Combined Weighted Method for TDOA-Based Localization

Efficient closed-form solution for target localization using TDOA measurements in the presence of sensor position errors

This paper focuses on the three-dimensional (3-D) source localization problem using one-dimensional (1-D) angle of arrival (AOA) and time difference of arrival (TDOA) measurements. A weighted least square (WLS) estimate is proposed by exploiting the pseudo-linearization technique for this novel scheme. This method provides a closed-form solution of the source position. Then, the Gauss-Newton method is applied to improve the accuracy further. The Cramer-Rao lower bound (CRLB) is derived to evaluate the performance. Simulation results show that the performance of the proposed localization algorithm is close to the CRLB in small measurement noise region.

Global navigation satellite systems (GNSS) have been widely used in our daily lives. Meanwhile, GNSS are facing a severe threat of spoofing attacks, which can mislead users by false position, velocity and time (PVT) solutions, resulting in serious consequences. Especially, with the development of vehicle navigation, autonomous driving and unmanned aerial vehicles (UAV), increasing spoofing attacks have emerged towards vehicles, where anti-spoofing techniques are desperately needed. Among these techniques, spoofer localization is a necessary step to fundamentally address the spoofing attacks. Most of the conventional spoofer localization methods are based on time of arrival (TOA) or time difference of arrival (TDOA). These methods require measurements observed by multiple receivers at the same time, where time synchronization is indispensable. However, it is hard to apply these methods to the existing vehicles equipped with independent commercial receivers. To cope with this problem, in this paper we develop a flexible method for vehicles to locate a stationary spoofer based on Doppler and clock drift double difference (DCDD). DCDD is calculated using Doppler measurements and clock drift solutions of one vehicular receiver and another reference receiver. Generally, Doppler frequency of GNSS signals changes slowly with time, which releases the restriction for high-precision time synchronization. DCDD represents the Doppler frequency caused by the vehicle motion, from which the spoofer position can be estimated. In addition, with the elimination of irrelevant components in Doppler measurements, DCDD minimizes the user complexity required for localization, where at least two receivers and one vehicle is needed. Utilizing DCDD, an iterative weighted least squares (WLS) algorithm is used to solve the spoofer position, and a closed-form initial value is given to improve the convergence rate. We evaluate the localization performance by the derived Cramér-Rao lower bound (CRLB) in the presence of vehicle position and velocity errors. Simulations and field vehicle experiments are conducted to verify the feasibility and effective of the proposed method. The experimental results show that with an average vehicle velocity of 8 m/s, the horizontal localization error can be less than 40 m and 20 m, using DCDD observations of 4 epochs and 8 epochs respectively.

This paper develops an efficient bi-iterative source location and propagation speed estimation method utilizing time difference of arrival (TDOA) measurements. The source location and propagation speed estimation is a nonlinear problem due to the nonlinearity in the TDOA measurement equations. The newly developed bi-iterative method computes the source location and propagation speed alternately. The asymptotic convergence of the new bi-iterative method is theoretically analyzed. First-order perturbation analysis is applied to the newly developed solution to derive its bias and variance. The first-order analytical results show that the proposed method provides approximately unbiased source position and propagation speed estimates for low noise levels and the accuracy of these estimates approaches the Cramer-Rao lower bound (CRLB). The extension of the new bi-iterative method to the more general situation where the sensor locations are subject to random errors is also presented. Simulation studies are given to show the good performance of the proposed method.

Utilizing the optical characteristics of the target for detection and localization does not require actively emitting signals and has the advantage of strong concealment. Once the optoelectronic platform mounted on the unmanned aerial vehicle (UAV) detects the target, the vector pointing to the target in the camera coordinate system can estimate the angle of arrival (AOA) of the target relative to the UAV in the Earth-centered Earth-fixed (ECEF) coordinate system through a series of rotation transformations. By employing two UAVs and the corresponding AOA measurements, passive localization of an unknown target is possible. To achieve high-precision target localization, this paper investigates the following three aspects. Firstly, two error transfer models are established to estimate the noise distributions of the AOA and the UAV position in the ECEF coordinate system. Next, to reduce estimation errors, a weighted least squares (WLS) estimator is designed. Theoretical analysis proves that the mean squared error (MSE) of the target position estimation can reach the Cramér–Rao lower bound (CRLB) under the condition of small noise. Finally, we study the optimal placement problem of two coplanar UAVs relative to the target based on the D-optimality criterion and provide explicit conclusions. Simulation experiments validate the effectiveness of the localization method.

We can only extract the differential time delay (DTD) measurements between the direct and reflected paths in multistatic localization when there is no synchronization in time between the transmitter and receiver and among the receivers. This paper first addresses the problem of multistatic localization of a fixed object when the transmitter position is not available by using the DTD measurements. We propose a two-step optimization method for jointly estimating the object and transmitter positions. In the first step, we formulate a non-convex constrained weighted least squares (CWLS) problem by transforming the DTD measurement model and introducing nuisance variables. Such a non-convex CWLS problem is then relaxed to a tractable convex semidefinite programming (SDP) problem by applying semidefinite relaxation. In the second step, the error coming from relaxation and approximation in the SDP solution is reduced iteratively through solving a generalized trust region subproblem (GTRS) in each iteration. If the receivers are synchronized such that the time difference of arrival (TDOA) measurements can be acquired in addition to DTD, we formulate a different CWLS problem by utilizing both DTD and TDOA measurements, which is solved by convex relaxation as well. The relaxed SDP problem can achieve the optimal solution of the CWLS problem, and further refinement is no longer needed. We conduct the mean square error (MSE) analysis to validate that both proposed methods are able to achieve the Cramer-Rao Lower Bound (CRLB) performance under small Gaussian noise, which is also validated by simulations.

In this paper, a weighted least squares (WLS) method is presented to find the location of a single target by the use of bistatic range (BR) measurements in multistatic passive radar. The proposed method is based on the idea of eliminating the nuisance parameter. Then, the method employs several WLS minimizations to obtain the target location estimate. The weighting matrix producing a significant improvement in the performance of the method is derived in two different conditions—one of them results in the maximum likelihood estimator (MLE) and the other one leads to the best linear unbiased estimator (BLUE). The accuracy properties of the method are analyzed, and the Cramer-Rao lower bound (CRLB) for target localization accuracy is also derived. In addition, in a two-dimensional space and in the case of localizing with one transmitter and the minimal number of receivers, the different placements of receivers are analytically determined, in each of which the location solution is not unique. Furthermore, from different aspects, the BR-based localization is compared to the hyperbolic localization derived from range difference (RD) measurements.

Source localization in a wireless sensor network (WSN) is sensitive to the sensors' positions. In practice, due to mobility, the receivers' positions may be known inaccurately, leading to non-negligible degradation in source localization estimation performance. The goal of this paper is to develop a semidefinite programming (SDP) method using time-difference-of arrival (TDOA) and frequency-difference-of-arrival (FDOA) by taking the sensor position uncertainties into account. Specifically, we transform the commonly used maximum likelihood estimator (MLE) problem into a convex optimization problem to obtain an initial estimation. To reduce the coupling between position and velocity estimator, we also propose an iterative method to obtain the velocity and position, by using weighted least squares (WLS) method and SDP method, respectively. Simulations show that the method can approach the Cramér-Rao lower bound (CRLB) under both mild and high noise levels.

This paper considers the problem of moving source localization by introducing an additional Doppler frequency shift (DFS) measurement into the previous location system using time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs). Because of the separation of associated data channels and the difference in the estimation techniques, the DFS is independent of TDOA and FDOA measurements. The Cramer-Rao lower bound (CRLB) analysis shows that introducing DFS can improve source localization accuracy comparing with that of the TDOA and FDOA localization system, especially for the velocity location accuracy. An improved two-step weighted least-squares (I-TSWLS) source localization method utilizing DFS as well as TDOA and FDOA is developed. The solution is shown analytically to achieve the CRLB over small noise region, and have a better performance than that of the TSWLS method, especially when the sensor station error noises become large. Simulation results demonstrate the good performance of the proposed method.

Direct solution for fixed source location using well-posed TDOA and FDOA measurements

In this paper, we introduce an integration method of Global Positioning System (GPS) pseudorange and time difference of arrival (TDOA) measurements from a pilot signal of a wireless communication network to solve the navigation problem in the case that the number of visible GPS satellites is not sufficient. TDOA measurements are known to have poor accuracy due to technical limitations of a communication system and propagation environments. The major error source of TDOA is the non-line-of-sight (NLOS). To estimate the TDOA error in a cellular network, we propose a method using a Kalman filter for each TDOA channel. In addition, a method to measure TDOA errors is proposed. Some simulations and real tests are conducted to evaluate the performance of the proposed methods. Real experiments are conducted near the city center of Seoul. Using a commercial cellular phone and a GPS receiver, pseudorange of GPS and TDOAs are measured. Test results show that the proposed methods continuously provide subscribers a reasonably accurate location in the case that GPS data are occasionally insufficient.

One main drawback of using time difference of arrival (TDOA) methods for source localization and navigation is that they require centralization of multiple copies of a signal. This paper considers blindly estimating the location of a cyclic prefix (CP) in an orthogonal frequency division multiplexing (OFDM) signal, enabling distributed TDOA computation up to an integer ambiguity. This ambiguity can be resolved using integer least-squares methods, if enough TDOAs are available, requiring only minimal cooperation between receivers. The contributions of this paper are development of an algorithm for simultaneously resolving the integer ambiguities and obtaining a position estimate; and derivation of the Cramer-Rao lower bound (CRLB) on locating the CP, and hence, on the underlying source localization and navigation problems.

Electromagnetic, acoustic, or seismic sources can be positioned using direction, time, frequency or time/frequency difference of their transmitted signals. Source coordinates are found using time difference of arrival (TDOA) method that depends on the time differences of spatially separated receivers. In these methods, two different TDOA sets are used, including independent/spherical and full. In this study, positioning with particle swarm optimization (PSO) is achieved using independent set and full sets and positioning error is compared with Cramer-Rao lower bound (CRLB). In addition, it is proven that CRLB can be exceeded using the PSO algorithm that uses the full TDOA set.

This paper focuses on underwater target tracking based on a multi-static sonar network composed of passive sonobuoys and an active ping. In the multi-static sonar network, the location of the target can be estimated using TDOA (Time Difference of Arrival) measurements. However, since the sensor network may obtain insufficient and inaccurate TDOA measurements due to ambient noise and other harsh underwater conditions, target tracking performance can be significantly degraded. We propose a robust target tracking algorithm designed to operate in such a scenario. First, track management with track splitting is applied to reduce performance degradation caused by insufficient measurements. Second, a target location is estimated by a fusion of multiple TDOA measurements using a Gaussian Mixture Model (GMM). In addition, the target trajectory is refined by conducting a stack-based data association method based on multiple-frames measurements in order to more accurately estimate target trajectory. The effectiveness of the proposed method is verified through simulations.

This paper considers the problem of tracking a moving target using the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements obtained at two unmanned aerial vehicles (UAVs) with varying baseline. Accumulation time is necessary because the emitter position cannot be estimated at each emission when there are only two UAVs as sensors. And the target position and velocity estimation accuracy often suffer from low convergence in conventional formation flight mode no matter what nonlinear filtering algorithm is used. Based on the analysis of the influence of different flight modes on the positioning performance, a moving target tracking system from two UAVs with varying baseline is proposed. The performance of unscented Kalman filter (UKF) under proposed system is analysed and compared with the Cramer-Rao lower bound (CRLB). Simulation results show that the proposed system can speed up the convergence.