Bayesian FDOA Positioning with Correlated Measurement Noise

In this paper, the problem of source localization using only frequency difference of arrival (FDOA) measurements is considered. A new FDOA-only localization technique is developed to determine the position of a narrow-band source. In this scenario, time difference of arrival (TDOA) measurements are not normally useful because they may have large errors due to the received signal having a small bandwidth. Conventional localization algorithms such as the two-stage weighted least squares (TSWLS) method, which jointly exploits TDOA and FDOA measurements for positioning, are thus no longer applicable since they will suffer from the thresholding effect and yield meaningless localization results. FDOA-only localization is non-trivial, mainly due to the high nonlinearity inherent in FDOA equations. Even with two FDOA measurements being available, FDOA-only localization still requires finding the roots of a high-order polynomial. For practical scenarios with more sensors, a divide-and-conquer (DAC) approach may be applied, but the positioning solution is suboptimal due to ignoring the correlation between FDOA measurements. To address these challenges, in this work, we propose a Bayesian approach for FDOA-only source positioning. The developed method, referred to as the Gaussian division method (GDM), first converts one FDOA measurement into a Gaussian mixture model (GMM) that specifies the prior distribution of the source position. Next, the GDM assumes uncorrelated FDOA measurements and fuses the remaining FDOAs sequentially by invoking nonlinear filtering techniques to obtain an initial positioning result. The GDM refines the solution by taking into account and compensating for the information loss caused by ignoring that the FDOAs are in fact correlated. Extensive simulations demonstrate that the proposed algorithm provides improved performance over existing methods and that it can attain the Cramér–Rao lower bound (CRLB) accuracy under moderate noise levels.