Posterior Cramer-Rao Lower Bound Research Articles

Simultaneous Spatiotemporal Bias Compensation and Data Fusion for Asynchronous Multisensor Systems

Bias estimation of sensors is an essential prerequisite for accurate data fusion. Neglect of temporal bias in general real systems prevents the existing algorithms from successful application. In this paper, both spatial and temporal biases in asynchronous multisensor systems are investigated and two novel methods for simultaneous spatiotemporal bias compensation and data fusion are presented. The general situation that the sensors sample at different times with different and varying periods is explored, and unknown time delays may exist between the time stamps and the true measurement times. Due to the time delays, the time stamp interval of the measurements from different sensors may be different from their true measurement interval, and the unknown difference between them is considered as the temporal bias and augmented into the state vector to be estimated. Multisensor measurements are collected in batch processing or sequential processing schemes to estimate the augmented state vector, results in two spatiotemporal bias compensation methods. In both processing schemes, the measurements are formulated as functions of both target states and spatiotemporal biases according to the time difference between the measurements and the states to be estimated. The Unscented Kalman Filter is used to handle the nonlinearity of the measurements and produce spatiotemporal bias and target state estimates simultaneously. The posterior Cramer-Rao lower bound (PCRLB) for spatiotemporal bias and state estimation is presented and simulations are conducted to demonstrate the effectiveness of the proposed methods.

Constrained Estimation Analysis and Physical Parameters Study for Chain Tethered Formation Systems

Accurate state estimation of the tethered formation systems (TFS) is challenging due to complex dynamic coupling and elastic tether constraints. In this paper, we develop a new fundamental tool to reveal the influence of tether physical param- eters on the estimation performance for a class of chain-type TFS. First, we model the elastic tether as a Kelvin-Voigt material to evaluate the influence of tether stiffness and damping on estimation performance. To facilitate analysis, equilibrium configurations are used to characterize the dynamic behavior of chain TFS. Then, we extend the posterior Crame´r-Rao lower bound (PCRB) to constrained subspace and derive a recursive form to calculate the constrained PCRB (C-PCRB) for discrete-time filtering with additive Gaussian noises and state equality constraints. Besides, a numerical approach using the particle filter is also developed to approximate the C-PCRB without directly calculating the complex mathematical expectations. Finally, we systematically investigate the influence of the tether stiffness and damping on estimation performance for a chain TFS by extensive simulations.

Joint Power and Bandwidth Allocation in Collocated MIMO Radar Based on the Quality of Service Framework

The simultaneous multi-beam working mode of the collocated multiple-input and multiple-output (MIMO) radar enables the radar to track multiple targets simultaneously. A joint power and bandwidth allocation algorithm in a collocated MIMO radar based on the quality of service (QoS) framework is proposed for the multi-target tracking problem with different threat levels. Firstly, a posterior Cramer–Rao lower bound (PCRLB) concerning the power and bandwidth is derived. In addition, the optimal objective functions of power and bandwidth are designed based on the QoS framework, and the problem is solved using the convex relaxation technique and the cyclical minimization algorithm. The numerical results show that the proposed algorithm has better tracking accuracy and achieves more reasonable resource allocation compared to strategies such as average allocation.

A Tightly Coupled Integration Approach for Cooperative Positioning Enhancement in DSRC Vehicular Networks

Intelligent transportation system significantly relies on accurate positioning information of land vehicles for both safety and non-safety related applications, such as hard-braking ahead warning and red-light violation warning. However, existing Global Navigation Satellite System (GNSS) based solutions suffer from positioning performance degradation in challenging environments, such as urban canyons and tunnels. In this paper, we focus on the positioning performance enhancement of land vehicles via cooperative positioning under a partial GNSS environment in a Vehicular Ad-hoc NETwork (VANET). The availability of Time-of-Flight (ToF) based inter-vehicle or vehicle-to-infrastructure ranges is verified via 5.9 GHz Dedicated Short-Range Communication (DSRC) vehicle-to-everything communication with RTS/CTS unicast mechanism. An inertial navigation sensor aided, tightly coupled integration approach for land vehicle cooperative positioning using DSRC ToF ranges and carrier frequency offset range-rates is proposed, where a digital map is used to constrain the position estimates. If available, the GNSS pseudorange and Doppler shift under partial GNSS environment can also be incorporated. A Rao–Blackwellized particle filter is utilized to estimate the unknown variables allowing for reduced computational complexity in comparison with the conventional particle filter. The posterior Cramer–Rao lower bound is also derived to give a theoretical performance guideline. Both simulation and experimental results show the validity of our proposed approach.

Gaussian processes based extended target tracking in polar coordinates with input uncertainty

Gaussian processes (GP) based extended target tracking (ETT) technique has attracted the attention of scientists since it can accurately estimate the kinematic states as well as the contour states of irregular-shape targets. Most traditional GP methods consider the ETT problem with a linear measurement model in Cartesian coordinates and ignore the input uncertainty of GP for simplicity. In many applications, however, measurements are generated with high-resolution sensors in polar coordinates. More importantly, the ignorance of input uncertainty in GP will exacerbate the tracking performance in these cases. In order to track an irregular-shape extended target in polar coordinates with input uncertainty, an improved GP-based probabilistic data association (IGP-PDA) algorithm is developed that includes the following enhancements: Firstly, a nonlinear measurement model is used and the unbiased converted measurement technique is invoked for the ETT problem. Secondly, the analytical form of statistical properties of the input uncertainty due to measurements noise and predicted error is derived. Thirdly, an IGP-PDA taking into account measurement origin uncertainty as well as input uncertainty is proposed, and three sub-optimal implementations are provided. Last, the posterior Cramer-Rao lower bound of ETT with input uncertainty is derived. Simulation results verify the effectiveness of the proposed method.

Converted state equation Kalman filter for nonlinear maneuvering target tracking

Handling the nonlinearity between the measurement and kinematic states is the core issue in target tracking based on radar or sonar. The main novelty of this paper is the proposal of a new filter with a linear structure to achieve nonlinear tracking by integrating information in the polar coordinate system. The state vectors composed of the range, bearing and their differentials are constructed to make the measurement equations linear. After discretizing the ordinary differential dynamic equations in the polar coordinate system, linear time-varying state transition matrixes are established for two common Cartesian motions: constant velocity (CV) and constant acceleration (CA). For the process noises converted from the Cartesian coordinate system to the polar coordinate system, the first and second moments are derived with a closed form. Consequently, the coordinate system of the conventional state equations is converted so that the tracking can be conducted by a standard Kalman filter. Several simulation scenarios show that such a new filter effectively improves the tracking accuracy. The reasons for the superior performance of the proposed method are analyzed and exemplified. In addition, different posterior Cramer–Rao lower bounds (PCRLBs) for fusion estimation in Cartesian coordinates and polar coordinates are given and compared.

Quasi-Real RFI Source Generation Using Orolia Skydel LEO Satellite Simulator for Accurate Geolocation and Tracking: Modeling and Experimental Analysis

Accurate geolocation and tracking of Radio-Frequency Interference (RFI) sources, which affect wireless and satellite systems such as Global Navigation Satellite Systems (GNSS) and Satellite Communication (SatCom) systems, are considered to be a significant issue. Several studies connected to civil and military operations on this issue have been investigated recently. The literature review has surveyed many algorithm simulations for optimizing geolocation and target-tracking estimation. Although most of these algorithms have their own advantages, they have weaknesses, such as accuracy, mathematical complexity, difficulties in implementation, and validation in the real environment, etc. This study has been concerned with investigating the accuracy of geolocation and tracking under high speed and powerful rotation using extracted data from the Orolia Skydel simulator, which simulates the space environment involving Low Earth Orbit (LEO) satellites as sensors and Unmanned Aerial Vehicles (UAV) as RFI emitters. Various scenarios modeled using the Orolia Simulator for quasi-real dynamic trajectories of LEO satellites have been created. The assumed approaches have been verified by Cramer–Rao Lower Bound (CRLB) and Posterior CRLB (PCRLB) to determine the increase in Root Mean Square Error (RMSE) value. The simulation scenarios have been performed using the Monte Carlo iteration. Eventually, the overall achieved results of the considered approaches using data acquired from the Orolia Simulator were presented and compared with theoretical simulation.

Ground Target Tracking Using an Airborne Angle-Only Sensor with Terrain Uncertainty and Sensor Biases.

Airborne angle-only sensors can be used to track stationary or mobile ground targets. In order to make the problem observable in 3-dimensions (3-D), the height of the target (i.e., the height of the terrain) from the sea-level is needed to be known. In most of the existing works, the terrain height is assumed to be known accurately. However, the terrain height is usually obtained from Digital Terrain Elevation Data (DTED), which has different resolution levels. Ignoring the terrain height uncertainty in a tracking algorithm will lead to a bias in the estimated states. In addition to the terrain uncertainty, another common source of uncertainty in angle-only sensors is the sensor biases. Both these uncertainties must be handled properly to obtain better tracking accuracy. In this paper, we propose algorithms to estimate the sensor biases with the target(s) of opportunity and algorithms to track targets with terrain and sensor bias uncertainties. Sensor bias uncertainties can be reduced by estimating the biases using the measurements from the target(s) of opportunity with known horizontal positions. This step can be an optional step in an angle-only tracking problem. In this work, we have proposed algorithms to pick optimal targets of opportunity to obtain better bias estimation and algorithms to estimate the biases with the selected target(s) of opportunity. Finally, we provide a filtering framework to track the targets with terrain and bias uncertainties. The Posterior Cramer–Rao Lower Bound (PCRLB), which provides the lower bound on achievable estimation error, is derived for the single target filtering with an angle-only sensor with terrain uncertainty and measurement biases. The effectiveness of the proposed algorithms is verified by Monte Carlo simulations. The simulation results show that sensor biases can be estimated accurately using the target(s) of opportunity and the tracking accuracies of the targets can be improved significantly using the proposed algorithms when the terrain and bias uncertainties are present.

Joint Detection Threshold Optimization and Illumination Time Allocation Strategy for Cognitive Tracking in a Networked Radar System

In this paper, a joint detection threshold optimization and illumination time allocation (JDTOITA) strategy was developed for multi-target tracking in an asynchronous networked radar system under cluttered background. The basis of this strategy is to facilitate detection and tracking using the prior target information in the tracking recursive cycle. The information reduction factor in the Bayesian detection framework is derived, optimized, and incorporated in the posterior Cramer-Rao lower bound (PCRLB), which is then utilized to serve as the optimization metric. Due to the asynchronous data and cluttered environment, the objective function needs to be recursively deduced and is nonlinear and nonconvex. We propose an efficient solver integrating the convex relaxation with the local search technique for this problem solving. Simulation results demonstrate the superiority of the JDTOITA strategy compared with the benchmarks with no optimization or optimization of either the illumination time allocation (ITA) or detection threshold alone. The results also imply that the target reflectivity and sampling interval of local radars are two important factors that influence the resource optimization.

A New Pseudolinear Filter for Bearings-Only Tracking without Requirement of Bias Compensation.

In bearings-only tracking systems, the pseudolinear Kalman filter (PLKF) has advantages in stability and computational complexity, but suffers from correlation problems. Existing solutions require bias compensation to reduce the correlation between the pseudomeasurement matrix and pseudolinear noise, but incomplete compensation may cause a loss of estimation accuracy. In this paper, a new pseudolinear filter is proposed under the minimum mean square error (MMSE) framework without requirement of bias compensation. The pseudolinear state-space model of bearings-only tracking is first developed. The correlation between the pseudomeasurement matrix and pseudolinear noise is thoroughly analyzed. By splitting the bearing noise term from the pseudomeasurement matrix and performing some algebraic manipulations, their cross-covariance can be calculated and incorporated into the filtering process to account for their effects on estimation. The target state estimation and its associated covariance can then be updated according to the MMSE update equation. The new pseudolinear filter has a stable performance and low computational complexity and handles the correlation problem implicitly under a unified MMSE framework, thus avoiding the severe bias problem of the PLKF. The posterior Cramer–Rao Lower Bound (PCRLB) for target state estimation is presented. Simulations are conducted to demonstrate the effectiveness of the proposed method.

Nearly constant acceleration model for state estimation in the range‐Doppler plane

The problem of motion modelling in the range-Doppler (R-D) plane as well as range and Doppler estimation for the Cartesian nearly constant acceleration motion, which is a common manoeuvering motion, is investigated. The temporal evolution equation is derived based on the state vector consisting of target range, Doppler and derivatives of the product of range and range rate versus time. In this way, the measurement equation of range and Doppler measurements can be maintained in a desirable linear-Gaussian structure. Based on the non-linear state equation and the linear measurement equation, the unscented Kalman filter is adopted to tackle the non-linear filtering problem. The corresponding filter initialisation method is developed based on the two-point differencing method. Explicit expressions of the initial state estimates and the initial covariance matrix are presented in analytic forms where the correlation among the state components is handled properly. The posterior Cramer–Rao lower bound (PCRLB) is provided for state estimation in the R-D plane. Comprehensive comparisons of the proposed method against the existing R-D state estimation methods using approximate models, Cartesian state estimator and PCRLB are carried out in simulations to demonstrate the validity and correctness of the proposed motion model and estimation method.

Motion modeling and state estimation in range-Doppler plane

The problem of motion modeling in range-Doppler (R-D) plane as well as range and Doppler estimation is investigated. The time evolving equations of target range and Doppler are formulated, for two common Cartesian motions, nearly constant velocity (NCV) and constant turn (CT). This builds the foundation for target tracking based on multisensor data fusion using only range and Doppler measurements and other applications. The R-D state vectors are constructed by range, Doppler and the derivative of the product of range and Doppler with respect to time. The state equations corresponding to these motions are derived by explicit substitutions according to the relationship between R-D states and Cartesian states. The unscented Kalman filter (UKF) is employed to extract R-D states from range and Doppler measurements. The filter initialization is derived by two-point differencing methodology where the correlation among the state components is handled properly. At last, the posterior Cramer-Rao Lower Bound (PCRLB) for state estimation in R-D plane is provided, and the performance of the proposed method is compared against the PCRLB and the existing methods using approximate models in the numerical experiments. The simulation results demonstrate the effectiveness of the proposed models and filtering methods. The enhancement of estimation accuracy benefits the data fusion using range and/or Dopper estimates.

Near-Field Tracking With Large Antenna Arrays: Fundamental Limits and Practical Algorithms

Applications towards 6G have brought a huge interest towards arrays with a high number of antennas and operating within the millimeter and sub-THz bandwidths for joint communication and localization. With such large arrays, the plane wave approximation is often not accurate because the system may operate in the near-field propagation region (Fresnel region) where the electromagnetic field wavefront is spherical. In this case, the curvature of arrival (CoA) is a measure of the spherical wavefront that can be used to infer the source position using only a single large array. In this paper, we study a near-field tracking problem for inferring the state (i.e., the position and velocity) of a moving source with an ad-hoc observation model that accounts for the phase profile of a large receiving array. For this tracking problem, we derive the posterior Cram\'er-Rao Lower Bound (P-CRLB) and show the effects when the source moves inside and outside the Fresnel region. We provide insights on how the loss of positioning information outside Fresnel comes from an increase of the ranging error rather than from inaccuracies of angular estimation. Then, we investigate the performance of different Bayesian tracking algorithms in the presence of model mismatches and abrupt trajectory changes. Our results demonstrate the feasibility and high accuracy for most of the tracking approaches without the need of wideband signals and of any synchronization scheme. signals and of any synchronization scheme.

Online sequential spatiotemporal bias compensation using multisensor multitarget measurements

In multisensor multitarget tracking systems, spatiotemporal bias compensation is essential for correct data fusion. However, waiting for the measurements of all targets from a sensor to estimate the biases in a batch processing scheme is computational prohibitive and infeasible in some real applications. In this paper, an online sequential spatiotemporal bias compensation method using multisensor multitarget measurements is presented. The spatiotemporal biases of the sensors are augmented into the state vector of each target to be estimated by the measurements of this target. When updating the estimates of a target, the spatiotemporal bias estimates of the previous target are used as linear pseudo-measurements. The correlation between the pseudo-measurement and the augmented state estimates of this target from the previous time step is analyzed. To handle the correlation problem, a novel estimator is derived under the minimum mean square error (MMSE) framework to obtain correct sequential estimation in this situation. The proposed method produces both target state and spatiotemporal bias estimates once the measurements of one target from a sensor are received, without requirement of waiting for all target measurements and calculation with high dimensional matrix. The posterior Cramer-Rao Lower Bound (PCRLB) for spatiotemporal bias and state estimation is presented. Simulation results demonstrate the effectiveness of the proposed method.

Power and Bandwidth Allocation for Multi-Target Tracking in Collocated MIMO Radar

A collocated multiple-input and multiple-output (MIMO) radar has the ability to track multiple targets under the simultaneous multi-beam mode, where each transmit beam is defocused to search the entire surveillance space, and then; a focused beam is formed at the receiver to achieve a higher resolution. A joint power and bandwidth allocation strategy is developed in this mode. The optimization model is established with the aim of improving the sum of weighted posterior Cramer-Rao lower bounds (PCRLBs) of multiple targets, which is predicted based on the feedback information in the tracking recursive cycle, subject to the limited power and bandwidth budget. By exploiting the convex relaxation technique and the cycle minimizer, this problem is transformed into a two-stage convex problem, and an efficient solution is provided using the cyclical optimization framework. The effectiveness of the proposed strategy is demonstrated by comparison with three uniform resource allocation methods in many cases.

Joint Target Assignment and Power Allocation in Multiple Distributed MIMO Radar Networks

The “defocused transmit-focused receive” (DTFR) mode in the distributed multiple-input multiple-output (MIMO) radar network is very effective in multitarget tracking. In this mode, a completely defocused beam is transmitted and a focused receive beam is synthesized so that the MIMO radar is capable of tracking targets independently. A joint target assignment and power allocation (TAPA) strategy is developed for multiple distributed MIMO radar networks in cluttered environment using the DTFR mode. Our aim is to achieve the better system tracking accuracy under the constraints of receive beam direction capability and power budget. We derive the posterior Cramer-Rao lower bound (PCRLB) and adopt it as the objective function, since it quantifies the precision of target state estimates. It is shown that the TAPA problem is a mixed integer programming and NP-hard problem, where two involved parameters, i.e., the target-radar assignment and power allocation, are both coupled in the objective and in the constraints. By introducing an intermediate variable, we propose an efficient two-step-based solution for solving this problem. The simulation results show the superior performance and adaptivity compared with existing algorithms.

Joint power and time width allocation in collocated MIMO radar for multi‐target tracking

Collocated multiple-input multiple-output radar can track multiple targets simultaneously by transmitting multiple orthogonal beams and adopting the digital beamforming technology. In this scenario, the authors propose a joint power and time width allocation approach, which combines a cognitive tracking model based on the posterior Cramer-Rao lower bound (PCRLB) and the square-root cubature Kalman filter. The aim of the optimisation model is to improve the velocity estimation accuracy by minimising the sum of the PCRLBs of the velocity of multiple targets, which are predicted based on the feedback information from the cognitive tracking model. However, there are two finite working resources in the optimisation model: the total transmit power of multiple beams and the total effective time width of each corresponding signal. The resource allocation problem can be transformed into a non-convex optimisation problem, which can be converted into a standard convex optimisation problem by the linear relationship between the optimal power and the optimal time width. In this way, the joint power and time width allocation scheme is established as an adaptive closed-loop system. Numerical results demonstrate that the velocity tracking accuracy can be improved efficiently by the proposed algorithm.

Use of Doppler and Doppler Rate for RF Geolocation Using a Single LEO Satellite

A passive radio frequency (RF) geolocation solution is provided that uses a single low earth orbit (LEO) satellite to find an uncooperative earth-bound emitter. For the first time, an unambiguous solution is available for real-time, single-pass, and time-constrained acquisition scenarios where single transmissions are expected. The geolocation algorithm rapidly maps Doppler and Doppler rate measurements to an RF emitter location. Corresponding theoretic performance bounds are provided for mission analysis and optimality comparison. The proposed algorithm is a variant of the constrained Unscented Kalman Filter (cUKF), chosen for computational efficiency, modeling abilities, and ease of constraint incorporation. A novel, low-computation method of sigma point projection upon the surface of the earth is derived that drastically improves search area capabilities and convergence rates. The theoretical performance bound is in the form of the recursive constrained Posterior Cramer-Rao Bound (rcPCRB), which is uniquely suited to gauge the mean squared error optimality of iterative nonlinear estimation algorithms. Numerical analysis over measurement noise, center frequency, slant angle, and initialization error exhibit the algorithm’s robustness over various mission types. The performance of the cUKF is demonstrated on raw IQ data acquired from the TDS-1 satellite operated by Surrey Satellites.

Cramér–Rao Bounds for Filtering Based on Gaussian Process State-Space Models

Posterior Cramer-Rao bounds (CRBs) are derived for the estimation performance of three Gaussian process-based state-space models. The parametric CRB is derived for the case with a parametric state transition and a Gaussian process-based measurement model. We illustrate the theory with a target tracking example and derive both parametric and posterior filtering CRBs for this specific application. Finally, the theory is illustrated with a positioning problem, with experimental data from an office environment where the obtained estimation performance is compared to the derived CRBs.

Comprehensive Time-Offset Estimation for Multisensor Target Tracking

Temporal registration of sensors is an essential preprocessing step in multisensor target tracking systems. A new approach for multisensor time-offset estimation is proposed in this article. First, the time offset pseudomeasurement equation is derived and calculated in both centralized and decentralized scenarios, where measurements and local tracks are available at the fusion center, respectively. The observability of time offset is analyzed theoretically, which shows that only relative time offsets between sensors are observable. Second, a two-sensor two-stage filtering method is developed with four different formulations corresponding to different time-offset statistical models and target dynamic models to obtain a relative time-offset estimate. A multisensor two-stage filter is also proposed to obtain a minimum-bias time-offset estimate. Furthermore, the interacting multiple model estimator is used to deal with temporal registration in the presence of target maneuvers. Finally, the posterior Cramer–Rao lower bound (PCRLB) is derived for relative time-offset estimation. Simulation results show that the proposed algorithm with two sensors yields an empirically unbiased estimate of the relative time offset, and that the root-mean-square errors (RMSEs) match the corresponding PCRLB. Simulation results for multisensor target tracking are also presented to demonstrate the validity of the proposed algorithms.