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.     

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。     

基于短时时差序列的无源定位方法

针对脉冲辐射源定位问题,提出了一种基于短时时差(TDOA)序列的多站无源定位新方法.首先建立了TDoA序列模型,然后推导了该方法定位误差的克拉美罗下限(CRLB),并提出了一种高效的解算方法--两步定位法(TSLM):第1步利用TDOA序列线性估计出TDOA及其变化率信息,第2步基于TDOA及其变化率信息进行定位.典型...     

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.     

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。     

布站方式对基于TDOA的干扰源定位影响分析

介绍了到达时间差(TDOA)算法的基本原理,并分析了定位存在模糊解区域和无解区域的原因。对二维不同布站条件下的模糊解区域进行仿真,得到了不同角度、不同基线长度布站方式下的模糊解区域分布,总结了布站方式对二维模糊解区域分布的影响。基于TDOA的几何精度因子(GDOP)方法分析了基线长度和布站形状对定位精度的影响,在四站三维条件下对不同布站形式下的定位精度进行仿真和分析,得出了不同布站条件下GDOP的分布规律,为实际工程应用中的接收机布站提供了参考。     

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.     

Least squares range difference location

An array of n sensors at known locations receives the signal from an emitter whose location is desired. By measuring the time differences of arrival (TDOAs) between pairs of sensors, the range differences (RDs) are available and it becomes possible to compute the emitter location. Traditionally geometric solutions have been based on intersections of hyperbolic lines of position (LOPs). Each measured TDOA provides one hyperbolic LOP. In the absence of measurement noise, the RDs taken around any closed circuit of sensors add to zero. A bivector is introduced from exterior algebra such that when noise is present, the measured bivector of RDs is generally infeasible in that there does not correspond any actual emitter position exhibiting them. A circuital sum trivector is also introduced to represent the infeasibility; a null trivector implies a feasible RD bivector. A 2-step RD Emitter Location algorithm is proposed which exploits this implicit structure. Given the observed noisy RD bivector Δ, (1) calculate the nearest feasible RD bivector Δˆ, and (2) calculate the nearest point to the ( ₃ⁿ) planes of position, one for each of the triads of elements of Δˆ. Both algorithmic steps are least squares (LS) and finite. Numerical comparisons in simulation show a substantial improvement in location error variances     

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...     

一种低空声源分布式定位系统及其算法与实验研究

低空声源目标具有较强的隐身性,为了实现对此类目标的定位,构建基于TMS320C6713平台的分布式麦克风阵列低空声源定位系统.麦克风阵列节点以DSP为核心处理器,通过A/D芯片采样麦克风阵列接收到的信号,通过到达时间差(TDOA)计算声源的角度,利用测向交叉法,融合多个节点的角度信息,计算出声源的位置.结果表明:在时延估计精确的情况下,对低空目标具有较好地估计;系统具有较高的精度和稳定性,能够满足对低空声源目标进行定位的实际需求.     

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)。     

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。     

基于两步最小二乘定位的偏差改进算法

针对传统最小二乘(LS)定位算法在噪声较大时会出现有偏估计的问题。首先详细推导了传统两步最小二乘算法在时差角度联合定位场景下的理论偏差,给出了出现偏差的原因;其次对误差均值加入二次约束条件,提出一种基于时差角度联合定位的改进算法,并详细推导新算法的理论偏差以及均方误差。相比于其他加限制条件的方法,新算法能有效降低估计偏差,另外由于其不需要进行特征值分解且能得到闭式解,计算复杂度较小。仿真结果表明,新算法在保持原有均方误差(MSE)的前提下能显著降低估计偏差,其定位偏差与最大似然估计器相当。     

Sensitivity analysis of dual-satellite geolocation

An analytic solution is presented for the linearized estimation problem arising from the dual-satellite geolocation of the source of a narrowband signal using time- and frequency-difference-of-arrival (TDOA and FDOA, respectively) observations and an altitude constraint. This solution is used to analyze the covariance matrix of the resultant source location estimate, along with the sensitivity of its mean to errors in the presumed source altitude and velocity, and the measured or predicted positions and velocities of the satellites. Following the practice of differential Global Positioning System, the technique of differential calibration is used to reduce or eliminate various sources of solution bias which would otherwise require tight, and consequently expensive, individual calibration     

传感器参数误差下的运动目标TDOA/FDOA无源定位算法

在运动目标无源定位系统中，许多算法的前提是精确已知传感器的位置以及速度，但实际情况下可利用的传感器的参数均会存在一些噪声扰动。针对这一问题，提出一种改进的两步加权最小二乘（TSWLS）时差（TDOA）与频差（FDOA）定位算法。该算法是一种闭式算法并且分为2步。第1步与经典的两步加权最小二乘算法相同，第2步进一步研究了额外变量与目标参数之间的关系并且建立了新的矩阵方程。随后，利用加权最小二乘技术给出了最终解。理论分析证明了在测量噪声较小时该算法能够达到克拉美罗界（CRLB）。所提算法具有计算复杂度低，实时性高的优点；另外，经过适当的维度调整，该算法同样适用于对多非相交源进行定位求解。计算机仿真进一步证明了理论分析的正确性。     

一种4站情况下基于TDOA/FDOA的无源定位方法

在基于时频差的三维运动目标无源定位系统中,针对在4个接收站的情况下搜索法实时性低的问题,提出了一种基于改进的加权最小二乘法（MWLS）与萤火虫算法（FA）相结合的无源定位方法（MWLS-FA）。该方法的第1步通过构造一组新的方程来对加权最小二乘（WLS）方法进行改进,使得改进后的WLS方法在4站情况下也能得到目标位置和速度的初始值,第2步利用这个初始值为FA方法提供一个动态的搜索区域,同时在约束条件的添加和参数选择两个方面针对性地对FA方法做出了调整和改进。仿真结果表明,该方法在4站情况下对目标的定位精度可以达到克拉美罗下限（CRLB）,而且在实时性和抗噪性方面优于传统的搜索法,同时该方法在5站情况下的抗噪性能优于两步加权最小二乘法（TSWLS）和约束加权最小二乘（CWLS）法。     

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。     

PASSIVE LOCATION AND ACCURACY ANALYSIS USING TDOA INFORMATION OF MULTI-STATIONS

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Emitter Location Accuracy Using TDOA and Differential Doppler

Two methods for locating a radio or sonar transmitter are briefly described: time difference of arrival (TDOA). d differential Doppler. Two formulas are derived which relate the accuracy of the time and frequency measurements the ?one-sigma widths? of the lines of constant TDOA onstant differential Doppler on the surface of the FDOA.