Efficient closed-form estimators in multistatic target localization and motion analysis
Authors:   , Zhang Yang 
Published: May 2022

Object localization is fast becoming an important research topic because of its wide applications. Often of the time, object localization is accomplished in two steps. The first step exploits the characteristics of the received signals and extracts certain localization information i.e. measurements. Some typical measurements include timeof-arrival (TOA), time-difference-of-arrival (TDOA), received signal strength (RSS) and angle-of-arrival (AOA). Together with the known receiver position information, the object location is then estimated in the second step from the obtained measurements. The localization of an object using a number of sensors is often challenged due to the highly nonlinear relationship between the measurements and the object location. This thesis focuses on the second step and considers designing novel and efficient localization algorithms to solve such a problem. This thesis first derives a new algebraic positioning solution using a minimum number of measurements, and from which to develop an object location estimator. Two measurements are sufficient in 2-D and three in 3-D to yield a solution if they are consistent. The derived minimum measurement solution is exact and reduces the computation to the roots of a quadratic equation. The solution derivation also leads to simple criteria to ascertain if the line of positions from two measurements intersects. By partitioning the overdetermined set of measurements first to obtain the individual minimum measurement solutions, we propose a best linear unbiased estimator to form the final location estimate. The analysis supports the proposed estimator in reaching the Cramer-Rao Lower Bound (CRLB) accuracy under Gaussian noise. A measurement partitioning scheme is developed to improve performance when the noise level becomes large. We mainly use elliptic time delay measurements for presentation, and the derived results apply to the hyperbolic time difference measurements as well. Both the 2-D and 3-D scenarios are considered. A multistatic system uses a transmitter to illuminate the object of interest and collects the reflected signal by several receivers to determine its location. In some scenarios such as passive coherent localization or for gaining flexibility, the position of the transmitter is not known. In this thesis, we investigate the use of the indirect path measurements reflected off the object alone, or together with the direct path measurements from the transmitter to receiver for locating the object in the absence of the transmitter position. We show that joint estimation of the object and transmitter positions from both the indirect and direct measurements can yield better object location estimate than using the indirect measurements only by eliminating the dependency of the transmitter position. An algebraic closed-form solution is developed for the nonlinear problem of joint estimation and is shown analytically to achieve the CRLB performance under Gaussian noise over the small error region. To complete the study and gain insight, the optimum receiver placement in the absence of transmitter position is derived, by minimizing the estimation confidence region or the estimation variance for the object location. The performance lost due to unknown transmitter position under the optimum geometries is quantified. Simulations confirm well with the theoretical developments. In practice, a more realistic localization scenario with the unknown transmitter is that the transmitter works non-cooperatively. In this situation, no timestamp is available in the transmitted signal so that the signal sent time is often not known. This thesis next considers the extension of the localization scenario to such a case. More generally, the motion potential of the unknown object and transmitter is considered in the analysis. When the transmitted signal has a well-defined pattern such as some standard synchronization or pilot sequence, it would still be able to estimate the indirect and direct time delays and Doppler frequency shifts but with unknown constant time delay and frequency offset added. In this thesis, we would like to estimate the object and transmitter positions and velocities, and the time and frequency offsets jointly. Both dynamic and partial dynamic localization scenarios based on the motion status of the object and the transmitter are considered in this thesis. By investigating the CRLB of the object location estimate, the improvement in position and velocity estimate accuracy through joint estimation comparing with the differencing approach using TDOA/FDOA measurements is evaluated. The degradation due to time and frequency offsets is also analyzed. Algebraic closed-form solutions to solve the highly nonlinear joint estimation problems are then proposed in this thesis, followed by the analysis showing that the CRLB performance can be achieved under Gaussian noise over the small error region. When the transmitted signal is not time-stamped and does not have a well-defined pattern such as some standard synchronization or pilot sequence, it is often impossible to obtain the indirect and direct measurements separately. Instead, a self-calculated TDOA between the indirect- and direct-path TOAs shall be considered which does not require any synchronization between the transmitter and a receiver, or among the receivers. A refinement method is developed to locate the object in the presence of the unknown transmitter position, where a hypothesized solution is needed for initialization. Analysis shows that the refinement method is able to achieve the CRLB performance under Gaussian noise. Three realizations of the hypothesized solution applying multistage processing to simplify the nonlinear estimation problem are derived. Simulations validate the effectiveness in initializing the refinement estimator.