CLEAR: A Closed-Form Minimal-Sensor TDOA/FDOA Estimator for Moving-Source IoT Localization

This paper presents CLEAR -- a closed-form localization estimator with a reduced sensor network. The proposed method is a computationally efficient, two-stage estimator that fuses time-difference-of-arrival (TDOA) and frequency-difference-of-arrival (FDOA) measurements with a minimal number of sensors. CLEAR localizes a moving source in N-dimensional space using only N+1 sensors, achieving the theoretical minimum sensor count. The first stage introduces auxiliary range and range-rate parameters to construct a set of pseudo-linear equations, solved via weighted least squares. An algebraic elimination using Sylvester's resultant then reduces the problem to a quartic equation, yielding closed-form estimates for the nuisance variables. A second, lightweight linear refinement stage is applied to mitigate residual bias. Under mild Gaussian noise assumptions, the estimator's position and velocity estimates are statistically efficient, closely approaching the Cramer-Rao lower bound (CRLB). Extensive Monte Carlo simulations in 2-D and 3-D scenarios demonstrate CRLB-level accuracy and consistent performance gains over representative two-stage and iterative baselines, confirming the method's high suitability for power-constrained, distributed Internet of Things (IoT) applications such as UAV tracking and smart transportation.