利用半定松弛的高斯混合模型RSS定位WSNs

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"RSS-Based Localization in WSNs Using Gaussian Mixture Model via Semidefinite Relaxation" 本文是一篇关于无线传感器网络（Wireless Sensor Networks, WSNs）中的定位技术的研究论文，作者包括Yueyue Zhang, Song Xing, Yaping Zhu, Feng Yan以及Lianfeng Shen。该论文提出了一种基于接收信号强度指示（Received Signal Strength Indicator, RSSI）的定位算法，利用高斯混合模型来处理实际环境中非高斯分布的噪声问题。在传统的能量基础源定位方法中，通常假设信道路径损耗模型中的噪声服从高斯分布。然而，实际情况中，噪声可能并不符合这一假设，而是呈现出更复杂的特性，如多径效应、阴影衰落等，导致非高斯分布。因此，作者引入了高斯混合模型来更准确地描述这种复杂噪声环境，以此提高定位精度。论文的核心贡献在于，他们设计了一种基于RSSI的定位算法，该算法旨在实现最大似然位置估计器。初始的非线性非凸估计器通过使用Jensen不等式和半定松弛（Semidefinite Relaxation, SDR）技术转化为一个凸优化问题。这种方法的优势在于，它能够被有效地解决，从而得到全局最优解，避免了传统方法可能陷入局部最优的问题。此外，论文还推导了相应的Cramer-Rao下界（CRLB），作为性能评估的标准。通过模拟和实验结果，作者证明了所提出的定位算法相比传统方法能显著提高定位性能，尤其是在应对非高斯噪声时表现出更强的鲁棒性。这篇论文为无线传感器网络的定位技术提供了新的思路，利用高斯混合模型和半定松弛优化策略，提高了在实际复杂环境下的定位精度，对于WSNs的部署和应用具有重要意义。

IEEE COMMUNICATIONS LETTERS, VOL. 21, NO. 6, JUNE 2017 1329

RSS-Based Localization in WSNs Using Gaussian Mixture

Model via Semideﬁnite Relaxation

Yueyue Zhang, Student Member, IEEE, Song Xing, Member, IEEE, Yaping Zhu, Student Member, IEEE,

Feng Yan, Member, IEEE, and Lianfeng Shen, Senior Member, IEEE

Abstract—Energy-based source localization methods are

normally developed according to the channel path-loss models

in which the noise is generally assumed to follow Gaussian

distributions. In this letter, we represent the practical additive

noise by the Gaussian mixture model, and develop a localization

algorithm based on the received signal strength to achieve a max-

imum likelihood location estimator. By using Jensen’s inequality

and semideﬁnite relaxation, the initially proposed nonlinear

and nonconvex estimator is relaxed into a convex optimization

problem, which can be efﬁciently solved to obtain the glob-

ally optimal solution. Besides, the corresponding Cramer–Rao

lower bound is derived for performance comparison. Simulation

and experimental results show a substantial performance gain

achieved by our proposed localization algorithm in wireless

sensor networks.

Index Terms— Received signal strength (RSS), Gaussian

mixture model (GMM), semideﬁnite relaxation.

I. INTRODUCTION

IRELESS source localization has attracted

considerable attentions over the past decades. Among

different localization methods, energy-based localization

via received signal strength (RSS) or received signal

strength difference (RSSD) enables a simple implementation

compared to other conventional technologies such as time

of arrival (TOA) [1], time difference of arrival (TDOA) [2],

and angle of arrival (AOA) [3]. Recent progress has made it

practical to realize the energy-based localization in various

networks including wireless sensor networks (WSNs) [4],

wireless local area network (WLAN) [5], and vehicular

ad-hoc networks (VANETs).

Nevertheless, RSS-based localization to achieve maximum

likelihood (ML) estimator of the coordinates of target nodes

leads to a nonlinear and nonconvex optimization problem.

Several methods have been proposed to address this issue.

In [6], for example, a weighed least squares (WLS) formu-

lation has been derived to jointly estimate the sensor node

location and the transmit power, based on the unscented trans-

formation (UT). In [4], a proposed estimator approximates the

Manuscript received January 10, 2017; accepted January 30, 2017. Date

of publication February 8, 2017; date of current version June 8, 2017.

This work was supported by the National Natural Science Foundation of

China (No. 61471164, 61601122), the Innovation Project of Jiangsu Province

(KYLX16 0222), and the Research Fund of National Mobile Communications

Research Laboratory (NCRL), Southeast University (SEU) (No. 2016B02).

The associate editor coordinating the review of this letter and approving it for

publication was L. Mucchi. (Corresponding author: Lianfeng Shen.)

Y. Zhang, Y. Zhu, F. Yan, and L. Shen are with the National

Mobile Communications Research Laboratory, Southeast University,

Nanjing 210096, China (e-mail: douleyue@seu.edu.cn; xyzzyp@seu.edu.cn;

feng.yan@seu.edu.cn; lfshen@seu.edu.cn).

S. Xing is with the Department of Information Systems, California

State University, Los Angeles, CA 90032 USA (e-mail: sxing@exchange.

calstatela.edu).

Digital Object Identiﬁer 10.1109/LCOMM.2017.2666157

ML estimate for low noise level ﬁrst, and then is relaxed by

applying the efﬁcient convex relaxations that are based on the

second-order cone programming (SOCP). And, a semideﬁnite

programming (SDP) estimator has been designed speciﬁcally

for the RSS-based localization problems [7].

Additionally, it is worth noting that traditional localiza-

tion algorithm designs in WSNs are based on the classical

channel path-loss model with the noise generally modeled by

Gaussian distribution. However, in reality the noise does not

always follow Gaussian distribution due to the heterogeneity

of multiple sources [8]. Hence, the conventional Gaussian

model cannot properly represents the measured noise, leading

to improper localization algorithms in WSNs. To the best of

our knowledge, no study has utilized a practical channel path-

loss model with non-Gaussian noise in the node localization

algorithms in WSNs.

Motivated by above observations, in this letter we proposed

an improved RSS-based node localization algorithm, called

Gaussian mixture-semideﬁnite programming (GM-SDP) esti-

mator, to achieve the ML estimation of the node positions

in WSNs. Speciﬁcally, the noise is represented by a non-

Gaussian model followed by an empirical parameter estimate

of the noise. Then we formulate the localization algorithm

as an optimization problem to achieve ML performance. The

subsequent nonlinear and nonconvex optimization problem is

solved by semideﬁnite relaxation to obtain the suboptimal

solution. Finally, both simulation and experimental results

illustrate the performance gain of our proposed GM-SDP

algorithm over the traditional localization algorithms.

Notation: In the paper, R

and S

denote the set of n vectors

and the n ×n symmetric matrix, respectively. In addition, for

any symmetric matrix A, A  0 means that A is positive

semideﬁnite. ·

, ·

and ·

∞

denote the 

, 

and 

∞

vector norms, respectively.

II. S

YSTEM MODEL AND PRELIMINARIES

Let’s denote the unknown coordinates of the jth target

node as ϕ

=[ϕ

,ϕ

]

(ϕ

∈ R

, j = 1, ···, M)andthe

known coordinates of the ith anchor node as α

=[α

,α

]

(α

∈ R

, i = 1, ···, N), where M and N are the total

number of targets and anchors, respectively. From [4] and [7],

the power received at the jth target from the ith anchor (or

vice versa) is typically modeled as

i, j

= P

− 10β log

d(ϕ

,α

)

+ n

i, j

, (1)

where P

denotes the transmitted power at reference

distance d

from the receiver, d(ϕ

,α

) =ϕ

−α



is the

Euclidean distance between the jth target and the ith anchor

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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