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 重庆邮电大学学报(自然科学版)  2018, Vol. 30 Issue (6): 768-775  DOI: 10.3979/j.issn.1673-825X.2018.06.006 0

### 引用本文

ZHENG Qian, HU Jiusong, LIU Hongli, XIAO Guoxuan, CHEN Liang, XU Kun. Localization performance analysis of indoor positioning system based on compressive sensing[J]. Journal of Chongqing University of Posts and Telecommunications (Natural Science Edition), 2018, 30(6): 768-775. DOI: 10.3979/j.issn.1673-825X.2018.06.006.

### 文章历史

1. 湖南大学 电气与信息工程学院，长沙 410082;
2. 国家电网 浙江乐清市供电公司，浙江 乐清 325600

Localization performance analysis of indoor positioning system based on compressive sensing
ZHENG Qian1 , HU Jiusong1 , LIU Hongli1 , XIAO Guoxuan2 , CHEN Liang1 , XU Kun1
1. College of Electrical and Information Engineering, Hunan University, Changsha 410082, P. R. China;
2. State Grid Yueqing Electric Power Supply Company, Zhejiang University, Yueqing, 325600, P. R. China
Foundation Items: The Central State-Owned Capital Management and Budget Project ([2013]470); The National Nature Science Foundation of China(61771191); The Fundamental Research Funds for the Central Universities(1053214004); The Natural Science Foundation of Hunan Province (2017JJ2052); The Ministry of Education Cooperative Education Project(201601004010, 201701056026); The Teaching Reform Project of Hunan Ordinary Colleges and Universities(XJT 2016-400); The Graduate Innovation Project of Hunan Province (CX2017B112)
Abstract: With the popularity of Wi-Fi technology, the WiFi-based indoor positioning technology has fostered growing interests. The compressive sensing (CS) theory has been proposed for the WiFi-based indoor positioning. In order to study the localization performance of the system with various CS algorithms, we construct a CS-based indoor location fingerprinting positioning system. In the system, we collect data and build a fingerprint database firstly in the offline phase, and then in the online phase we use different compressive sensing algorithms to compare the localization performance of the system. Experiments show that: first of all, the larger the reference point data set collected from multiple directions contain, the better the performance of location becomes. Secondly, the algorithm parameters of CS have certain influences on the positioning accuracy. Thirdly, the stage-wise weak orthogonal matching pursuit (SWOMP) algorithm in compressive sensing has a better localization accuracy than the k-nearest neighbor(KNN) algorithm by 21.9% in the experimental environment of the paper. And lastly, among all kinds of compressive sensing algorithms, the orthogonal matching pursuit (OMP) is worse than other CS algorithms, and this difference becomes even more obvious with the growth of reference point data.
Keywords: indoor positioning    compressive sensing    fingerprint database    clustering
0 引言

1 定位系统 1.1 室内定位框架

 图 1 室内定位系统框架 Figure 1 Indoor positioning system frame
1.2 离线阶段

 $\mathit{\boldsymbol{rs}}{\mathit{\boldsymbol{s}}_i} = {\left( {rs{s_{i1}},rs{s_{i2}}, \cdots ,rs{s_{iL}}} \right)^{\rm{T}}}$ (1)

N个指纹点的接收信号强度向量组合在一起便形成一张形如电子地图的矩阵，表示为

 $\mathit{\boldsymbol{ \boldsymbol{\varPsi} }} = {\left( {\begin{array}{*{20}{c}} {rs{s_{11}}}&{rs{s_{12}}}& \cdots &{rs{s_{1L}}}\\ {rs{s_{21}}}&{rs{s_{22}}}& \cdots &{rs{s_{2L}}}\\ \vdots&\vdots &{}& \vdots \\ {rs{s_{N1}}}&{rs{s_{N2}}}& \cdots &{rs{s_{NL}}} \end{array}} \right)^{\rm{T}}}$ (2)

Ψo表示设备面向方向o(oO={E, W, S, N}, 即东南西北4个方向)的电子地图。实验时，每个参考点采集40次，取RSS的平均值存入指纹库，而对于未接收到接入点APs信号的点，其强度统一用-100 dBm表示。

 $s\left( {i,j} \right) = - \sqrt {{{\left( {\mathit{\boldsymbol{rs}}{\mathit{\boldsymbol{s}}_i} - \mathit{\boldsymbol{rs}}{\mathit{\boldsymbol{s}}_j}} \right)}^2}}$ (3)

 $r\left( {i,k} \right) = s\left( {i,k} \right) - \mathop {\max }\limits_{k' \ne k} \left( {a\left( {i,k'} \right) + s\left( {i,k'} \right)} \right)$ (4)

 $\begin{array}{*{20}{c}} {a\left( {i,k} \right) = }\\ {\left\{ {\begin{array}{*{20}{c}} {\min \left\{ {0,r\left( {k,k} \right) + \sum\limits_{i' \notin \left\{ {i,k} \right\}} {\max \left( {0,r\left( {i',k} \right)} \right)} } \right\}}&{,i \ne k}\\ {\sum\limits_{i' \notin k} {\max \left( {0,r\left( {i',k} \right)} \right)} }&{,i = k} \end{array}} \right.} \end{array}$ (5)
1.3 在线阶段

1.3.1 信号的稀疏表示

 ${\mathit{\boldsymbol{\varphi }}_t} = {\left( {rs{s_{t1}},rs{s_{t2}}, \cdots ,rs{s_{tL}}} \right)^{\rm{T}}}$ (6)

 $\mathit{\boldsymbol{\theta }} = {\left[ {0, \cdots ,0,1,0, \cdots ,0} \right]^{\rm{T}}}$ (7)

(7) 式中，Ψ代表稀疏基，则原始信号可以表达为

 ${\mathit{\boldsymbol{\varphi }}_t} = \mathit{\boldsymbol{ \boldsymbol{\varPsi} \theta }}$

 $\mathit{\boldsymbol{\theta }} = {\left[ {0, \cdots ,0,{\lambda _1},0, \cdots ,0,{\lambda _2},0, \cdots ,0,{\lambda _K},0, \cdots ,0} \right]^{\rm{T}}}$ (9)

(9) 式中，匹配到的少数参考点的个数为向量θ的稀疏度K

1.3.2 非相干测量

CS理论中，原始信号通过测量基投影得到测量向量。定位系统中，定义测量基Φ(M×L)为接入点APs选择矩阵，通过最强接收信号强度的选择原则(MN)对原始信号投影得到测量向量y(M×1)为

 $\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}{\mathit{\boldsymbol{\varphi }}_t} = \mathit{\boldsymbol{ \boldsymbol{\varPhi} \boldsymbol{\varPsi} \theta }}$ (10)

 $\mathit{\boldsymbol{z}} = \mathit{\boldsymbol{Ty}}$ (11)

 $\mathit{\boldsymbol{z}} = \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{R}}^ * } \cdot \mathit{\boldsymbol{ \boldsymbol{\varPhi} \boldsymbol{\varPsi} \theta }} = \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{R}}^ * } \cdot \mathit{\boldsymbol{R\theta }} = \mathit{\boldsymbol{Q\theta }}$ (12)

 $\mathit{\boldsymbol{\hat \theta }} = \arg \min {\left\| \mathit{\boldsymbol{\theta }} \right\|_1}\;\;\;\;{\rm{s}}.{\rm{t}}.\;\;\;\mathit{\boldsymbol{z}} = \mathit{\boldsymbol{Q\theta }}$ (13)
1.3.3 重构算法

① 初始化r0=y, Λ0=∅, Q0=∅, t=1

② 找到索引，使得：${\lambda _t} = \arg \mathop {\max }\limits_{j = 1,2, \cdots ,N} \left| {\left\langle {{\mathit{\boldsymbol{r}}_{t - 1}},{\mathit{\boldsymbol{q}}_j}} \right\rangle } \right|$

③ 令Λt=Λt-1∪{λt}; Qt=Qt-1qλ

④ 求z = 的最小二乘解:

${{\mathit{\boldsymbol{\hat \theta }}}_t} = \arg \mathop {\max }\limits_{{\theta _t}} \left\| {\mathit{\boldsymbol{z}} - {\mathit{\boldsymbol{Q}}_t}{\mathit{\boldsymbol{\theta }}_t}} \right\| = {\left( {\mathit{\boldsymbol{Q}}_t^{\rm{T}}{\mathit{\boldsymbol{Q}}_t}} \right)^{ - 1}}\mathit{\boldsymbol{Q}}_t^{\rm{T}} \cdot \mathit{\boldsymbol{z}}$

⑤ 更新残差rt=z-Qt${{\mathit{\boldsymbol{\hat \theta }}}_t}$=z-Qt(QTtQt)-1QtT·z

t=t+1，如果tK，返回第②步

⑦ 重构所得的${{\mathit{\boldsymbol{\hat \theta }}}_t}$在Λt处有非零项，其值分别为最后一次迭代所得${{\mathit{\boldsymbol{\hat \theta }}}_t}$

2 实验与结果分析 2.1 实验环境

 图 2 定位区域环境平面图 Figure 2 Environmental plan of interest area

2.2 实验结果评估方法

 $error = \sqrt {{{\left( {x,y} \right)}^2} - {{\left( {x',y'} \right)}^2}}$ (14)

2.3 设备面向方向

 图 3 设备面向方向的定位误差 Figure 3 Positioning error of the different orientation of the device

2.4 不同参数K, M

 图 4 不同K值、M值的定位结果 Figure 4 Position results using different value of K, M
2.5 压缩感知算法 2.5.1 CS对比KNN及其性能分析

 图 5 不同算法的定位结果 Figure 5 Positioning results of different algorithms

 图 6 不同压缩感知算法的定位误差累积分布函数图 Figure 6 CDF of the location error of different compressed sensing algorithms
2.5.2 参考点数据量减少时各类压缩感知的比较

 图 7 不同压缩感知算法随参考点数量的变化 Figure 7 Location error of different compressed sensing algorithms vary with reference points
3 结束语