Improving Phase Difference Estimation by Optimizing the Condition Number

Phase difference estimation is a core component of measurement and calibration in multichannel systems. For two channels with the same nominal frequency, a joint sinusoidal model with seven parameters is employed to estimate the phase difference, thereby mitigating estimation bias from minor frequency mismatches. The parameters are obtained by solving the least-squares (LS) problem via singular value decomposition (SVD). Analysis shows that when the system matrix has a high condition number (CN), matrix computations are prone to ill-conditioning, leading to numerical instability in the iterative process. To alleviate this problem, this article derives a closed-form scaling factor that minimizes the system matrix CN while preserving the LS solution, thereby ensuring numerical stability. Based on different amplitude relationships between channels, the CN under coherent sampling can be optimized to the range of (14, 110). Simulation results show that the ill-conditioned risk of the original SVD-LS algorithm increases significantly as the CN enters the medium-to-high range. After introducing the scaling factor derived in this article, the CN of the system matrix decreases significantly, the iterative stability and convergence rate are systematically improved, and the LS optimal solution remains unchanged.