveness of the L-CRLB.