一种协同二维DOA和TDOA观测量的超视距短波辐射源定位新方法

王鼎 尹洁昕 高路 张莉

王鼎, 尹洁昕, 高路, 等. 一种协同二维DOA和TDOA观测量的超视距短波辐射源定位新方法[J]. 雷达学报(中英文), 待出版. doi: 10.12000/JR24136
引用本文: 王鼎, 尹洁昕, 高路, 等. 一种协同二维DOA和TDOA观测量的超视距短波辐射源定位新方法[J]. 雷达学报(中英文), 待出版. doi: 10.12000/JR24136
WANG Ding, YIN Jiexin, GAO Lu, et al. A novel cooperative positioning method for over-the-horizon shortwave emitter based on two-dimensional direction-of-arrival and time-difference-of-arrival measurements[J]. Journal of Radars, in press. doi: 10.12000/JR24136
Citation: WANG Ding, YIN Jiexin, GAO Lu, et al. A novel cooperative positioning method for over-the-horizon shortwave emitter based on two-dimensional direction-of-arrival and time-difference-of-arrival measurements[J]. Journal of Radars, in press. doi: 10.12000/JR24136

一种协同二维DOA和TDOA观测量的超视距短波辐射源定位新方法

DOI: 10.12000/JR24136
基金项目: 国家自然科学基金(62171469, 62071029)
详细信息
    作者简介:

    王 鼎,副教授,博士生导师,主要研究方向为无线定位、阵列信号处理等

    尹洁昕,副教授,硕士生导师,主要研究方向为无线定位、阵列信号处理等

    高 路,研究员,博士生导师,主要研究方向为电子对抗、智能处理等

    张 莉,教授,硕士生导师,主要研究方向为无线定位、阵列信号处理等

    通讯作者:

    尹洁昕 Cindyin0807@163.com

  • 责任主编:郭福成 Corresponding Editor: GUO Fucheng
  • 中图分类号: TN911.7

A Novel Cooperative Positioning Method for Over-the-horizon Shortwave Emitter Based on Two-dimensional Direction-of-arrival and Time-difference-of-arrival Measurements

Funds: The National Natural Science Foundation of China (62171469, 62071029)
More Information
  • 摘要: 针对超视距远距离短波辐射源定位误差较大的问题,该文在观测站同时获得二维到达角度和到达时间差参数的场景下,提出一种协同这两种观测量的定位新方法。首先,基于单跳电离层虚高模型构建面向短波辐射源的二维到达角度和到达时间差的非线性观测方程。然后,将超视距定位几何模型与代数模型相结合,并依次将两种非线性观测方程转化为伪线性观测方程,进而提出一种无需迭代的两阶段协同定位方法。阶段1通过求解一元六次多项式的根获得目标位置向量闭式解,阶段2通过构建等式约束优化模型对阶段1的估计误差进行改良,并利用拉格朗日乘子技术得到精度更高的定位结果。最后,利用约束误差扰动理论对新提出的协同定位方法的估计性能进行理论分析,证明新方法具有渐近统计最优性,同时还利用约束误差扰动理论定量分析短波辐射源高度信息误差对定位精度产生的影响,并推导能确保地球椭圆约束产生性能增益的短波辐射源高度信息误差最大门限值。仿真实验结果验证该文新方法能够获得显著的协同增益。

     

  • 图  1  短波辐射源2D-DOA和TDOA几何模型示意图

    Figure  1.  Schematic diagram of 2D-DOA and TDOA geometric models of shortwave radiation source

    图  2  在ECEF坐标系下的X-Y平面中的定位结果散布图与定位误差椭圆曲线

    Figure  2.  Scatter plot of positioning results and elliptical curve of positioning errors in the X-Y plane in the ECEF coordinate system

    图  3  正确的根、与正确的根最邻近的假根以及门限值$ - 1/{\kappa _{\max }}$

    Figure  3.  Correct root, the closest false root to the correct root and threshold value $ - 1/{\kappa _{\max }}$

    图  4  定位均方根误差随着参数δ1的变化曲线

    Figure  4.  Localization RMSE as a function of parameter δ1

    图  5  定位均方根误差随着参数δ2的变化曲线

    Figure  5.  Localization RMSE as a function of parameter δ2

    图  6  定位均方根误差随着参数δ3的变化曲线

    Figure  6.  Localization RMSE as a function of parameter δ3

    图  7  定位均方根误差随着相关系数cDT的变化曲线

    Figure  7.  Localization RMSE as a function of correlation coefficient cDT

    图  8  定位偏置随着高度he的变化曲线

    Figure  8.  Localization bias as a function of height he

    图  9  定位均方根误差随着高度he的变化曲线

    Figure  9.  Localization RMSE as a function of height he

    表  1  新方法的计算步骤及其所需的乘法次数

    Table  1.   Calculation steps of the new method and the number of multiplications required

    序号 计算内容 乘法次数
    步骤1 利用式(18)计算矩阵${{\boldsymbol{Q}}_\theta }$和向量${{\boldsymbol{p}}_\theta }$;利用式(22)计算矩阵${{\boldsymbol{Q}}_\beta }$和向量${{\boldsymbol{p}}_\beta }$;
    利用式(26)计算矩阵${{\boldsymbol{Q}}_t}$和向量${{\boldsymbol{p}}_t}$
    $27N - 8$
    步骤2 利用式(27)构造矩阵${\boldsymbol{Q}}({\boldsymbol{\hat z}},{\boldsymbol{\hat d}})$和向量${\boldsymbol{p}}({\boldsymbol{\hat z}},{\boldsymbol{\hat d}})$ 无实质乘法计算
    步骤3 利用式(30)计算矩阵${{\boldsymbol{C}}_z}$和${{\boldsymbol{C}}_d}$ $48{N^2} - 4N - 13$
    步骤4 利用式(34)计算协方差矩阵${\bf{COV}}({\boldsymbol{e}})$及其逆矩阵${({\bf{COV}}({\boldsymbol{e}}))^{ - 1}}$ $\begin{gathered} O[{(3N - 1)^3}] + 66{N^3} \\[-3pt] - 61{N^2} + 19N - 2 \end{gathered} $
    步骤5 利用式(38)计算矩阵${\boldsymbol{W}}({\boldsymbol{\hat z}},{\boldsymbol{\hat d}})$及其逆矩阵${({\boldsymbol{W}}({\boldsymbol{\hat z}},{\boldsymbol{\hat d}}))^{ - 1}}$ $\begin{gathered} O(64) + 36{N^2} \\[-3pt] + 24N - 12 \end{gathered} $
    步骤6 对矩阵${({\boldsymbol{W}}({\boldsymbol{\hat z}},{\boldsymbol{\hat d}}))^{ - 1}}{{\boldsymbol{\varOmega }}_1}$进行特征值分解得到特征值对角矩阵${{\boldsymbol{\varLambda }}_{\text{w}}}$和特征矩阵${{\boldsymbol{H}}_{\text{w}}}$ $O(64) + 4$
    步骤7 利用式(43)计算向量${{\boldsymbol{h}}_1}$和${{\boldsymbol{h}}_2}$ $\begin{gathered} O(64) + 9{N^2} \\ + 6N + 45 \\ \end{gathered} $
    步骤8 利用式(46)—式(50)计算多项式系数$ {\{ {\alpha _j}\} _{0 \le j \le 6}} $ $82$
    步骤9 求解式(45)中的一元六次多项式的根,并利用式(53)选择正确的根 $\begin{gathered} J \cdot O[{(3N - 1)^3}] + O(6) \\[-3pt] + J(12{N^3} + 2{N^2} - 2N) \end{gathered} $
    步骤10 利用式(37)计算向量${{\boldsymbol{\hat v}}_{{\text{o,f}}}}$,并进而构造向量$ {{{\hat {\bar {\boldsymbol{v}}}}}_{{\text{o,f}}}} $和${{\boldsymbol{\hat u}}_{{\text{o,f}}}}$ $O(64) + 16$
    步骤11 利用式(62)计算均方误差矩阵${\bf{MSE}}({{{\hat {\bar {\boldsymbol{v}}}}}_{{\text{o,f}}}})$及其逆矩阵${({\bf{MSE}}({{{\hat {\bar {\boldsymbol{v}}}}}_{{\text{o,f}}}}))^{ - 1}}$ $O(64) + O(27) + 145$
    步骤12 利用式(65)计算标量$ \hat \mu $和向量$ {\boldsymbol{\hat \eta }} $ 6
    步骤13 利用式(66)构造矩阵$ {\boldsymbol{\bar J}}({{\boldsymbol{\hat u}}_{{\text{o,f}}}}) $ 5
    步骤14 利用式(69)计算向量$ \Delta {{{\hat {\bar {\boldsymbol{v}}}}}_{{\text{o,f}}}} $ $28$
    步骤15 利用式(70)计算向量$ {{{\hat{ \bar {\boldsymbol{u}}}}}_{{\text{o,s}}}} $ 无实质乘法计算
    步骤16 利用式(71)计算最终定位结果$ {{\boldsymbol{\hat u}}_{{\text{o,s}}}} $ 3
    下载: 导出CSV

    表  2  观测站的经纬度与电离层虚高数值

    Table  2.   Longitude, latitude and ionospheric virtual height of shortwave observer

    观测站 经度(°) 纬度(°) 电离层虚高(km)
    1 123.04 40.94 273
    2 115.66 40.27 316
    3 114.61 34.98 358
    4 113.47 30.09 347
    5 115.75 26.22 255
    下载: 导出CSV

    表  3  多项式方程的根及其对应的MLE目标函数值

    Table  3.   Roots of polynomial equation and their corresponding MLE objective function values

    序号 一元六次多项式方程的根 与门限值$ - 1/{\kappa _{\max }} = - {\text{4}}{\text{.02}} \times {\text{1}}{{\text{0}}^{ - {\text{5}}}}$的关系 由式(53)给出的MLE目标函数值
    1 –8.12×10–2 小于 3.34×106
    2 –7.98×10–2 小于 3.22×106
    3 –1.81×10–2 小于 7.56×105
    4 –1.73×10–2 小于 6.90×105
    5 –8.08×10–5 小于 6.60×103
    6 2.89×10–7 唯一大于 1.23×101
    唯一的最小值
    下载: 导出CSV
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出版历程
  • 收稿日期:  2024-07-04
  • 修回日期:  2024-09-27
  • 网络出版日期:  2024-10-28

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