Spherical‐Interpolation‐Based Semidefinite Relaxation for Moving Emitter Localisation Using TDOA and FDOA Measurements

ABSTRACT

This paper develops a spherical‐interpolation‐based semidefinite relaxation method for joint position and velocity estimation of a moving emitter using time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements. A joint pseudo‐linear residual model is first constructed by introducing the range and range rate relative to a reference receiver as auxiliary variables. Spherical interpolation is then employed to obtain a closed‐form preliminary emitter‐state estimate, which is used to evaluate the unknown emitter‐dependent terms in the residual covariance. The resulting weighting matrix is incorporated into a semidefinite relaxation formulation that jointly refines the emitter position, velocity, reference range, and reference range rate. Receiver position and velocity uncertainties are further incorporated through first‐order covariance propagation. The Cramér–Rao lower bounds and a local covariance analysis are derived to characterise the estimation performance under small‐error conditions. Simulation results under near‐ and far‐field geometries show that the proposed method maintains favourable position and velocity estimation accuracy over a broad range of measurement and receiver‐state error levels, with improved robustness in several threshold regions compared with the considered algebraic and semidefinite estimators.