Constrained total least-squares for source location using TDOA measurements in the presence of sensor position errors

Modern location system often uses dynamic mobile platforms as receivers. The sensors' position may not be known exactly when using dynamic mobile platforms as receivers, and a slight error in sensor positions can lead to a big error in source localization estimation. In this paper, by utilizing the time difference of arrival (TDOA) measurements of a signal received at a number of sensors, a constrained total least-squares (CTLS) algorithms for estimating the position of a source with sensor position errors is proposed. By introducing an intermediate variable, the nonlinear TDOA location problem has been mathematically reformulated as a pseudo linear equations. And the CTLS method, as a natural extension of least-squares (LS) when noise occurs in all data and the noise components of the equations' coefficients are linearly dependent, is more appropriate than LS method for the above problem. On the basis of Newton's method, a numerical iterative solution can be obtained allowing real-time implementation. After the perturbation analysis, the bias and covariance of the proposed CTLS algorithm are also derived, indicating that the proposed CTLS algorithm is an unbiased estimator, and it could achieve the CRLB when the TDOA measurement noise and the sensor position errors are sufficiently small. Simulation results show that the proposed estimator achieves remarkably better performance than the total least-squares (TLS) and two-step weighted least squares (WLS) approach, which makes it possible that the Cramér-Rao lower bound (CRLB) is reached at a sufficiently high noise level before the threshold effect occurs.