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Article(id=1251226687665553832, tenantId=1146029695717560320, journalId=1251194772300279900, issueId=1251226682309423223, articleNumber=null, orderNo=null, doi=10.20079/j.issn.1001-893x.240807002, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1722960000000, receivedDateStr=2024-08-07, revisedDate=1731254400000, revisedDateStr=2024-11-11, acceptedDate=null, acceptedDateStr=null, onlineDate=1776245289007, onlineDateStr=2026-04-15, pubDate=1764259200000, pubDateStr=2025-11-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1776245289007, onlineIssueDateStr=2026-04-15, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1776245289007, creator=13041195026, updateTime=1776245289007, updator=13041195026, issue=Issue{id=1251226682309423223, tenantId=1146029695717560320, journalId=1251194772300279900, year='2025', volume='65', issue='11', pageStart='1729', pageEnd='1954', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1776245287729, creator=13041195026, updateTime=1776246742124, updator=13041195026, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1251232782568080068, tenantId=1146029695717560320, journalId=1251194772300279900, issueId=1251226682309423223, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1251232782568080069, tenantId=1146029695717560320, journalId=1251194772300279900, issueId=1251226682309423223, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1806, endPage=1811, ext={EN=ArticleExt(id=1251226688361808311, articleId=1251226687665553832, tenantId=1146029695717560320, journalId=1251194772300279900, language=EN, title=Indoor Fingerprint Positioning Based on Matrix Completion in 5G Ultra-dense Network, columnId=1251226683223781499, journalTitle=Telecommunication Engineering, columnName=Application Fundamental Research and Advanced Technology, runingTitle=null, highlight=null, articleAbstract=

In order to reduce complexity of construction of indoor positioning fingerprint database and improve the positioning accuracy, an indoor fingerprint positioning algorithm based on matrix completion under the 5G ultra-dense network is proposed. In the offline database construction stage,the algorithm first uses the K-nearest Neighbor(KNN) interpolation method to complete the matrix of part of the fingerprint database to construct a complete database. Secondly,the sparse auto-encoder is used to extract the sparse features of the fingerprint database, and the high-dimensional received signal strength indication (RSSI) signal is reduced. In the online fingerprint matching stage,the weighted KNN algorithm is used to estimate the coordinates of the point to be located. After experimental simulation,the average relative error of the algorithm to reconstruct the fingerprint database is 0.31% . Compared with that of the traditional KNN fingerprint matching algorithm,the average error is reduced by 24.41% .

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为降低室内定位指纹数据库构建的复杂度并提高定位精度，提出了一种5G超密集网络下的基于矩阵补全的室内指纹定位算法。在离线数据库构建阶段，算法首先采用K近邻（K-Nearest Neighbor，KNN）插值法对部分指纹库进行矩阵补全，构建完整的数据库；其次，采用稀疏自编码器提取指纹库的稀疏特征，对高维接收信号强度指示（Received Signal Strength Indication，RSSI）信号进行降维处理。在在线指纹匹配阶段，使用加权KNN算法估算出待定位点坐标。经过实验仿真分析，算法重构指纹数据库的平均相对误差为0.31%；与传统KNN指纹匹配算法相比，平均误差降低了24.41%。

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龙光利 Email:
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江海霞 女，2000年生于江苏淮安，2022年获学士学位，现为硕士研究生，主要研究方向为无线通信。

龙光利 男，1968年生于陕西汉中，2013年获硕士学位，现为教授，主要研究方向为无线通信、物联网技术及电子技术应用。

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江海霞 女，2000年生于江苏淮安，2022年获学士学位，现为硕士研究生，主要研究方向为无线通信。

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江海霞 女，2000年生于江苏淮安，2022年获学士学位，现为硕士研究生，主要研究方向为无线通信。

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龙光利 男，1968年生于陕西汉中，2013年获硕士学位，现为教授，主要研究方向为无线通信、物联网技术及电子技术应用。

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龙光利 男，1968年生于陕西汉中，2013年获硕士学位，现为教授，主要研究方向为无线通信、物联网技术及电子技术应用。

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指纹点RSSI值
123456
1-110.8305-109.5056-112.1806-109.9403-108.5113-113.3913
2-110.7548-109.3346-112.0386-109.7721-108.3548-113.2936
3-110.6823-109.1628-111.8951-109.6029-108.2004-113.1962
4-110.6131-108.9907-111.7503-109.4327-108.0485-113.0991
5-110.5475-108.8183-111.6040-109.2619-107.8996-113.0025
6-110.4855-108.6459-111.4564-109.0905-107.7543-112.9063
), ArticleFig(id=1251226701477397486, tenantId=1146029695717560320, journalId=1251194772300279900, articleId=1251226687665553832, language=CN, label=表1, caption=

原始指纹数据库

, figureFileSmall=null, figureFileBig=null, tableContent=
指纹点RSSI值
123456
1-110.8305-109.5056-112.1806-109.9403-108.5113-113.3913
2-110.7548-109.3346-112.0386-109.7721-108.3548-113.2936
3-110.6823-109.1628-111.8951-109.6029-108.2004-113.1962
4-110.6131-108.9907-111.7503-109.4327-108.0485-113.0991
5-110.5475-108.8183-111.6040-109.2619-107.8996-113.0025
6-110.4855-108.6459-111.4564-109.0905-107.7543-112.9063
), ArticleFig(id=1251226701565477875, tenantId=1146029695717560320, journalId=1251194772300279900, articleId=1251226687665553832, language=EN, label=null, caption=null, figureFileSmall=null, figureFileBig=null, tableContent=
指纹点RSSI值
123456
1-110.9620-109.5056-112.1806-109.9403-108.5113-113.3913
2-110.8547-109.3346-112.0386-109.7721-108.9692-113.2936
3-110.7709-109.1628-111.8951-109.6029-107.7454-113.1962
4-110.6453-108.6265-111.7503-109.4327-108.0485-113.0991
5-110.5475-108.8183-111.6040-108.8183-107.4451-113.0025
6-110.4855-108.6459-111.2473-109.0905-108.2856-112.9063
), ArticleFig(id=1251226701657752574, tenantId=1146029695717560320, journalId=1251194772300279900, articleId=1251226687665553832, language=CN, label=表2, caption=

矩阵补全重构指纹数据库

, figureFileSmall=null, figureFileBig=null, tableContent=
指纹点RSSI值
123456
1-110.9620-109.5056-112.1806-109.9403-108.5113-113.3913
2-110.8547-109.3346-112.0386-109.7721-108.9692-113.2936
3-110.7709-109.1628-111.8951-109.6029-107.7454-113.1962
4-110.6453-108.6265-111.7503-109.4327-108.0485-113.0991
5-110.5475-108.8183-111.6040-108.8183-107.4451-113.0025
6-110.4855-108.6459-111.2473-109.0905-108.2856-112.9063
), ArticleFig(id=1251226701745831939, tenantId=1146029695717560320, journalId=1251194772300279900, articleId=1251226687665553832, language=EN, label=null, caption=null, figureFileSmall=null, figureFileBig=null, tableContent=
指纹点相对误差/%
123456
10.120.000.000.000.000.00
20.090.000.000.000.570.00
30.080.000.000.000.420.00
40.030.330.000.000.000.00
50.000.000.000.410.420.00
60.000.000.190.000.490.00
), ArticleFig(id=1251226701875855373, tenantId=1146029695717560320, journalId=1251194772300279900, articleId=1251226687665553832, language=CN, label=表3, caption=

指纹数据库相对误差

, figureFileSmall=null, figureFileBig=null, tableContent=
指纹点相对误差/%
123456
10.120.000.000.000.000.00
20.090.000.000.000.570.00
30.080.000.000.000.420.00
40.030.330.000.000.000.00
50.000.000.000.410.420.00
60.000.000.190.000.490.00
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5G超密集网络下基于矩阵补全的室内指纹定位
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江海霞 , 龙光利
电讯技术 | 应用基础与前沿技术 2025,65(11): 1806-1811
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电讯技术 | 应用基础与前沿技术 2025, 65(11): 1806-1811
5G超密集网络下基于矩阵补全的室内指纹定位
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江海霞, 龙光利
作者信息
  • 陕西理工大学 物理与电信工程学院，陕西 汉中 723001

    • 江海霞 女，2000年生于江苏淮安，2022年获学士学位，现为硕士研究生，主要研究方向为无线通信。

    龙光利 男，1968年生于陕西汉中，2013年获硕士学位，现为教授，主要研究方向为无线通信、物联网技术及电子技术应用。

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龙光利 Email:
Indoor Fingerprint Positioning Based on Matrix Completion in 5G Ultra-dense Network
Haixia JIANG, Guangli LONG
Affiliations
  • School of Physics and Telecommunications Engineering，Shaanxi University of Technology，Hanzhong 723001，China
出版时间: 2025-11-28 doi: 10.20079/j.issn.1001-893x.240807002
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摘要
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为降低室内定位指纹数据库构建的复杂度并提高定位精度，提出了一种5G超密集网络下的基于矩阵补全的室内指纹定位算法。在离线数据库构建阶段，算法首先采用K近邻（K-Nearest Neighbor，KNN）插值法对部分指纹库进行矩阵补全，构建完整的数据库；其次，采用稀疏自编码器提取指纹库的稀疏特征，对高维接收信号强度指示（Received Signal Strength Indication，RSSI）信号进行降维处理。在在线指纹匹配阶段，使用加权KNN算法估算出待定位点坐标。经过实验仿真分析，算法重构指纹数据库的平均相对误差为0.31%；与传统KNN指纹匹配算法相比，平均误差降低了24.41%。

5G超密集网络  /  室内指纹定位  /  矩阵补全  /  稀疏自编码

In order to reduce complexity of construction of indoor positioning fingerprint database and improve the positioning accuracy, an indoor fingerprint positioning algorithm based on matrix completion under the 5G ultra-dense network is proposed. In the offline database construction stage,the algorithm first uses the K-nearest Neighbor(KNN) interpolation method to complete the matrix of part of the fingerprint database to construct a complete database. Secondly,the sparse auto-encoder is used to extract the sparse features of the fingerprint database, and the high-dimensional received signal strength indication (RSSI) signal is reduced. In the online fingerprint matching stage,the weighted KNN algorithm is used to estimate the coordinates of the point to be located. After experimental simulation,the average relative error of the algorithm to reconstruct the fingerprint database is 0.31% . Compared with that of the traditional KNN fingerprint matching algorithm,the average error is reduced by 24.41% .

5G ultra-dense network  /  indoor fingerprint positioning  /  matrix completion  /  sparse auto-encoding
江海霞, 龙光利. 5G超密集网络下基于矩阵补全的室内指纹定位. 电讯技术, 2025 , 65 (11) : 1806 -1811 . DOI: 10.20079/j.issn.1001-893x.240807002
Haixia JIANG, Guangli LONG. Indoor Fingerprint Positioning Based on Matrix Completion in 5G Ultra-dense Network[J]. Telecommunication Engineering, 2025 , 65 (11) : 1806 -1811 . DOI: 10.20079/j.issn.1001-893x.240807002
正文
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0　引言
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随着信息技术的发展，依赖位置的服务逐渐成为日常生活中不可或缺的[1]。在室外环境中，定位主要通过全球定位和北斗卫星定位系统等卫星定位系统完成，这些系统可以获得精确到10 m以内的位置[2-3]。然而，在室内环境中，由于建筑物遮挡、卫星信号差、定位精度误差大或无法定位，定位有时无法满足人们的室内定位需求[4]。相比基于移动热点（Wireless Fidelity，WiFi）定位系统的无线局域网（Wireless Local Area，WLAN）只能达到几米到十几米的精度，5G提供的超高频段和更大带宽使其具备亚米级的定位精度。同时，5G技术提供超密集网络（Ultra-dense Network，UDN）部署，大大增加了基站的密度。这有利于多个基站的协调定位，更好地满足人们的需求[5-6]。
许多学者对指纹定位算法进行了研究。Horsmanheimo等人[7]设计并实现了基于5G的室内定位平台，采用精细定时测量技术，通过多边计算估计位置，实现基于图像的定位和接收信号强度指示（Received Signal Strength Indication，RSSI）指纹识别。Liu等人[8]提出了一种改进的加权K近邻（Weighted K-Nearest Neighbors，WKNN）定位算法：在离线阶段，利用指纹数据的期望滤除无效数据，并选择接收信号的平均值和接入点的方差作为特征向量；在在线阶段，提出了加权距离来计算相似度，并根据接入点的差值获得位置。上述指纹匹配算法虽有不错的定位精度，但其指纹库数据相对较少，不易构建。5G UDN中的无线接入点数量显著增加，这意味着指纹数据库中每个指纹点的维度也将大大增加。如果定位维度从二维增加到三维，指纹数据库将变得更加繁琐和复杂[9]。
为此，本文提出了一种5G超密集网络下的基于矩阵补全的室内指纹定位算法。算法在离线指纹库构建阶段，首先采用K近邻（K-Nearest Neighbors，KNN）插值法对有缺失的部分指纹库进行矩阵补全，其次采用稀疏自编码器通过学习数据的稀疏表示，提取更具代表性的稀疏特征。在线匹配阶段，采用欧氏距离来计算相似度，并采用WKNN算法估算出待定位点坐标，实现指纹定位。
1　原理
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1.1　指纹定位算法类别
指纹定位算法的核心思想是利用从不同位置接收到的不同无线发射点的信号强度之差，在接收到的信号强度与位置坐标之间建立相应的关系[10]。指纹定位算法可分为两个阶段：在离线指纹库构建阶段，需要收集各指纹点接收到的不同信号源的接收信号强度，构建离线指纹数据库（离线指纹数据库包含待定位区域内所有点的坐标信息和接收信号强度信息）;在在线指纹匹配阶段，待定位目标首先需要获取自身接收到的不同信号源的接收信号强度，并与指纹数据库中接收到的信号强度信息进行指纹匹配，通过指纹匹配算法计算出自己的位置坐标。常用的指纹匹配算法有K最近邻法和加权K最近邻算法[11-12]等，测度指纹匹配相似度的常用原理是欧氏距离原理和切比雪夫距离原理。本文所指的传统指纹定位算法中使用的匹配算法和相似度测度原理分别是KNN算法和欧氏距离原理。
1.2　5G超密集网络
5G UDN的室内定位模型如图1所示。建立以O为原点的空间直角坐标系模型。模型的容积可以表示为A×B×C。假设5G基站部署在模型顶部，且所有基站的集合为BS，则BS可以表示为BS = [BS1，BS2，…，BSi，…，BSN]，其中N是基站总数。如果待定位用户点u位于空间中的任何位置，则u和BS之间使用5G通信技术，其坐标用Lu=（xu，yu，zu）表示。
假设基站分布服从齐次泊松点过程[13],5G定位模型顶部的面积为S=A×B,则基站密度为
根据齐次泊松点过程，基站的概率分布为
2　指纹定位算法
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本文提出的算法流程图如图2所示，可分为离线指纹库构建和在线指纹匹配两个阶段。在离线指纹库构建阶段，需要先确定定位区域的范围、基站数量、指纹数据库的规模、指纹点之间的距离等与定位相关的参数；其次，利用每个基站接收到的指纹点的RSSI来构建初始指纹数据库；然后，使用KNN插值法进行矩阵补全，构建完整的指纹数据库，之后使用稀疏自编码算法来构建新的数据库。在在线实时定位阶段，待定位点需要读取指纹数据库中的指纹信息，接收各基站的RSSI，使用欧氏距离来计算待定位点与参考指纹点的相似度，并采用WKNN算法对待定位点进行指纹匹配估计坐标。
2.1　离线指纹库构建阶段
2.1.1　KNN插值法
KNN插值法是一种基于最近邻的插值方法[14]，它根据空间上相邻位置的指纹数据进行插值，能够有效利用局部环境的相似性特征，根据邻近数据点的值进行合理补全，可以更好地保留实际信号特性。同时，由于指纹数据库中包含多个特征，KNN插值法可以同时处理这些多维数据，通过在所有特征维度上寻找相似样本进行插值，提升插值的整体准确性。图3是KNN插值法与其他算法的相对误差对比曲线[15]。
假设指纹数据库为矩阵X，其中某些值是缺失的。对于每个缺失值，可以通过以下步骤进行插值：使用欧氏距离计算待插值样本与所有其他样本之间的距离；根据计算的距离，选择距离最小的k个样本作为邻居；使用这k个邻居的值来计算缺失值，采用的方法是取这k个邻居的平均值：
式中：是填补后的值；xl,j是第l个邻居在第j个特征上的值。
2.1.2　稀疏自编码算法
稀疏自编码器是一种无监督学习模型，旨在学习数据的稀疏表示，即用少量的特征来表达输入数据[16]。在室内定位场景中，指纹数据库中的RSSI值是由多个5G基站采集的信号强度数据。通过稀疏自编码器，可以对这些高维的RSSI信号进行降维，并提取出能够有效区分位置的特征，从而使得提取的稀疏特征更加稳健，提升定位精度并提高定位的鲁棒性。
稀疏自编码器的优化目标是最小化损失函数，其损失函数是重构误差、L2正则化和稀疏正则化三部分的加权和，即
用xi表示输入数据，表示重构数据，n是样本数量，重构误差具体公式见式（5）:
L2正则化项用于防止过拟合，通过惩罚权重的平方和实现，见式（6）:
式中：λ是正则化参数；m、n是神经元数量；Wjk是权重矩阵的元素。
稀疏正则化项常用KL散度（Kullback-Leibler Divergence，KLD）来度量实际激活值与期望激活值之间的差异，见式（7）:
式中：β是稀疏正则化参数；ρ是期望的稀疏比例；是隐藏层第j个神经元的平均激活值。
通过训练好的稀疏编码器对输入数据进行编码，提取特征。编码过程可以表示为
式中：h是特征表示；W是权重矩阵；x是输入数据；b是偏置向量；σ是激活函数。在训练过程中，自编码器可以学习到原始高维数据的低维数据特征表示，从而达到降维的目的[16]，降低指纹数据复杂度。
2.2　在线指纹匹配阶段
新算法采用欧氏距离计算待定位点与参考指纹点的相似度，并采用WKNN算法对待定位点进行指纹匹配。对每个用户和每个位置，计算用户特征与位置特征之间的欧氏距离。欧氏距离公式为
式中：a、b均为三维坐标向量。
在WKNN算法中，对每个测试点（xi，yi，zi），取出其k个最近邻的指纹点位置，根据与测试点的距离计算权重，使用距离的倒数作为权重ωi，并计算这些点位置的平均值，作为测试点的估计位置（，，），如式（11）:
3　仿真结果与分析
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新算法的实验仿真平台为MATLAB，设房间模型长、宽、高分别为10 m、10 m、5 m，指纹点间距为d= 0.2 m[9-10]。设基站分布在室内顶部，服从齐次泊松过程。根据文献[17]中超密集网络基站密度λ≫0.1 cell/m2，设置模型基站密度为0.3 cell/m2。假设信号传播是视距传播，则天线增益是单位的，并且忽略系统损耗因子，那么，指纹点的RSSI数据可以用式（11）计算出来[9] :
式中：Ps表示基站的信号传输功率，单位为dB；Pr表示待定位点的信号接收功率，单位为dB；λ为信号波长；d是收发器和接收器之间的距离；n0是高斯白噪声。
设置指纹库缺失比例为30%，经过仿真，给出了部分指纹的数据，表1为原始指纹数据库，表2为使用KNN插值法进行矩阵补全后重构的指纹数据库，表3为重构指纹数据库与原始指纹数据库的相对误差。从表3中可以看出，重构后的指纹数据库与原始指纹数据库之间有些许误差，结合所有数据，可以得到指纹数据库的平均相对误差为0.31%，对定位误差的影响几乎可以忽略不计，表明算法能够很好地还原原始指纹数据库。
指纹匹配时间比较如图4所示，可以看出，随着位置区域体积的增加，新算法与传统KNN算法之间的匹配时间差距越来越大，表明新算法降低了构建指纹数据库的复杂度。
算法的定位结果如图5所示，圆点和星号分别表示要定位的点的真实位置和估计位置。从图5中可以看出，算法估计的待定位点位置与待定位点的实际位置基本吻合。与传统指纹算法和随机森林算法[18]的平均误差比较如图6所示，可以看出，本文提出的改进算法的平均误差为0.0590 m，传统KNN指纹算法的平均误差为0.0734 m，随机森林算法的平均误差为0.0934 m。与传统指纹算法相比，本文提出的新算法的平均误差降低了24.41%，提高了定位精度。图7为3种算法的标准差对比曲线，可以看出，新算法的定位结果稳定性较高于传统KNN算法，鲁棒性较强。
4　结束语
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本文提出了一种5G超密集网络下的基于矩阵补全的室内指纹定位算法。算法在构建指纹库时，首先建立初始指纹数据，采用KNN插值法对有缺失的部分指纹库进行矩阵补全，构建完整的数据库；之后通过稀疏自编码算法提取稀疏特征，对高维RSSI信号进行降维处理。在匹配定位时，采用WKNN算法使用估算出待定位点坐标。仿真实验表明，算法能够有效重构原始指纹数据库，降低了数据库构建的复杂度，同时还降低了定位误差，与传统的KNN指纹匹配算法相比，在定位精度上有了明显的提升。相比于以往的研究，新算法同时受益于稀疏特征提取的降维效果，以及KNN插值法对局部相似性的有效利用，表现出更高的定位精度和鲁棒性。然而，在现实世界中，室内空间环境更为复杂。在后续研究中，将考虑在一个复杂室内环境中进行仿真并考虑室内环境存在的非视距因素，以进一步验证所提出的定位方法的定位性能。
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2025年第65卷第11期
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doi: 10.20079/j.issn.1001-893x.240807002
  • 接收时间：2024-08-07
  • 首发时间：2026-04-15
  • 出版时间：2025-11-28
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  • 收稿日期：2024-08-07
  • 修回日期：2024-11-11
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    陕西理工大学 物理与电信工程学院，陕西 汉中 723001

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2种不同金属材料的力学参数
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Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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