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Source localization based on time of arrival (TOA) measurements in the presence of clock asynchronization and sensor position uncertainties is investigated in this paper. Different from the traditional numerical algorithms, a neural circuit named Lagrange programming neural network (LPNN) is employed to tackle the nonlinear and nonconvex constrained optimization problem of source localization. With the augmented term, two types of neural networks are developed from the original maximum likelihood functions based on the general framework provided by LPNN. The convergence and local stability of the proposed neural networks are analyzed in this paper. In addition, the Cramér-Rao lower bound is also derived as a benchmark in the presence of clock asynchronization and sensor position uncertainties. Simulation results verify the superior performance of the proposed LPNN over the traditional numerical algorithms and its robustness to resist the impact of a high level of measurement noise, clock asynchronization, as well as sensor position uncertainties.

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Time difference of arrival (TDOA) localization does not require time stamping of the source signal and is playing an increasingly important role in passive location. In addition to measurement noise, receiver position errors and synchronization clock bias are two important factors affecting the performance of TDOA positioning. This paper proposes a bias-reduced solution for passive source localization using TDOA measurements in the presence of receiver position errors and synchronization clock bias. Like the original two-step weighted least-squares solution, the new technique has two stages. In the first stage, the proposed method expands the parameter space in the weighted least-squares (WLS) formulation and imposes a quadratic constraint to suppress the bias. In the second stage, an effective WLS estimator is given to reduce the bias generated by nonlinear operations. With the aid of second-order error analysis, theoretical biases for the original solution and proposed bias-reduced solution are derived, and it is proved that the proposed bias-reduced method can achieve the Cramér-Rao lower bound performance under moderate Gaussian noise, while having smaller bias than the original algorithm. Simulation results exhibit smaller estimation bias and better robustness for all estimates, including those of the source position, refined receiver positions, and clock bias vector, when the measurement noise or receiver position error increases.

Multiple sources localization based on time difference of arrival (TDOA) measurements is investigated in this paper. Different from the traditional methods, a novel and practical multisource localization algorithm is proposed by adopting a priori information of relative distance among emitting sources. Since the maximum likelihood (ML) cost function for multisource estimation is highly nonconvex, the semidefinite relaxation (SDR) is utilized to reformulate the ML cost function. A robust estimator is obtained, which can be solved by semidefinite programming (SDP). Moreover, the constrained Cramér-Rao bound is also derived as a benchmark by considering the range constraints between sources. Simulation results verify the superior performance of the proposed algorithm over the traditional methods.