Constrained total least-squares localisation algorithm using time difference of arrival and frequency difference of arrival measurements with sensor location uncertainties

In this study, a constrained total least-squares (CTLS) algorithm for estimating the position and velocity of a moving source with sensor location uncertainties that uses the time difference of arrival and frequency difference of arrival measurements of a signal received at a number of sensors is proposed. The CTLS method, as a natural extension of LS when noise occurs in all the data and the noise components of the coefficients are linearly dependent, is more appropriate than the LS method for the above problem. By utilising the Lagrange multipliers technique, the known relation between the intermediate variables and the source localisation coordinates has been exploited to constrain the solution. In addition, the Lagrange multipliers can be obtained efficiently and robustly, which can allow real-time implementation as well as ensure global convergence. After a 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 Craméŕ Rao lower bound (CRLB) when the measurement noise and the sensor location errors are sufficiently small. The simulation results show that the proposed estimator achieves remarkably better performance than the TLS and two-step weighted least squares approach, which makes it possible that the CRLB is attained at a sufficiently high noise level before the threshold effect occurs.