A Closed-Form Localization Algorithm and GDOP Analysis for Multiple TDOAs and Single TOA Based Hybrid Positioning

Cellular communication systems support mobile phones positioning function for Enhanced-911 (E-911) location requirements, but the positioning accuracy is poor. The fifth-generation (5G) cellular communication system can use indoor distribution systems to provide accurate multiple time-difference-of-arrival (TDOA) and single time-of-arrival (TOA) measurements, which could significantly improve the indoor positioning ability. Unlike iterative localization algorithms for TDOA or TOA, the existing closed-form algorithms, such as the Chan-Ho algorithm, do not have convergence problems, but can only estimate position based on one kind of measurement. This paper proposes a closed-form localization algorithm for multiple TDOAs and single TOA measurements. The proposed algorithm estimates the final position result using three-step weighted least squares (WLSs). The first WLS provides an initial position for the last two steps. Then the algorithm uses two WLSs to estimate position based on heteroscedastic TDOA and TOA measurements. In addition, the geometric dilution of precision (GDOP) of the proposed hybrid TDOA and TOA positioning has been derived. The analysis of GDOP shows that the proposed hybrid positioning has lower GDOP than TDOA-only positioning, which means the proposed hybrid positioning has a higher accuracy limitation than TDOA-only positioning. The simulation shows that the proposed localization algorithm could have better performance than closed-form TDOA-only positioning methods, and the positioning accuracy could approximate Cramer-Rao lower bound (CRLB) when the TDOA measurement errors are small.