The subject of localization has received great deal attention in the past decades. Although it is perhaps a well-studied problem, there is still room for improvement. Traditional localization methods usually assume the number of sensors is sufficient for providing desired performance. However, this assumption is not always satisfied in practice. This paper studies the time of arrival (TOA)-based source positioning in the presence of sensor position errors. An error refined solution is developed for reducing the mean-squared-error (MSE) and bias in small sensor network (the number of sensors is fewer) when the noise or error level is relatively large. The MSE performance is analyzed theoretically and validated by simulations. Analytical and numerical results show the proposed method attains the Cramér-Rao lower bound (CRLB). It outperforms the existing closed-form methods with slightly raising computation complexity, especially in the larger noise/error case.

Energy-based localization techniques, such as received signal strength (RSS) and differential RSS (DRSS), have gained widespread use in both indoor and outdoor environments due to their high accuracy and low cost. Recently, a new model for DRSS has been introduced, accompanied by a suboptimal solution that fails to attain the Cramér Rao Bound (CRB). However, a comprehensive solution that achieves the CRB and the corresponding analysis is still lacking. To address this gap, this paper proposes a novel formulation for DRSS localization that takes into account sensor position errors from the perspective of nullspace projection. A close-form coarse estimate is obtained and subsequently refined by the constraint. Additionally, we enhance the accuracy of our approach in the presence of large noise and errors by utilizing Taylor expansion. We provide a theoretical performance analysis that establishes the achievement of the proposed solution to the CRB, which is verified through simulation. Our results demonstrate that the proposed solution outperforms existing algorithms in both mean-square error and bias.

Classical localization methods use Cartesian or Polar coordinates, which require a priori range information to determine whether to estimate position or to only find bearings. The modified polar representation (MPR) unifies near-field and far-field models, alleviating the thresholding effect. Current localization methods in MPR based on the angle of arrival (AOA) and time difference of arrival (TDOA) measurements resort to semidefinite relaxation (SDR) and Gauss-Newton iteration, which are computationally complex and face the possible diverge problem. This paper formulates a pseudo linear equation between the measurements and the unknown MPR position, which leads to a closed-form solution for the hybrid TDOA-AOA localization problem, namely hybrid constrained optimization (HCO). HCO attains Cramér-Rao bound (CRB)-level accuracy for mild Gaussian noise. Compared with the existing closed-form solutions for the hybrid TDOA-AOA case, HCO provides comparable performance to the hybrid generalized trust region subproblem (HGTRS) solution and is better than the hybrid successive unconstrained minimization (HSUM) solution in large noise region. Its computational complexity is lower than that of HGTRS. Simulations validate the performance of HCO achieves the CRB that the maximum likelihood estimator (MLE) attains if the noise is small, but the MLE deviates from CRB earlier.