Total least-squares (TLS) aims to estimate the unknown parameters of an errors-in-variables (EIV) model from noisy observations when the coefficients are also perturbed by errors. It is helpful to know whether more variates and coefficients lead to more accurate estimation for TLS. In this study, our motivation was to reveal the relationship between the estimation performance and the number of variates or coefficients. The challenge was how to observe a performance change when using additional variates/coefficients. First, the Cramér–Rao bound (CRB) and the hybrid Bhattacharyya–Barankin (HBB) bound were derived for the multivariate EIV model under Gaussian noise typical to most systems. Second, we theoretically affirmed the properties by validating the analytical CRB that additional variates/coefficients were favorable. Third, the quantity properties were verified by simulation about CRB and HBB, considering the source direction estimation using time difference of arrival as an example. We conclude that the estimation performance can be improved if additional variates/coefficients are available. The multivariate TLS should be superior to the univariate one. The conclusions have significance for guiding practical system development.

The paper examines the anchorless direction estimation problem using signal measurements of time difference of arrival (TDOA) from multiple far-field sources in 3D space. A systematic solution involving coordinate frame construction is proposed to jointly estimate source directions and sensor positions through low-rank matrix approximation. The number of unknown parameters to be estimated is reduced by QR decomposition of the mixing matrix to determine the relevant parameters. Pseudolinear and iterative estimators are developed for the solution through norm constraints. We study the property of TDOA-based anchorless direction estimation, discuss the necessary number of sources and sensors, and derive the constrained Crame ' r-Rao bound for performance evaluation. Additionally, an exception treatment strategy is proposed in the complex case for pseudolinear estimator. A simulation conducted to validate the proposed solution's effectiveness showed that the solution's estimation accuracy was high, indicating its feasibility and effectiveness.

The paper focuses on the time-difference-of-arrival (TDOA)-based direction estimation of plane wave from a signal-emitting source by spatial array. In order to avoid the potential impact of geometric dilution of precision (GDOP) on estimation accuracy, we propose that the minimum measurements (i.e. only two TDOAs) are used to deterministically calculate the source direction without involving numerical techniques. It is however challenging to algebraically obtain the common solution of two measurement equations in original nonlinear form. By making use of the TDOA invariance to coordinate system rotation, a novel closed-form method is proposed to tackle the above problem via a well-ordered coordinate rotation. Then, we develop a weighted-sum scheme to combine the minimum measurements based individual direction estimates with their weights determined by GDOP. The theoretical estimation mean and variance of the proposed geometry-aided scheme have been derived. The equation curve of TDOA measurement is also geometrically interpreted in the paper. Computer simulations demonstrate the superior performance of this scheme over other similar estimators in the literature, which involve the grid-scan technique, the sub-minimum-measurement scheme, and the linear least squares method. It is validated that the proposed scheme considerably outperforms the other closed-form estimators and achieves estimation accuracy similar to numerical technique but with significantly reduced computational complexity. (c) 2021 Elsevier B.V. All rights reserved.

针对传统的基于到达时间差的曲线交叉点测向精度不高、权值确定困难的问题,提出一种采用旋转坐标系求出交叉点,根据几何精度因子GDOP来加权不同交叉点参量的方法;通过坐标系旋转,获得两个非线性方程关于两个未知变量的解析解,即方位角和俯仰角,改善了曲线交叉点测向精度不高的问题;与传统的加权方法相比,采用GDOP指标综合加权估计的WI法对目标位置的估计更加准确;仿真验证了采用3种不同加权方式下算法的性能,根据实验结果得出:采用综合加权估计的WI法更能接近克拉美罗下界,提高了对交叉点估计的精度,具有较高的定位性能.

This paper focuses on the time-difference-of-arrival (TDOA)-based direction finding for a low-altitude source with an acoustic sensor array. The source bearing in space must be estimated by the joint azimuth-elevation estimation. The azimuth may be the only concern for some scenarios, where the elevation is trivial; however, an extra parameter about elevation needs to be estimated in the joint estimation. When a planar array is introduced to replace the spatial array, the azimuth-only estimation solution with the original TDOA measurements is proposed. This solution is simpler than the joint estimation by excluding the elevation calculation. The mathematical relation between the TDOAs of spatial and planar arrays is derived, and their azimuth Cramer–Rao lower bounds (CRLBs) are compared. The CRLBs together with the pseudo-linear estimation and Taylor series least squares method are evaluated for estimating the source azimuth by simulations as well as real-life data. The results demonstrate that the azimuth-only estimation would create a modest precision improvement over the joint azimuth-elevation estimation. The effect of array altitude variation is also examined, suggesting that the relative heights among sensors have little influence on the estimation of source azimuth.

针对到达时间差的声源测向问题,如果能够知道声源现场的速度参量,结合线性最小二乘法(Linear Least Squares,LLS)的波达方向(Direction of Arrival,DOA)的估计结果,则可以采用基于最大似然原理的迭代式算法进行求解.但在某些环境下,声信号传播的速度参量无法获得,或测量偏差较大,势必会影响DOA的估计精度.根据LLS无需声速的特点,以及基于最大似然原理的迭代式算法可直接求解声源方位角与俯仰角的特点,文中提出免声速迭代式模型,并采用泰勒级数展开法以及莱温伯格-马夸特法进行求解,该方法无需预先知道声音信号的速度参量.另外,文中推导了泰勒级数展开法、莱温伯格-马夸特法的测向均方差以及免声速迭代式模型的克拉姆-拉奥下界.模拟实验结果表明,在存在声速测量误差的情况下,所提方法明显优于现有方法.

为提高短基线传感器阵列测向性能,在传统的方位角/俯仰角联合估计基础上提出一种纯方位角估计方案,将三维空间阵列映射为平面阵列,利用平面关系只对三维空间瞬时声源的方位角进行估计.基于克拉姆-拉奥下界理论的分析表明纯方位角估计性能优于联合估计的方位角结果,并讨论了纯方位角估计的使用时机.另外本文分析认为,阵列传感器之间高差只影响俯仰角的测向精度,对方位角估计精度不产生影响.仿真结果验证了纯方位角估计的有效性和优越性,为进一步工程应用奠定了理论基础.

In the Time Difference of Arrival(TDOA) positioning technology,when the position coordinates of the sensor have errors,the estimated value obtained by the Least Squares(LS) method no longer has the best unbiasedness,resulting in a decrease in the direction finding accuracy.Aiming at this problem,a direction finding method based on Total Least Squares(TLS) is proposed.By transforming the nonlinear observation equation into pseudo-linear equation,the augmented matrix is constructed and the singular value decomposition is carried out to achieve the target position.Theoretical analysis and simulation results show that this method is more accurate than the classical LS method and LS-Tylar method.

This paper focuses on the bearing estimation problem of far-field signal source via time-difference-of-arrival (TDOA) with a synchronized array in 3-D space. It is usually assumed that the propagation speed (PS) is perfectly known in localization. In reality, only an imperfect knowledge of PS could be obtained. The traditional closed-form solutions without involving PS have the advantage of low complexity, but suffer from low estimation accuracy. A measurement-division model is proposed to offer an alternative solution without the need of the propagation speed. This speed-free model combines two original TDOA measurement equations into a division formula, whose Cramer-Rao lower bound (CRLB) is derived for the observed data. A typical optimization method, i.e. the Levenberg-Marquardt (LM) algorithm is adopted to resolve the nonlinear measurement-division model, resulting in an estimation accuracy improvement because of its iterative search behavior. The theoretical performance of this solution is evaluated in terms of bias and covariance. Simulations are conducted to demonstrate an accuracy advantage of the solution over the related methods.

This paper focuses on estimating the azimuth and elevation of a sound emitter in 3-D space based on time-difference-of-arrival (TDOA) measurements with an array of acoustic sensors. The TDOA-based direction finding problem is significant because in a range of scenarios the source only emits a transient signal and only TDOA measurements can be used to find the direction. The linear least squares estimator provides a suboptimal solution, since there is nontrivial information loss in the linearization of the nonlinear observation equation. To avoid the information loss, the Lagrange multiplier method is usually used to realize the constrained optimization, but the computational complexity is rather high. This paper proposes a constrained least squares estimator to deal with the direction finding problem. The proposed method makes use of both Lagrange multiplier and quadratic constraints to form the cost function. The resultant estimator is shown to be approximate closed-form so that the computational complexity is reduced greatly, but contributes little under a small noise level. Mathematical formula is derived to evaluate the theoretical accuracy of the proposed estimator in terms of mean square error. Both simulation and field experimental results demonstrate that the proposed estimator can outperform the traditional linear and nonlinear estimators.

In view of the direction finding problem based on time difference of arrival (TDOA), when the TDOA measurement noise was high, the direction finding accuracy of linear least squares (LLS) algorithm decreased. Iterative Taylor Series (Taylor) algorithm had high accuracy, but a dependency on the initial value. In order to compensate both shortcomings, a cooperative algorithm was proposed firstly, which combined the LLS with the Taylor. Then, the mean square error of direction finding about cooperative algorithm and the Cramer-Rao Lower Bound (CRLB) was derived. By comparing them, thus to prove that the cooperative algorithm can reach CRLB. Finally, the simulation results are shown that when the TDOA measurement noise is high, the cooperative direction finding algorithm based on LLS and Taylor can better improve the accuracy of direction finding and has strong robustness.

This chapter focuses on estimating the azimuth and elevation angles of a sound emitter based on time-difference-of-arrival (TDOA) measurements using an array of acoustic sensors. The TDOA-based direction-finding problem is appropriate because in a range of scenarios the source only emits a transient signal and TDOA measurements provide a simple method of finding the direction of the received signal. Given the measurement of TDOA, three methods for calculating the actual bearing of an acoustic source are considered—algebraic calculations based on trigonometric functions, linear least squares, and nonlinear least squares—and these results are also compared with the Cramer-Rao lower bound (CRLB). In this chapter, a comprehensive analysis of TDOA-based direction-finding methods is presented with regard to different application conditions, while their estimation performances are analysed with both simulation and field experimental results produced by 3-D microphone array.

A direction finding algorithm based on Levenberg-Marquardt(LM) is proposed to reduce the interference of Time Difference of Arrival(TDOA) measurement noise on lateral accuracy and avoid the phenomenon that algorithm results are not convergent.The closed-form solution obtained by the Linear Least Squares(LLS) algorithm is used as the initial bearing estimation of the algorithm,and the azimuth of the radiator is obtained by iterative operation,so as to achieve a high accuracy estimation of the acoustic source azimuth.Experimental results show that,compared with LLS algorithm and Taylor algorithm,this algorithm can reach the Cramer-Rao Lower Bound(CRLB),improve the direction finding accuracy and robustness while ensuring the convergence of the results.

Since the shock wave of supersonic projectile is difficult to be captured due to its complexity and susceptible interference,the miss distance test is a difficult problem.The testing scene of miss distance is described,the formation mechanism of shock wave is analyzed,and the parameter relations before and after shock wave forming are identified.The test model of miss distance and the linear and nonlinear interference models based on simplified shock wave environment are established.The time duration test principle is used for field experiment,and the cumulative effect of pressure of shock wave field is used for simulating inversion in order to verify the accuracy and effectiveness of test methods and engineering test.

A sound localization system consisting of a number of spatially distributed sensors can be employed to estimate the source bearing by measuring the relative time-difference-of-arrival (TDOA) of the transient acoustic signal. However, sensor measurements may contain nonnegligible consistent systematic biases, resulting in significant direction finding estimation error. In this paper, an efficient expectation-maximization (EM) algorithm is proposed for accurately estimating the direction-of-arrival (DOA) of the signal from a far field source in the presence of biased TDOA measurements. The unknown biases are treated as hidden variables, and nonlinear least square estimators are developed to jointly estimate the biases and DOA parameters for both the reference-free mode and the reference mode. TDOA error distribution is investigated and four different distributions [Gaussian distribution, Laplace distribution, Gaussian-Laplace distribution, and generalized normal distribution (GND)] are employed to fit the experimental data. It is observed that GND is the best for modeling the biased TDOA errors in terms of cumulative distribution function fitting root mean square error, while Laplace distribution offers a good tradeoff between the accuracy and complexity. Both the simulation and field experimental results demonstrate that the proposed EM-based estimators can considerably outperform the existing algorithms.

This paper focuses on direction finding of a signal source using time-difference-of-arrival (TDOA) measurements at a 3-D acoustic sensor array. Two evolutionary computation methods are proposed to solve the direction finding problem, which are the genetic algorithm and the particle swarm optimization algorithm. Sound speed is used in the development of the algorithms, which is estimated based on observed weather parameters and initial direction estimation results from the least square (LS) estimator which is presently the key method in TDOA-based direction finding. All reference-free TDOA measurements are adopted in defining cost function to improve performance. To guarantee fast convergence, an LS estimator is also utilized to provide initial direction estimates for the two swarm intelligent algorithms. Simulation results demonstrate that the proposed methods with a full TDOA set are superior to the Cramer-Rao lower bound with a limited set of reference-based TDOA measurement, significantly outperforming the LS estimator. Extensive field experiments were conducted and there is good agreement between the experimental results and simulation results.

As the traditional direction-finding method of passive sound source has the problem of low precision and fixed structure,a direction-finding method based on short baseline sensor network was proposed.According to the time difference of arrival (TDoA)measurements of a sound signal among any sensor nodes with random locations,the least square technique was adopted to compute the bearing of source souce.Practical experimental data of the artillery fire and burst point in the true environment showed that the short baseline sensor network had a better direction-finding ability compared with the traditional array of fixed shapes.

In order to solve the non uniform environment of the target area influencing the variance of DF error, the weighted maximum likelihood estimation (WMLE) algorithm was proposed. In this algorithm, the effect of the target distance was introduced into MLE. we construct the weighted vector to make up for the effect when the target distance increase the variance of the DF error become worse. Theoretical analysis showed that the algorithm of WMLE could further improve the accuracy of the multi-station DF crossing localization.