CN102170658B

CN102170658B - Geometric positioning improvement method under NLOS (non-line-of-sight) environment

Info

Publication number: CN102170658B
Application number: CN 201110108493
Authority: CN
Inventors: 赵军辉; 赵聪; 李秀萍
Original assignee: Beijing Jiaotong University
Current assignee: Beijing Jiaotong University
Priority date: 2011-04-28
Filing date: 2011-04-28
Publication date: 2013-10-30
Anticipated expiration: 2031-04-28
Also published as: CN102170658A

Abstract

The invention relates to an improved geometric positioning method under the NLOS environment, which adopts a Gauss-Newton iterative method with fast convergence, and optimizes its iterative initial coordinate value in combination with a grid search method. Compared with the existing geometric positioning method, the present invention has the following advantages: moderate complexity, fast and stable convergence, and higher positioning accuracy of the mobile station.

Description

An Improved Method of Geometric Positioning in NLOS Environment

技术领域 technical field

本发明涉及一种NLOS环境下的几何定位改进方法，应用于蜂窝网无线定位技术领域。The invention relates to an improved geometric positioning method under the NLOS environment, which is applied to the technical field of cellular network wireless positioning.

背景技术 Background technique

几何定位方法是根据一个或多个测量参数以及基站与移动台的几何分布关系对移动台进行位置估计。The geometric positioning method is to estimate the position of the mobile station according to one or more measurement parameters and the geometric distribution relationship between the base station and the mobile station.

用于进行位置计算的参数TOA和AOA的测量误差主要由两部分组成，即系统测量误差和NLOS传播产生的误差。系统测量误差服从高斯分布，随着技术的不断发展会逐渐减小，而NLOS传播误差受电波传播环境的影响会始终存在，并且成为测量误差的主要组成部分。在针对减小NLOS传播对混合TOA/AOA方法定位精度的影响的众多方法中，散射模型是其中一种。在使用单次反射模型的前提下，现有的几何定位方法有传统TOA(Time OfArrival)/AOA(Angle Of Arrival)方法及其改进方法、LOP(linear lineof position)方法及其改进方法HLOP(hybrid LOP)方法等，这几种方法利用移动台、散射体和基站之间的几何位置关系估计移动台的位置，虽然方法简单、运算量小，但是定位精度却并不高。一般情况下，对移动台的定位问题都归结为解非线性最小二乘问题，其中最基本的一种方法是高斯-牛顿迭代法，它的特点是在拥有较好的迭代初始值时，在具有很好的收敛性，而如果选择了较差的初始值，收敛性就会变差。The measurement error of the parameters TOA and AOA used for position calculation is mainly composed of two parts, that is, the system measurement error and the error caused by NLOS propagation. The system measurement error obeys the Gaussian distribution and will gradually decrease with the continuous development of technology, while the NLOS propagation error will always exist due to the influence of the radio wave propagation environment and become the main component of the measurement error. Scattering models are one of the methods aimed at reducing the impact of NLOS propagation on the positioning accuracy of hybrid TOA/AOA methods. Under the premise of using the single reflection model, the existing geometric positioning methods include the traditional TOA (Time Of Arrival)/AOA (Angle Of Arrival) method and its improved method, the LOP (linear lineof position) method and its improved method HLOP (hybrid LOP) method, etc. These methods use the geometric position relationship among the mobile station, scatterers and base stations to estimate the position of the mobile station. Although the method is simple and the amount of calculation is small, the positioning accuracy is not high. In general, the positioning problem of the mobile station comes down to solving the nonlinear least squares problem. The most basic method is the Gauss-Newton iterative method. Convergence is very good, and if a poor initial value is chosen, the convergence will be poor.

发明内容 Contents of the invention

为避免以上现有技术的不足，本发明提出一种NLOS环境下的几何定位改进方法，以解决定位精度不高的问题。本发明使用网格搜索方法优化了移动台初始迭代坐标值，使得高斯-牛顿方法获得更好的收敛性。In order to avoid the above shortcomings of the prior art, the present invention proposes a method for improving geometric positioning in an NLOS environment to solve the problem of low positioning accuracy. The invention uses a grid search method to optimize the initial iteration coordinate value of the mobile station, so that the Gauss-Newton method can obtain better convergence.

本发明的目的通过以下技术方案来实现：The purpose of the present invention is achieved through the following technical solutions:

一种NLOS环境下的几何定位改进方法，该方法包括参数测量和移动台位置估算两个阶段，具体方法如下：An improved method for geometric positioning under NLOS environment, the method includes two stages of parameter measurement and mobile station position estimation, the specific method is as follows:

1)参数测量阶段：1) Parameter measurement stage:

根据移动台测量得到的基站信号参数，信号到达时间

和信号到达角度

以及移动台、散射体与基站的几何位置关系有：According to the base station signal parameters measured by the mobile station, the signal arrival time

and the angle of arrival of the signal

And the geometric relationship between the mobile station, the scatterer and the base station is:

${\overset{~ ~}{α α}}_{i i} - - arctan arctan ((\frac{{y the y}_{i i} - - {y the y}_{b b}^{i i}}{{x x}_{i i} - - {x x}_{b b}^{i i}})) = = 00 - - - - - - ((11))$

${\overset{~ ~}{L L}}_{i i} - - {r r}_{b b}^{i i} - - {r r}_{m m}^{i i} = = {\overset{~ ~}{L L}}_{i i} - - \sqrt{{(({x x}_{i i} - - {x x}_{b b}^{i i}))}^{22} + + {(({y the y}_{i i} - - {y the y}_{b b}^{i i}))}^{22}} - - \sqrt{{(({x x}_{i i} - - {x x}_{m m}))}^{22} + + {(({y the y}_{i i} - - {y the y}_{m m}))}^{22}} = = 00 - - - - - - ((22))$

其中

和

分别代表基站信号传播距离和到达角度，(x_i，y_i)和(x_m，y_m)分别表示基站、散射体和移动台的位置坐标，信号传播距离

可以由测量参数信号传播时间

与光速C相乘得到：in

and

represent the base station signal propagation distance and arrival angle, respectively, ( _xi , y _i ) and (x _m , y _m ) denote the position coordinates of base station, scatterer and mobile station respectively, and the signal propagation distance

The signal propagation time can be measured by the parameter

Multiplied by the speed of light C to get:

${\overset{~ ~}{L L}}_{i i} = = {\overset{~ ~}{τ τ}}_{i i} \times \times C C - - - - - - ((33))$

2)移动台位置估算阶段：根据基站信号参数

以及采用高斯牛顿迭代方法并且结合网格搜索方法优化迭代初始坐标值，解算得到移动台的位置，具体包括以下步骤：2) The stage of mobile station position estimation: according to the base station signal parameters

as well as Using the Gauss-Newton iterative method and combining the grid search method to optimize the iterative initial coordinate value, the position of the mobile station is obtained by solving, which specifically includes the following steps:

(1)缩小可行域(1) Narrow down the feasible region

在NLOS环境下，由于移动台(x_m，y_m)与各个基站

之间的距离

由散射体到移动台的距离

加上散射体到基站间的距离构成，即

则根据参数和

粗略估计移动台的位置：In the NLOS environment, since the mobile station (x _m , y _m ) and each base station

the distance between

Distance from scatterer to mobile station

Add the distance from the scatterer to the base station constitute, namely

Then according to the parameter and

Roughly estimate the position of the mobile station:

$\{\begin{matrix} {\overset{^^}{x x}}_{ml ml}^{i i} = = {x x}_{b b}^{i i} + + {\overset{~ ~}{L L}}_{i i} cos cos {\overset{~ ~}{α α}}_{i i} \\ {\overset{^^}{y the y}}_{m m 11}^{i i} = = {y the y}_{b b}^{i i} + + {\overset{~ ~}{L L}}_{i i} sin sin {\overset{~ ~}{α α}}_{i i} \end{matrix} - - - - - - ((44))$

$\{\begin{matrix} {\overset{^^}{x x}}_{m m 22}^{i i} = = {x x}_{b b}^{i i} + + (({\overset{~ ~}{L L}}_{i i} - - {R R}_{d d}^{i i})) cos cos {\overset{~ ~}{α α}}_{i i} \\ {\overset{^^}{y the y}}_{m m 22}^{i i} = = {y the y}_{b b}^{i i} + + (({\overset{~ ~}{L L}}_{i i} - - {R R}_{d d}^{i i})) sin sin {\overset{~ ~}{α α}}_{i i} \end{matrix} - - - - - - ((55))$

根据GBSBCM模型所提出的约束条件

得到算法的可行域：According to the constraints proposed by the GBSBCM model

Get the feasible region of the algorithm:

$0 \leq x_{m} \leq (\sqrt{3} / 2) R$ (6) $0 \leq x_{m} \leq (\sqrt{3} / 2) R$ (6)

$00 \leq \leq {y the y}_{m m} \leq \leq min min {{\sqrt{33} {x x}_{m m},, - - ((\sqrt{33} / / 33)) {x x}_{m m} + + R R}}$

$00 \leq \leq {r r}_{m m}^{i i} = = | | {x x}_{i i} + + {jy jy}_{i i} - - (({x x}_{m m} + + {jy jy}_{m m})) | | \leq \leq {R R}_{d d}^{i i},, i i = = 1,2,3 1,2,3 - - - - - - ((77))$

其中，R为小区半径，为各个基站对应的散射体分布半径的最大值，i＝1，2，3。Among them, R is the cell radius, is the maximum value of the scatterer distribution radius corresponding to each base station, i=1, 2, 3.

从以上(4)、(5)的坐标中选取满足(6)、(7)的最大、最小坐标值，得到一个缩小的可行域[x_min，x_max]×[y_min，y_max]。Select the maximum and minimum coordinate values satisfying (6) and (7) from the coordinates of (4) and (5) above, and obtain a reduced feasible region [x _min , x _max ]×[y _min , y _max ].

(2)网格搜索，确定迭代初始坐标值(2) Grid search, determine the initial coordinate value of iteration

在此缩小的可行域中进行网格搜索，选取满足约束条件(7)的网格点构成候选点集CPS；Carry out a grid search in this narrowed feasible region, and select grid points satisfying the constraint condition (7) to form a candidate point set CPS;

对CPS中所有候选点的坐标值求平均，得到优化的迭代初始值x₀；The coordinate values of all candidate points in the CPS are averaged to obtain the optimized iteration initial value x ₀ ;

(3)采用高斯-牛顿迭代法估算移动台的位置坐标(3) Using the Gauss-Newton iterative method to estimate the position coordinates of the mobile station

理论上，各个基站信号到达移动台的传播距离与到达角度可以表示为：Theoretically, the propagation distance and angle of arrival of each base station signal to the mobile station can be expressed as:

L(x)＝[L₁(x)，L₂(x)，L₃(x)，α₁(x)，α₂(x)，α₃(x)] (8)L(x)=[L ₁ (x), L ₂ (x), L ₃ (x), α ₁ (x), α ₂ (x), α ₃ (x)] (8)

其中x＝[x_m，y_m]， $L_{i} (x) = \sqrt{{(x - x_{b}^{i})}^{2} + {(y - y_{b}^{i})}^{2}},$ $α_{i} (x) = \arctan \frac{y - y_{b}^{i}}{x - x_{b}^{i}}$ where x = [x _m , y _m ], $L_{i} (x) = \sqrt{{(x - x_{b}^{i})}^{2} + {(the y - {the y}_{b}^{i})}^{2}},$ $α_{i} (x) = \arctan \frac{the y - {the y}_{b}^{i}}{x - x_{b}^{i}}$

而各个基站信号实际上的传播距离与到达角度为：The actual propagation distance and arrival angle of each base station signal are:

$\overset{~ ~}{L L} = = L L ((x x)) + + n no - - - - - - ((99))$

其中n为NLOS传播引起的误差以及服从均值为零的高斯分布的系统测量误差；where n is the error caused by NLOS propagation and the systematic measurement error that obeys the Gaussian distribution with zero mean;

由于误差的存在，(1)、(2)并不总是能够得到满足，由此得到目标函数：Due to the existence of errors, (1) and (2) are not always satisfied, and thus the objective function is obtained:

$ϵ ϵ ((x x)) = = {((\overset{~ ~}{L L} - - L L ((x x))))}^{T T} {Σ Σ}_{n no}^{- - 11} ((\overset{~ ~}{L L} - - L L ((x x)))) - - - - - - ((1010))$

其中∑_n为噪声n的协方差矩阵：where ∑ _n is the covariance matrix of noise n:

∑_n＝E{nn^T} (11)∑ _n = E{nn ^T } (11)

则满足下式的坐标即可作为移动台的位置估计值：Then the coordinates satisfying the following formula can be used as the estimated position of the mobile station:

$\overset{^^}{x x} = = \underset{x x}{arg arg min min} ϵ ϵ ((x x)) - - - - - - ((1212))$

对(9)式在迭代初始值x₀处进行线性化：Linearize (9) at the iteration initial value x ₀ :

$L L ((x x)) = = L L (({x x}_{00})) + + φ φ ((x x)) {| |}_{x x = = {x x}_{00}} ((x x - - {x x}_{00})) - - - - - - ((1313))$

其中in

$φ φ ((x x)) = = {&dtri; &dtri;}_{x x}^{T T} &CircleTimes; &CircleTimes; L L ((x x)) = = [\begin{matrix} ((x x - - {x x}_{11})) / / {r r}_{11},, (({y the y - - y the y}_{11})) / / {r r}_{11} \\ (({x x - - x x}_{22})) / / {r r}_{22},, (({y the y - - y the y}_{22})) / / {r r}_{22} \\ ((x x - - {x x}_{33})) / / {r r}_{33},, (({y the y - - y the y}_{33})) / / {r r}_{33} \\ ((y the y - - {y the y}_{11})) / / {r r}_{11}^{22},, ((x x - - {x x}_{11})) / / {r r}_{11}^{22} \\ (({y the y - - y the y}_{22})) / / {r r}_{22}^{22},, (({x x - - x x}_{22})) / / {r r}_{22}^{22} \\ (({y the y - - y the y}_{33})) / / {r r}_{33}^{22},, ((x x - - {x x}_{33})) / / {r r}_{33}^{22} \end{matrix}],, - - - - - - ((1414))$

根据(10)、(13)式，对下式进行迭代求解：

According to equations (10) and (13), the following equations are solved iteratively:

${x x}^{((k k + + 11))} = = {x x}^{((k k))} + + {(({φ φ}^{T T} (({x x}^{((k k))})) {Σ Σ}_{n no}^{- - 11} φ φ (({x x}^{((k k))}))))}^{- - 11}$

$. . . . {φ φ}^{T T} (({x x}^{((k k))})) {Σ Σ}_{n no}^{- - 11} ((\overset{~ ~}{L L} - - L L (({x x}^{((k k))})))) - - - - - - ((1515))$

$= = {x x}^{((k k))} + + {A A}^{((k k)),, - - 11} \cdot &Center Dot; {φ φ}^{T T} (({x x}^{((k k))})) {Σ Σ}_{n no}^{- - 11} ((\overset{~ ~}{L L} - - L L (({x x}^{((k k))}))))$

当两次迭代结果的差值小于一个任意小的正数时，迭代中止，得到最终的移动台估计坐标 When the difference between the two iteration results is less than an arbitrarily small positive number, the iteration is terminated, and the final estimated coordinates of the mobile station are obtained

本发明的优点在于：The advantages of the present invention are:

采用缩小可行域、网格搜索两个步骤优化了高斯-牛顿方法的初始迭代坐标值，获得更加稳定、快速的收敛性，复杂度适中，能够获得更高的移动台定位精度。The initial iterative coordinate value of the Gauss-Newton method is optimized by two steps of narrowing the feasible region and grid search, obtaining more stable and fast convergence, moderate complexity, and higher positioning accuracy of the mobile station.

附图说明 Description of drawings

图1是本发明的具体方法的流程图；Fig. 1 is the flowchart of the concrete method of the present invention;

图2是GBSBCM模型；Figure 2 is the GBSBCM model;

图3是移动台、基站和散射体的几何关系图。Fig. 3 is a geometric relationship diagram of a mobile station, a base station and a scatterer.

图4是小区布局示意图；Figure 4 is a schematic diagram of the layout of the community;

图5(a)是在不同的散射体分布最大半径情况下的平均定位误差比较；Figure 5(a) is a comparison of the average positioning error in the case of different maximum radii of scatterer distribution;

图5(b)显示了各个算法的累积分布函数曲线。Figure 5(b) shows the cumulative distribution function curves of each algorithm.

具体实施方式 Detailed ways

如图1所示，为本发明实现方法的具体流程图。本发明改进的混合TOA/AOA几何定位方法按如下步骤实现：As shown in FIG. 1 , it is a specific flow chart of the implementation method of the present invention. The improved hybrid TOA/AOA geometric positioning method of the present invention is realized in the following steps:

(1)参数测量(1) Parameter measurement

移动台测量基站的无线信号参数，即信号到达时间(Time Of Arrival，TOA)

和信号到达角度(Angle Of Arrival，AOA)

并且结合移动台与基站的几何位置关系，可以推算出移动台的位置。The mobile station measures the wireless signal parameters of the base station, that is, the signal time of arrival (Time Of Arrival, TOA)

and signal angle of arrival (Angle Of Arrival, AOA)

And combining the geometric position relationship between the mobile station and the base station, the position of the mobile station can be calculated.

在NLOS(non-line-of-sight，非视距)环境中，由于存在阻挡物或者散射体，移动台测量到的TOA和AOA参数值存在较大误差。如图2所示，用单次反射圆模型(geometrically based single bounce macrocell circularmodel，GBSBCM)模型来模拟NLOS传播环境。该模型利用了与实际相符的一个假设：在宏蜂窝环境中，基站天线较高，基站附近反射物不产生反射信号。散射体在以移动台为中心、R_d为半径的圆内呈高斯分布。In an NLOS (non-line-of-sight, non-line-of-sight) environment, due to the existence of obstacles or scatterers, there are large errors in the TOA and AOA parameter values measured by the mobile station. As shown in FIG. 2 , a geometrically based single bounce macrocell circular model (GBSBCM) model is used to simulate the NLOS propagation environment. The model utilizes an assumption that is consistent with reality: In a macro-cellular environment, the base station antenna is high, and reflectors near the base station do not produce reflected signals. The scatterers are Gaussian distributed in a circle with the mobile station as the center and _Rd as the radius.

如图3所示，根据移动台、散射体与基站的几何位置关系有：As shown in Figure 3, according to the geometric positional relationship between the mobile station, the scatterer and the base station:

其中

和

分别代表基站信号到达移动台的传播距离和到达角度，

(x_i，y_i)和(x_m，y_m)分别代表基站、散射体和移动台的位置坐标。信号传播距离

可以由测量参数信号传播时间与光速C相乘得到：in

and

Represent the propagation distance and angle of arrival of the base station signal to the mobile station, respectively,

( _xi , y _i ) and (x _m , y _m ) represent the position coordinates of the base station, the scatterer and the mobile station, respectively. signal propagation distance

The signal propagation time can be measured by the parameter Multiplied by the speed of light C to get:

(2)缩小可行域(2) Narrow down the feasible region

在NLOS环境下，由于移动台(x_m，y_m)与各个基站

之间的距离

由散射体到移动台的距离

加上散射体到基站间的距离

构成，即

则根据参数

和

the distance between

Distance from scatterer to mobile station

Add the distance from the scatterer to the base station

constitute, namely

Then according to the parameter

and

Roughly estimate the position of the mobile station:

而在如图4所示的小区布局示意图中，参与定位的基站数目为3，其中基站1为服务基站，移动台处于OABC包围的灰色区域中。另外根据GBSBCM模型所提出的约束条件

我们得到算法的可行域：In the cell layout diagram shown in FIG. 4 , the number of base stations participating in positioning is 3, of which base station 1 is the serving base station, and the mobile station is in the gray area surrounded by OABC. In addition, according to the constraints proposed by the GBSBCM model

We get the feasible region of the algorithm:

其中，R为小区半径，为各个基站对应的散射体分布半径最大值，i＝1，2，3。Among them, R is the cell radius, is the maximum distribution radius of scatterers corresponding to each base station, i=1, 2, 3.

(3)网格搜索，确定迭代初始坐标值(3) Grid search, determine the initial coordinate value of iteration

在此缩小的可行域中进行网格搜索，选取满足约束条件(7)的网格点构成候选点集CPS(Candidate Point Set)。The grid search is carried out in this narrowed feasible area, and the grid points satisfying the constraint condition (7) are selected to form the candidate point set CPS (Candidate Point Set).

对CPS中所有候选点的坐标值求平均，得到优化的迭代初始值x₀。The coordinate values of all candidate points in the CPS are averaged to obtain the optimized iteration initial value x ₀ .

(4)采用高斯-牛顿迭代方法估算移动台的位置坐标(4) Estimate the position coordinates of the mobile station using the Gauss-Newton iterative method

其中x＝[x_m，y_m]， $L_{i} (x) = \sqrt{{(x - x_{b}^{i})}^{2} + {(y - y_{b}^{i})}^{2}},$ $α_{i} (x) = \arctan \frac{y - y_{b}^{i}}{x - x_{b}^{i}} .$ where x = [x _m , y _m ], $L_{i} (x) = \sqrt{{(x - x_{b}^{i})}^{2} + {(the y - {the y}_{b}^{i})}^{2}},$ $α_{i} (x) = \arctan \frac{the y - {the y}_{b}^{i}}{x - x_{b}^{i}} .$

$\overset{~ ~}{L L} = = L L ((x x)) + + n no - - - - - - ((99))$

其中n为NLOS传播引起的误差以及服从均值为零的高斯分布的系统测量误差。where n is the error caused by NLOS propagation and the systematic measurement error following a Gaussian distribution with zero mean.

∑_n＝E{nn^T} (11)∑ _n = E{nn ^T } (11)

其中in

根据(10)、(13)式，对下式进行迭代求解：

$= = {x x}^{((k k))} + + {A A}^{((k k)),, - - 11} \cdot \cdot {φ φ}^{T T} (({x x}^{((k k))})) {Σ Σ}_{n no}^{- - 11} ((\overset{~ ~}{L L} - - L L (({x x}^{((k k))}))))$

当两次迭代结果的差值小于一个任意小的正数时，迭代中止，得到最终的移动台估计坐标

When the difference between the two iteration results is less than an arbitrarily small positive number, the iteration is terminated, and the final estimated coordinates of the mobile station are obtained

对改进的方法和现有方法进行计算机仿真比较，如图5(a)和5(b)所示。The improved method is compared with the existing method by computer simulation, as shown in Fig. 5(a) and 5(b).

图5(a)是在不同的散射体分布最大半径情况下的平均定位误差比较，对于服务基站1，散射体分布最大半径值固定为0.15km。可以看出，本文方法平均误差均低于现有定位方法，其定位精度有了一定的提高。Fig. 5(a) is a comparison of average positioning errors under different maximum radii of scatterer distribution. For serving base station 1, the maximum scatterer distribution radius is fixed at 0.15km. It can be seen that the average error of the method in this paper is lower than that of the existing positioning methods, and its positioning accuracy has been improved to a certain extent.

图5(b)显示了各个算法的累积分布函数曲线。对于3个基站，移动台周围散射体分布半径分别为0.15km，0.25km和0.25km。其中本文方法的定位误差小于0.1km的概率为83.7％，而现有定位方法中的改进的传统TOA/AOA算法、传统TOA/AOA算法、LOP算法以及改进的HLOP算法分别为73％，47.1％，31.3％和62.7％。由此可知，本发明提出的几何定位改进方法的定位性能优于上述现有定位方法。Figure 5(b) shows the cumulative distribution function curves of each algorithm. For the three base stations, the distribution radii of scatterers around the mobile station are 0.15km, 0.25km and 0.25km respectively. Among them, the probability of the positioning error of the method in this paper is less than 0.1km is 83.7%, while the improved traditional TOA/AOA algorithm, traditional TOA/AOA algorithm, LOP algorithm and improved HLOP algorithm in the existing positioning methods are 73% and 47.1% respectively. , 31.3% and 62.7%. It can be seen that the positioning performance of the improved geometric positioning method proposed by the present invention is better than that of the above-mentioned existing positioning methods.

Claims

1. a geometric positioning improvement method under the NLOS environment, it is characterized in that, the method comprises two stages of parameter measurement and mobile station position estimation, and concrete method is as follows:

1) Parameter measurement stage:

A single reflection circle model is used to simulate the NLOS propagation environment. According to the base station signal parameters measured by the mobile station, the signal arrival time

and the angle of arrival of the signal

{\overset{~ ~}{α α}}_{i i} - - arctan arctan ((\frac{{y the y}_{i i} - - {y the y}_{b b}^{i i}}{{x x}_{i i} - - {x x}_{b b}^{i i}})) = = 00 - - - - - - ((11))

{\overset{~ ~}{L L}}_{i i} - - {r r}_{b b}^{i i} - - {r r}_{m m}^{i i} = = \overset{~ ~}{{L L}_{i i}} - - \sqrt{{(({x x}_{i i} - - {x x}_{b b}^{i i}))}^{22} + + {(({y the y}_{i i} - - {y the y}_{b b}^{i i}))}^{22}} - - \sqrt{{(({x x}_{i i} - - {x x}_{m m}))}^{22} + + {(({y the y}_{i i} - - {y the y}_{m m}))}^{22}} = = 00 - - - - - - ((22))

in

and

Represent the base station signal propagation distance and signal arrival angle, respectively,

and (x _m , y _m ) denote the position coordinates of base station, scatterer and mobile station respectively, and the signal propagation distance

signal arrival time

Multiplied by the speed of light C to get:

{\overset{~ ~}{L L}}_{i i} = = {\overset{~ ~}{τ τ}}_{i i} \times \times C C - - - - - - ((33))

2) The stage of mobile station position estimation: according to the signal arrival time

and the angle of arrival of the signal

Using the Gauss-Newton iterative method and combining the grid search method to optimize the iterative initial coordinate value, the position of the mobile station is obtained by solving, which specifically includes the following steps:

(1) Narrow down the feasible region

In the NLOS environment, since the mobile station (x _m ,y _m ) and each base station

the distance between

Distance from scatterer to mobile station

Add the distance from the scatterer to the base station

constitute, namely

Then according to the parameter

and

Roughly estimate the position of the mobile station:

\{\begin{matrix} {\overset{^^}{x x}}_{m m 11}^{i i} = = {x x}_{b b}^{i i} + + {\overset{~ ~}{L L}}_{i i} cos cos {\overset{~ ~}{α α}}_{i i} \\ {\overset{^^}{y the y}}_{m m 11}^{i i} = = {y the y}_{b b}^{i i} + + {\overset{~ ~}{L L}}_{i i} sin sin {\overset{~ ~}{α α}}_{i i} \end{matrix} - - - - - - ((44))

\{\begin{matrix} {\overset{^^}{x x}}_{m m 22}^{i i} = = {x x}_{b b}^{i i} + + (({\overset{~ ~}{L L}}_{i i} - - {R R}_{d d}^{i i})) cos cos {\overset{~ ~}{α α}}_{i i} \\ {\overset{^^}{y the y}}_{m m 22}^{i i} = = {y the y}_{b b}^{i i} + + (({\overset{~ ~}{L L}}_{i i} - - {R R}_{d d}^{i i})) sin sin {\overset{~ ~}{α α}}_{i i} \end{matrix} - - - - - - ((55))

According to the constraints proposed by the GBSBCM model

Get the feasible region of the algorithm:

00 \leq \leq {x x}_{m m} \leq \leq ((\sqrt{33} / / 22)) R R

0 \leq {the y}_{m} \leq \min {\sqrt{3} x_{m}, - (\sqrt{3} / 3) x_{m} + R}

(6)

0 \leq r_{m}^{i} = | x_{i} + {jy}_{i} - (x_{m} + {jy}_{m}) | \leq R_{d}^{i}, i = 1,2,3

(7)

Among them, R is the cell radius,

is the maximum value of the scatterer distribution radius corresponding to each base station, i=1,2,3;

Select the maximum and minimum coordinate values satisfying (6) and (7) from the coordinates of (4) and (5) above to obtain a reduced feasible region [x _min , x _max ]×[y _min ,y _max ];

(2) Grid search, determine the initial coordinate value of iteration

Carry out a grid search in this narrowed feasible region, and select grid points satisfying the constraint condition (7) to form a candidate point set CPS;

Calculate the average of the coordinate values of all candidate points in the CPS to obtain the optimized iteration initial value x ₀ ;

(3) Using the Gauss-Newton iterative method to estimate the position coordinates of the mobile station

Theoretically, the propagation distance and angle of arrival of each base station signal to the mobile station can be expressed as:

L(x)=[L ₁ (x), L ₂ (x), L ₃ (x), α ₁ (x), α ₂ (x), α ₃ (x)] (8)

in

x = [x_{m}, {the y}_{m}], L_{i} (x) = \sqrt{{(x - x_{b}^{i})}^{2} + {(the y - {the y}_{b}^{i})}^{2}}, α_{i} (x) = \arctan \frac{the y - {the y}_{b}^{i}}{x - x_{b}^{i}}

The actual propagation distance and arrival angle of each base station signal are:

\overset{~ ~}{L L} = = L L ((x x)) + + n no - - - - - - ((99))

where n is the error caused by NLOS propagation and the systematic measurement error that obeys the Gaussian distribution with zero mean;

Due to the existence of errors, (1) and (2) are not always satisfied, and thus the objective function is obtained:

ϵ ϵ ((x x)) {((\overset{~ ~}{L L} - - L L ((x x))))}^{T T} {Σ Σ}_{n no}^{- - 11} ((\overset{~ ~}{L L} - - L L ((x x)))) - - - - - - ((1010))

where ∑ _n is the covariance matrix of noise n:

∑ _n =E{nn ^T }(11)

Then the coordinates satisfying the following formula can be used as the estimated position of the mobile station:

\overset{^^}{x x} = = \underset{x x}{arg arg min min} ϵ ϵ ((x x)) - - - - - - ((1212))

Linearize (9) at the iteration initial value x ₀ :

L L ((x x)) = = L L (({x x}_{00})) + + φ φ ((x x)) {| |}_{x x = = {x x}_{00}} ((x x - - {x x}_{00})) - - - - - - ((1313))

in

φ φ ((x x)) = = {&dtri; &dtri;}_{x x}^{T T} &CircleTimes; &CircleTimes; L L ((x x)) = = [\begin{matrix} ((x x - - {x x}_{11})) / / {r r}_{11},, ((y the y - - {y the y}_{11})) / / {r r}_{11} \\ ((x x - - {x x}_{22})) / / {r r}_{22},, ((y the y - - {y the y}_{22})) / / {r r}_{22} \\ ((x x - - {x x}_{33})) / / {r r}_{33},, ((y the y - - {y the y}_{33})) / / {r r}_{33} \\ ((y the y - - {y the y}_{11})) / / {r r}_{11}^{22},, ((x x - - {x x}_{11})) / / {r r}_{11}^{22} \\ ((y the y - - {y the y}_{22})) / / {r r}_{22}^{22},, ((x x - - {x x}_{22})) / / {r r}_{22}^{22} \\ ((y the y - - {y the y}_{33})) / / {r r}_{33}^{22},, ((x x - - {x x}_{33})) / / {r r}_{33}^{22} \end{matrix}],, - - - - - - ((1414))

{x x}^{((k k + + 11))} = = {x x}^{((k k))} + + {(({φ φ}^{T T} (({x x}^{((k k))})) {Σ Σ}_{n no}^{- - 11} φ φ (({x x}^{((k k))}))))}^{- - 11}

\cdot &Center Dot; \cdot &Center Dot; {φ φ}^{T T} (({x x}^{((k k))})) {Σ Σ}_{n no}^{- - 11} ((\overset{~ ~}{L L} - - L L (({x x}^{((k k))})))) - - - - - - ((1515))

= = {x x}^{((k k))} + + {A A}^{((k k)),, - - 11} \cdot \cdot {φ φ}^{T T} (({x x}^{((k k))})) {Σ Σ}_{n no}^{- - 11} ((\overset{~ ~}{L L} - - L L (({x x}^{((k k))}))))

.