Bias Reduction for an Explicit Solution of Source Localization Using TDOA

This paper proposes two methods to reduce the bias of the well-known algebraic explicit solution (Chan and Ho, “A simple and efficient estimator for hyperbolic location,” IEEE Trans. Signal Process., vol. 42, pp. 1905–1915, Aug. 1994) for source localization using TDOA. Bias of a source location estimate is significant when the measurement noise is large and the geolocation geometry is poor. Bias also dominates performance when multiple times of independent measurements are available such as in UWB localization or in target tracking. The paper starts by deriving the bias of the source location estimate from Chan and Ho. The bias is found to be considerably larger than that of the Maximum Likelihood Estimator. Two methods, called BiasSub and BiasRed, are developed to reduce the bias. The BiasSub method subtracts the expected bias from the solution of Chan and Ho's work, where the expected bias is approximated by the theoretical bias using the estimated source location and noisy data measurements. The BiasRed method augments the equation error formulation and imposes a constraint to improve the source location estimate. The BiasSub method requires the exact knowledge of the noise covariance matrix and BiasRed only needs the structure of it. Analysis shows that both methods reduce the bias considerably and achieve the CRLB performance for distant source when the noise is Gaussian and small. The BiasSub method can nearly eliminate the bias and the BiasRed method is able to lower the bias to the same level as the Maximum Likelihood Estimator. The BiasRed method is extended for TDOA and FDOA positioning. Simulations corroborate the performance of the proposed methods.