Cramér–Rao lower bound in nonlinear filtering problems under noises and measurement errors dependent on estimated parameters

This paper derives recurrent expressions for the maximum attainable estimation accuracy calculated using the Cramér–Rao inequality (Cramér–Rao lower bound) in the discretetime nonlinear filtering problem under conditions when generating noises in the state vector and measurement error equations depend on estimated parameters and the state vector incorporates a constant subvector. We establish a connection to similar expressions in the case of no such dependence. An example illustrates application of the obtained algorithms to lowerbound accuracy calculation in a parameter estimation problem often arising in navigation data processing within a model described by the sum of a Wiener sequence and discrete-time white noise of an unknown variance.     

Cross-ambiguity function domain multipath channel parameter estimation

A new array signal processing technique is proposed to estimate the direction-of-arrivals (DOAs), time delays, Doppler shifts and amplitudes of a known waveform impinging on an array of antennas from several distinct paths. The proposed technique detects the presence of multipath components by integrating cross-ambiguity functions (CAF) of array outputs, hence, it is called as the cross-ambiguity function direction finding (CAF-DF). The performance of the CAF-DF technique is compared with the space-alternating generalized expectation–maximization (SAGE) and the multiple signal classification (MUSIC) techniques as well as the Cramér–Rao lower bound. The CAF-DF technique is found to be superior in terms of root-mean-squared-error (rMSE) to the SAGE and MUSIC techniques.     

Asymptotically efficient estimators for the fittings of coupled circles and ellipses

Fitting a pair of coupled geometric objects to a number of coordinate points is a challenging and important problem in many applications including coordinate metrology, petroleum engineering and image processing. This paper derives two asymptotically efficient estimators, one for concentric circles fitting and the other for concentric ellipses fitting, based on the weighted equation error formulation and non-linear parameter transformation. The Kanatani–Cramér–Rao (KCR) lower bounds for the parameter estimates of the concentric circles and concentric ellipses under zero-mean Gaussian noise are provided to serve as the performance benchmark. Small-noise analysis shows that the proposed estimators reach the KCR lower bound performance asymptotically. The accuracy of the proposed estimators is corroborated by experiments with synthetic data and realistic images.     

A study of two-dimensional sensor placement using time-difference-of-arrival measurements

Finding the position of a radiative source based on time-difference-of-arrival (TDOA) measurements from spatially separated receivers has important applications in sonar, radar, mobile communications and sensor networks. Each TDOA defines a hyperbolic locus on which the source must lie and the position estimate can then be determined with the knowledge of the sensor array geometry. While extensive research works have been performed on algorithm development for TDOA estimation and TDOA-based localization, limited attention has been paid in sensor array geometry design. In this paper, an optimum two-dimensional sensor placement strategy is derived with the use of optimum TDOA measurements, assuming that each sensor receives a white signal source in the presence of additive white noise. The minimum achievable Cramér–Rao lower bound is also produced.     

Efficient frequency estimation of a single real tone based on principal singular value decomposition

The problem of single-tone frequency estimation for a discrete-time real sinusoid in white Gaussian noise is addressed. We first show that the frequency information is embedded in the principal singular vectors of a matrix which stores the observed data with no repeated entry. The technique of weighted least squares is then utilized for finding the frequency from the singular vectors. It is proved that the variance of the frequency estimate approaches Cramér–Rao lower bound when the data observation length tends to infinity. The computational efficiency and estimation accuracy are demonstrated via computer simulations.     

A cramér-rao bound for multidimensional discrete-time dynamical systems

In this note a mean-square error lower bound for the discrete-time nonlinear filtering problem is derived based on Cramér-Rao theory. The lower bound is applicable to multidimensional nonlinear dynamical systems and is tighter than others that have appeared in the literature. The case of singular process noise covariance is considered. A smoothing lower bound for the multidimensional case is also obtained. It is shown that all lower bounds derived can be conveniently evaluated by Monte Carlo simulation techniques.     

Analytic performance investigation of signal level estimator based on empirical characteristic function in impulsive noise

In this paper, we evaluate the mean square error (MSE) performance of empirical characteristic function (ECF) based signal level estimator in a binary communication system. By calculating Cramér-Rao lower bound (CRLB) we investigate the performance of the ECF based estimator in the presence of Laplace and Gaussian mixture noises. We have derived an analytic expression for the variance of the ECF based estimator which shows that it is asymptotically unbiased and consistent. Simulation and analytic results indicate that the ECF based level estimator outperforms the previously proposed estimators in some signal to noise ratio (SNR) regions when the observation noise distribution is unknown.     

Modified particle filter and Gaussian filter with packet dropouts

This technical note is concerned with the nonlinear filtering for networked control systems. First, the modified particle filter algorithm with intermittent observations is proposed and the conditional Cramér‐Rao lower (CRL) bound with packet dropouts for nonlinear non‐Gaussian system is derived. Second, an upper bound for the CRL bound of the Gaussian filter with packet losses is obtained by constructing a linear Gaussian‐Markovian networked system because of the complexity in direct analysis and computation. Third, a sufficient condition is given for the bounded expectation of the CRL bound, which is the necessary condition for bounded mean‐square error covariance. Finally, an example illustrates the effectiveness of the proposed filter.     

Near optimum sampling design and an efficient algorithm for single tone frequency estimation

Work on single tone frequency estimation has focused on uniformly sampled data. However, it has been shown that, for a given number of samples, more information on the frequency of a signal can be gained by non-uniform sampling schemes [M. Wieler, S. Trittler, F.A. Hamprecht, Optimal design for single tone frequency estimation, Digital Signal Process., in press]. Unfortunately, an optimum sampling pattern (that, for example, minimizes the Cramér–Rao bound) does not automatically have a fast and simple algorithm for frequency estimation associated with it. For application in an interferometric measurement system, an algorithm is needed that gathers as much information as possible from a low number of samples, while at the same time keeping the computational effort sufficiently low to process millions of time series in a few seconds. This paper proposes a simple approximation to the optimum sampling pattern by using uniformly sampled blocks of data and further proposes to estimate phase and frequency in each of these blocks and to exploit these intermediate results in the final estimation. An approach to do so is investigated in detail. Results are compared to the Cramér–Rao bound (CRB), and it is shown that this algorithm almost reaches this limit on the variance of unbiased estimators, at a computational complexity lower than that of a typical FFT-based approach. For $M=32$ samples and a signal-to-noise ratio of 10, the standard deviation of the frequency estimate is lower by more than 50% compared to uniform sampling. In addition, the algorithm can easily be applied to poorly characterized systems, e.g. systems for which the noise is not known exactly. Finally, we demonstrate that the proposed strategy yields results that are within 3% of the theoretically achievable accuracy for the theoretically optimum sampling pattern.     

Lagrange programming neural networks for time-of-arrival-based source localization

Finding the location of a mobile source from a number of separated sensors is an important problem in global positioning systems and wireless sensor networks. This problem can be achieved by making use of the time-of-arrival (TOA) measurements. However, solving this problem is not a trivial task because the TOA measurements have nonlinear relationships with the source location. This paper adopts an analog neural network technique, namely Lagrange programming neural network, to locate a mobile source. We also investigate the stability of the proposed neural model. Simulation results demonstrate that the mean-square error performance of our devised location estimator approaches the Cramér–Rao lower bound in the presence of uncorrelated Gaussian measurement noise.     

A statistical analysis of the Delogne–Kåsa method for fitting circles

In this paper, we examine the problem of fitting a circle to a set of noisy measurements of points on the circle's circumference. Delogne [Proc. IMEKO-Symp. Microwave Measurements, 1972, pp. 117–123] has proposed an estimator which has been shown by Kåsa [IEEE Trans. Instrum. Meas. 25 (1976) 8–14] to be convenient for its ease of analysis and computation. Using Chan's circular functional model to describe the distribution of points, we perform a statistical analysis of the estimate of the circle's centre, assuming independent, identically distributed Gaussian measurement errors. We examine the existence of the mean and variance of the estimator for fixed sample sizes. We find that the mean exists when the number of sample points is greater than 3 and the variance exists when this number is greater than 4. We also derive approximations for the mean and variance for fixed sample sizes when the noise variance is small. We find that the bias approaches zero as the noise variance diminishes and that the variance approaches the Cramér–Rao lower bound.     

Cramér–Rao bound and statistical resolution limit investigation for near-field source localization

In this paper, we investigate the performance analysis for near-field source localization in terms of the mean square error and resolvability. We first derive and analyze non-matrix, closed-form expressions of the deterministic Cramér–Rao bound for two closely spaced, time-varying near-field sources in the context of linear arrays. Numerical simulations confirm the validity of the obtained expressions. Using these expressions and based on Smith's criterion, we discuss the behavior of the statistical resolution limit with respect to some features of interest, namely, the correlation factor, the central frequency, the minimum resolution limit boundary and the array geometry. Finally, to avoid the complexity of the optimal geometry design procedure, we propose a fast, nearly optimal, array design scheme to enhance the capacity of resolvability under given constraints (e.g., number of sensors and array aperture).     

Two-dimensional angle estimation for monostatic MIMO arbitrary array with velocity receive sensors and unknown locations

In this paper, the problem of two-dimensional angle estimation for monostatic multi-input multi-output (MIMO) array is studied, and an algorithm based on the usage of velocity receive sensors is proposed. The algorithm applies the estimation method of signal parameters via rotational invariance technique (ESPRIT) algorithm to obtain automatically paired two-dimensional angle estimation. By utilizing the relationship within the outputs of velocity sensors, the rotational invariance property of ESPRIT does not depend on the array geometry any more. Hence, the proposed algorithm can provide two-dimensional DOA estimation for the MIMO array without the knowledge of sensor locations in the array. The algorithm requires no peak searches, so it has low complexity. Furthermore, it has better angle estimation performance than propagator method using the same sensor configuration. Error analysis and Cramér–Rao bound (CRB) of angle estimation in MIMO radar are derived. Simulation results verify the usefulness of the algorithm.     

Multidimensional scaling approach for node localization using received signal strength measurements

Node localization has played an important role in wireless sensor networks. In this paper, cooperative localization using received signal strength (RSS) measurements is addressed. The technique of weighted multidimensional scaling (WMDS) which relies on pairwise distance information between nodes is utilized in our algorithm development. Assuming that the transmit power is available, we first convert the original nonlinear localization problem to a system of linear equations, leading to computational attractiveness. It is also proved that the positioning accuracy of the WMDS solution attains the Cramér–Rao lower bound at sufficiently small noise conditions. Furthermore, the proposed method is extended to the unknown transmit power case by exploiting the ratio of squared distance estimates extracted from the RSS information. The effectiveness of the WMDS approach is demonstrated via comparison with several conventional RSS-based positioning methods.     

Intrinsic Cramér–Rao bounds for distributed Bayesian estimator

This paper addresses the intrinsic Cramér–Rao bounds (CRBs) for a distributed Bayesian estimator whose states and measurements are on Riemannian manifolds. As Euclidean-based CRBs for recursive Bayesian estimator are no longer applicable to general Riemannian manifolds, the bounds need redesigning to accommodate the non-zero Riemannian curvature. To derive the intrinsic CRBs, we append a coordination step to the recursive Bayesian procedure, where the proposed sequential steps are prediction, measurement and coordination updates. In the coordination step, the estimator minimises the Kullback–Liebler divergence to obtain the consensus of multiple probability density functions (PDFs). Employing the PDFs from those steps together with the affine connection on manifolds the Fisher Information Matrix (FIM) and the curvature terms of the corresponding intrinsic bounds are derived. Subsequently, the design of a distributed estimator for Riemannian information manifold with Gaussian distribution – referred to as distributed Riemannian Kalman filter – is also presented to exemplify the application of the proposed intrinsic bounds. Finally, simulations utilising the designed filter for a distributed quaternionic estimation problem verifies that the covariance matrices of the filter are never below the formulated intrinsic CRBs.     

A Computationally Efficient Received Signal Strength Based Localization Algorithm in Closed-Form for Wireless Sensor Network

Ranging error is known to degrade significantly the target node localization accuracy. This paper investigates the use of computationally efficient positioning solution of least square (LS) in closed-form, to reduce localization accuracy loss caused by ranging error. For range-based node localization, the LS solution based on least square criterion has been confirmed to exhibit capability of optimum estimation but extensively achieve at a very complex calculation. In this paper we consider the problem how to acquire such LS solution provided with estimation performance at low complex calculation. In this paper, we use the Gauss noise model and use the weighted least squares criterion and the effective calculation method to solve the linearized equation derived from the RSS measurement, and put forward a new approach to estimate the performance of the target node location estimation. Based on the Fisher information matrix, the Cramér–Rao lower bound of target position estimation is derived based on received signal strength. We obviously indicate that the proposed algorithm can approximately achieve the LS solution in estimation performance at a markedly low complex calculation. Simulations are performed to show the improvement of the proposed algorithm.     

Received signal strength based positioning for multiple nodes in wireless sensor networks

We address the problem of locating multiple nodes in a wireless sensor network with the use of received signal strength (RSS) measurements. In RSS based positioning, transmit power and path-loss factor are two environment dependent parameters which may be uncertain or unknown. For unknown transmit powers, we devise two-step weighted least squares (WLS) and maximum likelihood (ML) algorithms for node localization. The mean square error of the former is analyzed in the presence of zero-mean white Gaussian disturbances. When both transmit powers and path-loss factors are unavailable, two nonlinear least squares estimators, namely, the direct ML approach and combination of linear least squares and ML algorithm, are developed. Numerical examples are also included to evaluate the localization accuracy of the proposed estimators by comparing with two existing node positioning methods and the Cramér–Rao lower bound.     

Iterative constrained weighted least squares estimator for TDOA and FDOA positioning of multiple disjoint sources in the presence of sensor position and velocity uncertainties

Sensor position and velocity uncertainties are known to be able to degrade the source localization accuracy significantly. This paper focuses on the problem of locating multiple disjoint sources using time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs) in the presence of sensor position and velocity errors. First, the explicit Cramér–Rao bound (CRB) expression for joint estimation of source and sensor positions and velocities is derived under the Gaussian noise assumption. Subsequently, we compare the localization accuracy when multiple-source positions and velocities are determined jointly and individually based on the obtained CRB results. The performance gain resulted from multiple-target cooperative positioning is also quantified using the orthogonal projection matrix. Next, the paper proposes a new estimator that formulates the localization problem as a quadratic programming with some indefinite quadratic equality constraints. Due to the non-convex nature of the optimization problem, an iterative constrained weighted least squares (ICWLS) method is developed based on matrix QR decomposition, which can be achieved through some simple and efficient numerical algorithms. The newly proposed iterative method uses a set of linear equality constraints instead of the quadratic constraints to produce a closed-form solution in each iteration. Theoretical analysis demonstrates that the proposed method, if converges, can provide the optimal solution of the formulated non-convex minimization problem. Moreover, its estimation mean-square-error (MSE) is able to reach the corresponding CRB under moderate noise level. Simulations are included to corroborate and support the theoretical development in this paper.     

Uncertainty analysis for optimum plane extraction from noisy 3D range-sensor point-clouds

We utilize a more accurate range noise model for 3D sensors to derive from scratch the expressions for the optimum plane fitting a set of noisy points and for the combined covariance matrix of the plane’s parameters, viz. its normal and its distance to the origin. The range error model used by us is a quadratic function of the true range and also the incidence angle. Closed-form expressions for the Cramér–Rao uncertainty bound are derived and utilized for analyzing four methods of covariance computation: exact maximum likelihood, renormalization, approximate least-squares, and eigenvector perturbation. The effect of the simplifying assumptions inherent in these methods are compared with respect to accuracy, speed, and ease of interpretation of terms. The approximate least-squares covariance matrix is shown to possess a number of desirable properties, e.g., the optimal solution forms its null-space and its components are functions of easily understood terms like the planar-patch’s weighted centroid and scatter. It is also fast to compute and accurate enough in practice. Its experimental application to real-time range-image registration and plane fusion is shown by using a commercially available 3D range sensor.     

The Cramér-Rao estimation error lower bound computation for deterministic nonlinear systems

For continuous-time nonlinear deterministic system models with discrete nonlinear measurements in additive Ganssian white noise, the extended Kalman filter (EKF) convariance propagation equations linearized about the true unknown trajectory provide the Cramér-Rao lower bound to the estimation error covariance matrix. A useful application is establishing the optimum filter performance for a given nonlinear estimation problem by developing a simulation of the nonlinear system and an EKF linearized about the true trajectory.