Closed-form two-step weighted-least-squares-based time-of-arrival source localisation using invariance property of maximum likelihood estimator in multiple-sample environment

In this study, the authors propose a closed-form time-of-arrival source localisation method and justify the employment of the invariance property of the maximum likelihood (ML) estimator in the source localisation context with multiple samples. The magnitude of the bias of the proposed sample vector function (the statistic that consists of the multiple observations set) using the invariance property of the ML estimator is smaller than that based on the sample mean. Therefore, the mean squared error (MSE) of the weighted least squares estimate using the proposed sample vector function is smaller than that based on the sample mean when the variances of both sample vector functions are the same. Furthermore, the authors investigate a situation in which sensors have erroneous position information. The simulation results show that the averaged MSE performance of the proposed method is superior to that of the existing methods irrespective of the number of samples.