A new iterative algorithm for geolocating a known altitude target using TDOA and FDOA measurements in the presence of satellite location uncertainty

AbstractThis paper considers the problem of geolocating a target on the Earth surface whose altitude is known previously using the target signal time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements obtained at satellites. The number of satellites available for the geolocation task is more than sufficient and their locations are subject to random errors. This paper derives the constrained Cramér-Rao lower bound (CCRLB) of the target position, and on the basis of the CCRLB analysis, an approximately efficient constrained maximum likelihood estimator (CMLE) for geolocating the target is established. A new iterative algorithm for solving the CMLE is then proposed, where the updated target position estimate is shown to be the globally optimal solution to a generalized trust region sub-problem (GTRS) which can be found via a simple bisection search. First-order mean square error (MSE) analysis is conducted to quantify the performance degradation when the known target altitude is assumed to be precise but indeed has an unknown but deterministic error. Computer simulations are used to compare the performance of the proposed iterative geolocation technique with those of two benchmark algorithms. They verify the approximate efficiency of the proposed algorithm and the validity of the MSE analysis

An efficient constrained weighted least squares method with bias reduction for TDOA-based localization

This paper addresses the source location problem by using time-difference-of-arrival (TDOA) measurements. The two-stage weighted least squares (TWLS) algorithm has been widely used in the TDOA location. However, the estimation accuracy of the source location is poor and the bias is significant when the measurement noise is large. Owing to the nonlinear nature of the system model, we reformulate the localization problem as a constrained weighted least squares problem and derive the theoretical bias of the source location estimate from the maximum-likelihood (ML) estimation. To reduce the location bias and improve location accuracy, a novel bias-reduced method is developed based on an iterative constrained weighted least squares algorithm. The new method imposes a set of linear equality constraints instead of the quadratic constraints to suppress the bias. Numerical simulations demonstrate the significant performance improvement of the proposed method over the traditional methods. The bias is reduced significantly and the Cramér–Rao lower bound accuracy can also be achieve

Consistent and Asymptotically Efficient Localization from Range-Difference Measurements

We consider signal source localization from range-difference measurements. First, we give some readily-checked conditions on measurement noises and sensor deployment to guarantee the asymptotic identifiability of the model and show the consistency and asymptotic normality of the maximum likelihood (ML) estimator. Then, we devise an estimator that owns the same asymptotic property as the ML one. Specifically, we prove that the negative log-likelihood function converges to a function, which has a unique minimum and positive definite Hessian at the true source's position. Hence, it is promising to execute local iterations, e.g., the Gauss-Newton (GN) algorithm, following a consistent estimate. The main issue involved is obtaining a preliminary consistent estimate. To this aim, we construct a linear least-squares problem via algebraic operation and constraint relaxation and obtain a closed-form solution. We then focus on deriving and eliminating the bias of the linear least-squares estimator, which yields an asymptotically unbiased (thus consistent) estimate. Noting that the bias is a function of the noise variance, we further devise a consistent noise variance estimator that involves $3$-order polynomial rooting. Based on the preliminary consistent location estimate, a one-step GN iteration suffices to achieve the same asymptotic property as the ML estimator. Simulation results demonstrate the superiority of our proposed algorithm in the large sample case