3-D Localization Integrating 1-D SA and TDOA Measurements: Analysis and Bias-Reduced Solutions
Tianyi Xing, Yimao Sun, Kehao Zhang, Xiangyu Peng, Lihua Ni, Ziqiang Wang, Ning Liu, Qun Wan · IEEE Transactions on Vehicular Technology · 2025
Using bearing measurements obtained by linear arrays (LAs), referred to as space angles (SAs), to determine a three-dimensional (3-D) source is attractive due to its low cost and simplicity. Recently, the focus has expanded to the hybrid SA and time difference of arrival (TDOA) positioning, which offers the potential of higher accuracy and closed-form solutions for unrestricted LA placement. Closed-form solutions have explored this topic but behaved with extremely large biases compared to the maximum likelihood estimator (MLE). The presence of bias may degrade the performance, especially in high-noise environments or situations with poor geometry. This paper seeks to address these bias-related challenges in existing closed-form approaches by introducing two novel bias-reduced methods: bias-subtracted deviation compensation (BSDC) and bias-reduced weighted total least squares (BRWTLS) solutions. The BSDC method starts with a pseudo-linearized formulation from measurements to gain a coarse solution, and then applies a closed-form correction to enhance the accuracy of this preliminary result, producing the deviation compensation (DC) solution. Subsequently, BSDC subtracts the expected bias from the DC solution to further mitigate bias. The BRWTLS method employs a weighted total least squares (WTLS) technique to reduce an error correlation that contributes to additional bias, followed by a closed-form mapping to improve accuracy. BSDC can even achieve lower bias than MLE while BRWTLS keeps a comparable bias level as MLE. However, BSDC requires the accurate value of the noise correlation matrix to compute the analytical bias, whereas BRWTLS, more engineering practically, has less dependence on high-precision covariance measurement. Mean square error (MSE) and bias analysis of the proposed methods are demonstrated in detail and validated through simulations, revealing that both methods can uphold the Cramér-Rao lower bound (CRLB) with significantly reducing bias compared to existing closed-form estimators.