Chan's Algorithm for TDOA Localization

Chan's algorithm is a classic TDOA (Time Difference of Arrival) positioning method primarily used to solve signal source localization problems. Its core advantage lies in transforming nonlinear TDOA equations into linear problems through two-step weighted least squares (WLS) calculations, significantly improving positioning accuracy and stability. Key Algorithm Implementation Steps First WLS Calculation: Constructs pseudo-linear equations relating TDOA measurements to signal source position and distance. Due to measurement noise, initial estimates are obtained using a weighted matrix (typically the inverse of measurement error covariance matrix). In code implementation, this involves building matrices from TDOA measurements and solving (A^T W A)^-1 A^T W b where W is the weight matrix. Error Correction and Second WLS: Utilizes error statistics from the first estimation to reconstruct the weight matrix for secondary optimization. The second WLS corrects covariance relationships of measurement noise, effectively suppressing bias from the initial estimate through improved matrix conditioning. Technical Implementation Considerations Overdetermined System Handling: When base stations exceed spatial dimensions (e.g., >4 stations for 3D positioning), Chan's algorithm solves overdetermined equations via least squares minimization in matrix operations. Weight Matrix Design: Both WLS stages require careful weight matrix selection based on error models (e.g., inverse covariance matrix for Gaussian noise), implemented through proper matrix normalization. Geometric Constraint Utilization: The algorithm implicitly leverages spatial geometry between signal sources and base stations, avoiding computational complexity of direct nonlinear equation solving through clever linearization techniques. Application Scenarios The algorithm is widely used in wireless positioning systems (UWB, cellular networks), achieving performance near the Cramér-Rao Lower Bound (CRLB) with proper base station deployment and known noise statistics. Practical implementations must address engineering challenges like measurement synchronization errors and base station calibration, often requiring preprocessing steps in actual code deployments.