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TDOA/FDOA passive localization algorithm for moving target with sensor parameter errors
SUN Ting, DONG Chunxi
School of Electronic Engineering, Xidian University, Xi'an 710071, China
Abstract: In a moving target passive localization system, the premise of many algorithms is that the position and velocity of the sensors are accurately known. However, there exists some noise disturbances in the parameters of available sensors. Aiming to solve this problem, an improved Two-Step Weighted Least Squares (TSWLS) localization algorithm using Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) is proposed. The proposed algorithm is a closed-form solution and is divided into two steps. While the first step is the same as that of the typical TSWLS method, the second step further studies the relationship between the nuisances and the target parameters, establishing a new matrix equation. Then, the final solution is given via Weighted Least Squares (WLS) technique. Theoretical analysis proves that this method can reach the Cramér-Rao Lower Bound (CRLB) at a low noise level. The proposed algorithm in this paper has the advantages of low computational complexity and high real-time performance. In addition, this method is also suitable for locating multiple disjoint sources after appropriate dimensional adjustment. Simulations further demonstrate the effectiveness of the theoretical analysis.
Keywords: passive localization    time difference of arrival    frequency difference of arrival    sensor position errors    Cramér-Rao lower bound

1 定位场景

 $\Delta \mathit{\boldsymbol{\beta }} = \mathit{\boldsymbol{\beta }} - {\mathit{\boldsymbol{\beta }}^{\rm{o}}} = {\left[ {\Delta {\mathit{\boldsymbol{s}}^{\rm{T}}},\Delta {{\mathit{\boldsymbol{\dot s}}}^{\rm{T}}}} \right]^{\rm{T}}}$ （3）

 $r_{i1}^{\rm{o}} = r_i^{\rm{o}} - r_1^{\rm{o}}\;\;\;i = 2,3, \cdots ,M$ （4）

 $r_i^{\rm{o}} = \left\| {{\mathit{\boldsymbol{u}}^{\rm{o}}} - \mathit{\boldsymbol{s}}_i^{\rm{o}}} \right\|\;\;\;\;i = 2,3, \cdots ,M$ （5）

 $\dot r_{i1}^{\rm{o}} = \dot r_i^{\rm{o}} - \dot r_1^{\rm{o}}\;\;\;i = 2,3, \cdots ,M$ （6）

 $\dot r_i^{\rm{o}} = {\left( {{\mathit{\boldsymbol{u}}^{\rm{o}}} - \mathit{\boldsymbol{s}}_i^{\rm{o}}} \right)^{\rm{T}}}\left( {{{\mathit{\boldsymbol{\dot u}}}^{\rm{o}}} - \mathit{\boldsymbol{\dot s}}_i^{\rm{o}}} \right)/r_i^{\rm{o}}$ （7）

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{r}} = {\left[ {{r_{21}},{r_{31}}, \cdots ,{r_{M1}}} \right]^{\rm{T}}} = {\mathit{\boldsymbol{r}}^{\rm{o}}} + \Delta \mathit{\boldsymbol{r}}\\ \mathit{\boldsymbol{\dot r}} = {\left[ {{{\dot r}_{21}},{{\dot r}_{31}}, \cdots ,{{\dot r}_{M1}}} \right]^{\rm{T}}} = {{\mathit{\boldsymbol{\dot r}}}^{\rm{o}}} + \Delta \mathit{\boldsymbol{\dot r}} \end{array} \right.$ （8）

$\partial \boldsymbol{\alpha}^{\circ} / \partial \boldsymbol{\varphi}_{2}^{\circ}$写为分块矩阵可以表示为

 $\frac{{\partial {\mathit{\boldsymbol{\alpha }}^{\rm{o}}}}}{{\partial \mathit{\boldsymbol{\varphi }}_2^{\rm{o}}}} = \left[ {\begin{array}{*{20}{l}} {\frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{u}}^{\rm{o}}}}}}&{\frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}}}{{\partial {{\mathit{\boldsymbol{\dot u}}}^{\rm{o}}}}}}\\ {\frac{{\partial {{\mathit{\boldsymbol{\dot r}}}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{u}}^{\rm{o}}}}}}&{\frac{{\partial {{\mathit{\boldsymbol{\dot r}}}^{\rm{o}}}}}{{\partial {{\mathit{\boldsymbol{\dot u}}}^{\rm{o}}}}}} \end{array}} \right]$ （A1）

 $\frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{u}}^{\rm{o}}}}}\left( {i - 1,:} \right) = \mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_i^{\rm{o}}}^{\rm{T}} - \mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_1^{\rm{o}}}^{\rm{T}}$ （A2）
 $\frac{{\partial {{\mathit{\boldsymbol{\dot r}}}^{\rm{o}}}}}{{\partial {{\mathit{\boldsymbol{\dot u}}}^{\rm{o}}}}}\left( {i - 1,:} \right) = \mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_i^{\rm{o}}}^{\rm{T}} - \mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_1^{\rm{o}}}^{\rm{T}}$ （A3）
 $\frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}\left( \mathit{\boldsymbol{\theta }} \right)}}{{\partial {{\mathit{\boldsymbol{\dot u}}}^{\rm{o}}}}}\left( {i - 1,:} \right) = \mathit{\boldsymbol{\gamma }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_i^{\rm{o}}}^{\rm{T}} - \mathit{\boldsymbol{\gamma }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_1^{\rm{o}}}^{\rm{T}}$ （A4）

 $\frac{{\partial {\mathit{\boldsymbol{\alpha }}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{\beta }}^{\rm{o}}}}} = \left[ {\begin{array}{*{20}{l}} {\frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{s}}^{\rm{o}}}}}}&{\frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}}}{{\partial {{\mathit{\boldsymbol{\dot s}}}^{\rm{o}}}}}}\\ {\frac{{\partial {{\mathit{\boldsymbol{\dot r}}}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{s}}^{\rm{o}}}}}}&{\frac{{\partial {{\mathit{\boldsymbol{\dot r}}}^{\rm{o}}}}}{{\partial {{\mathit{\boldsymbol{\dot s}}}^{\rm{o}}}}}} \end{array}} \right]$ （A5）

 $\begin{array}{l} \frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{s}}^{\rm{o}}}}}\left( {i - 1,:} \right) = \frac{{\partial {\mathit{\boldsymbol{r}}^{\rm{o}}}}}{{\partial {{\mathit{\boldsymbol{\dot s}}}^{\rm{o}}}}}\left( {i - 1,:} \right)\\ \;\;\;\;\;\; = \left[ {\mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_1^{\rm{o}}}^{\rm{T}},{\bf{0}}_{3\left( {i - 2} \right) \times 1}^{\rm{T}}, - \mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_i^{\rm{o}}}^{\rm{T}},{\bf{0}}_{3\left( {M - i} \right) \times 1}^{\rm{T}}} \right] \end{array}$ （A6）
 $\frac{{\partial {{\mathit{\boldsymbol{\dot r}}}^{\rm{o}}}}}{{\partial {\mathit{\boldsymbol{s}}^{\rm{o}}}}}\left( {i - 1,:} \right) = \left[ {\mathit{\boldsymbol{\gamma }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_1^{\rm{o}}}^{\rm{T}},{\bf{0}}_{3\left( {i - 2} \right) \times 1}^{\rm{T}}, - \mathit{\boldsymbol{\gamma }}_{{\mathit{\boldsymbol{u}}^{\rm{o}}},\mathit{\boldsymbol{s}}_i^{\rm{o}}}^{\rm{T}},{\bf{0}}_{3\left( {M - i} \right) \times 1}^{\rm{T}}} \right]$ （A7）

i=2, 3, …, M－1。

 ${\mathit{\boldsymbol{G}}_3} = \partial {\mathit{\boldsymbol{\alpha }}^{\rm{o}}}/\partial \mathit{\boldsymbol{\varphi }}_2^{\rm{o}},{\mathit{\boldsymbol{G}}_4} = - \partial {\mathit{\boldsymbol{\alpha }}^{\rm{o}}}/\partial {\mathit{\boldsymbol{\beta }}^{\rm{o}}}$ （A8）

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http://dx.doi.org/10.7527/S1000-7527.2019.23317

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#### 文章信息

SUN Ting, DONG Chunxi

TDOA/FDOA passive localization algorithm for moving target with sensor parameter errors

Acta Aeronautica et Astronautica Sinica, 2020, 41(2): 323317.
http://dx.doi.org/10.7527/S1000-7527.2019.23317