基于定位误差修正的运动目标TDOA/FDOA无源定位方法

针对时差(TDOA)、频差(FDOA)无源定位的两步加权最小二乘(TSWLS)方法定位均方根误差(RMSE)和定位偏差适应测量噪声能力差的问题,在分析了影响两步法定位性能的因素基础上提出一种基于一阶泰勒级数展开的定位误差修正方法。该方法的第1步和两步法相同;其第2步避免了两步法第2步中引入估计偏差的平方运算,利用一阶泰勒级数展开得到第1步定位误差的线性最小均方估计,修正第1步定位结果得到目标位置和速度的最终估计,从理论上证明了该方法可以达到定位的克拉美罗下限(CRLB)。计算机仿真对比了新方法和TSWLS方法、基于泰勒级数(TS)展开的迭代极大似然(ML)方法以及约束总体最小二乘(CTLS)方法的定位性能,新算法复杂度和两步法相当,且均方误差和定位偏差低于两步法、泰勒级数法和CTLS方法。     

传感器位置误差条件下的约束总体最小二乘时差定位算法

 现代定位系统中,传感器往往被安放在运动平台上,其位置无法精确得知,存在估计误差,将严重影响对目标的定位精度。针对这一问题,提出基于约束总体最小二乘(CTLS)的到达时差(TDOA)定位算法。首先通过引入中间变量,将非线性TDOA定位方程转化为伪线性方程,再利用CTLS技术,全面考虑伪线性方程所有系数中的噪声。在此基础上推导了定位方程的目标函数,再根据牛顿迭代方法,进行数值迭代,快速得到精确解。采用一阶小噪声扰动分析方法,对该算法的理论性能进行了分析,证明了算法的无偏性和逼近克拉美-罗下限(CRLB)。仿真实验表明,该算法克服了现有总体最小二乘(TLS)算法不能达到CRLB、两步加权最小二乘(two-step WLS)算法在较高噪声时性能发散的缺陷,在较高噪声时定位精度仍然能达到CRLB。     

传感器位置误差条件下仅用到达频率差的无源定位性能分析

传感器自身的位置误差对辐射源的无源定位精度可能有较大影响,详细分析并推导了在存在传感器位置误差条件下,仅使用到达频率差(FDOA)参数对辐射源定位时,定位精度的克拉美罗下限(CRLB)和均方误差(MSE).在传感器位置误差条件下,提出仅用FDOA来同时估计辐射源和传感器位置的泰勒级数方法,并证明该定位方法理论上的MSE...     

Approximate Maximum Likelihood Algorithm for Moving Source Localization Using TDOA and FDOA Measurements

 A closed-form approximate maximum likelihood (AML) algorithm for estimating the position and velocity of a moving source is proposed by utilizing the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements of a signal received at a number of receivers. The maximum likelihood (ML) technique is a powerful tool to solve this problem. But a direct approach that uses the ML estimator to solve the localization problem is exhaustive search in the solution space, and it is very computationally expensive, and prohibits real-time processing. On the basis of ML function, a closed-form approximate solution to the ML equations can be obtained, which can allow real-time implementation as well as global convergence. Simulation results show that the proposed estimator achieves better performance than the two-step weighted least squares (WLS) approach, which makes it possible to attain the Cram閞-Rao lower bound (CRLB) at a sufficiently high noise level before the threshold effect occurs.     

基于最大似然估计的TDOA/FDOA无源定位偏差补偿算法

在到达时差/到达频差(TDOA/FDOA)无源定位系统中,定位问题的非线性使得定位的结果存在偏差,特别是在噪声较大或者接收站布站不合理的情况下,定位的偏差尤其显著。针对这一问题,提出了一种基于最大似然估计的偏差补偿算法。该方法分为3步:首先,利用最大似然估计器对目标的位置和速度进行求解;其次,通过利用目标定位的估计值和含噪的测量值,对目标的位置和速度偏差值进行理论分析和推导;最后,将最大似然估计解减去理论偏差值,得到经过偏差补偿的新的目标定位解。理论分析和实验仿真证明,在一定噪声的情况下,所推导的目标位置和速度的理论偏差值与实际偏差值相符,并且经过偏差补偿后的定位算法,在保持目标定位的均方根误差(RMSE)与原最大似然算法一致的情况下,目标的位置和速度偏差值远远小于原最大似然算法的偏差值,目标定位精度得到了有效的提高。     

利用重要性采样的时差-频差联合估计算法

针对无源定位中参考信号真实值未知的时差(TDOA)-频差(FDOA)联合估计问题,构建了一种新的时差-频差最大似然(ML)估计模型,并采用重要性采样(IS)方法求解似然函数极大值,得到时差-频差联合估计。算法通过生成时差-频差样本,并统计样本加权均值得到估计值,克服了传统互模糊函数(CAF)算法只能得到时域和频域采样间隔整数倍估计值的问题,且不存在期望最大化(EM)等迭代算法的初值依赖和收敛问题。推导了时差-频差联合估计的克拉美罗下界(CRLB),并通过仿真实验表明,算法的计算复杂度适中,估计精度优于CAF算法和EM算法,在不同信噪比条件下估计误差接近CRLB。     

一种新的基于TDOA与GROA的无源定位算法

针对基于到达时间差(TDOA)与到达增益比(GROA)的辐射源无源定位问题,提出了一种新的定位算法。首先通过引入一个中间变量,根据TDOA和GROA测量模型,构造一个约束加权最小二乘(CWLS)估计。由于这个CWLS问题是非凸的优化问题,现有的方法不能很好地求解。为此,提出了一种有效的方法可以求解到其全局最优解。最后,所提算法被推广到观测站存在自定位误差时的定位求解。计算机仿真结果验证了所提算法能够获得优于传统两步加权最小二乘法(2WLS)的定位性能,能够在更大的噪声条件下达到克拉美罗下界(CRLB)。     

Geolocation of a known altitude object from TDOA and FDOA measurements

Most satellite systems for locating an object on Earth use only time difference of arrival (TDOA) measurements. When there are relative motions between an emitter and receivers, frequency difference of arrival (FDOA) measurements can be used as well. Often, the altitude of an object is known (it is zero, for example) or can be measured with an altimeter. Two sets of geolocation solutions are proposed which exploit the altitude constraint to improve the localization accuracy. One is for TDOAs alone and the other for the combination of TDOA and FDOA measurements. The additional complexity by imposing the constraint is a one-dimensional Newton's search and the rooting of a polynomial. The covariance matrices of the new estimators are derived under a small measurement noise assumption and shown to attain the constrained Cramer-Rao lower bound (CRLB). When there is a bias error in the assumed altitude, using the altitude constraint will introduce a bias to the solution. Since applying the constraint decreases the variance, there is a tradeoff between variance and bias in the mean square error (MSE). The maximum allowable altitude error such that the constraint solution will remain superior to the unconstraint is given. Simulation results are included to corroborate the theoretical development.     

Moving target localization in distributed MIMO radar with transmitter and receiver location uncertainties

In this paper, the problem of moving target localization from Bistatic Range(BR) and Bistatic Range Rate(BRR) measurements in a Multiple-Input Multiple-Output(MIMO) radar system having widely separated antennas is investigated. We consider a practically motivated scenario,where the accurate knowledge of transmitter and receiver locations is not known and only the nominal values are available for processing. With the transmitter and receiver location uncertainties,which are usually neglected in MIMO radar systems by prior studies, taken into account in the measurement model, we develop a novel algebraic solution to reduce the estimation error for moving target localization. The proposed algorithm is based on the pseudolinear set of equations and two-step weighted least squares estimation. The Cramer-Rao Lower Bound(CRLB) is derived in the presence of transmitter and receiver location uncertainties. Theoretical accuracy analysis demonstrates that the proposed solution attains the CRLB, and numerical examples show that the proposed solution achieves significant performance improvement over the existing algorithms.     

Accurate single-observer passive coherent location estimation based on TDOA and DOA

This paper investigates the problem of target position estimation with a single-observer passive coherent location(PCL) system. An approach that combines angle with time difference of arrival(ATDOA) is used to estimate the location of a target. Compared with the TDOA-only method which needs two steps, the proposed method estimates the target position more directly. The constrained total least squares(CTLS) technique is applied in this approach. It achieves the Cramer–Rao lower bound(CRLB) when the parameter measurements are subject to small Gaussian-distributed errors. Performance analysis and the CRLB of this approach are also studied. Theory verifies that the ATDOA method gets a lower CRLB than the TDOA-only method with the same TDOA measuring error. It can also be seen that the position of the target affects estimating precision.At the same time, the locations of transmitters affect the precision and its gradient direction.Compared with the TDOA, the ATDOA method can obtain more precise target position estimation.Furthermore, the proposed method accomplishes target position estimation with a single transmitter,while the TDOA-only method needs at least four transmitters to get the target position. Furthermore,the transmitters' position errors also affect precision of estimation regularly.     

A closed-form solution for moving source localization using LBI changing rate of phase difference only

Due to the deficiencies in the conventional multiple-receiver localization systems based on direction of arrival(DOA) such as system complexity of interferometer or array and amplitude/phase unbalance between multiple receiving channels and constraint on antenna configuration,a new radiated source localization method using the changing rate of phase difference(CRPD)measured by a long baseline interferometer(LBI) only is studied. To solve the strictly nonlinear problem, a two-stage closed-form solution is proposed. In the first stage, the DOA and its changing rate are estimated from the CRPD of each observer by the pseudolinear least square(PLS) method,and then in the second stage, the source position and velocity are found by another PLS minimization. The bias of the algorithm caused by the correlation between the measurement matrix and the noise in the second stage is analyzed. To reduce this bias, an instrumental variable(IV) method is derived. A weighted IV estimator is given in order to reduce the estimation variance. The proposed method does not need any initial guess and the computation is small. The Cramer–Rao lower bound(CRLB) and mean square error(MSE) are also analyzed. Simulation results show that the proposed method can be close to the CRLB with moderate Gaussian measurement noise.     

A direct position determination method with combined TDOA and FDOA based on particle filter

The localization of a stationary transmitter using moving receivers is considered. The original Direct Position Determination (DPD) methods, with combined Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA), do not perform well under low Signal-to-Noise Ratio (SNR), and worse still, the computation cost is difficult to accept when the computational capabilities are limited. To get better positioning performance, we present a new DPD algorithm that proves to be more computationally efficient and more precise for weak signals than the conventional approach. The algorithm partitions the signal received with the same receiver into multiple non-overlapping short-time signal segments, and then uses the TDOA, the FDOA and the coherency among the short-time signals to locate the target. The fast maximum likelihood estimation, one iterative method based on particle filter, is designed to solve the problem of high computation load. A secondary but important result is a derivation of closed-form expressions of the Cramer-Rao Lower Bound (CRLB). The simulation results show that the algorithm proposed in this paper outperforms the traditional DPD algorithms with more accurate results and higher computational efficiency, and especially at low SNR, it is more close to the CRLB.     

一种基于观测站数目最小化的TDOA/FDOA无源定位算法

在三维（3D）运动目标无源定位系统中，无模糊定位最少需要4个观测站。而传统的两步加权最小二乘（TSWLS）及其改进的闭式算法至少需要5个观测站进行求解，当减少一个观测站时，这些闭式算法往往无法提供可靠解。针对这一问题，提出一种最小化观测站数目的到达时间差（TDOA）与到达频率差（FDOA）定位算法。该算法是一种闭式解法并且能够在三维场景下仅使用4个观测站进行定位。该算法分为两步：第1步分离传统的TSWLS算法中未知参数空间，建立了一组新的等式，并且利用加权最小二乘（WLS）算法得到目标位置与速度的初始值；第2步利用泰勒级数展开算法将中间变量线性化，对目标位置和速度初始值进一步校正。理论分析证明了在适当的噪声水平下该算法能够达到克拉美罗界（CRLB）。此外，计算机仿真表明仅使用4个观测站时，该算法对于近场以及远场目标参数的估计精度在测量噪声较小时可以实现CRLB；并且还表明使用5个观测站估计时，该算法比TSWLS及其改进算法能更好地适应大的测量噪声。     

Ballistic missile track initiation from satellite observations

An algorithm is presented to initiate tracks of a ballistic missile in the initial exoatmospheric phase, using line of sight (LOS) measurements from one or more moving platforms (typically satellites). The major feature of this problem is the poor target motion observability which results in a very ill-conditioned estimation problem. The Gauss-Newton iterative least squares minimization algorithm for estimating the state of a nonlinear deterministic system with nonlinear noisy measurements has been previously applied to the problem of angles-only orbit determination using more than three observations. A major shortcoming of this approach is that convergence of the algorithm depends strongly on the initial guess. By using the more sophisticated Levenberg-Marquardt method in place of the simpler Gauss-Newton algorithm and by developing robust new methods for obtaining the initial guess in both single and multiple satellite scenarios, the above mentioned difficulties have been overcome. In addition, an expression for the Cramer-Rao lower bound (CRLB) on the error covariance matrix of the estimate is derived. We also incorporate additional partial information as an extra pseudomeasurement and determine a modified maximum likelihood (ML) estimate of the target state and the associated bound on the covariance matrix. In most practical situations, probabilistic models of the target altitude and/or speed at the initial point constitute the most useful additional information. Monte Carlo simulation studies on some typical scenarios were performed, and the results indicate that the estimation errors are commensurate with the theoretical lower bounds, thus illustrating that the proposed estimators are efficient     

Moving scanning emitter tracking by a single observer using time of interception: Observability analysis and algorithm

The target motion analysis (TMA) for a moving scanning emitter with known fixed scan rate by a single observer using the time of interception (TOI) measurements only is investigated in this paper.By transforming the TOI of multiple scan cycles into the direction difference of arrival (DDOA) model,the observability analysis for the TMA problem is performed.Some necessary conditions for uniquely identifying the scanning emitter trajectory are obtained.This paper also proposes a weighted instrumental variable (WIV) estimator for the scanning emitter TMA,which does not require any initial solution guess and is closed-form and computationally attractive.More importantly,simulations show that the proposed algorithm can provide estimation mean square error close to the Cramer-Rao lower bound (CRLB) at moderate noise levels with significantly lower estimation bias than the conventional pseudo-linear least square (PLS) estimator.     

Passive ranging of a low observable ballistic missile in agravitational field

In this paper we present an estimation algorithm for tracking the motion of a low-observable target in a gravitational field, for example, an incoming ballistic missile (BM), using angle-only measurements. The measurements, which are obtained from a single stationary sensor, are available only for a short time. Also, the low target detection probability and high false alarm density present a difficult low-observable environment. The algorithm uses the probabilistic data association (PDA) algorithm in conjunction with maximum likelihood (ML) estimation to handle the false alarms and the less-than-unity target detection probability. The Cramer-Rao lower bound (CRLB) in clutter, which quantifies the best achievable estimator accuracy for this problem in the presence of false alarms and nonunity detection probability, is also presented. The proposed estimator is shown to be efficient, that is, it meets the CRLB, even for low-observable fluctuating targets with 6 dB average signal-to-noise ratio (SNR). For a BM in free flight with 0.6 single-scan detection probability, one can achieve a track detection probability of 0.99 with a negligible probability of false track acceptance     

Joint TDOA,FDOA and differential Doppler rate estimation: Method and its performance analysis

Considering the estimation accuracy reduction of Frequency Difference of Arrival(FDOA) caused by relative Doppler companding, a joint Time Difference of Arrival(TDOA),FDOA and differential Doppler rate estimation method is proposed and its Cramer-Rao low bound is derived in this paper.Firstly, second-order ambiguity function is utilized to reduce the dimensionality and estimate initial TDOA and differential Doppler rate.Secondly, the TDOA estimation is updated and FDOA is obtained using cross ambiguity function, in which relative Doppler companding is compensated by the existing differential Doppler rate.Thirdly, differential Doppler rate estimation is updated using cross estimator.Theoretical analysis on estimation variance and Cramer-Rao low bound shows that the final estimation of TDOA, FDOA and differential Doppler rate performs well at both low and high signal–noise ratio, although the initial estimation accuracy of TDOA and differential Doppler rate is relatively poor under low signal–noise ratio conditions.Simulation results finally verify the theoretical analysis and show that the proposed method can overcome relative Doppler companding problem and performs well for all TDOA, FDOA and differential Doppler rate estimation.     

Low observable target motion analysis using amplitude information

In conventional passive and active sonar system, target amplitude information (AI) at the output of the signal processor is used only to declare detections and provide measurements. We show that the AI can be used in passive sonar system, with or without frequency measurements, in the estimation process itself to enhance the performance in the presence of clutter where the target-originated measurements cannot be identified with certainty, i.e., for “low observable” or “dim” (low signal-to-noise ratio (SNR)) targets. A probabilistic data association (PDA) based maximum likelihood (ML) estimator for target motion analysis (TMA) that uses amplitude information is derived. A track formation algorithm and the Cramer-Rao lower bound (CRLB) in the presence of false measurements, which is met by the estimator even under low SNR conditions, are also given. The CRLB is met by the proposed estimator even at 6 dB in a cell (which corresponds to 0 dB for 1 Hz bandwidth in the case of a 0.25 Hz frequency cell) whereas the estimator without AI works only down to 9 dB. Results demonstrate improved accuracy and superior global convergence when compared with the estimator without AI. The same methodology can be used for bistatic radar     

WGS-84模型下时差频差半定规划定位算法

利用多颗卫星的时差频差对辐射源进行位置和速度的测量,其本质意义上是一个含有噪声项的高度非线性方程组求解问题,针对地面目标而言,可以采用基于WGS-84地球模型作为目标位置和速度约束,更进一步的增加了定位系统的复杂性。提出了一种基于半定规划(SDP)的定位解算法,将非线性方程求解问题通过适当的松弛,转化为半定优化(SDO)的问题,借助于业界较为成熟的CVX等优化软件进行定位求解,并研究了该模型条件下的克拉美罗下界(CRLB)。仿真结果表明,该算法能够较好地逼近克拉美罗下界。     

基于ECEF的广义最小二乘误差配准技术

 雷达组网数据处理首先要进行误差配准,来准确地估计和消除系统误差。传统的误差配准技术多基于球极投影,当雷达之间距离较远时,给配准结果引入一定的误差。基于地球中心坐标系(ECEF),提出了一种广义最小二乘的ECEF-GLS误差配准技术,较好地解决了远距离误差配准问题,误差分析表明,如果忽略模型线性化引入的误差,配准结果达到了CRLB下限。最后,使用仿真数据验证了算法的性能,并和Zhou提出的基于ECEF坐标系的最小二乘ECEF-LS误差配准算法进行了比较。