Accuracy Analysis of 2D Three-Station TDOA Passive Localization Using Chan's Algorithm

Two-dimensional three-station Time Difference of Arrival (TDOA) passive localization is a positioning method based on signal arrival time differences, widely applied in radar, sonar, and wireless communication systems. Chan's algorithm serves as an analytical solution method that enhances positioning accuracy through least squares estimation. Compared to iterative algorithms, it offers higher computational efficiency with reduced processing load. In code implementation, the algorithm typically involves matrix operations for solving linearized equations, where MATLAB's pinv() function or equivalent matrix inversion routines can be applied for efficient computation.

In 2D localization scenarios, three receiving stations calculate target positions by measuring TDOA of signals transmitted from the target. Chan's algorithm linearizes the TDOA equations and employs weighted least squares solutions to mitigate measurement errors. The implementation requires careful handling of covariance matrices to weight measurements according to their uncertainty, often achieved through Cholesky decomposition or similar numerical techniques.

Geometric Dilution of Precision (GDOP) serves as a crucial metric for evaluating localization accuracy, reflecting how receiver station geometry affects positioning errors. When station layouts approach collinearity, GDOP increases while positioning accuracy deteriorates. Optimal station placement strategies can minimize GDOP and enhance overall system performance. In practical implementations, GDOP calculation involves matrix determinants and eigenvalues, which can be computed using linear algebra libraries like NumPy or Eigen.

This methodology can be extended to 3D localization scenarios by adding at least one more receiving station and adapting Chan's algorithm solution process. For 3D applications, GDOP analysis remains applicable but requires consideration of spatial distribution among receivers to ensure optimal accuracy. The algorithm extension involves expanding the measurement matrix to accommodate z-coordinate parameters and adjusting the weighting matrix accordingly.