RSS-Based Target Localization With Unknown Model Parameters and Sensor Position Errors

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.

We present a novel hybrid localization algorithm for wireless sensor networks in the absence of knowledge regarding the transmit power and path-loss exponent. Transmit power and the path-loss exponent are critical parameters for target localization algorithms in wireless sensor networks, which help extract target position information from the received signal strength. In the absence of information on transmit power and path-loss exponent, it is critical to estimate them for reliable deployment of conventional target localization algorithms. In this paper, we propose a simultaneous estimation of transmit power and path-loss exponent based on Kalman filter. The unknown transmit power and path-loss exponent are estimated using a Kalman filter with the tentatively estimated target position based solely on angle information. Subsequently, the target position is refined using a hybrid method incorporating received signal strength measurements based on the estimated transmit power and path-loss exponent. Our proposed algorithm accurately estimates transmit power and path-loss exponent and yields almost the same target position accuracy as the simulation results confirm, as the hybrid target localization algorithms with known transmit power and path-loss exponent. Simulation results confirm the proposed algorithm achieves 99.7% accuracy of the target localization performance with known transmit power and path-loss exponent, even in the presence of severe received signal strength measurement noise.

In this paper, we present an estimation scheme for the transmit power(TP) and path loss exponent(PLE) for hybrid target localization in wireless sensor networks. In the absence of knowledge on the TP and PLE, hybrid target localization schemes that utilize both received signal strength (RSS) and angle of arrival(AOA) measurements have significantly reduced performances. To estimate the TP and PLE in the presence of a severe measurement noise, moving-averaged RSS measurements are applied to a least-squares algorithm utilizing a tentatively estimated target position using an AOA-only algorithm.

We pose the estimation of the parameters of multiple superimposed exponential signals in additive Gaussian noise problem as a maximum likelihood (ML) estimation problem. The ML problem is non linear and hard to solve. Some previous works focused on finding alternative estimation procedures, for example by denoising. In contrast, we tackle the ML estimation problem directly. First, we use the same transformation as the first step of iterative quadratic maximum likelihood (IQML) and transform the ML problem into another optimization problem that gets rid of the amplitude coefficients. Second, we solve the remaining optimization problem with a gradient descent approach ("pseudo-quadratic maximum likelihood"). We also use this algorithm for ultra-wideband channel estimation and estimate ranging in non-line of sight environment.

Received signal strength (RSS)-based localization techniques have gained much attention in location-based services (LBSs). However, the coexistence of unknown path loss exponent (PLE), uncertain sensor positions, and impulsive noise poses serious challenges to localization accuracy. To address the problem, we first model the impulsive noise as a Mixture of Gaussian (MoG) distribution with unknown parameters. Thus, the noise model and the channel model can be refined using the observed data under the variational Bayesian inference (VBI) framework, which is defined as the model refinement learning. We then propose a corresponding online target localization procedure with the refined noise distribution, PLE and sensor positions. The Bayesian Cramer-Rao bound (BCRB) is finally derived in terms of all unknown parameters. Simulation results together with real experiment demonstrate that the proposed VBI algorithm can effectively learn the true noise distribution, and the developed localization method exhibits robust localization performance in various scenarios.

The problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise. These results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise-free digital communications to a fusion center. We first discuss the case in which all the quantizers use an identical threshold. The parameter identifiability condition is given, and, surprisingly, it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval, the parameter will not generally be identifiable, in contrast to the additive noise case. The best achievable mean square error (MSE) performance is characterized by deriving the corresponding Cramer-Rao Lower Bound (CRLB). A closed-form expression describing the corresponding maximum likelihood (ML) estimator is presented. The stability of the performance of the ML estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter. The thresholds are designed by maximizing the minimum asymptotic relative efficiency (ARE) between quantized and unquantized ML estimators. Although the ML estimation problem is nonconvex, it is shown that one can relax the optimization to make it convex. The solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem. Next, the case where both the variances of the additive noise and multiplicative noise are unknown is studied. The corresponding CRLB is obtained, and the ML estimation problem is transformed to a convex optimization, which can be solved efficiently. Finally, numerical simulations are performed to substantiate the theoretical analysis.

A weighted hybrid indoor positioning method based on path loss exponent estimation

Decaying with the increasing of signal propagation distance, Received Signal Strength (RSS) is used in the wireless localization due to its low cost and easily implementation.When the transmit power is unavailable, two convex optimization algorithms including semi definite programming (SDP), second order cone and semi definite programming (SOC/SDP) are designed to estimate the source locations by relaxing the non-convex problem as convex optimization.The corresponding Cramér-Rao lower bound (CRLB) of the problem is derived.The simulations demonstrate that the SOC/SDP algorithm provides the similar accuracy performance compared with the SDP algorithm.However, the computational complexity of SOC/SDP is lower than that of the SDP due to the less variables and equality constraints.When perfect knowledge of the path loss exponent is available, the simulations also show that the accuracy performance of the proposed convex optimization algorithms degrades as the path loss exponent increases.

This letter addresses the problem of source localization in non-line of sight (NLOS) environment using received signal strength (RSS) and time of arrival (TOA) measurements when both transmission time and power are unknown. To mitigate the NLOS bias, a robust weighted least square (RWLS) problem is formulated from original non-convex maximum-likelihood (ML) estimation problem using a set of tight approximations. The RWLS problem is further tightened by a novel convex-hull constraint and is ultimately solved using second-order cone programming (SOCP). Simulation results confirm the optimal performance of the proposed robust method in various network scenarios and also validate its excellence over state-of-art methods.

Source localization is an important field of research and the received signal strength (RSS)-based method is of particular interest. Further, by exploiting the sparsity of localization problem, the compressive sensing (CS) can be applied to considerably decrease the number of RSS measurements, especially in multi-source scenario. However, most existing CS-based localization methods usually neglect some practical issues. In particular, the sensor positions are assumed be known exactly, while in practice they may not be accurate. Additionally, a uniform Gaussian noise assumption is made that the noise variances of sensors are identical, but in practice the noise should be nonuniform. When these assumptions are violated, the localization performance will deteriorate dramatically. To address such issues, in this paper, we formulate the source localization based on superimposed RSS measurements as a sparse signal recovery problem. Moreover, by regarding the sensor positions as adjustable parameters, the inaccurate sensor positions can be refined through the adjustment of parameters. Following this idea, we develop a novel iterative algorithm for joint signal recovery and parameter optimization based on the variational expectation-maximization algorithm. Consequently, the sensor position uncertainty can be alleviated and thus the signal recovery performance will be improved greatly. Meanwhile, it is also capable of learning the variance of nonuniform noise. Extensive simulation results demonstrate the superiority of the proposed method.

We consider the problem of single source localization using time-difference-of-arrival (TDOA) measurements. By analyzing the maximum-likelihood (ML) formulation of the problem, we show that under certain mild assumptions on the measurement noise, the estimation errors of both the closed-form least-squares estimate proposed in [1] and the ML estimate, as measured by their distances to the true source location, are of the same order. We then use this to establish the curious result that the objective function of the ML estimation problem is actually locally strongly convex at an optimal solution. This implies that some lightweight solution methods, such as the gradient descent (GD) and Levenberg-Marquardt (LM) methods, will converge to an optimal solution to the ML estimation problem when properly initialized, and the convergence rates can be determined by standard arguments. To the best of our knowledge, these results are new and contribute to the growing literature on the effectiveness of lightweight solution methods for structured non-convex optimization problems. Lastly, we demonstrate via simulations that the GD and LM methods can indeed produce more accurate estimates of the source location than some existing methods, including the widely used semidefinite relaxation-based methods.

Using the classical received signal strength (RSS)-distance log-normal model in wireless sensor network (WSN) applications poses a series of characteristic challenges derived from (a) the model’s structural limitations when it comes to explaining real observations, (b) the inherent hardware (HW) variability typically encountered in the low-cost nodes of WSNs, and (c) the inhomogeneity of the deployment environment. The main goal of this article is to better characterize how these factors impact the model parameters, an issue that has received little attention in the literature. For that matter, I qualitatively elaborate on their effects and interplay, and present the results of two quantitative empirical studies showing how much the parameters can vary depending on (a) the nodes used in the model identification and their position in the environment, and (b) the antenna directionality. I further show that the path loss exponent and the reference power can be highly correlated. In view of all this, I argue that real WSN deployments are better represented by random model parameters jointly accounting for HW and local environmental characteristics, rather than by deterministic independent ones. I further argue that taking this variability into account results in more realistic models and plausible results derived from their usage. The article contains example values of the mean and standard deviation of the model parameters, and of the correlation between the path loss exponent and the reference power. These can be used as a guideline in other studies. Given the sensitivity of localization algorithms to the proper model selection and identification demonstrated in the literature, the structural limitations of the log-normal model, the variability of its parameters and their interrelation are all relevant aspects that practitioners need to be aware of when devising optimal localization algorithms for real WSNs that rely on this popular model.

The oscillator phase hit (PH) results in a temporary link loss in the communication system and bears an expensive cost on the operator companies. In this work, we propose a double-stage solution to mitigate the PH. In the first stage, we use Neyman-Pearson binary hypothesis testing to develop a low-cost PH detection algorithm, in which a likelihood ratio test is designed, and the optimal detection threshold is analytically calculated. The proposed PH detection scheme entails small real-time computations while using the existing pilot symbols in the system. Hence, no extra pilot overhead is required. In the second stage, the maximum likelihood (ML) estimation is used to develop a PH correction scheme. In particular, the phase noise and the PH are jointly mitigated by solving an ML estimation problem. By applying the proposed correction scheme, the number of remaining erroneous symbols due to PH (if any) is within the error correction capability of modern forward error-correcting codes. Numerical results verify the effectiveness of the proposed scheme in detecting and correcting the PH.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

The sensor position uncertainties and synchronization offsets can cause substantial performance degradation if the sensors are not properly calibrated. This paper investigates the localization of a constant velocity moving target and the self-calibration of sensors using a sequence of range and azimuth measurements observed at successive instants. A theoretical study by the Cramer-Rao Lower Bound (CRLB) reveals that the sensor positions can only be self-calibrated when there are at least two sensors and synchronization offsets can be handled by joint estimation. A low complexity sequential closed-form solution is proposed to estimate the target position and velocity first, and the coordinates of each sensor and synchronization offset afterward. While less intuitive, the analysis shows that the closed-form solutions for both the target and sensor parameters can reach the CRLB accuracy under small Gaussian noise. We also develop a semidefinite programming (SDP) solution by semidefinite relaxation (SDR) for joint localization and calibration from the Maximum Likelihood formulation, which exhibits higher noise tolerance than the closed-form solution. Simulations validate the analysis and the performance of the proposed methods.