Cramér-Rao Bound Analysis of UWB based Localization Approaches

The problem of localizing one impulse radio (IR) Ultra-wideband (UWB) transmitter from the signals received at several anchors is assessed. In the direct position estimation (DPE) approach the target position is estimated in one step from the received signals from all anchors, while in two steps positioning (2SP) estimation, the times of arrival (TOAs) at different anchors are estimated in a first step, and the associated position estimate is found in a second step. This paper derives the Cramér-Rao Bound (CRB) for both DPE and 2SP in a single-path additive white Gaussian noise (AWGN) ultra-wideband (UWB) channel. In 2SP the amplitudes received at each anchor need to be estimated in order to generate the optimal weighting matrix. Considering the received amplitudes as a new unknown parameter for the 2SP approach gives rise to an expression relating the CRBs for DPE and 2SP. We find that DPE approach always outperforms 2SP approach. The derived bounds are compared with simulations using the corresponding Maximum Likelihood Estimators (MLE).