Algorithm and Theory for Robust Wireless Sensor ...

International Journal of Distributed Sensor Networks

Algorithm and Theory for Robust Wireless Sensor Networks 2014 Guest Editors: Chang Wu Yu, Qin Xin, Naveen Chilamkurti, Shengming Jiang, S. Kami Makki, and Guowei Wu

Algorithm and Theory for Robust Wireless Sensor Networks 2014


Algorithm and Theory for Robust Wireless Sensor Networks 2014 Guest Editors: Chang Wu Yu, Qin Xin, Naveen Chilamkurti, Shengming Jiang, S. Kami Makki, and Guowei Wu

Copyright © 2015 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in “International Journal of Distributed Sensor Networks.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Editorial Board Jemal H. Abawajy, Australia Miguel Acevedo, USA Cristina Alcaraz, Spain Ana Alejos, Spain Mohammod Ali, USA Giuseppe Amato, Italy Habib M. Ammari, USA Michele Amoretti, Italy Christos Anagnostopoulos, UK Li-Minn Ang, Australia Nabil Aouf, UK Francesco Archetti, Italy Masoud Ardakani, Canada Miguel Ardid, Spain Muhammad Asim, UK Stefano Avallone, Italy Jose L. Ayala, Spain Javier Bajo, Spain N. Balakrishnan, India Prabir Barooah, USA Federico Barrero, Spain Paolo Barsocchi, Italy Paolo Bellavista, Italy Olivier Berder, France Roc Berenguer, Spain Juan A. Besada, Spain Gennaro Boggia, Italy Alessandro Bogliolo, Italy Eleonora Borgia, Italy Janos Botzheim, Japan Farid Boussaid, Australia Arnold K. Bregt, The Netherlands Rob Brennan, Canada Richard R. Brooks, USA Ted Brown, USA Davide Brunelli, Italy James Brusey, UK Carlos T. Calafate, Spain Tiziana Calamoneri, Italy José Camacho, Spain Juan Carlos Cano, Spain Xianghui Cao, USA João Paulo Carmo, Brazil Roberto Casas, Spain Luca Catarinucci, Italy

Michelangelo Ceci, Italy Yao-Jen Chang, Taiwan Naveen Chilamkurti, Australia Wook Choi, Republic of Korea H. Choo, Republic of Korea Kim-Kwang R. Choo, Australia Chengfu Chou, Taiwan Mashrur A. Chowdhury, USA Tae-Sun Chung, Republic of Korea Marcello Cinque, Italy Sesh Commuri, USA Mauro Conti, Italy Iñigo Cuiñas, Spain Alfredo Cuzzocrea, Italy Donatella Darsena, Italy Dinesh Datla, USA Amitava Datta, Australia Iyad Dayoub, France Danilo De Donno, Italy Luca De Nardis, Italy Floriano De Rango, Italy Paula de Toledo, Spain Marco Di Felice, Italy Salvatore Distefano, Italy Longjun Dong, China Nicola Dragoni, Denmark George P. Efthymoglou, Greece Frank Ehlers, Italy Melike Erol-Kantarci, Canada Farid Farahmand, USA Michael Farmer, USA Florentino Fdez-Riverola, Spain Silvia Ferrari, USA Gianluigi Ferrari, Italy Giancarlo Fortino, Italy Luca Foschini, Italy Jean Y. Fourniols, France David Galindo, Spain Ennio Gambi, Italy Weihua Gao, USA Preetam Ghosh, USA Athanasios Gkelias, UK Iqbal Gondal, Australia Francesco Grimaccia, Italy Jayavardhana Gubbi, Australia

Song Guo, Japan Andrei Gurtov, Finland Mohamed A. Haleem, USA Qi Han, USA Kijun Han, Republic of Korea Zdenek Hanzalek, Czech Republic Shinsuke Hara, Japan Wenbo He, Canada Paul Honeine, France Feng Hong, China Haiping Huang, China Xinming Huang, USA Chin-Tser Huang, USA Mohamed Ibnkahla, Canada Syed K. Islam, USA Lillykutty Jacob, India Won-Suk Jang, Republic of Korea Antonio Jara, Switzerland Shengming Jiang, China Yingtao Jiang, USA Ning Jin, China Raja Jurdak, Australia Konstantinos Kalpakis, USA Ibrahim Kamel, UAE Joarder Kamruzzaman, Australia Rajgopal Kannan, USA Johannes M. Karlsson, Sweden Gour C. Karmakar, Australia Marcos D. Katz, Finland Jamil Y. Khan, Australia Sherif Khattab, Egypt Sungsuk Kim, Republic of Korea Hyungshin Kim, Republic of Korea Andreas König, Germany Gurhan Kucuk, Turkey Sandeep S. Kumar, The Netherlands Juan A. L. Riquelme, Spain Yee W. Law, Australia Antonio Lazaro, Spain Didier Le Ruyet, France Yong Lee, USA Seokcheon Lee, USA Joo-Ho Lee, Japan Stefano Lenzi, Italy Pierre Leone, Switzerland

Shuai Li, USA Shancang Li, UK Weifa Liang, Australia Yao Liang, USA Qilian Liang, USA I-En Liao, Taiwan Jiun-Jian Liaw, Taiwan Alvin S. Lim, USA Antonio Liotta, The Netherlands Hai Liu, Hong Kong Donggang Liu, USA Yonghe Liu, USA Leonardo Lizzi, France Jaime Lloret, Spain Kenneth J. Loh, USA Juan Carlos Lpez, Spain Manel López, Spain Pascal Lorenz, France Chun-Shien Lu, Taiwan Jun Luo, Singapore Michele Magno, Italy Sabato Manfredi, Italy Athanassios Manikas, UK Pietro Manzoni, Spain fhlvaro Marco, Spain Jose R. Martinez-de Dios, Spain Ahmed Mehaoua, France Nirvana Meratnia, The Netherlands Christian Micheloni, Italy Lyudmila Mihaylova, UK Paul Mitchell, UK Mihael Mohorcic, Slovenia José Molina, Spain Antonella Molinaro, Italy Jose I. Moreno, Spain Salvatore Morgera, USA Kazuo Mori, Japan Leonardo Mostarda, Italy V. Muthukkumarasamy, Australia Kshirasagar Naik, Canada Kamesh Namuduri, USA Amiya Nayak, Canada

George Nikolakopoulos, Sweden Alessandro Nordio, Italy Michael J. O’Grady, Ireland Gregory O’Hare, Ireland Giacomo Oliveri, Italy Saeed Olyaee, Iran Luis Orozco-Barbosa, Spain Suat Ozdemir, Turkey Vincenzo Paciello, Italy Sangheon Pack, Republic of Korea Marimuthu Palaniswami, Australia Meng-Shiuan Pan, Taiwan Seung-Jong J. Park, USA Miguel A. Patricio, Spain Luigi Patrono, Italy Rosa A. Perez-Herrera, Spain Pedro Peris-Lopez, Spain Janez Perˇs, Slovenia Dirk Pesch, Ireland Shashi Phoha, USA Robert Plana, France Carlos Pomalaza-Ráez, Finland Neeli R. Prasad, Denmark Antonio Puliafito, Italy Hairong Qi, USA Meikang Qiu, USA Veselin Rakocevic, UK Nageswara S.V. Rao, USA Luca Reggiani, Italy Eric Renault, France Joel Rodrigues, Portugal Pedro P. Rodrigues, Portugal Luis Ruiz-Garcia, Spain Mohamed Saad, UAE Stefano Savazzi, Italy Marco Scarpa, Italy Arunabha Sen, USA Olivier Sentieys, France Salvatore Serrano, Italy Zhong Shen, China Chin-Shiuh Shieh, Taiwan Minho Shin, Korea

Pietro Siciliano, Italy Olli Silven, Finland Hichem Snoussi, France Guangming Song, China Antonino Staiano, Italy Muhammad A. Tahir, Pakistan Jindong Tan, USA Shaojie Tang, USA Luciano Tarricone, Italy Kerry Taylor, Australia Sameer S. Tilak, USA Chuan-Kang Ting, Taiwan Sergio Toral, Spain Vicente Traver, Spain Ioan Tudosa, Italy Anthony Tzes, Greece Bernard Uguen, France Francisco Vasques, Portugal Khan A. Wahid, Canada Agustinus B. Waluyo, Australia Yu Wang, USA Jianxin Wang, China Ju Wang, USA Honggang Wang, USA Thomas Wettergren, USA Ran Wolff, Israel Chase Wu, USA Na Xia, China Qin Xin, Faroe Islands Yuan Xue, USA Chun J. Xue, Hong Kong Geng Yang, China Theodore Zahariadis, Greece Miguel A. Zamora, Spain Hongke Zhang, China Xing Zhang, China Jiliang Zhou, China Xiaojun Zhu, China Ting L. Zhu, USA Yifeng Zhu, USA Daniele Zonta, Italy Antonio-Javier Garc´ıa-Sánchez, Spain

Contents Algorithm and Theory for Robust Wireless Sensor Networks 2014, Chang Wu Yu, Qin Xin, Naveen Chilamkurti, Shengming Jiang, S. Kami Makki, and Guowei Wu Volume 2015, Article ID 489718, 1 page Data Collection for Time-Critical Applications in the Low-Duty-Cycle Wireless Sensor Networks, Shuyun Luo, Yongmei Sun, and Yuefeng Ji Volume 2015, Article ID 931913, 15 pages A Key Management Method Based on Dynamic Clustering for Sensor Networks, Ying Zhang, Bingxin Zheng, Pengfei Ji, and Jinde Cao Volume 2015, Article ID 763675, 9 pages An Improved PSO Algorithm for Distributed Localization in Wireless Sensor Networks, Dan Li and Xian bin Wen Volume 2015, Article ID 970272, 8 pages Two Novel DV-Hop Localization Algorithms for Randomly Deployed Wireless Sensor Networks, Guozhi Song and Dayuan Tam Volume 2015, Article ID 187670, 9 pages Amorphous Localization Algorithm Based on BP Artificial Neural Network, Lin-zhe Zhao, Xian-bin Wen, and Dan Li Volume 2015, Article ID 657241, 9 pages Detecting the Boundary of Sensor Networks from Limited Cyclic Information, Carlos Lara-Alvarez, Juan J. Flores, and Chieh-Chih Wang Volume 2015, Article ID 401838, 7 pages Optimal Energy Allocation Scheme in Distributed Estimation for Wireless Sensor Networks over Rayleigh Fading Channels, Eni D. Wardihani, Wirawan, and Gamantyo Hendrantoro Volume 2015, Article ID 437239, 10 pages Decentralized Kalman Filtering with Multilevel Quantized Innovation in Wireless Sensor Networks, Zhi Zhang and Jianxun Li Volume 2015, Article ID 323980, 13 pages A Hierarchical Scheduling Scheme in WSNs Based on Node-Failure Pretreatment, Hai-yuan Liu, Yi-nan Guo, Mei-rong Chen, and Yuan-shun Zhu Volume 2015, Article ID 397615, 12 pages

Hindawi Publishing Corporation International Journal of Distributed Sensor Networks Volume 2015, Article ID 489718, 1 page http://dx.doi.org/10.1155/2015/489718

Editorial Algorithm and Theory for Robust Wireless Sensor Networks 2014 Chang Wu Yu,1 Qin Xin,2 Naveen Chilamkurti,3 Shengming Jiang,4 S. Kami Makki,5 and Guowei Wu6 1

Department of Computer Science and Information Engineering, Chung Hua University, No. 707, Section 2, Wufu Road, HsinChu 300, Taiwan 2 Faculty of Science and Technology, University of the Faroe Islands, Noatun 3, 100 Torshavn, Faroe Islands 3 Department of Computer Science and Computer Engineering, La Trobe University, Melbourne, Australia 4 College of Information Engineering, Shanghai Maritime University, China 5 Department of Computer Science, Lamar University, USA 6 School of Software, Dalian University of Technology, China Correspondence should be addressed to Chang Wu Yu; [email protected] Received 31 March 2015; Accepted 31 March 2015 Copyright © 2015 Chang Wu Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The special issue “Algorithm and Theory for Robust Wireless Sensor Networks” was published earlier in 2013. Based on the success of 2013’s special issue, we are honored to be invited again to make this special issue (a new issue) as 2014’s annual special issue, which means that it will become the first issue in a series of special issues which will be published each year in this journal. We hope that such a series can have a longterm impact and in time gather a wireless sensor network community around it. The main focus of the series of these special issues from 2013 to 2014 is devoted to a deeper understanding of the fundamental protocols, algorithms, modeling, and analysis techniques for robust wireless sensor networks, which exhibit substantial vulnerability when compared to other networks. Recently there are so many papers developing new techniques in wireless sensor networks. However, we believe that the algorithmic and theoretical issues in the robust wireless sensor networks were not fully explored. In this annual special issue, totally we have received 25 submissions, coming from different countries all around the globe in response to call for paper. We also invited some extended best papers from 2014 IET International Conference on Frontiers of Internet of Things (FIT), which was held in Hsinchu, Taiwan, on December 4–6, 2014. Each accepted article has been reviewed by at least three reviewers. In the

end, nine articles are revised and selected for publishing in this annual special issue. We believe that the accepted papers present the most up-to-date progress in algorithms and theory for robust wireless sensor networks with respect to different networking problems.

Acknowledgments First of all, we would like to thank our guest editors: Professor Qin Xin, Professor Naveen Chilamkurti, Professor Shengming Jiang, Professor Kami Makki, and Professor Guowei Wu, who have provided their remarkable contributions to this special issue. We also appreciate the outstanding review work performed by the referees of this special issue for providing valuable comments to the authors. Chang Wu Yu Qin Xin Naveen Chilamkurti Shengming Jiang S. Kami Makki Guowei Wu

Hindawi Publishing Corporation International Journal of Distributed Sensor Networks Volume 2015, Article ID 931913, 15 pages http://dx.doi.org/10.1155/2015/931913

Research Article Data Collection for Time-Critical Applications in the Low-Duty-Cycle Wireless Sensor Networks Shuyun Luo, Yongmei Sun, and Yuefeng Ji State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China Correspondence should be addressed to Shuyun Luo; [email protected] Received 17 October 2014; Revised 14 January 2015; Accepted 18 January 2015 Academic Editor: Qin Xin Copyright © 2015 Shuyun Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In low-duty-cycle wireless sensor networks, wireless nodes usually have two states: active state and dormant state. The necessary condition for a successful wireless transmission is that both the sender and the receiver are awake. In this paper, we study the problem: How fast can raw data be collected from all source nodes to a sink in low-duty-cycle WSNs with general topology? Both the lower and upper tight bounds are given for this problem. We use TDMA scheduling on the same frequency channel and present centralized and distributed fast data collection algorithms to find an optimal solution in polynomial time when no interfering links happen. If interfering links happen, multichannel scheduling is introduced to eliminate them. We next propose a novel Receiverbased Channel and Time Scheduling (RCTS) algorithm to obtain the optimal solution. Based on real trace, extensive simulations are conducted and the results show that the proposed RCTS algorithm is significantly more efficient than the link schedule on one channel and achieves the lower bound. We also evaluate the proposed data collection algorithms and find that RCTS is time-efficient and suffices to eliminate most of the interference in both indoor and outdoor environment for moderate size networks.

1. Introduction In most wireless sensor networks (WSNs), how to improve the network robustness for data collection is an important issue. Two seemingly contradictory, yet related objectives of performance exist: prolonging the network lifetime due to limited power resources and reducing data latency. On the one hand, researchers have put much effort to save or harvest energy for wireless sensor nodes in order to prolong the lifetime of individual nodes since most wireless sensor nodes are powered by batteries. On the other hand, some application scenarios like surveillance [1] require a bound on the data collection latency to guarantee availability of accurate information about the sensing field; otherwise, collected data information may become irrelevant or even useless [2, 3]. In many emergency applications, the collected sensor data are usually useful during a finite amount of time, after which the information may become irrelevant. The requirement is special in some time-critical application scenarios using WSNs, such as the application of monitoring the escapees’ current locations and expanding situation of hazard areas in

the emergency navigation system [2]. In network diagnosis, it is necessary to collect metrics periodically, such as residual energy levels of nodes and instant network conditions, which are required to be collected in real time [3]. In the above situations, sensor data and metric information are both asked to be collected from nodes to the sink as soon as possible. Hence, the maximum end-to-end delay for each bit must be limited within some acceptable time-efficient requirement. One of simple but efficient methods to prolong the lifetime of WSNs is to decrease the duty cycle of individual nodes [4–6] such that they finish coping with sensing and transmission operations when they are awake while falling into sleep for the rest of time in a time period. Hence, the working mode of sensor nodes is normally designed as periodic active in most monitoring applications [7, 8]. However, this brings forth another challenge for minimizing end-to-end delay since any individual node existing on some end-to-end transmission path makes the path unfeasible if it goes into dormant state such that the end-to-end delay cannot be controlled easily. In addition, the low-duty-cycle mechanism causes more severe collision because the available

2


Input: The given VGN over the time expanded. Output: The optimal data collection paths with the minimum delay. 𝑖 (1) Initialization: Δ = max𝑛−1 𝑖=1 {𝑇min }, 𝑓𝑚 = 0, 𝑘 = ⌊Δ/𝑇𝜋 ⌋ (2) while 𝑓𝑚 < 𝑛 − 1 do (3) Update the sub-graph of VGN under the time threshold 𝑘𝑇𝜋 . (4) while there is an 𝑠-𝑑 path in the residual graph 𝐺𝑓 (Δ) do (5) 𝑃 be a simple 𝑠-𝑑 path in 𝐺𝑓 (Δ) (6) 𝑓󸀠 = augment(𝑓, 𝑃) (7) if 𝑓󸀠 satisfies the constraints in the above max-flow problem then (8) Update 𝑓 to be 𝑓󸀠 (9) Update the residual graph 𝐺𝑓 (Δ) to be 𝐺𝑓󸀠 (Δ) (10) Obtain the maximum flow 𝑓𝑚 of (𝐺, 𝑠, 𝑑, 𝑐, Δ) and optimal flow paths 𝑃. (11) 𝑘=𝑘+1 (12) Among the 𝑁𝑖,𝑡 through flow in the 𝑘th 𝑇𝜋 period, find the the largest 𝑡 of 𝑁𝑖,𝑡 . (13) 𝑇min = the largest 𝑡. Algorithm 1: Centralized Fast Data Collection algorithm.

time for nodes to receive packets is notably reduced [6]. Several attempts have been applied to decrease the end-toend latency for WSNs. For instance, [9] utilized collisionavoidance method to reduce interference among wireless transmission since the latter causes much retransmission due to packet loss, especially when a contention-based MAC protocol (e.g., CSMA) is explored. Some other work [10] also proved that one of the main reasons of traffic latency is the “wasting” sleeping time of the receiver. In this paper, we focus on how to design Fast Data Collection algorithms to reach the minimum data collection delay in low-duty-cycle WSNs. We apply contention-free MAC protocols, that is, TDMA, to eliminate collisions and make the best of the limited link resources in compliance with node working schedules. In addition, since the network protocols can affect the end-to-end delay of each bit [10], we use a joint routing and scheduling cross-layer design to search the minimum data collection delay paths. To address fast data collection problem, we present both centralized and Distributed Fast Data Collection algorithms when no interfering links happen. If the interfering links happen, the multichannel scheduling is introduced to eliminate interfering links. Furthermore, a simple but efficient distributed channel and time scheduling method is proposed to search the minimum data collection delay paths. The main intellectual contributions of this work are summarized as follows. (1) For the raw data collection, we define a minimum data collection delay (MDCD) problem and give both the upper and lower tight bounds on the data collection delay. (2) We present a novel Centralized Fast Data Collection (CFDC) algorithm (Algorithm 1), adopted to any general topology, which achieves the lower bound

and returns the optimal collection paths with the minimum delay in polynomial time. (3) We further propose a Distributed Fast Data Collection (DFDC) algorithm based on the node’s local information. It is proved that the DFDC algorithm can also obtain the optimal solution. (4) In order to eliminate the interfering links and guarantee network robustness, we present a Receiver-based Channel and Time Scheduling (RCTS) algorithm to the search for minimum data collection delay paths. The data trace based evaluations show the use of RCTS can suffice to eliminate most of the interference in both the indoor and outdoor environment. The rest of this paper is organized as follows. Section 2 briefly presents the related work. We present the network model and formulate our problem formally in Section 3. In Section 4, both the upper and lower bounds are given on the MDCD problem. We propose a CFDC algorithm to solve the fast data collection problem when no interfering links happen, which obtains the optimal solution. In Section 5, a DFDC algorithm is further presented when no interfering links happen. When interfering links happen, combining with multiple channel scheduling, a RCTS algorithm is proposed to search for the minimum data collection delay paths in Section 6. Based on the data trace from the Berkeley Lab, extensive simulations are conducted in outdoor and indoor scenarios and further present and analyze the results in detail in Section 7. We conclude our paper and discuss future work in Section 8.


2. Related Work The contribution of our work lies in the intersection of two important cutting-edge research topics: (1) the function performance of data collection and (2) the low-duty-cycle working modes of sensor nodes. We review some related work in both topics as follows. Data Collection. The objective of most WSN-related applications is to collect data from physical world, such as environmental surveillance [8, 11] and vehicle and structural monitoring [12–15]. Some work [9, 16, 17] concentrated on how to improve the energy efficiency, time efficiency, and (or) reliability of deployed WSNs. In [16], the authors presented a practical, energy efficient, and reliable solution to the problem of periodic data collection. Reference [9] proposed a link scheduling algorithm finding the minimum delay schedule when the slot lengths of link are given. Basically, they investigated the tradeoff between the energy consumption and delay. By developing a novel and efficient TDMA schedule, [17] studied the effect of dynamic traffic patterns for data collection. However, for all the aforementioned work, the authors assumed that the data collection was applied in the case with constant availability of connectivity such that the proposed algorithms may lead to very low efficiency in low-duty-cycle sensor networks with sleep latency. Low-Duty-Cycle. One of simple but efficient methods to prolong the lifetime of WSNs is to decrease the duty cycle of individual nodes such that they finish coping with sensing and transmission operations when they are awake while falling into sleep for the rest of time in a time period since a node almost costs no energy in dormant state. Some other work studied the low-duty-cycle related topics from different points of view, such as data flooding and broadcasting [10, 18], sleep scheduling [5, 19], and multiple tasks scheduling [6]. In [10], the authors detailed an opportunistic flooding method in a low-duty-cycle WSN with unreliable links when the working schedules of all wireless sensor nodes were given. Reference [18] aimed to solve the latencyoptimal broadcasting problem in the duty-cycled multihop wireless networks and proposed three algorithms to improve the approximation ratios. They also proposed a forwarder selection method to alleviate the hidden terminal problem. The work [19] modified the random duty cycling to solve slow diffusion problem. Reference [5] introduced energyefficient sleep scheduling in order to minimize latency. The authors of [6] provided two efficient scheduling algorithms in order to balance network traffic. However, most of above work regarding low-duty-cycle WSNs does not concern data collection function. Some work, like [4, 20], has similar objectives with our problem. In [4], the authors presented a dynamic data forwarding (DSF) scheme in extremely low-duty-cycle sensor networks with unreliable links. They studied the impact of both lossy radio link and sleep latency at the network layer. Although the DSF scheme can be used to data collection scenario in low-duty-cycle networks, it adopted some multiple paths routing strategy which suffered packet duplicated issue

3 inevitably. In addition, exchanging traffic statistic frequently for routing cutover may introduce nonnegligible communication overhead. Reference [20] provided cross-layer analysis framework for end-to-end delay distribution in WSNs. Some MAC protocols, such as B-MAC [21], exploited network information like packet transmission or routing paths to optimize their link scheduling. Reference [4] considered the MAC layer information (like link quality) into the network layer, which leads to better link selections. For the interference elimination, the use of multiple channels has been studied extensively in both cellular and ad hoc networks. However, in the domain of wireless sensor networks, there exist some research that utilize multiple channels [22]. To the best of our knowledge, there is no prior work that has thoroughly researched on the optimal delay routing algorithm for the data collection scenario in low-duty-cycle sensor networks without topology constraints. In this work, by taking the available time of link resources (working schedules) into consideration, we reveal that the cross-layer design is to obtain a delay-optimal data collection path for each wireless sensor node and exploit multiple-channel scheduling to eliminate the interfering links.

3. Network Model and Problem Formulation In this section, we first focus on a TDMA scheduling crosslayer design when the nodes communicate on the same channel without any interfering link present. In the following Section 6, the case where interfering links happen is considered. We will combine with the multichannel scheduling to eliminate the interfering links without compromising the time efficiency of data collection. 3.1. Network Model. In this work, it is considered that a connected network with 𝑛 wireless sensor nodes works in a low-duty-cycle mode. Basically, each sensor node has two states: active and dormant. When a node is in active state, it can sense or receive packets from neighbor nodes. For the purpose of energy conservation, a node in dormant turns off all its functional models except a timer to wake itself up. Under the low-duty-cycle working mode, if a node only delivers packets in active state, the packets will hardly be sent out since the active time may not overlap with that of its neighbors. To solve this problem, each node still receive packets when it is in active state, but the transmission rule is changed as follows: if a node has packets to forward beyond its active time slots, it wakes up its transceiver and transmits packets when its next hop neighbor turns into active state [23]. To simplify the discussion, it is assumed that all nodes in the network work in the asynchronous duty-cycle mode and are sources except the sink. The objective is to minimize the whole time 𝑇 required to complete data collection. In TDMA MAC protocol, 𝑇 is divided into a number of equal-length time slots 𝜏, each of which is long enough to transmit one packet successfully. We also assume that each node wakes up to receive the packet for only one time slot in every cycle,

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International Journal of Distributed Sensor Networks j

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m

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Intersecting links (b)

(a)

1

4

2

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Interfering links (c)

Figure 1: The constraints for links scheduling.

which is reasonable since the function of low-duty-cycle aims to reduce the waking time of individual node [24]. To symbolize the working schedule more clearly, we describe it as a binary string for individual node, in which 1 means the node in active state, while 0 indicates that in dormant state [4]. The cycle period of working schedule is denoted as 𝑇𝜋 . For example, the working schedule of node 𝑛𝑖 can be extended to 𝜋𝑖𝑇 = 0100010001, if the network collecting time 𝑇 is 10 time slots, 𝜋𝑖 = 0100 and 𝑇𝜋 is 4 time slots. The above assumption is the same as the previous work [4, 10, 23]. There are two common types of interference models: the graph based protocol model and the SINR-based physical model. In the protocol model, the transmission from node 𝑖 to a node 𝑗 is successful, if for every other 𝑘, simultaneously transmitting, the following condition holds: 𝑑 (𝑘, 𝑗) ≥ (1 + Δ) 𝑑 (𝑖, 𝑗) ,

(Δ > 0) ,

(1)

where 𝑑(𝑖, 𝑗) is the distance between node 𝑖 and node 𝑗 and 𝛿 is a parameter that promise that the concurrently transmitting nodes are enough far away from the receiver to prevent interference. The physical model considers the accumulative effect from multiple concurrent transmissions. The transmission of a packet from node 𝑖 to node 𝑗 is successful when the ratio between the received signal strength at node 𝑗 and the cumulative interference caused by all other concurrent transmissions and the background noise is greater than a certain threshold 𝛽; that is, SINR𝑖𝑗 =

𝑃𝑖 ⋅ 𝑔 (𝑑𝑖𝑗 ) ∑𝑘=𝑖̸ 𝑃𝑘 ⋅ 𝑔 (𝑑𝑘𝑗 ) + N

,

path-loss exponents, and the physical interference model is closer to the actual situation than the protocol model. Hence, we use the physical model in all the following evaluation cases. 3.2. Problem Formulation. We define the MDCD problem as follows. Given a WSN with 𝑛 sensor nodes, if the sink node is the 𝑛th node, the other 𝑛 − 1 nodes are source nodes, each of which generates one packet. Collection delay is defined as the time elapsed from packets being generated until all 𝑛 − 1 packets reach the sink. The goal is to find data collection paths to the sink from each source with the minimum collection delay. Symbol EED(V𝑖 ) is used to denote the end-to-end delay (EED) for source node V𝑖 to the sink within either one or multiple hops. The objective is to minimize 𝑇 = max𝑛−1 𝑖=1 EED(V𝑖 ), ∀V𝑖 ∈ 𝑉. Formally, let 𝑥(𝑖, 𝑗, 𝑡) be a 1 − 0 integer variable indicating whether the node V𝑗 at time 𝑡 receives data from V𝑖 . 𝐻𝑗 indicates the active time of node V𝑗 and 𝑐(𝑝𝑖 , 𝑗, 𝑡) is also a Boolean variable denoting whether the packet 𝑝𝑖 is delivered by node V𝑗 at time 𝑡. Thus, the MDCD problem can be described as follows. Problem: Minimizing Data Collection Delay Objective: Minimize T Subject to: 𝑛−1

𝑛−1

𝑖=1,𝑖=𝑗̸

𝑚=1,𝑚=𝑗̸

∑ 𝑥 (𝑖, 𝑗, 𝑡1 ) +

∑ 𝑥 (𝑗, 𝑚, 𝑡2 ) ≤ 1,

(3)

where 𝑡1 ∈ 𝐻𝑗 , 𝑡2 ∈ 𝐻𝑚 (2)

where 𝑃𝑖 is the transmission power at node 𝑖, SINR𝑖𝑗 indicates the signal-to-interference-plus-noise ratio, N is the background noise level, 𝑔(⋅) is the signal attenuation function, and 𝑑𝑖𝑗 is the distance between nodes 𝑖 and 𝑗. Since all nodes use the same constant transmission power, we set 𝑃𝑖 = 𝑃𝑘 = 1. For simplicity of exposition, we use a simple distance dependent −𝛼 path-loss model as 𝑔𝑖𝑗 = 𝑑𝑖𝑗 𝑖𝑗 , where the path-loss exponent 𝛼𝑖𝑗 is a factor between 2 and 6, depending on the external wireless environment, such as humidity, temperature, and obstacles. It is assumed that the interference level in both models is static and does not change over time. Reference [25] found that the use of the graph based model fails most in sparse network deployments with higher

𝑇

∑𝑐 (𝑝𝑖 , 𝑗, 𝑡) ≤ 1,

𝑡=1

𝑥 (𝑖, 𝑗, 𝑡3 ) = 0,

where 𝑡 ∈ [0, 𝑇]

where 𝑡3 ∈ [0, 𝑇] , 𝑡3 ∉ 𝐻𝑗 .

(4) (5)

Here, inequality (3) ensures that each node in active time can only receive one packet from its neighbor node and cannot receive or send data simultaneously. The links in Figure 1(a) can not be scheduled simultaneously. Inequality (4) guarantees that the same packet can only be delivered once by each node, which can restrict routing loops in the network. Equation (5) restricts the ability of each node to receive data when it is in dormant state. In terms of collision type, the links for interference are divided into two categories: intersecting links and interfering

International Journal of Distributed Sensor Networks links [25]. The intersecting links, defined as the links with a common destination shown in Figure 1(b), cannot transmit on the same time slot since there is only one half-duplex radio transmitter in a sensor node. The constraint described by inequality (3) ensures that the intersecting links will not exist in the data collection paths. The interfering links are the links which create or face interference if they are scheduled simultaneously, which happens when two nodes send data simultaneously, and the SINR of any receiver is not greater than the predefined threshold. Figure 1(c) shows an example where the dotted line represents interference. Interfering links should not get the same time slot and channel. Since our goal is to minimize the number of time slots, the best option is to assign the same time slot on nonconflicting channels. Hence, the interfering links can be avoided to exploit the orthogonal frequencies [24, 25], that is, using multiple frequency channels to enable more concurrent transmissions. In the following study cases, we first consider MDCD problem in a case where nodes communicate on the same channel with the goal of minimizing the data collection delay. Next, we combine with the multichannel scheduling to avoid the interfering links.

4. Centralized Fast Data Collection Algorithm In this section the goal is to address the MDCD problem. We first analyze the lower and upper bounds on the delay for data collection and give the procedure of proofs. Furthermore, we introduce a novel concept, the VGN to convert the MDCD problem into max-flow problem with special constraints. Based on VGN, we propose a Centralized Fast Data Collection algorithm (CFDC) to obtain the optimal collection paths with minimum delay in polynomial time when no interfering links happen. 4.1. The Lower and Upper Bounds on the Data Collection Delay Lemma 1. The delay for data collection is tightly lower bounded by 𝑆(𝑠𝑖𝑛𝑘) + (𝑁 − 1)𝑇𝜋 , where 𝑇𝜋 denotes cycle period of working schedule, 𝑆(𝑠𝑖𝑛𝑘) is the active start time of the sink in 𝑇𝜋 , and 𝑁 is the number of source nodes. Proof. The detailed proof can be seen in our previous work [26]. Lemma 2. The delay for data collection is tightly upper 𝑖 bounded by ∑𝑁 𝑖=1 𝑇min , where 𝑁 is the number of source nodes 𝑖 and 𝑇min denotes the minimum time that source node 𝑖 requires to deliver a packet to the sink without the effect of other packets. Proof. The detailed proof can also be seen in our previous work [26]. 4.2. Virtual Grid Network. To make the node states and packet transmission process more clear, we define and detail a concept, VGN inspired by the time-expanded network proposed in [4]. The edge construction regulations of VGN are different from the time-expanded network. We first show

5 how to construct the VGN based on the original directed communication graph 𝐺, which is the foundation of the data collection method proposed later. For simple presentation, we first classify the nodes in 𝐺 into three types by different roles. (i) Leaf node: only acts as a source node; that is, it transmits its own sensor reading. (ii) Intermediate node: acts as double roles of a source node and a relay node; that is, not only does it transmit its sensor reading, but also receives a packet and tries to forward it when its neighbors awake. The own generated packet has higher priority to transmit than the forwarded packets. (iii) Sink node: responsible for receiving packets. A deployed WSN is transferred into a communication graph 𝐺 = (𝑉, 𝐸), where 𝑉 denotes the node set with cardinality 𝑛 and 𝐸 is the edge set. Assume that the working schedule for each wireless node 𝑛𝑖 ∈ 𝑉 is given; we build a VGN according to the following rules. (1) For each node 𝑛𝑖 active at time 𝑡 ∈ [0, 𝑇], we build a virtual active node 𝑁𝑖,𝑡 in VGN. (2) When node 𝑛𝑖 is a leaf node and has a directed edge to the node 𝑛𝑗 in 𝐺, if the first active time of 𝑛𝑖 is 𝑡 and the active time of node 𝑛𝑗 is 𝑝 after time 𝑡, we add a directed edge from 𝑁𝑖,𝑡 to 𝑁𝑗,𝑝 in VGN. Since the leaf node only transmits its own sensor reading, it is not possible for 𝑛𝑖 to have new arrived packet at the other active time except the first one. (3) When node 𝑛𝑖 is an intermediate node and has a directed edge to the node 𝑛𝑗 in 𝐺, if the active time of node 𝑛𝑖 and 𝑛𝑗 are 𝑡 and 𝑝 (𝑝, 𝑡 ∈ [0, 𝑇]), respectively, and 𝑝 > 𝑡, we add a directed edge from 𝑁𝑖,𝑡 to 𝑁𝑗,𝑝 in VGN. For the raw data collection, the MDCD problem can obtain an optimal result by reducing into the max-flow problem. In order to formulate a max-flow problem over the VGN, we introduce two virtual vertices, the super source and the super sink, symbolized by 𝑠 and 𝑑, respectively, to represent the source and destination of the total flow over the graph. We complement the rules to build the connection from 𝑠 and 𝑑 to the virtual active nodes. (4) When node 𝑛𝑖 is a sink, we connect all its corresponding virtual active nodes to the super sink 𝑑. (5) We build the edges from the super source 𝑠 to the corresponding first active virtual nodes of all source nodes. Here, the first rule illustrates the regulations to establish all nodes of VGN. The remainder rules depict how to build edges to connect nodes in VGN according to 𝐺. We take the simple network in the best case mentioned in Figure 2 as an example to give a walk-through of VGN construction (shown in Figure 4). In the following, we detail how to construct VGN graph in terms of original network topology and working schedule

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International Journal of Distributed Sensor Networks 4 (0010) 3

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of each node. First, we set the nodes in VGN based on the working schedule of each node in 𝐺 by Rule 1 and the red nodes in VGN represent the virtual active nodes. Next, we describe how to construct corresponding edges in VGN according to Rules 2–5. For the leaf node V1 , it can send a packet to its neighbor V3 when V3 is active. Thus, we establish the edges of 𝑁1,1 → 𝑁3,2 , 𝑁1,1 → 𝑁3,6 , and 𝑁1,1 → 𝑁3,10 in VGN by Rule 2. For another leaf node V2 , it can also send a packet to its neighbor V3 when V3 is active. According to the same rule, we build the edges of 𝑁2,4 → 𝑁3,6 and 𝑁2,4 → 𝑁3,10 in VGN. For the intermediate node V3 , it can send a packet to its neighbor V4 when V4 is active. Hence, we establish the edges from 𝑁3,2 → 𝑁4,3 , 𝑁3,2 → 𝑁4,7 , and 𝑁3,2 → 𝑁4,11 in VGN. At the same time, as a relay node, node V3 can receive packets when it turns to be active and forwards packets when its neighbor is active. Accordingly, we establish the edges 𝑁3,6 → 𝑁4,7 , 𝑁3,6 → 𝑁4,11 , and 𝑁3,10 → 𝑁4,11 in VGN. For the sink V4 , we build the edges from the corresponding virtual active nodes to the super sink 𝑑 by Rule 4, that is, 𝑁4,3 → 𝑑, 𝑁4,7 → 𝑑, and 𝑁4,11 → 𝑑. Finally, we build the connection from the super source 𝑠 to the corresponding first active virtual nodes of all source nodes by Rule 5, that is, 𝑠 → 𝑁1,1 , 𝑠 → 𝑁2,4 , and 𝑠 → 𝑁3,2 . The whole mapping process for one of the best cases is completed, as shown in Figure 4. 4.3. The Max-Flow Problem with Transmission Constraints. Given the VGN, our next step is the formulation of an optimization problem whose objective is to maximize the flow from 𝑠 to 𝑑, that is, the most number of source nodes from which the sink can collect data successfully under the given time duration 𝑇. 𝑓(∗, ∗) indicates the binary traffic flow over an edge connecting two vertices in VGN. Denoting by 𝐹(∗, ∗) the total flow from the source to the destination, our objective can be described as max {𝐹 (𝑠, 𝑑)} .

(6)

The data collection delay is defined as the total time required to complete data collection, which depends on the maximum EED for each source node. Minimizing the maximum EED problem is equal to the max-flow problem in VGN, which needs to be solved taking into account several constraints due to, for example, interference influence and

half-duplex transceiver limitation. We detail such constraints below. Constraints Nonnegative Flow and Flow Conservation. The flow on every existing edge must be greater than or equal to zero. In the meantime, for any vertex in the VGN, the amount of flow entering the vertex must be equal to the amount of outgoing flow. Mapping to the VGN, the constraint can be expressed as ∑ 𝑓 (𝑁𝑝,𝑡𝑝 , 𝑁𝑖,𝑡 ) = ∑ 𝑓 (𝑁𝑖,𝑡 , 𝑁𝑞,𝑡𝑞 ) ,

𝑝∈𝑁(𝑖)

𝑞∈𝑁(𝑖)

(7)

where 𝑡𝑝 < 𝑡 < 𝑡𝑞 ∈ [0, 𝑇] . The function 𝑁(𝑖) indicates the set of node 𝑖’s neighbors in the original network 𝐺. Half-Duplex Transceiver Limitation. Due to the hardware function limitation, a node cannot transmit and receive packets simultaneously shown in Figure 1(a). Mapping to the VGN, the constraint can be expressed as 𝑓 (𝑁𝑖,𝑡1 , 𝑁𝑗,𝑡2 ) + 𝑓 (𝑁𝑗,𝑡3 , 𝑁𝑚,𝑡2 ) ≤ 1 where 𝑡1 , 𝑡2 , 𝑡3 ∈ [0, 𝑇] ,

(8)

where the binary function 𝑓(∗, ∗) is equal to 1 if the edge carries a flow; otherwise it is 0. We notice that the case shown in Figure 1(a) happens only when 𝑗 and 𝑚 are the neighbors and active at the same time in the original topology. Meanwhile, a node cannot receive from more than one neighbor at the same time, shown in Figure 1(b). For this constraint, we set the node capacity in VGN to one unit, that is, 𝐶𝑁𝑖,𝑡 = 1. We can deduce that the edge capacity in VGN satisfies the conditions below: ∑ 𝑓 (𝑁𝑝,𝑡𝑝 , 𝑁𝑖,𝑡 ) ≤ 𝐶𝑁𝑖,𝑡 = 1,

where 𝑡𝑝 ≤ 𝑡 ∈ [0, 𝑇] ,

∑ 𝑓 (𝑁𝑖,𝑡 , 𝑁𝑞,𝑡𝑞 ) ≤ 𝐶𝑁𝑖,𝑡 = 1,

where 𝑡 ≤ 𝑡𝑞 ∈ [0, 𝑇] .

𝑝∈𝑁(𝑖)

𝑞∈𝑁(𝑖)

(9) Since we assume that the interfering links are eliminated, the max-flow problem is completed to formulate.


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Figure 4: VGN transformation. (a) A simple network. (b) The Virtual Grid Network.

4.4. Minimum Data Collection Delay Algorithm. After converting the MDCD problem into a max-flow problem, we present a novel CFDC algorithm inspired by the FordFulkerson max-flow method to solve the MDCD problem and obtain the optimal solution in polynomial time. In CFDC algorithm, the working schedule for each node needs to be known by the sink in the network initialization, which can be easily achieved through exchange of the hello messages. The initial expanding time of VGN is set to be the maximum value of the set of minimum time for each packet delivered to the sink by Dijkstra Algorithm. The maximum flow 𝑓𝑚 is set to zero at the initialization step. Each iteration of the outmost while-loop corresponds to one phase. In each phase, the time threshold 𝑇 is monotonously increased with the step of cycle period 𝑇𝜋 . The number of phases stops to increase until the maximum flow satisfied by all constraints is equal to the number of source nodes. This is an indication that all packets can be scheduled within 𝑇 without collision. Obviously, the result 𝑃 is the optimal collection paths for all source nodes. Theorem 3. If all the interfering links are eliminated, CFDC algorithm can obtain the optimal data collection paths with the minimum delay, that is, 𝑆(𝑠𝑖𝑛𝑘) + (𝑁 − 1)𝑇𝜋 . Proof. The max-flow problem is to find the max flow from the source 𝑠 to the destination 𝑑. In the VGN, the neighbors of super sink come from the corresponding virtual active nodes of the sink. Therefore, in order to make the flow maximum from 𝑠 to 𝑑, the corresponding virtual active nodes

of the sink try to carry flow as much as possible. When all corresponding virtual active nodes of the sink carry flow, the sink keeps receiving packets in all active time. At this moment, CFDC algorithm returns the optimal collection path with the minimum delay, that is, 𝑆(sink) + (𝑁 − 1)𝑇𝜋 . In the best case illustrated above, the optimal collection paths returned by CFDC algorithm are shown in bold in Figure 4, and the minimum collection delay achieves the lower bound. Thus, the CFDC algorithm can obtain the optimal collection paths with the minimum delay. In addition, due to the half-duplex transceiver constraints, it sometimes fails to make all corresponding virtual active nodes of the sink carried flow, shown in the worst case mentioned above (Figure 3). But it can also obtain the optimal solution. Figure 5 shows the optimal collection paths returned by CFDC algorithm. However, the returned minimum delay will not achieve the lower bound, since it applies in the best case. Lemma 4. The total time complexity for CFDC algorithm is 𝑂(𝑇3 𝑛3 ). Here, 𝑇 is the minimum time threshold when the maximum flow is equal to the number of source nodes. Proof. The detailed proof can be seen in [26].

5. Distributed Fast Data Collection Algorithm Since the CFDC needs to obtain all nodes cycling information, it is not practical to implement the CFDC in the large

8


Input: The original network topology Output: The optimal link schedule solution with the minimum delay. (1) source-node.buffer = full (2) Compute 𝑝(𝑖) for each node 𝑖 ∈ 𝑛 (3) Sort nodes in the ascending order of 𝑝(𝑖) (4) while Any node’s buffer is full do (5) for Each node 𝑖 in the sorted order of 𝑉 do (6) if node.buffer == 𝑓𝑢𝑙𝑙 then (7) for Each node ∈ 𝑖’s neighbors who are active do (8) if 𝑝(𝑖.neighbor)< 𝑝(𝑖) and 𝑖.neighbor.buffer == 𝑒𝑚𝑝𝑡𝑦 then (9) 𝑖.neighbor is node 𝑖’s next hop (10) 𝑖.neighbor.buffer = 𝑓𝑢𝑙𝑙 (11) 𝑖.buffer = 𝑒𝑚𝑝𝑡𝑦

Node ID

Algorithm 2: Distributed Fast Data Collection algorithm.

Since the nodes far away from the sink are more difficult to send packets to the sink, they have higher pressure to deliver their packets.

1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 t

Figure 5: The optimal data collection paths of the worst case returned by the CFDC algorithm.

scale networks. In this section, we present the distributed version of our Fast Data Collection algorithm. We first analyze the generic principle for link scheduling and then we propose a Distributed Fast Data Collection (DFDC) algorithm to reduce the beacon overhead. Finally, we present the theoretical analysis of DFDC. 5.1. Design Philosophy. Such CFDC is a link-based method and is difficult to be performed in a distributed environment. This is because it needs the information of all potential links to construct VGN, which brings out large overheads. We aim to design a node-based method to solve MDCD problem, by which the node make a decision by itself. That means the node decides locally what its next hop is, and when the data is delivered. The main idea are as follows: (1) Keep the sink busy in receiving packets for as many active time slots as possible. Because the sink has duty cycle, we need to guarantee the nodes within one hop to the sink always waiting for sending packets. (2) Give higher priority to the node to send data that is nearer to the sink. The number of hops from node to the sink is represented as the distance to the sink by Dijkstra shortest path algorithm. Definition 5. The pressure index of node 𝑖 ∈ 𝑛, denoted by 𝑝(𝑖), is the minimum number of hops from node 𝑖 to the sink by Dijkstra shortest path algorithm, which represents the distance to the sink.

5.2. Algorithm Description. The formal description of DFDC is shown in Algorithm 2. Each source node keeps a buffer and its corresponding state, which is logical, either full or empty. The proposed DFDC algorithm does not require large buffers, because it is guaranteed that at any time the buffer will store not more than one packet. We initialize that all source nodes’ buffers are full and keep the sink’s buffer always empty for ease of explanation. We first sort all nodes based on the pressure index. For each time slot, packets then go through each node 𝑖 ∈ 𝑉 in the sorted order and are pushed from nodes with higher 𝑝(𝑖) to ones with lower 𝑝(𝑖) step by step, until all packets are delivered to the sink. When node 𝑖 decides its next hop, it will communicate with its neighbors and find the ones that satisfy two constraints: (1) the neighbors’ 𝑝(𝑖) is lower than the current node’s; (2) the neighbors’ buffers are empty. For the result of finding conditional neighbors, there are three cases: (1) There is only one neighbor meeting both of constraints and then this neighbor is selected as the next hop. (2) There is more than one neighbor satisfying the above two conditions and then we choose one of them at random as the next hop. (3) There are no conditional neighbors; then the node will wait for another time slot until it has conditional neighbors to send its packet. 5.3. A Walk-Through Example. In Figures 6(a) and 6(b), we show the results of link scheduling in one of the best cases and one of the worst cases. Both of the original networks shown in Figure 6 contain four source nodes. The solid lines represent potential links when the receivers are active. The numbers beside the links represent the time slots at which the links are scheduled to send packets, and the numbers inside the circles denote nodes’ IDs.


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Figure 6: Link schedule using DFDC algorithm: (a) One of the best cases. (b) One of the worst cases.

We run through an example shown in Figure 6(a) to explain the DFDC algorithm. We first compute the pressure index of each node and obtain that 𝑝(1) = 𝑝(2) = 2, 𝑝(3) = 𝑝(4) = 1, and 𝑝(5) = 0. In the first duty cycle, since only the sink (node 5)’s buffer is empty and 𝑝(5) is smaller than 𝑝(4), we schedule the link (4, 5) when node 5 is active. Thus, the sink receives a packet from node 4 in slot 3. In the second duty cycle, the buffers of node 3 and node 5 are empty and 𝑝(3) < 𝑝(1), 𝑝(5) < 𝑝(4); thus we schedule the links (1, 3) and (4, 5) in slots 6 and 7, respectively. Then the buffers of nodes 1 and 4 are empty. In the third duty cycle, the node 2’s buffer is full and one of its neighbors, node 4’s buffer, is empty. Thus we schedule the link (2, 4) in slot 9. For the same reason, we schedule the link (3, 5) in slot 11. Hence, the buffers of nodes 2 and 3 are empty. In the fourth duty cycle, only node 4’s buffer is full. Hence we schedule the link (4, 5) in slot 15. This process continues until all the packets are delivered to the sink, yielding a link schedule that requires 15 time slots. In Figure 6(b), we show an assignment in one of the worst cases when all the interfering links are eliminated, yielding a schedule length of 27 time slots. 5.4. Algorithm Analysis. In the following, we prove that the DFDC algorithm can obtain the optimal solution when no interfering links happen. Before giving the detailed proof, we first highlight the two main principles of the algorithm: (1) The sink is kept in receiving packets for as many time slots as possible. (2) A node’s buffer is not empty for two or more consecutive duty cycles as long as the buffers of one or more nodes with higher 𝑝(𝑖) are full. The first one is easy to prove by the scheduling rules of DFDC algorithm. We prove the second one in the following lemma. Lemma 6. At least one node with 𝑝(𝑖) = 1 has full buffer during two consecutive duty cycles if one or more nodes with 𝑝(𝑖) > 1 have full buffers. Proof. We prove it by induction on time slot 𝑡 = 𝑘𝑇𝜋 + 1. We focus on the first slot of each duty cycle. At 𝑡 = 1, the theorem is intuitively true because all buffers of the source nodes are full. Suppose the lemma holds for 𝑡 = 𝑘𝑇𝜋 + 1; that is, for slot 𝑡 = 𝑘𝑇𝜋 + 1 the buffers of all nodes with 𝑝(𝑖) = 1 are empty,

and for slots 𝑡 = (𝑘 − 1)𝑇𝜋 + 1 and 𝑡 = (𝑘 + 1)𝑇𝜋 + 1 there exists at least one node with 𝑝(𝑖) = 1 whose buffer is full. At 𝑡 = (𝑘+1)𝑇𝜋 +1, one node 𝑝(𝑖) = 1 whose buffer is full sends a packet to the sink, and its buffer becomes empty. There exist the following two cases: (1) all nodes with 𝑝(𝑖) = 1 whose buffers are empty and at least one node with 𝑝(𝑖) = 1 whose buffer is full at the following slot 𝑡 = (𝑘 + 2)𝑇𝜋 + 1. (2) another node with 𝑝(𝑖) = 1 whose buffer is full, which is noted as node 𝑗. The first case means that, for slot 𝑡 = 𝑘𝑇𝜋 + 1, there is only one node with 𝑝(𝑖) = 1, whose buffer is full. Since there is one or more nodes with 𝑝(𝑖) > 1 in full buffers, during the next duty cycle, the nodes with 𝑝(𝑖) = 1 will receive at least one packet. The second case means that, for slot 𝑡 = 𝑘𝑇𝜋 + 1, there are more than one node with 𝑝(𝑖) = 1, whose buffers are full. For slot 𝑡 = (𝑘 + 1)𝑇𝜋 + 1, one node with 𝑝(𝑖) = 1 whose buffer is full sends a packet to the sink. Since the sink only receives one packet during each duty cycle, there is another node with 𝑝(𝑖) = 1, whose buffer is full. Therefore, the lemma holds for 𝑡 = (𝑘 + 1)𝑇𝜋 + 1, and the proof follows. Theorem 7. DFDC algorithm can obtain the optimal data collection paths with the minimum delay, that is, 𝑆(𝑠𝑖𝑛𝑘)+(𝑁− 1)𝑇𝜋 . Proof. The principle of DFDC algorithm is to keep the nodes with 𝑝(𝑖) = 1 always waiting for sending a packet. From Lemma 6, we know that the sink keeps receiving a packet during two consecutive duty cycles. Therefore, the DFDC algorithm can obtain the optimal data collection paths with the minimum delay. Note that the lower bound of the best case illustrated in Figure 6 is 𝑆(sink) + (𝑁 − 1)𝑇𝜋 = 3 + 3 × 4 = 15, the same as the result of the proposed DFCD algorithm.

6. Impact of Interference So far, we consider the scheduling methods which assign the same channel to all the receivers. Since all communications on the same channel can not avoid the interfering links, the collision will happen if the SINR value at any receiver is not

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International Journal of Distributed Sensor Networks 1

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Figure 7: Delivery Ratio under different duty cycle.

7. Performance Evaluation greater than the predefined threshold, especially when two or more transmissions are launched at the same time slot. In this section, we combine the channel with time scheduling to eliminate the interfering links. As shown in Figure 6(b), links (4, 5) and (1, 3) at slot 11 as well as links (4, 5) and (2, 3) at slot 19 are interfering links. If the interfering links are scheduled at the same channel, it will result in serious data loss. We give the experimental results to show how serious it is when the density is on the rise. In the 100 m square field, varied number of nodes are deployed randomly and we evaluate the delivery ratio along with the increase of node density. From Figure 7, we observe that the delivery ratio deteriorates when all communications are scheduled on the same channel, especially in high network density. The main reason is that interfering links cause massive collisions. The easy way to avoid the interfering links is to assign extra time slots, however it results in the large delay of data collection. If all the interfering links are present, the total schedule length will extend to 35 slots for the case in Figure 6(b). Without compromising the time efficiency of data collection, we use multiple frequency channels to avoid the interfering links. We propose a simple Receiver-based Channel and Time Scheduling (RCTS) algorithm which assigns the time slots and channels based on DFDC algorithm. Since the data transmission depends on the time slot when the receiver wakes up, we first use DFDC algorithm to assign links on the same channel. Next, for receivers working at the same slot, it is checked whether any interfered receiver based on SINR thresholds exists and assigns the next available channel iteratively starting from the interfered receiver with the lowest 𝑝(𝑖). Note that we assume there are adequate channels available to be allocated to the interfered receivers.

In this section, we evaluate the impact of duty cycle, multiple channels for both the outdoor and indoor environment under the physical interference model. Simulation Setup. A sink is positioned in the center of deployment field, and each sensor node sends its packet to the sink over single or multiple hops. The working schedules of all the nodes are predefined by picking their one active slot randomly in a cycle period, which will be fixed through the data collection process. We investigate the network performance with different duty cycles for the proposed RCTS algorithm and the centralized MDCD algorithm which assigns extra time slots to eliminate interfering links (noted as MDCD-IL), compared with one of well-known SPR algorithms, that is, Dijkstra Algorithm [27]. Since the Dijkstra Algorithm aims to determine the path with minimum delay for end-to-end communications, collision will occur when it is extended to many-to-one data collection. Thus, retransmission is adopted as collision recovery strategy for the Dijkstra Algorithm in compared simulations. In order to evaluate the general case, we set the pathloss exponent 𝛼 to be 2 in the cases under the outdoor environment. The transmission power is fixed at the highest level, which is 31 in the TelosB sensor nodes with MSP430 MCU and CC2420 RFIC integrated on board. Since IEEE 802.15.4 ZigBee radios, used on Telosb and TmoteSky motes, are capable of operating on 16 different frequencies with nonoverlapping channels with a fixed bandwidth of 2 MHz, the maximum number of available channels is set to be 16. Based on the physical model, we find that the path-loss exponent 𝛼 can be set as 1.8 and 2.3 to simulate the real outdoor environment without or with obstacles, respectively, according to the real RSSI measurement experiment [28], as demonstrated in Figure 8.


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Figure 10: Link quality of data trace. (a) Link quality distribution. (b) CDF of link quality.

Outdoor Scenario Settings. Sensor nodes are randomly deployed in a fixed 100 m × 100 m square field by the Waxman random network topology generator [29]. The reason to choose the random deployment is that sensors are usually scattered randomly in the wild. The number of sensor nodes are varied between 20 and 60 to simulate different levels of density. Indoor Scenario Settings. We use the real data trace from the 54 sensors deployed in Intel Berkeley Research lab, shown in Figure 9, monitoring the building environment information,

such as temperature, light, humidity, and voltage values [30]. This data trace contains the aggregate connectivity data averaged over all time for 37 days, whose distribution and Cumulative Distribution Function (CDF) are shown in Figure 10. We establish the links with link quality above 0.5 in our indoor scenario simulations. Performance Metrics. We exploit the following metrics to evaluate the network performance. (1) Collection delay is measured as the time elapsed from packets being generated until all (𝑛 − 1) packets reach the sink, reported by the

12

(1) Results in Outdoor Scenario. Each simulation is repeated for 100 times, and we report the average values as statistical results. In order to evaluate the impact of duty cycles, the duty cycles are set to be 0.05, 0.1, 0.15, 0.2, and 0.25, respectively. 40 nodes are randomly deployed in 100 × 100 square field. In order to show the improvement of time efficiency for eliminating all interfering links, RCTS is compared with MDCD-IL, which exploits extra time slots to eliminate interfering links. We present the results comparing our RCTS algorithm with MDCD-IL and the Dijkstra Algorithm as well as the upper and lower bounds on the data collection delay. Figures 11 and 12 show the data collection delay and the number of transmissions under different duty cycles, respectively. From Figure 11, we can see that RCTS algorithm has the smallest delay than the compared Dijkstra Algorithm under different duty cycles while retaining low energy cost, since the number of transmissions remains small and stable over different duty cycles shown in Figure 12. As Figure 11 depicts, the delay of RCTS algorithm outperforms that of Dijkstra Algorithm by 32% to 11% when duty cycle increases from 0.05 to 0.25. Since the available time for nodes to receive packets is notably reduced, lower duty cycle has high probability to incur collisions. The channel and time scheduling of RCTS algorithm can largely reduce collisions, and the advantage of RCTS algorithm is more obvious in lower duty cycle. Compared with RCTS and MDCD-IL, we observe that the use of multiple channels can efficiently improve the data collection delay when the interfering links are avoided by multiple channels. It is also observed that the collection delay of RCTS algorithm almost overlaps its lower bound. Thus, it validates that RCTS algorithm can obtain the optimal solution. From Figure 12, we find that the duty cycle has slight effect on the number of transmission in RCTS algorithm, which outperforms the Dijkstra Algorithm up to 67%. Figure 13 shows that the number of transmissions is positively correlated with the total number of nodes in the network, while little is related with the duty cycle. Thus, it is a good way to save energy by reducing the duty cycle, which will not cause increased transmission times. The reason is that RCTS algorithm is scheduled well without collision to retransmission. We can also infer that the delay in low-dutycycle WSNs is mainly caused by sleep latency instead of retransmission. We use 𝑈 and 𝐷 to represent the upbound of collection delay and the time distance from the upbound to the optimal latency, respectively. We define the ratio of 𝐷 and 𝑈 to evaluate the effectiveness of RCTS algorithm. From Figure 14, we also find that the effectiveness of RCTS algorithm has

1200 Number of time slots

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Figure 11: Delay versus duty cycles.

700 600 Number of transmissions

total number of time slots to complete data collection. (2) Energy consumption is measured by the total number of transmissions for all packets delivered to the sink. Since the power level is fixed, the power consumption of each transmission is equal. (3) Delivery ratio is measured as the ratio of successful transmission times compared to the total number of transmissions for all packets delivered to the sink.


500 400 300 200 100 0.05

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Figure 12: Number of transmissions versus duty cycles.

little impact on the number of nodes and duty cycle and stays between the range of 30% and 50%. From Figure 15, we can observe that the delivery ratio raises rapidly, when the duty cycle decreases to 0.2. The concurrent transmissions on the same channel are alleviated, since the probability of nodes active simultaneously is reduced. When the duty cycle is extremely low (less than 0.1), we observe that the single channel is enough to deliver packets with few interfering links as the deployment gets sparser (less than 0.4). This happens because at low densities the interference is less and less concurrent transmissions take place. We also find that the delivery ratio can be over 0.9 when the duty cycle and density are both low, which can satisfy the requirements in most wireless networks.


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Figure 13: Number of transmissions versus duty cycles.

0.55 0.5 Effectiveness

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.05

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20 30 40

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Figure 14: The effectiveness of RCTS algorithm under different cases.

The number of occupied channels distribution

Figure 15: Delivery ratio under different duty cycle. 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

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4 5 6 7 8 9 10 11 12 13 14 The number of occupied channels 50 nodes 60 nodes

Figure 16: Occupied channels distribution.

Figure 16 shows the number of assigned channels by Algorithm 3 with different node densities when the duty cycle is set to be 0.25. With the increase of node density, the channel resources are needed more to eliminate the interfering links. As we can see from Figure 16, when the number of nodes augments to 60, we need at most 13 channels to avoid interfering links. We find that channel scheduling can suffice to eliminate most interference for moderate size of about 60 nodes in the outdoor environment, even when the duty cycle is not extremely low (0.25). (2) Results in the Indoors Scenario. We also use the exponential path-loss model for signal propagation with the path-loss

exponent 𝛼 varying between 3 and 4, which is typical for the indoor environment. Based on the physical interference model, the transmission power is also set to be the highest level and SINR threshold is set as 𝛽 = −3 dB. The above parameters setting is the same as [24]. From Figure 7, we see that the delivery ratio is relatively high when all nodes communicate on the same channel in the indoor environment. Since the signal strength rapidly degrades with the increase of communication distance, the interference among neighbors will decrease accordingly.

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Input: The original network topology. Output: The optimal data collection paths and channel assignment with the minimum delay. (1) Call the DFDC algorithm to obtain the optimal link schedule on the same channel. (2) The receivers check the SINR condition among the concurrently transmitting senders (3) if There exist receivers with SINR ≤ 𝛽 then (4) The receiver with lower 𝑝(𝑖) assigns one available channel.

Mean number of assigned max channels

Algorithm 3: Receiver-based Channel and Time Scheduling algorithm.

We next proposed a novel Receiver-based Channel and Time Scheduling (RCTS) algorithm to obtain the optimal solution. Based on real trace, extensive simulations were conducted and the results showed that the proposed distributed RCTS algorithm is significantly more efficient than the link schedule on the same channel and achieved the lower bound. We also evaluated the multichannel scheduling method and found that multichannel scheduling can suffice to eliminate most of the interference in both the indoor and outdoor environment for moderate size networks. In the future, we shall extend this work into the case of unreliable links.

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1 0.05

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𝛼=3 𝛼 = 3.5 𝛼=4

Figure 17: Number of channels versus duty cycles.

Thus, the appearance of interfering links will reduce in large amount. Although the delivery ratio of CFDC algorithm is in good performance, in order to ensure the stable communication, we can also use the multichannel scheduling to improve the performance of CFDC algorithm further. Figure 17 shows the mean number of assigned maximum channels in different path-loss exponent (𝛼) by Algorithm 3. The performance improvement of RCTS is more obvious when the duty cycle is highly compared with CFDC algorithm.

8. Conclusion In this paper, we studied the problem: How fast can raw data be collected from all source nodes to a sink in low-dutycycle WSNs with general topology? Both lower and upper tight bounds were given for this problem. We presented both centralized and Distributed Fast Data Collection algorithms to address this problem, both of which are able to find an optimal solution in polynomial time when no interfering links happen. When interfering links happen, multichannel scheduling is introduced to eliminate the interfering links.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment This research was supported by major Program of National Natural Science Foundation of China (no. 61190114). Preliminary results of this work have been published on the International Conference of Globecom’12.

References [1] S. Gandham, Y. Zhang, and Q. Huang, “Distributed minimal time convergecast scheduling in wireless sensor networks,” in Proceedings of the 26th IEEE International Conference on Distributed Computing Systems (ICDCS ’06), p. 50, IEEE, July 2006. [2] X. Mao, S. Tang, X.-Y. Li, and M. Gu, “Mens: multi-user emergency navigation system using wireless sensor networks,” Ad-Hoc and Sensor Wireless Networks, vol. 12, no. 1-2, pp. 23–53, 2011. [3] Y. Revah and M. Segal, “Improved algorithms for data-gathering time in sensor networks II: ring, tree, and grid topologies,” International Journal of Distributed Sensor Networks, vol. 5, no. 5, pp. 463–479, 2009. [4] Y. Gu and T. He, “Dynamic switching-based data forwarding for low-duty-cycle wireless sensor networks,” IEEE Transactions on Mobile Computing, vol. 10, no. 12, pp. 1741–1754, 2011. [5] Z. Yuan, Y. Zhang, and C. J. Xue, “Sleep-aware mode assignment in wireless embedded systems,” Journal of Parallel and Distributed Computing, vol. 71, no. 7, pp. 1002–1010, 2011.

International Journal of Distributed Sensor Networks [6] S. Xiong, J. Li, M. Li, J. Wang, and Y. Liu, “Multiple task scheduling for low-duty-cycled wireless sensor networks,” in Proceedings of the IEEE INFOCOM, pp. 1323–1331, IEEE, April 2011. [7] T. He, S. Krishnamurthy, L. Luo et al., “VigilNet: an integrated sensor network system for energy-efficient surveillance,” ACM Transactions on Sensor Networks, vol. 2, no. 1, pp. 1–38, 2006. [8] G. Tolle, J. Polastre, R. Szewczyk et al., “A macroscope in the redwoods,” in Proceedings of the 3rd International Conference on Embedded Networked Sensor Systems (SenSys ’05), pp. 51–63, San Diego, Calif, USA, November 2005. [9] S. Cui, R. Madan, A. Goldsmith, and S. Lall, “Energy-delay tradeoffs for data collection in TDMA-based sensor networks,” in Proceedings of the IEEE International Conference on Communications (ICC ’05), pp. 3278–3284, May 2005. [10] S. Guo, Y. Gu, B. Jiang, and T. He, “Opportunistic flooding in low-duty-cycle wireless sensor networks with unreliable links,” in Proceedings of the 15th Annual International Conference on Mobile Computing and Networking (MobiCom ’09), pp. 133–144, Association for Computing Machinery, September 2009. [11] L. Mo, Y. He, Y. Liu et al., “Canopy closure estimates with GreenOrbs: sustainable sensing in the forest,” in Proceedings of the 7th ACM Conference on Embedded Networked Sensor Systems (SenSys ’09), pp. 99–112, ACM, Berkeley, Calif, USA, November 2009. [12] C. Sharp, S. Schaffert, A. Woo et al., “Design and implementation of a sensor network system for vehicle tracking and autonomous interception,” in Proceeedings of the 2nd European Workshop on Wireless Sensor Networks, pp. 93–107, 2005. [13] N. Xu, S. Rangwala, K. K. Chintalapudi et al., “A wireless sensor network for structural monitoring,” in Proceedings of the 2nd International Conference on Embedded Networked Sensor Systems (SenSys ’04), pp. 13–24, ACM, November 2004. [14] S. Ji, A. S. Uluagac, R. Beyah, J. S. He, and Y. Li, “Cell-based snapshot and continuous data collection in wireless sensor networks,” ACM Transactions on Sensor Networks, vol. 9, no. 4, article 47, 2013. [15] S. Ji and Z. Cai, “Distributed data collection in large-scale asynchronous wireless sensor networks under the generalized physical interference model,” IEEE/ACM Transactions on Networking, vol. 21, no. 4, pp. 1270–1283, 2013. [16] H. Lee and A. Keshavarzian, “Towards energy-optimal and reliable data collection via collision-free scheduling in wireless sensor networks,” in Proceedings of the 27th IEEE Communications Society Conference on Computer Communications (INFOCOM ’08), pp. 116–120, IEEE, April 2008. [17] W. Zhao and X. Tang, “Scheduling sensor data collection with dynamic traffic patterns,” IEEE Transactions on Parallel and Distributed Systems, vol. 24, no. 4, pp. 789–802, 2013. [18] X. Jiao, W. Lou, J. Ma, J. Cao, X. Wang, and X. Zhou, “Minimum latency broadcast scheduling in duty-cycled multihop wireless networks,” IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 1, pp. 110–117, 2012. [19] C.-H. Lee and D. Y. Eun, “Smart sleep: sleep more to reduce delay in duty-cycled wireless sensor networks,” in Proceedings of the IEEE INFOCOM, pp. 611–615, IEEE, Shanghai, China, April 2011. [20] Y. Wang, M. C. Vuran, and S. Goddard, “Cross-layer analysis of the end-to-end delay distribution in wireless sensor networks,” in Proceedings of the 30th IEEE Real-Time Systems Symposium (RTSS ’09), pp. 138–147, IEEE, December 2009.

15 [21] J. Polastre, J. Hill, and D. Culler, “Versatile low power media access for wireless sensor networks,” in Proceedings of the 2nd International Conference on Embedded Networked Sensor Systems (SenSys ’04), pp. 95–107, ACM, 2004. [22] O. Incel, A. Ghosh, and B. Krishnamachari, “Scheduling algorithms for tree-based data collection in wireless sensor networks,” in Theoretical Aspects of Distributed Computing in Sensor Networks, Monographs in Theoretical Computer Science, pp. 407–445, 2011. [23] L. Su, C. Liu, H. Song, and G. Cao, “Routing in intermittently connected sensor networks,” in Proceedings of the 16th IEEE International Conference on Network Protocols (ICNP ’08), pp. 278–287, Orlando, Fla, USA, October 2008. [24] O. Durmaz Incel, A. Ghosh, B. Krishnamachari, and K. Chintalapudi, “Fast data collection in tree-based wireless sensor networks,” IEEE Transactions on Mobile Computing, vol. 11, no. 1, pp. 86–99, 2012. ¨ D. Incei and B. Krishnamachari, “Enhancing the data collec[25] O. tion rate of tree-based aggregation in wireless sensor networks,” in Proceedings of the 5th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON ’08), pp. 569–577, IEEE, San Francisco, Calif, USA, June 2008. [26] S. Luo, X. Mao, Y. Sun, Y. Ji, and S. Tang, “Delay minimum data collection in the low-duty-cycle wireless sensor networks,” in Proceedings of the IEEE Global Communications Conference (GLOBECOM ’12), pp. 232–237, IEEE, December 2012. [27] J. Kleinberg, Algorithm Design, Pearson Education India, 2006. [28] C. Bo, D. Ren, S. Tang et al., “Locating sensors in the forest: a case study in GreenOrbs,” in Proceedings of the IEEE Conference on Computer Communications (INFOCOM ’12), pp. 1026–1034, IEEE, March 2012. [29] H. Tangmunarunkit, R. Govindan, S. Jamin, S. Shenker, and W. Willinger, “Network topology generators: degree-based vs. structural,” ACM SIGCOMM Computer Communication Review, vol. 32, no. 4, pp. 147–159, 2002. [30] Intel lab data, http://select.cs.cmu.edu/data/labapp3/index.html.


Research Article A Key Management Method Based on Dynamic Clustering for Sensor Networks Ying Zhang,1 Bingxin Zheng,1 Pengfei Ji,1 and Jinde Cao2,3 1

College of Information Engineering, Shanghai Maritime University, Shanghai 201306, China Department of Mathematics, Southeast University, Nanjing 210096, China 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2

Correspondence should be addressed to Ying Zhang; [email protected] Received 6 August 2014; Revised 23 November 2014; Accepted 24 November 2014 Academic Editor: Qin Xin Copyright © 2015 Ying Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Many cluster-based routing protocols had been proposed which had rarely considered the network security issues so far. The existing key management methods have imperfection when they combine with cluster-based routing protocols. Normally cluster-based key management method has better performance than the distributed key management method, but most of the layer-cluster key management methods do not consider the problem of key updating and being captured for cluster heads. Considering the nodes’ capture probability, particle swarm optimization algorithm was used to optimize the clustering of sensor networks. A dynamic key management method was proposed to achieve key updating regularly and provided a security strategy for sensor networks to solve the problem of being captured for cluster heads. The simulation illustrates that the proposed key management method can achieve better security performance.

1. Introduction Sensor network security issues are becoming the focus of the industry recently [1, 2]. Because of limited storage space and computing power, lacking of a priori knowledge for later nodes deployed, and inability to guarantee the physical security in deployment region for sensor network, the traditional network security method is not suitable for sensor networks. Lightweight key management method, which aims to secure communication, becomes the most important and basic aspect of security research for wireless sensor networks [3–5]. According to the differences of topological structure, sensor networks can be divided into flat networks and layercluster networks. Layer-cluster wireless sensor networks have the advantage of high-energy efficiency. Recently, many scholars had proposed cluster-based routing protocols [6–8], which only considered the energy factor but paid no attention to security issues. Key management is an important way to protect the safety of clustering. Nodes are often deployed in the enemy area for monitoring, so they could be captured by the enemies. In particular in layer-cluster networks, cluster

heads play an important role in the network. Once being captured, they will reveal more keys and information. It will threaten the safety of the whole network. This paper presents a kind of capture probability model of network nodes. The model considers the capture probability as network clustering, and the nodes which hold lower capture probability will be the cluster heads first. On the basis of this model, a dynamic key management method (DKMM) was proposed. The key management method can achieve dynamic key update and solve the problem of system security defense when the cluster heads are captured. The system simulation indicates that the proposed method based on clustering routing protocol has the features of lower storage consumption and strong ability of resistance to capture.

2. Related Work There are many kinds of key management methods proposed [9–12]. In the field of distributed key management method, Eschenauer and Gligor (E-G) [13, 14] first presented a random key predistribution method. In this method, each node randomly selects 𝑚 keys from the key pools before

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deployment. If the adjacent nodes at least have one same key, they can directly establish a session key. Chan et al. [15] also proposed a method based on the E-G method which is called 𝑞-composite key management method. In this scheme, the adjacent nodes can establish communication if they at least have 𝑞 same keys. The connection rate of these two methods is lower, and the cost of keys storage is higher. In the field of cluster-based key management methods, Zhu et al. [16] devised a method called LEAP. This method not only can support the processing inside the network, but also is a kind of key management method with fine ability of resistance to capture. In order to meet the different security requirements, LEAP supports the establishment of four types of keys. They are individual key, group key, clustered key, and pair key, respectively. It also provides the network node authentication based on one-way key chain. But its mechanisms of key update, revocation, nodes canceling, and nodes adding are not perfect, and clusters will dynamically be changed in practical applications. Jolly et al. [17] proposed a low-energy key management protocol that supports revocation for the attacked nodes. Since each node can only communicate with the cluster heads or base station, each sensor node only needs to store two symmetric keys in the key predistribution process. This method adopts the multilayer network architecture, which greatly reduces the energy consumption caused by the key management. However, this method has poorer network expansibility. The key update is not supported, so it increases the chance of being cracked by the enemy when using one kind of key in a long period of time. Du et al. [18] devised a key predistribution for heterogeneous sensor networks. This method takes the high-energy nodes as cluster heads, which can store a lot of keys and be equipped with tamper-proof hardware devices. The ordinary nodes will only be preloaded with a small number of keys. Compared with the existing random key predistribution methods, this method enhances the anticapture capabilities. But its storage overhead of keys is larger and the cluster heads are unable to be changed dynamically. Once being captured, the clusters will not work properly.

3. Network Model The proposed DKMM method adopts the model of layercluster wireless sensor network, which is shown in Figure 1. We make some assumptions as follows.

BS

CH

CH

CH

CH

Figure 1: Schematic diagram of sensor network model (BS is base station; CH is cluster head).

(6) Data fusion technology can be used to reduce the amount of data transferred. (7) The base station (BS) is trusted and has sufficient energy, and it can communicate with all the nodes in monitoring area. Once a node is captured, it will be detected immediately by BS. This paper also presents a model of capture probability of the network nodes shown as follows: 0 { { { { 𝑑 − 𝑑0 2 𝐺 = {𝐺max × ( ) { 𝑑𝐻 − 𝑑0 { { {𝐺max

0 ≤ 𝑑 < 𝑑0 𝑑0 ≤ 𝑑 < 𝑑𝐻

(1)

𝑑𝐻 ≤ 𝑑.

𝑑𝐻 is a threshold value of monitoring area; 𝑑0 is the safety value of monitoring area. 𝐺max is the maximum probability of being captured. The values of 𝑑𝐻, 𝑑0 , and 𝐺max are determined by the specific monitoring area; 𝑑 is the distance from the node to the base station. Assuming the base station is the safest position, when the distance to base station is less than 𝑑0 , owing to being far away from the enemy activity area, the probability of enemy appearance is almost zero. Therefore, the capture probability of the nodes in this area is the lowest. When the distance is more than 𝑑0 , the chance of the enemy appearing grows, and the capture probability of the nodes also increases. When the distance to the base station exceeds 𝑑𝐻, the nodes can monitor the main activity area of the enemies, and the capture probability of the nodes is the maximum.

(1) The node has a unique ID and is randomly deployed in the monitoring area.

4. Clustering Based on PSO

(2) After being deployed, all the nodes are stationary and energy-constrained.

Particle swarm optimization (PSO) is an algorithm based on iterative optimization [19]. PSO initializes particle swarm as a group of random solutions and then searches optimal solution in the solution space by following the current optimum particles. In the process of iteration, the particles are updated according to the two extreme values which are called the individual extreme value and the global extreme value. Thereinto, the individual extreme value is the optimal solution which is founded by each particle, and the global extreme value is the optimal solution which is founded by

(3) All nodes are with the same capacity, equal status. Once being captured, they would reveal the keys they had stored. (4) Nodes can adjust the transmission power according to the distances. (5) All nodes are aware of their position and perform the data collection tasks periodically.


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Table 1: The implication of parameters in PSO. Variable V 𝑥 𝑡 𝑐1 , 𝑐2 𝜑1 , 𝜑2 𝑝id 𝑝gd 𝑝𝑏𝑒𝑠𝑡 𝑔𝑏𝑒𝑠𝑡 w

Variable name The particle velocity The particle position The rounds Learning factors Random number between 0 and 1 Individual best position Global best position Fitness value of 𝑝id Fitness value of 𝑝gd Inertia weight

Initialize position and velocity of each particle

Calculate the fitness of each particle to get pid and pgd Update and x

Limitation on and x Map the particle’s location to cluster nodes’ location Calculate the fitness of each particle

global particle swarm. Particle position and velocity can be updated by

cost < pbest ?

Vid (𝑡) = 𝑤 × Vid (𝑡 − 1) + 𝑐1 × 𝜑1 (𝑝id − 𝑥id (𝑡 − 1)) + 𝑐2 × 𝜑2 (𝑝gd − 𝑥id (𝑡 − 1)) 𝑥id (𝑡) = 𝑥id (𝑡 − 1) + Vid (𝑡) ,

cos 𝑡 = 𝛽1 × 𝑓1 + 𝛽2 × 𝑓2 + 𝛽3 × 𝑓3 ,

pbest < gbest ?

(3)

where 𝑑 (𝑛𝑖 , CH𝑝,𝑘 ) } { 𝑓1 = max { ∑ 󵄨󵄨 󵄨 } 𝑘=1,2,...,𝐾 󵄨󵄨𝐶𝑝,𝑘 󵄨󵄨󵄨 󵄨 󵄨 } {∀𝑛𝑖 ∈𝐶𝑝,𝑘 𝑓2 =

∑𝑁 𝑖=1 𝐸 (𝑛𝑖 ) 𝐾 ∑𝑘=1 𝐸 (CH𝑝,𝑘 )

(4)

𝐾

𝐴 × 𝐺𝑘 . 𝐾 𝑘=1

𝑓3 = ∑

𝑓1 is the evaluation factor of cluster’s compactness, which is the maximum average Euclidean distance of nodes to their

Update pid and pbest

No

(2)

where the parameters in the formula are defined as in Table 1. The clustering is based on centralized control strategy and realized by base station with unlimited energy. Polling mechanism is applied to protocol implementation, and each round includes two stages: the setting-up of clustering and steady state of clustering. After the nodes deployment, they will report the base station about the information of their position and energy. Because the base station knows the initial energy of all the nodes, it can estimate the energy consumption of the nodes by each round clustering information and gets node energy information after each round. The node’s location is fixed. Therefore, nodes do not need to send the information of position and energy to the base station subsequently. According to application requirements and the circumstance of network operation, they will resend the information of position and energy after a long interval of cycle. The probability of nodes being captured will be considered in the process of clustering. The fitness function is set as follows:

Yes

Yes Update pgd and gbest

No No Iterations == Max? Yes End

Figure 2: Flowchart of PSO clustering.

associated cluster heads. 𝑑(𝑛𝑖 , CH𝑝,𝑘 ) is the distance between node 𝑛𝑖 and cluster head. |𝐶𝑝,𝑘 | is the number of the nodes which belong to cluster 𝐶𝑘 in particle 𝑃. 𝑓2 is the evaluation factor of cluster head’s energy, which is the proportion of total initial energy of all nodes 𝑛𝑖 (𝑖 = 1, 2, 3, . . . , 𝑁) in the network and the total current energy of the cluster heads in the current round. 𝑓3 is the evaluation factor of probability of node being captured, and 𝐴 is a constant. 𝑓1 , 𝑓2 , 𝑓3 are in the same magnitude. 𝐺𝑘 is the probability of each cluster head being captured. 𝛽1 , 𝛽2 , 𝛽3 are the weight coefficients of each evaluation factor, and 𝛽1 + 𝛽2 + 𝛽3 = 1. Figure 2 is the flowchart of PSO clustering, and the specific steps of PSO clustering are as follows. (a) Initialize 𝑆 particles, and each particle contains 𝐾 cluster heads randomly selected from the eligible candidates of cluster head. (b) Node 𝑛𝑖 is assigned to the nearest cluster head, and the fitness function of each particle 𝑃 (𝑃 = 1, 2, . . . , 𝑆) can be calculated according to formula (3)∼(4). (c) Find the individual and global best solution for each particle. (d) Update the particle’s velocity and position.

4


Random initialization for velocity V𝑖 (0) and position 𝑥𝑖 (0) of particles, 𝑖 ∈ [1, . . . , 𝑆] Map the particle’s location to the nodes’ location according to the distance between the position of each cluster heads and the position of the nodes for each particle 𝑖 ∈ [1, . . . , 𝑆]: Calculate fitness 𝑓(𝑥𝑖 (0)) of particle using formula (3) 𝑝id ← 𝑥𝑖 (0) 𝑝𝑏𝑒𝑠𝑡𝑖 ← 𝑓(𝑥𝑖 (0)) end for 𝑔𝑏𝑒𝑠𝑡 ← min{𝑓(𝑥1 (0)), 𝑓(𝑥2 (0)), . . . , 𝑓(𝑥𝑆 (0))} for iterations 𝑡 ∈ [1, . . . , 𝑀𝑎𝑥𝐼𝑡𝑒𝑟]: for each particle 𝑖 ∈ [1, . . . , 𝑆]: Update V𝑖 (𝑡) and 𝑥𝑖 (𝑡) Map the particle location to the location of the eligible candidates of cluster head Calculate fitness of particles if 𝑓(𝑥𝑖 (𝑡)) < 𝑝𝑏𝑒𝑠𝑡𝑖 then 𝑝𝑏𝑒𝑠𝑡𝑖 ← 𝑓(𝑥𝑖 (𝑡)), 𝑝id ← 𝑥𝑖 (𝑡) if 𝑝𝑏𝑒𝑠𝑡𝑖 < 𝑔𝑏𝑒𝑠𝑡 then 𝑔𝑏𝑒𝑠𝑡 ← 𝑝𝑏𝑒𝑠𝑡, 𝑝gd ← 𝑝id end for end for Pseudocode 1

(e) Map the particle’s location to cluster heads’ location. (f) Repeat the above steps until achieving the maximum number of iterations. It will get into the steady phase after building cluster, and then the cluster heads will complete the tasks as data collection and fusion. After a period of time, in order to ensure the safety of the system, it should update the clustering to repeat the process above. The pseudocode of this PSO clustering is shown in Pseudocode 1.

5. The Key Management Method for Sensor Networks Based on Dynamic Clustering 5.1. Key Establishment. Suppose 𝑁 nodes are safe for a period of time after deployment, and they could not be captured by the enemy. Each node stores the initial key 𝐾0 and one-way hash function 𝐻. Nodes are randomly deployed in monitored area, and each node 𝑛𝑖 sent their own position 𝐿 𝑖 , energy 𝐸𝑖 , and id𝑖 information to the base station by key 𝐾0 : 𝑛𝑖 󳨀→ BS : 𝐸𝐾0 (id1 ‖ ⋅ ⋅ ⋅ id𝑖 ‖𝐿 𝑖 ‖ ⋅ ⋅ ⋅ 𝐿 𝑖 ‖𝐸1 ‖ ⋅ ⋅ ⋅ 𝐸𝑖 ) 𝑖 = 1, 2, . . . , 𝑁.

(5)

After receiving the information from each node, base station considers the probability of nodes being captured and makes the clustering by the method above. Then it will broadcast id𝐶𝑗 and location information 𝐿 𝑗 of cluster heads after selecting 𝐾 cluster heads: BS 󳨀→ ∗ : 𝐸𝐾0 (id1 ‖ ⋅ ⋅ ⋅ id𝐶𝑗 ‖𝐿 1 ‖ ⋅ ⋅ ⋅ 𝐿 𝑗 ) 𝑗 = 1, 2, . . . , 𝐾.

(6)

Each node, which is not the cluster head, decrypts the received information and then chooses the nearest cluster head to join in according to the location information. They also calculate the session key and a cluster key which are used for communicating with cluster head and base station, respectively. After establishing three kinds of session keys, the initial key 𝐾0 will be deleted. Assume node 𝑢 wants to establish a session key with the cluster head 𝐶𝑗 : 𝐾𝑢𝐶𝑗 = 𝐻 (𝐾0 ‖id𝐶𝑗 ‖id𝑢 )

(7)

𝐾𝑢𝐵 = 𝐻 (𝐾0 ‖id𝑢 )

(8)

𝐾𝐶𝑗 = 𝐻 (𝐾0 ‖id𝐶𝑗 ) .

(9)

After receiving the information, cluster head will calculate the session key for communicating with base station: 𝐾𝐶𝑗 𝐵 = 𝐻 (𝐾0 ‖id𝐶𝑗 ) .

(10)

Then 𝐾0 will be deleted. The cluster head will not store cluster keys. After calculating according to formula (7)∼(10), the base station will obtain the session keys used for communication with all the nodes and the cluster keys of all the clusters, and then it will tell the cluster heads about all the information of its corresponding nodes assigned to them by 𝐾𝐶𝑗 𝐵 . After getting the information of every node in the cluster, each cluster head will calculate the session key for communicating with nodes in the cluster according to formula (7); then 𝐾0 will be deleted. So the establishment of the entire network key is completed. 5.2. Key Update. In the stable transmission phase, if the key was used for a long time, it will have the risk to be cracked by


5

the enemies; so the key needs to be updated. Key update will be launched by the cluster head, and each cluster head needs to calculate the formula 𝑃(𝑥) = (𝑥 − 𝐾1 )(𝑥 − 𝐾2 ) ⋅ ⋅ ⋅ (𝑥 − 𝐾𝑎 ), where 𝐾𝑎 is the session key between the cluster head and the nodes. The cluster head broadcasts 𝑔(𝑥) = 𝑃(𝑥) + 𝑆 in the cluster. When the nodes put the session keys into 𝑔(𝑥), they can get key update parameter 𝑆, and then they can get the new session keys: 󸀠 = 𝐻 (𝐾𝑢𝐶𝑗 ‖𝑆) . 𝐾𝑢𝐶 𝑗

(11)

5.3. Key Establishment in Clustering by Polling. Suppose time 𝑇 of establishing the cluster is less than time 𝑇min which is the time for nodes being captured. In the clustering phase, the base station elects the cluster heads with PSO algorithm and then sends the new cluster head information and key parameter 𝑆󸀠 to the cluster members of last round, respectively, by the cluster key of the last round. Because the cluster head does not have the cluster key, the base station will transfer the new cluster header information and 𝑆󸀠 to the cluster head of last round by 𝐾𝐶𝑗 𝐵 with unicast. Each noncluster head node decrypts the received information and then chooses the nearest cluster head to join in according to the location information. They also calculate the cluster key and the session key which are used for communicating with cluster head, while updating the session key used for communicating with the base station. After establishing the session key, 𝑆󸀠 will be deleted. Assume node 𝑢 wants to establish a session key with the cluster head 𝐶𝑗 : 󸀠 𝐾𝑢𝐶 𝑗

󸀠

= 𝐻 (𝑆 ‖id𝐶𝑗 ‖id𝑢 )

(12)

After the nodes in cluster received the message, the session key for communicating with old cluster head will be deleted, and a new session key for communicating with the new cluster head will be created: 󸀠 = 𝐻 (𝑆󸀠󸀠 ‖id𝐶𝑗 ‖id𝑢 ) . 𝐾𝑢𝐶 𝑗

(16)

The base station passes the information about the nodes in cluster to the new cluster head by 𝐾𝑢𝐵 , and then the new cluster head would calculate and get the session key within the cluster according to formula (12).

6. Performance Evaluation We would evaluate the performance of our proposed key management method from the aspects of connectivity, storage overhead, communication overhead, and security by system simulation. In the simulation experiment, 100 nodes are deployed in the area of 100 × 100 m2 . We adopt the wireless communication model proposed in literature [10]. If we transmit 𝑘-bit message and the transfer distance is 𝑑, the energy consumption of transmitter can be expressed as 𝐸𝑇𝑥 (𝑘, 𝑑) = 𝐸𝑇𝑥 elec (𝑘) + 𝐸𝑇𝑥 amp (𝑘, 𝑑) 𝑘𝐸 + 𝑘𝜀𝑓𝑠 𝑑2 = { elec 𝑘𝐸elec + 𝑘𝜀𝑚𝑝 𝑑4

𝑑 < 𝑑0 𝑑 ≥ 𝑑0 ,

(17)

where 𝜀𝑓𝑠 , 𝜀𝑚𝑝 are the energy consumption coefficient of power amplification circuit. Consider 𝑑0 = √𝜀𝑓𝑠 /𝜀𝑚𝑝 . When receiving 𝑘-bit data, the energy can be consumed as

󸀠 𝐾𝑢𝐵 = 𝐻 (𝐾𝑢𝐵 ‖𝑆󸀠 )

(13)

𝐸𝑅𝑥 (𝑘) = 𝐸𝑅𝑥 elec (𝑘) = 𝑘𝐸elec ,

𝐾𝐶󸀠 𝑗 = 𝐻 (𝑆󸀠 ‖id𝐶𝑗 ) .

(14)

where 𝐸elec = 50 nJ/bit, 𝜀𝑓𝑠 = 10 pJ/bit/m2 , and 𝜀𝑚𝑝 = 0.0013 pJ/bit/m4 .

The base station tells cluster head the information of its corresponding nodes assigned in its cluster by 𝐾𝑢𝐵 . After getting the information of all the nodes in cluster, each cluster head can calculate the session key used for communicating with the nodes according to formula (12). Then the establishment of the session key by polling is completed, and 𝑆󸀠 will be deleted. 5.4. Key Revocation. If the base station found a noncluster head node captured, it would tell cluster head to delete the keys related to the captured node by the key used to communicate with the cluster head. If the captured node was cluster head, the base station would find the node whose energy is greater than the average energy of the nodes in the cluster and its probability of being caught is the smallest, and let it be the new cluster head. As the cluster head does not store the cluster key in the key establishment process, the base station would transfer the information 𝑚 about the captured cluster head and the information of the new cluster head, as well as the key update parameter 𝑆󸀠󸀠 to the cluster members with multicast: BS 󳨀→ CH : 𝐸𝐾𝐶 (𝑚‖id𝐶𝑗 ‖𝑆󸀠󸀠 ) . 𝑗

(15)

(18)

6.1. Comparison of the Capture Probability of Cluster Heads. DKMM was proposed based on the type of layer-cluster network structure. Cluster heads are the key nodes in the network which contain a large number of keys. The keys will be revealed when the cluster heads are captured, and rebuilding the cluster needs a great deal of energy consumption. Therefore, nodes with lower capture probability should be selected as cluster heads in the clustering process. The clustering method with particle swarm algorithm considering the probability of nodes being captured (PSO-G) will be compared with the clustering method with particle swarm algorithm without considering the probability of nodes being captured (PSO-C). The simulation parameters are shown in Table 2, and the curves of capture probability model of the nodes are shown in Figure 3. In PSO-C method, nodes whose energy is greater than the average energy of the nodes can be a candidate of the cluster heads. PSO-G2 is a kind of clustering method based on PSO-C which considers the capture probability of the nodes. PSO-G1 is also a kind of clustering method based on PSOC which not only considers the capture probability of the

International Journal of Distributed Sensor Networks Table 2: Simulation parameters definition. Parameter values

The number of particles 𝑆 Learning factors 𝑐1 , 𝑐2 Inertia weight w

Capture probability

Evaluation factor 𝛽1 Evaluation factor 𝛽2 Evaluation factor 𝛽3 The number of cluster heads 𝐾 Constant 𝐴 The maximum capture probability 𝐺max The threshold value of the monitoring area 𝑑𝐻 The safety value of monitoring area 𝑑0 The size of the message data Initial energy of node 𝐸0 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

20 2 Decrease from 0.9 to 0.4 linearly 0.25 0.15 0.6 5 100 0.1

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

20

40

60

80

100 120 140 160 180 200 220 Rounds

PSO-G1 PSO-C

Figure 4: The average capture probability of cluster heads in each round.

65 m 5m 4000 bits 0.1 J

The average probability of cluster head captured in each round

Parameter

The average probability of cluster head captured in each round

6

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

2

4

6

8

10

12

14

16

18

20

22

Rounds × 10

0

10 20 30 40 50 60 70 80 90 The distance between nodes and base stations (m)

100

Figure 3: The curves of capture probability model of the nodes.

nodes, but also makes the nodes whose energies are more than half of the average energy become the candidates of cluster head. Normally the time duration from the starting to the time when the first dead node appears is defined as the survival time of the network. Figure 6 shows that the first dead node in PSO-G1 appears on the 203rd round, PSO-G2’s first death node appears on the 216th round, and the PSOC’s first death node appears on the 218th round. The average capture probability of cluster heads in PSO-G1 and PSOC is shown in Figure 4. In order to analyze better, capture probability value of a cluster head could be taken in every ten rounds, and it is shown in Figure 5. As shown in Figures 4 and 5, in front of the 203rd round, we can find that the average capture probability of cluster heads in PSO-G1 is lower than PSO-C’s in the process of nearly 90% rounds, and the average capture probability of cluster heads in PSO-G2 is lower than PSO-C’s in the process of more than 60% rounds. It indicates that the cluster heads will be safer if they are selected by considering the capture probability of the nodes. In PSO-G1 method, we can choose the nodes with lower capture probability as cluster heads, since it reduces the influence of energy for electing the cluster

PSO-G1 PSO-G2 PSO-C

Figure 5: The average capture probability plot of cluster heads of PSO-G1 and PSO-G2 compared with PSO-C (get the average capture probability of cluster heads in every 10 rounds).

heads. There exist more candidate cluster heads for electing. However in PSO-G2 method, only the nodes whose energies are greater than the average energy can be the candidate cluster heads. This shrinks the range of selecting for cluster heads. Thus we can draw a conclusion that the security of cluster heads which are selected by PSO-G1 is better than the thing in PSO-G2. In Figure 6, the network lifetime and energy balance of PSO-G1 are all worse than PSO-G2 and PSO-C. Its first dead node appears only ahead of 15 rounds compared to the others. PSO-G1 has greatly improved the security of the network by sacrificing a little network survival time, so it can be acceptable from the view of the whole process. 6.2. Storage Overhead. The storage space of sensor network nodes is limited, so it is necessary to reduce consumption of node’s key storing under the condition of safety. In DKMM, each node needs to store one cluster key and two session keys which are used for communicating with base station and cluster heads. At the same time, cluster heads need to store keys which are used for communicating with nodes in its

100 90 80 70 60 50 40 30 20 10 0

7 5000 4500 4000

0

50

100

150

200 250 Rounds

300

350

400

Total number of keys

Number of nodes alive


3500 3000 2500 2000 1500 1000

PSO-G1 PSO-G2 PSO-C

500 0

20

30

40

Figure 6: The curves of the network lifetime for three methods. E-G m = 50 PKAS

6.3. Energy Consumption. In the strategy of DKMM, the nodes only need to receive the updated parameters of secret keys transmitted from base station to get their session secret keys by the one-way hash function. So its energy consumption is lower. Because the energy consumption of calculation is smaller compared with the energy consumption of communication, we only consider the total communication energy consumption of the nodes while setting up the keys. Based on this circumstance, the comparison of energy consumption for DKMM, E-G, and CECC [22] is shown in Figure 8. It is clearly observed that the communication energy consumption for setting up keys of DKMM is the lowest one.

7. Security Analysis of the Scheme The cluster head is the key node in the layer-cluster network structure, which contains a large number of keys. It will reveal a lot of information once being physically captured by the enemies. Rebuilding clusters will also consume a lot of energy. So in order to guarantee the security and effectiveness of the network, we should select the nodes with lower capture probability as the cluster heads when building clusters.

80

90

100

MUQAMI DKMM

Figure 7: The comparison of key storage consumption regarding DKMM with E-G, PKAS, and MUQAMI.

Communication energy consumption (J)

cluster and base station. Assuming that there are 𝑁 ordinary nodes and 𝑀 number of cluster heads in the network, so it needs to store 4 × 𝑁 + 𝑀 keys in DKMM. In the E-G method, each node needs to store a large number of keys in advance. In order to guarantee the connectivity rate reaching 0.5, it is necessary to store 75 keys for each node when the size of the key pool reaches 10000. In 𝑞-composite method based on E-G, it will store more keys to keep the same connectivity rate compared with E-G. If each node selects 𝑚 keys from the key pool randomly, the number of keys of storage in E-G will be equal to 𝑚 × (𝑁 + 𝑀). In the PKAS [20] and MUQAMI [21] schemes, the storage space requirement will be equal to (4𝑀 + 6𝑁), 𝑀(3 + 2(log2 𝑁)2 ) + 𝑁(5 + log2 𝑁), respectively. Figure 7 shows the comparison of key storage consumption regarding DKMM with the methods of E-G, PKAS, and MUQAMI.

50 60 70 Total number of nodes

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 20

30

40

50 60 70 Total number of nodes

80

90

100

E-G m = 50 CECC DKMM

Figure 8: Comparison of communication energy consumption for setting up keys regarding DKMM with E-G and CECC.

Clustering by PSO with consideration of the capture probability factors of the nodes can ensure the nodes with lower capture probability to become the cluster heads. As illustrated in Figures 4 and 5, we can find that the cluster heads of PSO-G1 and PSO-G2 have lower capture probabilities than PSO-C in the most part of the iteration process. After the first dead node appearing, the average capture probabilities of cluster heads of PSO-G1 and PSO-G2 start to be higher than PSO-C. The reason is that the nodes with lower capture probability will be the cluster heads with many times, and their energy consumption will be greater, so most

8 of the dead nodes will be this kind of secure nodes. With energy reducing of these nodes, energy becomes the major factor of electing the head clusters. That is why the cluster heads’ average capture probabilities of PSO-G1 and PSO-G2 are higher than PSO-C after the first dead node appearing, and it goes up quickly. Fortunately, after the first dead node appearing, it is out of the network’s normal life time; so, in the normal life cycle of the network, clustering with PSO-G1 and PSO-G2 is more secure than PSO-C. As Figure 5 illustrates, the security of cluster heads elected with PSO-G1 is higher than PSO-G2 from an overall perspective. The energy threshold of PSO-G1 for selection of candidate cluster heads is lower than PSO-G2; it reduces the influence of energy for cluster heads’ election in PSOG1, so it leads to emerging more candidate cluster heads in PSO-G1. This brings the chance to select more nodes with lower capture probability to become the cluster heads. But, in PSO-G2, only the nodes whose energies are greater than the average energy of the nodes could be the candidate cluster heads, this shrinks the scope of cluster heads’ selection. So the security of cluster heads elected by PSO-G2 is lower than PSO-G1. Although the network survival time and the balance of energy consumption of PSO-G1 are not as good as PSOG2 and PSO-C and the round of its first dead node appears ahead of 13 rounds and 15 rounds, respectively, compared to PSO-G2 and PSO-C, the network survival time and the balance of energy consumption of PSO-G1 are still better than LEACH protocol. PSO-G1 greatly improves the security of the network by sacrificing a little balance of energy consumption of the network. This kind of sacrifice is acceptable and valuable indeed. In the scheme of DKMM, the keys of each node communicating with base station or cluster heads are different from each other. When a node is captured, it will not reveal the communication keys between other nodes. Cluster heads, as the key nodes in the network, will be selected from the more secure nodes as clustering. When a cluster head is captured, the base station will select a new cluster head according to the residual energy and capture probability of the nodes within cluster, and it will inform the nodes within cluster by cluster keys. Therefore the antidestroying ability of DKMM is better. When the node is captured, it will not influence the secure communication between other nodes. For the widely used dynamic key management scheme of EBS matrix method [23], when multiple nodes are captured within cluster at the same time, the captured nodes could obtain the whole EBS key set of the cluster by sharing their respective keys. This will make the whole cluster lose the security and seriously threaten the safety of whole network. But in the scheme of DKMM, even though multiple nodes are captured within a cluster at the same time, this will not reveal the keys between other nodes. So DKMM has higher security. For LEAP key management method, it also has fine antidestroying ability, but each node needs to store cluster key which is shared by the whole network. Once a node is captured, the update of keys will consume a lot of energies.

8. Conclusion This paper puts forward a probability model of nodes being captured. We not only consider the energy in the case of

International Journal of Distributed Sensor Networks dynamic clustering, but also take the capture probability of nodes into consideration. A kind key management method named DKMM was proposed, and this key management strategy makes more secure nodes most likely have the chance to become the cluster heads. Although considering the safety of the cluster heads while clustering, the cluster heads may still be possible to be captured. So DKMM method considers the possibility that the cluster heads may be captured. By the realization of the cluster head’s reselection mechanisms and the dynamical updating mechanisms of the keys, we can minimize the risk of information leaking due to the cluster heads’ capture.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This work was supported by the National Nature Science Foundation of China (no. 61273068), Nature Science Foundation of Shanghai (no. 12ZR1412600), and Scientific Research Innovation Project of Shanghai Education Committee (no. 13YZ084).

References [1] K. H. Kalita and A. Kar, “Wireless sensor network security analysis,” International Journal of Next-Generation Networks (IJNGN), vol. 1, no. 1, pp. 1–10, 2009. [2] X. He, M. Niedermeier, and H. De Meer, “Dynamic key management in wireless sensor networks: a survey,” Journal of Network and Computer Applications, vol. 36, no. 2, pp. 611–622, 2013. [3] G. D. Anand, G. H. Chandrakanth, and M. N. Giriprasad, “Security threats & issues in wireless sensor networks,” International Journal of Engineering Research and Applications, vol. 2, no. 1, pp. 911–916, 2012. [4] A. S. Reegan and E. Baburaj, “Key management schemes in wireless sensor networks: a survey,” in Proceedings of the IEEE International Conference on Circuit, Power and Computing Technologies (ICCPCT ’13), pp. 813–820, March 2013. [5] J. C. Lee, V. C. M. Leung, K. H. Wong, J. Cao, and H. C. B. Chan, “Key management issues in wireless sensor networks: current proposals and future developments,” IEEE Wireless Communications, vol. 14, no. 5, pp. 76–84, 2007. [6] A. A. Abbasi and M. Younis, “A survey on clustering algorithms for wireless sensor networks,” Computer Communications, vol. 30, no. 14-15, pp. 2826–2841, 2007. [7] R. Azarderakhsh, A. Reyhani-Masoleh, and Z. E. Abid, “A key management scheme for cluster based wireless sensor networks,” in Proceedings of the 5th International Conference on Embedded and Ubiquitous Computing (EUC ’08), vol. 2, pp. 222– 227, December 2008. [8] M. F. Younis, K. Ghumman, and M. Eltoweissy, “Locationaware combinatorial key management scheme for clustered sensor networks,” IEEE Transactions on Parallel and Distributed Systems, vol. 17, no. 8, pp. 865–882, 2006.

International Journal of Distributed Sensor Networks [9] W. B. Heinzelman, A. P. Chandrakasan, and H. Balakrishnan, “An application-specific protocol architecture for wireless microsensor networks,” IEEE Transactions on Wireless Communications, vol. 1, no. 4, pp. 660–670, 2002. [10] O. Younis and S. Fahmy, “HEED: a hybrid, energy-efficient, distributed clustering approach for ad hoc sensor networks,” IEEE Transactions on Mobile Computing, vol. 3, no. 4, pp. 366– 379, 2004. [11] J. Zhang and V. Varadharajan, “Wireless sensor network key management survey and taxonomy,” Journal of Network and Computer Applications, vol. 33, no. 2, pp. 63–75, 2010. [12] X. Chen, K. Makki, K. Yen, and N. Pissinou, “Sensor network security: a survey,” IEEE Communications Surveys and Tutorials, vol. 11, no. 2, pp. 52–73, 2009. [13] Y. Xiao, V. K. Rayi, B. Sun, X. Du, F. Hu, and M. Galloway, “A survey of key management schemes in wireless sensor networks,” Computer Communications, vol. 30, no. 11-12, pp. 2314–2341, 2007. [14] L. Eschenauer and V. D. Gligor, “A key-management scheme for distributed sensor networks,” in Proceedings of the 9th ACM Conference on Computer and Communications Security (CCS ’02), pp. 41–47, New York, NY, USA, November 2002. [15] H. Chan, A. Perrig, and D. Song, “Random key predistribution schemes for sensor networks,” in Proceedings of the IEEE Symposium on Security And Privacy, pp. 197–213, Washington, DC, USA, May 2003. [16] S. Zhu, S. Setia, and S. Jajodia, “LEAP: efficient security mechanisms for large-scale distributed sensor networks,” in Proceedings of the 10th ACM Conference on Computer and Communications Security (CCS ’03), pp. 62–72, Washington, DC, USA, October 2003. [17] G. Jolly, M. C. Kuscu, P. Kokate, and M. Younis, “A lowenergy key management protocol for wireless sensor networks,” in Proceedings of the 8th IEEE International Symposium on Computers and Communication (ISCC ’03), pp. 335–340, July 2003. [18] X. Du, Y. Xiao, M. Guizani, and H.-H. Chen, “An effective key management scheme for heterogeneous sensor networks,” Ad Hoc Networks, vol. 5, no. 1, pp. 24–34, 2007. [19] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948, Perth, Australia, December 1995. [20] S. M. Rahman and K. El-Khatib, “Private key agreement and secure communication for heterogeneous sensor networks,” Journal of Parallel and Distributed Computing, vol. 70, no. 8, pp. 858–870, 2010. [21] M. K. R. R. Syed, H. Lee, S. Lee, and Y. K. Lee, “MUQAMI+: a scalable and locally distributed key management scheme for clustered sensor networks,” Annales des TelecommunicationsAnnals of Telecommunications, vol. 65, no. 1-2, pp. 101–116, 2010. [22] J. Mu, A Novel Cluster-based Isomorphic Key Management Scheme for Wireless Sensor Networks, Xidian University, 2011. [23] M. Eltoweissy, M. H. Heydari, L. Morales, and I. H. Sudborough, “Combinatorial optimization of group key management,” Journal of Network and Systems Management, vol. 12, no. 1, pp. 33– 50, 2004.

9


Research Article An Improved PSO Algorithm for Distributed Localization in Wireless Sensor Networks Dan Li1 and Xian bin Wen2 1

School of Computer Science and Technology, Tianjin University, Tianjin 300072, China Key Laboratory of Computer Vision and System, Ministry of Education, Tianjin University of Technology, Tianjin 300384, China

2

Correspondence should be addressed to Dan Li; [email protected] Received 17 October 2014; Accepted 8 January 2015 Academic Editor: Chang Wu Yu Copyright © 2015 D. Li and X. B. Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Accurate and quick localization of randomly deployed nodes is required by many applications in wireless sensor networks and always formulated as a multidimensional optimization problem. Particle swarm optimization (PSO) is feasible for the localization problem because of its quick convergence and moderate demand for computing resources. This paper proposes a distributed twophase PSO algorithm to solve the flip ambiguity problem, and improve the efficiency and precision. In this work, the initial search space is defined by bounding box method and a refinement phase is put forward to correct the error due to flip ambiguity. Moreover, the unknown nodes which only have two references or three near-collinear references are tried to be localized in our research. Simulation results indicate that the proposed distributed localization algorithm is superior to the previous algorithms.

1. Introduction Networks of distributed autonomous nodes that can sense their environment cooperatively are called wireless sensor networks (WSNs) which have great potential for diverse applications in scenarios such as hazardous environment exploration, military target tracking and surveillance, health monitoring, and home automation. In most of these applications, location information of sensor nodes which supports many other network services is critically important. So far, a wide variety of localization methods have been proposed according to diverse application requirements which can be roughly divided into two types: range-based and range-free. The range-free algorithms mainly use the connectivity between sensor nodes to estimate the locations of unknown nodes without any ranging techniques. The rangebased algorithms use some ranging techniques [1] such as Time of Arrival (TOA), Angle of Arrival (AOA), Time Difference of Arrival (TDOA), and Radio Signal Strength (RSS) to measure the distances or angles between unknown nodes and their neighbouring anchor nodes and formulate the localization problem as a multidimensional optimization problem after the ranging information is measured. At the same time, it provides more accurate localization results

than range-free algorithms. Compared with the classical optimization methods, bioinspired optimization algorithms such as simulated annealing algorithm (SAA), genetic algorithm (GA), and particle swarm optimization (PSO) solve the localization problem though population-based techniques. They are easier to implement, have faster convergence, and require less computing resources. Hereinto, PSO is an evolutionary computation method based on swarm intelligence without using evolution operators such as crossover and mutation. It is an efficient global optimization algorithm with the advantages of simple operation and avoiding local minima. A lot of literatures apply bioinspired algorithms to implement nodes localization in WSN. A simulated annealing based localization algorithm in wireless sensor network is proposed in [2] which uses lots of anchor nodes to localize all unknown nodes. The accuracy of this approach is not as good as the PSO algorithm in [3] according to the performance evaluation. A two-phase localization algorithm for WSN presented in [4, 5] is centralized and not feasible for large-scale sensor networks. The localization algorithm which uses a combination of SAA and GA proposed in [4] solves the flip ambiguity problem. An iterative PSO-based algorithm proposed in [6–8] localizes the unknown nodes with three or more neighboring anchors.

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A new objective function is presented in [9] to evaluate the fitness of particles based on the probabilistic distribution of ranging error. Literature [10] extends the node localization problem to three-dimensional space. In this paper, we propose a distributed localization algorithm based on PSO which can also extend to large-scale sensor networks. The main contributions of this paper have the following two aspects. (1) It reduces the initial search space by bounding box method to speed up the convergence. (2) It proposes a distributed two-phase PSO algorithm to solve the flip ambiguity problem and localize more target nodes. The rest of this paper is organized as follows. Section 2 overviews the theory and application of PSO algorithm. Section 3 introduces our improved PSO algorithm for distributed localization in WSNs. Simulation and analysis are presented in Section 4. Section 5 concludes this paper and look forward to the future work in the end.

2. Particle Swarm Optimization Algorithm 2.1. The Theory of PSO Algorithm. Particle swarm optimization (PSO) is a popular bioinspired stochastic global search algorithm proposed by Kennedy and Eberhart [11] that models social behavior of a flock of birds. In this algorithm, all individuals in a population are seen as particles in a multidimensional solution space. First of all, randomly initialize a group of particles in a population. Each of them is a feasible solution and its fitness value is determined by its position in the search space. Each particle moves in the solution space towards the randomly weighted average of the historical personal best position and the historical global best position and finds the current global solution. Assume that, in an 𝑛-dimension objective search space, the population consists of 𝑠 particles. The position and velocity of the particle 𝑖 are 𝑋𝑖𝑑 and 𝜐𝑖𝑑 , respectively, 1 ≤ 𝑖 ≤ 𝑠, 1 ≤ 𝑑 ≤ 𝑛. Fitness of a particle is evaluated by an objective function. A particle having smaller fitness value is closer to the global solution than the other particle that is far away. 𝑝𝑏𝑒𝑠𝑡𝑖𝑑 is the historical personal best position of particle 𝑖 where it had the smallest fitness, and 𝑔𝑏𝑒𝑠𝑡𝑑 is the smallest of all 𝑝𝑏𝑒𝑠𝑡𝑖𝑑 , 1 ≤ 𝑖 ≤ 𝑠. At each iteration 𝑘, the velocity 𝜐𝑖𝑑 and position 𝑋𝑖𝑑 of each particle are updated using

between the sensor nodes with known locations called beacons and the other sensor nodes which need to be localized, named target nodes. We consider the whole sensor network and sensor nodes as a two-dimensional coordinate system and coordinate points, respectively. Let (𝑥, 𝑦) be the coordinate of a target node 𝑇, (𝑥𝑖 , 𝑦𝑖 ) the position of its 𝑖th neighboring beacon 𝐵𝑖 , and 𝑑̂𝑖 the measured distance between 𝑇 and 𝐵𝑖 (𝑖 = 1, 2, . . . , 𝑃). Then, the problem of localization is regarded as an optimization problem whose purpose is to find the optimal solution (𝑥, 𝑦) to minimize the value of the objective function defined in 𝑓 (𝑥, 𝑦) =

2 1 𝑃 √ 2 2 ∑ ( (𝑥 − 𝑥𝑖 ) + (𝑦 − 𝑦𝑖 ) − 𝑑̂𝑖 ) . 𝑃 𝑖=1

(2)

The target nodes get settled and serve as beacons in the next iteration and the localization process is repeated until either all target nodes get settled or no more nodes can be localized. The main problem in the process is the flip ambiguity due to near-collinear beacons which have been solved very well in the next section. Moreover, the calculation time has been decreased by reducing the initial search space on the premise of guarantee accuracy.

3. An Improved PSO Algorithm In this section, a range-based distributed localization algorithm for wireless sensor networks is described. Differences between classical PSO algorithm and our proposed algorithm are reflected in the following four points.

Here, 𝜔 is the inertia weight which denotes the inertia of the particle, 𝑐1 and 𝑐2 are acceleration constants, and rand1 and rand2 are random numbers uniformly distributed in [0, 1]. In order to understand this process more clearly, pseudocode is shown in Algorithm 1.

3.1. Define the Initial Search Space by Bounding Box Method. The classical PSO algorithm employs a set of feasible solutions within the search space, called a swarm of particles with random initial locations. In our proposed algorithm, we reduced the initial search space by using a bounding box method. To facilitate understanding, an example is described in Figure 1. Node 𝑛 is an unknown node with three beacons or settled nodes 𝑛1 , 𝑛2 , and 𝑛3 in its transmission range. Instead of circles with radius 𝑟, we use squares with side length 2𝑟 to restrict the initial search space of node 𝑛 in order to avoid floating point operations. The centers of these squares are the nodes 𝑛1 , 𝑛2 , and 𝑛3 , respectively. Therefore, the overlapping portion of these squares (the shaded portion in Figure 1) describes the initial search space of node 𝑛. Suppose the coordinate of node 𝑛𝑖 is (𝑥𝑖 , 𝑦𝑖 ); then the shaded portion can be expressed as ([max(𝑥𝑖 − 𝑟), min(𝑥𝑖 + 𝑟)], [max(𝑦𝑖 − 𝑟), min(𝑦𝑖 + 𝑟)]). In our proposed algorithm, every target node has its unique search space determined by the references (beacons or settled nodes) in its transmission range. Comparing with the classical PSO algorithm using the whole solution space as the initial search space for all of the target nodes, the proposed algorithm reduces the computational time and the possibility of flip ambiguity.

2.2. The Application of PSO in WSN Localization. PSO is a kind of optimization algorithm which belongs to rangebased methods. That means it needs to measure the distance

3.2. Anticipation of Nearly Collinear References. An example of flip ambiguity phenomenon is showed in Figure 2. Node 𝐴 is located by using references 𝐵, 𝐶, 𝐷, and 𝐸 which are almost

𝜐𝑖𝑑 (𝑘 + 1) = 𝜔 × 𝜐𝑖𝑑 (𝑘) + 𝑐1 × rand1 × (𝑝𝑏𝑒𝑠𝑡𝑖𝑑 − 𝑋𝑖𝑑 ) + 𝑐2 × rand2 × (𝑔𝑏𝑒𝑠𝑡𝑑 − 𝑋𝑖𝑑 )

(1)

𝑋𝑖𝑑 (𝑘 + 1) = 𝑋𝑖𝑑 (𝑘) + 𝜐𝑖𝑑 (𝑘 + 1) .


3

PSO algorithm for minimize the objective function 𝑓(⋅) (1) Initialize 𝜔, 𝑐1 , 𝑐2 , 𝑋max , 𝑋min , 𝜐max , 𝜐min (2) Initialize the maximum number of iteration 𝑘max (3) Initialize the target fitness 𝐹 (4) Initialize gbest: 𝑓(𝑔𝑏𝑒𝑠𝑡) as close to ∞ as possible (5) for each particle 𝑖 do (6) for each dimension 𝑑 do (7) Initialize 𝑋𝑖𝑑 randomly in [𝑋min , 𝑋max ] (8) Initialize 𝜐𝑖𝑑 randomly in [𝜐min , 𝜐max ] (9) 𝑝𝑏𝑒𝑠𝑡𝑖𝑑 = 𝑋𝑖𝑑 (10) end for (11) if 𝑓(𝑋𝑖 ) < 𝑓(𝑔𝑏𝑒𝑠𝑡) then (12) for each dimension d do (13) 𝑔𝑏𝑒𝑠𝑡𝑑 = 𝑋𝑖𝑑 (14) end for (15) end if (16) end for (17) Initialize 𝑘 = 1 (18) while (𝑘 ≤ 𝑘max ) AND (𝑓(𝑔𝑏𝑒𝑠𝑡) > 𝐹) do (19) for each particle 𝑖 do (20) for each dimension 𝑑 do (21) if 𝜐𝑖𝑑 (𝑘) < 𝜐min (22) 𝜐𝑖𝑑 (𝑘) = 𝜐min (23) elseif 𝜐𝑖𝑑 (𝑘) > 𝜐max (24) 𝜐𝑖𝑑 (𝑘) = 𝜐max (25) end if (26) if 𝑋𝑖𝑑 (𝑘) < 𝑋min (27) 𝑋𝑖𝑑 (𝑘) = 𝑋min (28) elseif 𝑋𝑖𝑑 (𝑘) > 𝑋max (29) 𝑋𝑖𝑑 (𝑘) = 𝑋max (30) end if (31) end for (32) if 𝑓(𝑋𝑖 ) < 𝑓(𝑝𝑏𝑒𝑠𝑡𝑖 ) then (33) for each dimension 𝑑 do (34) 𝑝𝑏𝑒𝑠𝑡𝑖𝑑 = 𝑋𝑖𝑑 (35) end for (36) end if (37) if 𝑓(𝑋𝑖 ) < 𝑓(𝑔𝑏𝑒𝑠𝑡) then (38) for each dimension 𝑑 do (39) 𝑔𝑏𝑒𝑠𝑡𝑑 = 𝑋𝑖𝑑 (40) end for (41) end if (42) end for (43) 𝑘=𝑘+1 (44) end while Algorithm 1: Pseudocode of PSO algorithm.

collinear and it is possible to estimate 𝐴 as its flipped location 𝐴󸀠 without making obvious change in the objective function. The more the references are, the less the probability of flip ambiguity occurrence is. Therefore, when the number of references is small, we should have anticipation about whether these references are near-collinear by using the following rules.

a predefined threshold, these references are judged to be near-collinear. (ii) Calculate the distance from any one reference to the straight line which is determined by any two of the rest references. When the value is less than a predefined threshold, these references are judged to be near-collinear.

(i) If there are only three references, any two of them should not be too close to each other. When the distance between any two references is less than

The above rules can be used in combination to prevent the occurrence of flip ambiguity. When the references of a target node are judged to be near-collinear, the target node will not be localized in this iteration. In the following iteration,

4

International Journal of Distributed Sensor Networks 3.4. Localize the Target Nodes That Only Have Two References or Three Near-Collinear References. After those target nodes which have at least three non-near-collinear references are all localized, the remaining ones that only have two references or three near-collinear references can be localized by using the refinement phase at last.

n1

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3.5. Algorithm Steps. The main purpose of WSN node localization is to estimate the positions of 𝑁 target nodes as much as possible by using 𝑀 beacons with known locations in a distributed range-based way. The process of our proposed approach is as follows.

n2

(i) 𝑁 target nodes and 𝑀 beacons are randomly deployed in a two-dimensional sensor field. The transmission radii of target nodes and beacons are both 𝑟. Beacons possess location awareness and transmit their coordinates frequently. The target nodes that get settled at the end of iteration serve as beacons during the next iteration.

Figure 1: Sketch map of bounding box method.

H

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(ii) The target node that falls within the transmission range of three or more non-near-collinear references is considered as a localizable node.

A C

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(iii) The measured distance from a localizable node to each of its neighboring references is simulated as the actual distance plus a Gaussian additive white noise which can be expressed as 𝑑̂𝑖 = 𝑑𝑖 + 𝑛𝑖 . Here, 𝑑𝑖 = √(𝑥 − 𝑥𝑖 )2 + (𝑦 − 𝑦𝑖 )2 is the actual distance and (𝑥, 𝑦) and (𝑥𝑖 , 𝑦𝑖 ) are the locations of the target node and the 𝑖th references, respectively. The measurement noise 𝑛𝑖 has a random value uniformly distributed in the range 𝑑𝑖 (1 ± 𝑃𝑛 /100). 𝑃𝑛 is the percentage noise that affects the result of localization.

A󳰀 R

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Figure 2: Flip ambiguity phenomenon.

the target node may have more references and will be localized until its new references are judged to be non-nearcollinear. 3.3. A Refinement Phase to Correct the Error due to Flip Ambiguity. The refinement phase is to relocalize those nodes which still have flip ambiguity problem after the anticipation phase. To inspect whether the flip ambiguity happened, we analyze the relationship between the settled node and its nonneighbor nodes. As seen in Figure 2, if node 𝐴 is localized in the wrong location 𝐴󸀠 , it will be discovered because 𝐴󸀠 lies within the transmission range of nonneighbor nodes 𝐹 and 𝐺. In the refinement phase, for those localized nodes falls into the wrong neighborhood, an extra term (3) is added to the objective function (2) in order to introduce extra cost for a new estimate. Here, 𝛿(⋅) is unit step function; 𝑄 is the number of nonneighbor references whose transmission range the estimated position falls within. Consider 𝑄

2

∑ 𝛿 (𝑟 − 𝑑̂𝑗 ) (𝑟 − 𝑑̂𝑗 ) .

𝑗=1

(3)

(iv) Each localizable node runs the improved PSO algorithm to localize itself by finding the coordinates (𝑥, 𝑦) that minimize the value of the objective function (2). (v) In the refinement phase, some of the nodes localized in the previous step should be relocalized by adding an extra term (3) to the objective function (2) owing to their estimated position falls into the wrong neighborhood. (vi) Above steps are repeated until either all target nodes are settled or no more nodes can be localized. (vii) After those target nodes which have at least three nonnear-collinear references are all localized, the remaining ones that only have two references or three nearcollinear references can be localized by using the refinement phase at last. (viii) The total localization error 𝐸𝑙 is computed as the mean of squares of distances between the actual node locations (𝑥𝑖 , 𝑦𝑖 ) and the computed locations (𝑥̂𝑖 , 𝑦̂𝑖 ) (𝑖 = 1, 2, . . . , 𝑁𝐿 ) determined by the improved


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𝐸𝑙 =

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((𝑥𝑖 − 𝑥̂𝑖 ) + (𝑦𝑖 − 𝑦̂𝑖 ) )

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Figure 4: Result without flip ambiguity phenomenon.

PSO algorithm, as in (4). 𝑁𝐿 is the number of the localized nodes. Consider 𝑁 ∑𝑖=1𝐿

0

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4. Simulation and Analysis 4.1. Compared with the Classical PSO Algorithm. Simulation was carried out in MATLAB environment. 𝑁 = 40 target nodes and 𝑀 = 8 beacons are randomly deployed in a sensor field of 100 × 100 square units. Each node has a transmission radius of 𝑟 = 30 units. The parameters of the proposed node localization algorithm are set as follows: (i) population = 30, iterations = 150; (iii) inertia weight 𝜔 is decreased linearly from 𝜔max = 0.9 in the first iteration to 𝜔min = 0.4 in the last iteration, as described in (5). Here, 𝑘 is the number of iterations and 𝑘max is the maximum number of iterations; 𝜔max − 𝜔min × 𝑘; 𝑘max

0.4 0.3 0.2 0.1 0

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Figure 5: Localization error 𝐸𝑙 with the number of settled nodes 𝑁𝐿 .

(ii) acceleration constants 𝑐1 = 𝑐2 = 2.0;

𝜔 = 𝜔max −

0.5

(5)

(iv) limits on particle positions: 𝑋max = 100 and 𝑋min = 0. In order to show the advantage of our algorithm more intuitively, we made some contrast figures to analyze each step independently as follows. Firstly, we use methods described in Sections 3.2 and 3.3 to avoid the flip ambiguity phenomenon. An example in a trial shows the different localization results between the experiments with and without methods 3.2 and 3.3. As shown in Figure 3 (the red circle represents the location of beacon node, the blue circle represents the location of target node, the blue asterisk represents the location estimated by the PSO algorithm, and the blue straight line represents the localization error, that is, the distance between

the actual node location and the estimated location), flip ambiguity phenomenon leads to large localization errors. To avoid such problems, we use methods described in Sections 3.2 and 3.3 to reposition. As a result, large localization errors have been corrected in Figure 4. Figure 5 directly shows the difference from a purely numerical perspective. The blue curve and red curve which represent the variation tendency of the localization error defined as 𝐸𝑙 in (4) with the number of settled nodes 𝑁𝐿 gradually increase in a trial with and without methods 3.2 and 3.3, respectively. Obviously, the localization error 𝐸𝑙 produced by the first 26 settled nodes is reasonable and almost the same in both curves. However, the localization of the 27th node causes flip ambiguity which leads to the huge localization error in point 𝐵. Easy to see, the localization error 𝐸𝑙 on the red curve suddenly changes from 0.14 to 98.39. But, on the blue curve, we avoid the huge and unreasonable error with methods 3.2 and 3.3 and receive an acceptable localization error 0.20 in point 𝐴 instead of 𝐵.


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Figure 6: Computing time for 𝑃𝑛 = 2 and 𝑃𝑛 = 5.

Figure 7: Localizable nodes percentage for 𝑃𝑛 = 2 and 𝑃𝑛 = 5.

Secondly, we reduced the initial search space by using a bounding box method which is described in 3.1. 30 trial experiments are conducted for 𝑃𝑛 = 2 and 𝑃𝑛 = 5, respectively. Average of computing time for all 30 runs with and without method 3.1 is computed and analyzed comparatively in Figure 6. From this chart we can see that experiments with method 3.1 reduce the computing time for both 𝑃𝑛 = 2 and 𝑃𝑛 = 5. Finally, in our algorithm, we localize the target nodes that only have two references or three near-collinear references by using the refinement phase at last which is described in 3.4. 30 trial experiments are conducted for 𝑃𝑛 = 2 and 𝑃𝑛 = 5, respectively. The percentage of localizable nodes for all 30 runs with and without method 3.4 is computed and the mean value is analyzed comparatively in Figure 7. As we see in this chart, experiments with method 3.4 localize more target nodes for both 𝑃𝑛 = 2 and 𝑃𝑛 = 5.

The parameters of our proposed node localization algorithm are set as follows:

4.2. Compared with the HPSO Algorithm. The HPSO method [8] divides the whole swarm into subswarms and the particle with the best fitness in the local swarm is termed as 𝑙𝑏𝑒𝑠𝑡𝑖𝑑 . At each iteration 𝑘, the velocity 𝜐𝑖𝑑 and position 𝑋𝑖𝑑 of each particle are updated using 𝜐𝑖𝑑 (𝑘 + 1) = 𝜔 × 𝜐𝑖𝑑 (𝑘) + 𝑐1 × rand1 × (𝑝𝑏𝑒𝑠𝑡𝑖𝑑 − 𝑋𝑖𝑑 ) + 𝑐2 × rand2 × (𝑔𝑏𝑒𝑠𝑡𝑑 − 𝑋𝑖𝑑 ) + 𝑐3 × rand3 × (𝑙𝑏𝑒𝑠𝑡𝑖𝑑 − 𝑋𝑖𝑑 )

(6)

𝑋𝑖𝑑 (𝑘 + 1) = 𝑋𝑖𝑑 (𝑘) + 𝜐𝑖𝑑 (𝑘 + 1) . The parameters of HPSO node localization algorithm are set as follows: (i) population = 50, iterations = 150; (ii) acceleration constants 𝑐1 = 𝑐2 = 𝑐3 = 1.494; (iii) inertia weight 𝜔 = 0.7;

(i) population = 50, iterations = 150; (ii) acceleration constants 𝑐1 = 𝑐2 = 2.0; (iii) inertia weight 𝜔 is decreased linearly from 𝜔max = 0.9 in the first iteration to 𝜔min = 0.4 in the last iteration, as described in (5). In order to compare the three different algorithms, ten trial experiments for each algorithm are conducted in the MATLAB environment. 𝑁 = 100 target nodes are randomly deployed in a sensor field of 100 × 100 square units. Each node has a transmission radius of 𝑟 = 25 units and the percentage noise 𝑃𝑛 = 2. Average of localization error and computing time for 𝑀 = 20, 30, 40 beacons of the three different algorithms are shown in Figures 8 and 9, respectively. Easy to see, the mean localization error reduces with the increasing number of beacons 𝑀 whatever the method is. Having more beacons is advantageous because it gives more references for the target nodes to localize themselves. However, our method is more precise than the other two methods and saves more computing time. Average of localization error and computing time for 𝑃𝑛 = 1, 2, . . . , 5 are computed and analyzed comparatively in Figures 10 and 11, respectively. As shown in Figures 10 and 11, the effect of 𝑃𝑛 , percentage noise in distance measurement, can be clearly seen. The localization error increases with the increasing noise 𝑃𝑛 whatever the method is. However, our method is more precise than the other two methods which required significantly more computing time. That is to say, our proposed PSO algorithm provides higher accuracy in less computing time.

5. Conclusion and Future Work In this paper, an improved PSO algorithm is proposed to solve the distributed localization problem in WSN, which can


7 1.4

0.7 0.65

1.2 Mean localization error

Mean localization error

0.6 0.55 0.5 0.45 0.4 0.35

1 0.8 0.6 0.4

0.3 0.2

0.25 0.2

20

30

0

40

1

2

Number of beacons

4

5

PSO HPSO Proposed PSO


Figure 10: Localization error 𝐸𝑙 with percentage noise 𝑃𝑛 .

Figure 8: Localization error 𝐸𝑙 with number of beacons 𝑀. 30

45

28

40

26

35

24

Computing time (s)

Computing time (s)

3 Pn

22 20 18 16

30 25 20 15

14 10

12 10

5 20

30 Number of beacons

40


1

2

3

4

5

Pn


Figure 9: Computing time 𝑡 with number of beacons 𝑀.

Figure 11: Computing time 𝑡 with percentage noise 𝑃𝑛 .

be regarded as a multidimensional optimization problem. To improve the localization precision and algorithm efficiency, we present some methods in Section 3 whose effectiveness has been proved by the simulation results. In order to show the advantage of our algorithm more intuitively, we made some contrast figures to analyze each method independently. The experimental results show that our method which can localize more unknown nodes with higher precision in less computing time is superior to the other two methods. Further, the control of iterative error propagation and energy consumption are potential and significant directions. In this paper, we considered the nodes localization problem

in two-dimensional space. However, in real environment, sensor nodes are distributed in three-dimensional space. Some research will be taken on our proposed algorithm in three-dimensional space. Moreover, the proposed algorithm may be extended for nodes localization in mobile wireless sensor network. The effectiveness of these planning works will be validated in experiments in the future.


8

Acknowledgments This work is supported by National Natural Science Foundation of China (nos. 61472278, 60872064, and 61102125) and the Natural Science Foundation of Tianjin (no. 12JCYBJC10200).

References [1] G. Mao, B. Fidan, and B. D. O. Anderson, “Wireless sensor network localization techniques,” Computer Networks, vol. 51, no. 10, pp. 2529–2553, 2007. [2] A. A. Kannan, G. Mao, and B. Vucetic, “Simulated annealing based localization in wireless sensor network,” in Proceedings of the IEEE 30th Anniversary Conference on Local Computer Networks (LCN ’05), pp. 513–514, Sydney, Australia, November 2005. [3] A. Gopakumar and L. Jacob, “Localization in wireless sensor networks using particle swarm optimization,” in Proceedings of the IET Conference on Wireless, Mobile and Multimedia Networks, pp. 227–230, January 2008. [4] Q. Zhang, J. Huang, J. Wang et al., “A two-phase localization algorithm for wireless sensor network,” in Proceedings of the IEEE International Conference on Information and Automation (ICIA ’08), pp. 59–64, Changsha, China, June 2008. [5] P. H. Namin and M. A. Tinati, “Node localization using particle swarm optimization,” in Proceedings of the 7th International Conference on Intelligent Sensors, Sensor Networks and Information Processing (ISSNIP ’11), pp. 288–293, IEEE, Adelaide, Australia, December 2011. [6] R. V. Kulkarni, G. K. Venayagamoorthy, and M. X. Cheng, “Bio-inspired node localization in wireless sensor networks,” in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (SMC ’09), pp. 205–210, October 2009. [7] R. V. Kulkarni and G. K. Venayagamoorthy, “Bio-inspired algorithms for autonomous deployment and localization of sensor nodes,” IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, vol. 40, no. 6, pp. 663–675, 2010. [8] A. Kumar, A. Khosla, J. S. Saini, and S. Singh, “Computational intelligence based algorithm for node localization in Wireless Sensor Networks,” in Proceedings of the 6th IEEE International Conference Intelligent Systems (IS ’12), pp. 431–438, Sofia, Bulgaria, September 2012. [9] J. Lv, H. Cui, and M. Yang, “Distribute localization for wireless sensor networks using particle swarm optimization,” in Proceedings of the IEEE 3rd International Conference on Software Engineering and Service Science (ICSESS ’12), pp. 355–358, June 2012. [10] E. Dong, Y. Chai, and X. Liu, “A novel three-dimensional localization algorithm for wireless sensor networks based on particle swarm optimization,” in Proceedings of the 18th IEEE International Conference on Telecommunications (ICT ’11), pp. 55–60, 2011. [11] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948, Perth, Australia, December 1995.



Research Article Two Novel DV-Hop Localization Algorithms for Randomly Deployed Wireless Sensor Networks Guozhi Song and Dayuan Tam School of Computer Science and Software Engineering, Tianjin Polytechnic University, Tianjin 300387, China Correspondence should be addressed to Dayuan Tam; [email protected] Received 17 October 2014; Accepted 10 January 2015 Academic Editor: Chang Wu Yu Copyright © 2015 G. Song and D. Tam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Aiming at solving the problem of low accuracy in traditional distance vector-hop (DV-hop) algorithm used for wireless sensor networks node location, two refined localization algorithms, that is, hyperbolic-DV-hop localization algorithm and improved weighted centroid DV-hop localization algorithm (IWC-DV-hop), are proposed in this paper. Instead of taking the average HopSize of BeaconNode nearest to the UnknownNode, hyperbolic-DV-hop algorithm chooses the average of average HopSizes of all BeaconNodes, as the average HopSize of the UnknownNode. The ML localization algorithm is also replaced by hyperbolic location algorithm. IWC-DV-hop algorithm improves the accuracy by selecting appropriate BeaconNodes and replacing ML localization with centroid localization. And a weighted scheme is used in IWC-DV-hop so that the influence of different anchors is taken into consideration. Simulations have been conducted to prove that the accuracy is improved by both of the two algorithms and IWCDV-hop is the best choice.

1. Introduction Wireless sensor network (WSN) is an increasingly important technology attracting considerable research interest. WSNs consist of thousands of nodes with built-in sensors which can collect a variety of information from surrounding environment. Localization is one of the most important problems in WSNs, for position information is needed in all kinds of situations [1]. In environmental monitoring applications such as bush fire surveillance, water quality monitoring, and precision agriculture, the measurement data are meaningless without knowing the location from where the data are obtained. Moreover, location estimation may enable a myriad of applications such as inventory management, intrusion detection, road traffic monitoring, health monitoring, reconnaissance, and surveillance. Sensor network localization algorithms estimate the locations of sensors with initially unknown location information by using knowledge of the absolute positions of a few sensors and intersensor measurements such as distance and bearing measurements [2]. GPS is widely used for localization, but it is impossible

for WSNs to be used considering the cost limitation. So we need some special localization algorithm for WSNs. Currently, there are many kinds of self-localization algorithms [3], which can be divided into two categories, that is, range-based and range-free [4]. Range-based algorithms use absolute point-to-point distance or angle information to calculate distance between neighbouring sensors [5]. Although the use of range measurements results in a fine-grained localization scheme, range-based algorithms require the sensor to contain additional hardware to make range measurements. There are many common range-based localization algorithms such as received signal strength indicator (RSSI) [6], time of arrival (TOA) [7], time difference of arrival (TDOA) [8], and angle of arrival (AOA) [9]. Range-free algorithms do not need absolute range information. The accuracy is much lower than the range-based but it can still satisfy many applications’ requirements. Hardware requirement of range-free algorithms is little; it does not need actual measurements of physical properties like radio signal

2 strengths, angle of arrival of signal, and distance, so the rangefree algorithms are more economical, cost-effective, and feasible for the large-scale wireless sensor networks. There are many range-free localization algorithms, such as centroid algorithm [10], DV-hop [11, 12], amorphous [13], APIT [14], and MDS-MAP [15, 16]. WSN devices have a hardware limitation that is why solutions of range-free localization are being pursued as a costeffective method compared to other more expensive rangebased approaches [17]. In this paper we focus on DV-hop, one of the range-free localization algorithms. Extensive research has been done on the application of DV-hop algorithm since the algorithm was first proposed by Niculescu and Nath in 2001 [6, 7]. It is still very popular due to its simplicity, cost-effectiveness, and robustness. Therefore researchers keep working on its improvement. In [18], Dulman and Havinga discussed the influence of the underlying random topology, and based on that they derived a distance estimation function of the hop count by which two nodes are separated and characterized the precision of these estimations. And they obtained the improvement of roughly 26% for DV-hop localization algorithm in position accuracy. Checkout DV-hop algorithm and selective 3-anchor DVhop algorithm were proposed by Gui et al. A calculation step, called checkout step, was added to DV-hop in checkout DVhop [19]. In selective 3-anchor DV-hop algorithm, the normal node first selects any three anchors to form a 3-anchor group; then it calculates the candidate position based on each 3anchor group, and finally according to the relation between candidate positions and the minimum hop counts to anchors, the normal node chooses the best candidate position [20]. DV-hop + Lastdist algorithm was proposed by Zhang et al. in [21]. A distance variable of final hop was added to the DV-hop communication message. The distance from the node to the anchor node can properly reduce the error distance caused by the final hop. WD-DV-hop was proposed by Lin et al. in [22]. The centralized and distributed calculation method is replaced by a weighted calculation method. The distance between unknown nodes and anchor nodes is calculated by a method, in which the average one-hop distance of each anchor node multiplies by hops count between unknown nodes and anchor node. There are also some researchers working on centroid localization algorithm. In [23] Deng et al. applied his improved centroid localization algorithm in WSN. He proposed a centroid algorithm with selected anchor node localization algorithm (SA-Centroid) for WSNs. The triangle centroid and polygon centroid were used and the method of selecting nearest anchor node was adopted. However, the SACentroid is complex and the amount of calculation is huge. Furthermore the accuracy improvement is limited. Based on the related work above, we proposed our DV-hop algorithm, which is simpler and provides better localization accuracy. The rest of this paper is organized as follows. In Section 2, we briefly review two other researchers’ study on DVhop’s improvement which we have realized. In Section 3,

International Journal of Distributed Sensor Networks original DV-hop localization algorithm is briefly introduced. In Section 4, we introduce how we refine DV-hop, and the simulation results are illustrated in Section 5. We also compare and analyze the difference between original DV-hop and our two algorithms, as well as the difference between our two algorithms and other improved DV-hop algorithms. Finally, we present our conclusions in Section 6.

2. Related Works In [24], Li et al. proposed an improved DV-hop algorithm, named weighted DV-hop algorithm (WDV-hop). Jian Li’s algorithm is derived from DV-hop algorithm and uses weight of anchors to improve localization accuracy without any additional hardware device. Jian Li’s algorithm gives every anchor a weight which reflects the effects of anchor. In Jian Li’s algorithm an unknown node computes the hop-size based on all the hop-size values it receives from the anchors, instead of just taking the first received hop-size value as in original DVhop algorithm. Using this strategy, the position of unknown nodes is calculated by using weighted least square method. In [25], Zhang et al. analyse the drawback of DV-hop and centroid location algorithm and proposed a new weighted centroid algorithm based on DV-hop, named weighted centroid localization algorithm (WCL). Their algorithm uses weight of anchors to improve accuracy. Every anchor node is weighted by a value reflecting the effect of the anchor node on the unknown node. We have realized these two algorithms by programming with MATLAB so that we can compare our proposed algorithms with them from lots of aspects. The comparison and analysis will be made after we introduce our algorithms in following sections.

3. Theoretical Background In this paper, we focus on static networks, where nodes do not move, since this is already a challenging condition for distributed localization. We assume that some anchor nodes, named BeaconNode, have a priori knowledge of their own positions. Note that anchor nodes have the same capabilities (processing, communication, energy consumption, etc.) as all other sensor nodes with unknown positions, which are named UnknownNode. 3.1. DV-Hop Localization Algorithm. The traditional DV-hop algorithm was developed by Niculescu and Nath [6, 7]. It can be summarized as in the following three steps. 3.1.1. Finding the Minimum HopCount. In the first step, all BeaconNodes broadcast beacon messages to other nodes; the format of the beacon message is {id, 𝑥𝑖 , 𝑦𝑖 , HopCount}. The initial value of HopCount is 0. Each receiving node maintains a BeaconNode information table and keeps the minimum HopCount value from other nodes. After a period of time, all nodes in the network will have the minimum HopCount to other nodes.


3

3.1.2. Calculation of the Distance between BeaconNodes and UnknownNodes. In the second step, the BeaconNodes get minimum HopCount value to other BeaconNodes according to the result in the first step. Then it can estimate average HopSize to other BeaconNodes on the basis of the minimum HopCount and the distance between BeaconNodes. The 𝑖󸀠 th BeaconNode’s average HopSize can be obtained by 2

HopSize𝑖 =

𝑦1 − 𝑦𝑁 𝑥1 − 𝑥𝑁 [ 𝑥2 − 𝑥𝑁 𝑦2 − 𝑦𝑁 ] ] [ 2[ ], .. .. ] [ . . − 𝑥 𝑦 − 𝑦 𝑥 𝑁 𝑁−1 𝑁] [ 𝑁−1

(5)

𝑥 [ 𝑖] , 𝑦𝑖

(6)

𝑋 is

2

√ ∑𝑁 𝑗=1,𝑗=𝑖̸ (𝑥𝑖 − 𝑥𝑗 ) + (𝑦𝑖 − 𝑦𝑗 ) ∑𝑁 𝑗=1,𝑗=𝑖̸ ℎ𝑖𝑗

,

(1) and 𝐵 is

where (𝑥𝑖 , 𝑦𝑖 ) and (𝑥𝑗 , 𝑦𝑗 ) are coordinates of BeaconNodes 𝑖 and 𝑗, ℎ𝑖𝑗 is the minimum HopCount value between BeaconNodes 𝑖 and 𝑗, 𝑁 presents the quantity of BeaconNodes. Every BeaconNode broadcasts its average HopSize to the whole network. Each UnknownNode receives all BeaconNode’s average HopSize and selects the HopSize of a BeaconNode, which has the minimum HopCount value to this UnknownNode, as its average HopSize. In the end, we can calculate the distance of every UnknownNode to BeaconNode by 𝑑𝑖𝑗 = HopSize𝑖 ∗ HopCount𝑖𝑗 ,

(2)

󸀠

where 𝑑𝑖𝑗 is the distance between the 𝑖 th UnknownNode and 𝑗󸀠 th BeaconNode. HopCount𝑖𝑗 indicates the minimum HopCount between the 𝑖󸀠 th UnknownNode and 𝑗󸀠 th BeaconNode. 3.1.3. Calculation of Estimated Location. In the third step, the UnknownNode can calculate their locations by maximum likelihood (ML) estimation [26] or trilateration. In this paper we choose ML estimation and it will be introduced in the next section. 3.2. Maximum Likelihood Estimation. ML estimation is a good way to calculate UnknownNodes’ locations [4, 5, 27, 28]. With its help, we can obtain the following set of the equations: √(𝑥𝑖 − 𝑥1 )2 + (𝑦𝑖 − 𝑦1 )2 = 𝑑𝑖1 √(𝑥𝑖 − 𝑥2 )2 + (𝑦𝑖 − 𝑦2 )2 = 𝑑𝑖2 .. .

[ [ [ [ [ [

(3)

2 2 2 2 + 𝑦12 − 𝑦𝑁 − 𝑑𝑖1 + 𝑑𝑖𝑁 𝑥12 − 𝑥𝑁

2

2 2 2 2 𝑥22 − 𝑥𝑁 + 𝑦22 − 𝑦𝑁 − 𝑑𝑖2 + 𝑑𝑖𝑁 .. . 2

2

2

2

2

] ] ] ]. ] ]

(7)

[𝑥𝑁−1 − 𝑥𝑁 + 𝑦𝑁−1 − 𝑦𝑁 − 𝑑𝑖(𝑁−1) + 𝑑𝑖𝑁] According to the least square method, the solution of (4) should be ̂ = (𝐴𝑇 𝐴)−1 𝐴𝑇 𝐵. 𝑋

(8)

3.3. Centroid Localization Algorithm. This section is devoted to the centroid algorithm. It is the most basic schemes, and it comprises three stages. First, BeaconNodes broadcast the localization reference to its neighbourhoods. In the second stage, the UnknownNodes receive and restore the information transmitted by at least three different BeaconNodes and then, classical arithmetic mean is employed to calculate the position of UnknownNode. Each UnknownNode maintains a table {beacon’s id, position of the beacon (𝑥, 𝑦)}. In stage 3, when obtaining its coordinate, the UnknownNode becomes a BeaconNode that propagates packets to its vicinity. Figure 1 shows the detailed interpretation of the centroid algorithm. In centroid algorithm based positioning, the UnknownNode’s estimation coordinate is the centroid of the polygon formed by BeaconNodes. Suppose that there are five BeaconNodes around the UnknownNodes 𝐴 1 (𝑥1 , 𝑦1 ), 𝐴 2 (𝑥2 , 𝑦2 ), 𝐴 3 (𝑥3 , 𝑦3 ), 𝐴 4 (𝑥4 , 𝑦4 ), and 𝐴 5 (𝑥5 , 𝑦5 ) distributed around the UnknownNode ({𝑆(𝑥, 𝑦)}). Thus the position of UnknownNode 𝑆 can be calculated using: (𝑥, 𝑦) = (

𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 𝑦1 + 𝑦2 + 𝑦3 + 𝑦4 + 𝑦5 , ). 5 5 (9)

The general algorithm can be summarized using (10), where 𝑁 presents the quantity of BeaconNode:

√(𝑥𝑖 − 𝑥𝑁)2 + (𝑦𝑖 − 𝑦𝑁)2 = 𝑑𝑖𝑁, where 𝑑𝑖1 represents the distance of 𝑖󸀠 th UnknownNode to the first BeaconNode, 𝑁 presents the quantity of BeaconNode, (𝑥𝑖 , 𝑦𝑖 ) is the coordinate of the 𝑖󸀠 th UnknownNode, and (𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ) ⋅ ⋅ ⋅ (𝑥𝑁, 𝑦𝑁) are the coordinates of BeaconNode. Then (3) can be transformed by 𝐴𝑋 = 𝐵,

where 𝐴 is

(4)

𝑥𝑒𝑠𝑡 =

∑𝑁 𝑖=1 𝑥𝑖 , 𝑁

𝑦𝑒𝑠𝑡 =

∑𝑁 𝑖=1 𝑦𝑖 . 𝑁

(10)

4. Improvement of DV-Hop Localization Algorithm In this section, we present two enhanced DV-hop algorithms, that is, hyperbolic-DV-hop localization algorithm

4

International Journal of Distributed Sensor Networks We calculate the average HopSizes of all BeaconNodes by 2

HopSize𝑖 =

∑𝑖=𝑗̸ √ (𝑥𝑖 − 𝑥𝑗 ) + (𝑦𝑖 − 𝑦𝑗 )

2

∑𝑖=𝑗̸ ℎ𝑖𝑗

,

(11)

where (𝑥𝑖 , 𝑦𝑖 ) and (𝑥𝑗 , 𝑦𝑗 ) are coordinates of BeaconNodes 𝑖 and 𝑗 and ℎ𝑖𝑗 is the minimum HopCount value between BeaconNodes 𝑖 and 𝑗. Then we can calculate the average of the average HopSize of BeaconNodes and take this as the average HopSize of every UnknownNode—HopSizeUN : HopSizeUN = HopSizeavg =

BeaconNode Centroid UnknownNode

Figure 1: The concept of centroid algorithm.

and IWC-DV-hop localization algorithm. The original DVhop takes the average HopSize of BeaconNode nearest to the UnknownNode, which is believed to be the reason leading to large errors. So we replace it with the average of average HopSizes of all BeaconNodes, as the average HopSize of the UnknownNode. The ML localization algorithm is replaced by Hyperbolic location algorithm, which has better performance results. In IWC-DV-hop algorithm, centroid algorithm is combined with DV-hop algorithm, and a weighted scheme is used in IWC-DV-hop so that the influence of different anchors is taken into consideration. 4.1. Our Hyperbolic-DV-Hop Localization Algorithm. After a thorough study of the principle of original DV-hop localization algorithm, we found that the errors of original DV-hop localization algorithm are caused by HopCounts, averaging HopSizes, and localization algorithm. So we managed to improve it through working on the last two steps, that is, the way to calculate distance between BeaconNode and UnknownNode and the localization algorithm. 4.1.1. Refinement on Second Step—Distance Estimation. As what we have introduced above, UnknownNodes, in original DV-hop algorithm, take the average HopSize of its nearest BeaconNode as their own average HopSize. But the nodes in the WSN are disposed stochastically, which means the path between nodes may be not straight, which is the reason why the average HopSizes are always bigger than the actual value. When the distance between UnknownNode and BeaconNode is calculated using the average HopSize, the error is always large. The nearest BeaconNode is unique and cannot be used to represent the whole BeaconNodes. Since ML in the third step will make use of the distance between UnknownNodes and every BeaconNode, it is a good way to take the average HopSize of all BeaconNodes to replace that of the nearest BeaconNode for UnknownNodes.

∑ HopSize𝑖 . 𝑛

(12)

4.1.2. Refinement on Third Step—Location Algorithm. In the original DV-hop algorithm, ML or trilateration is used for physical distances estimations. In this paper, we use hyperbolic location algorithm [29, 30] to replace ML. We derive the following expression: 2 2 2 + 𝑌𝑁 − 2𝑋𝑁𝑋𝑖 − 2𝑌𝑁𝑌𝑖 + 𝑋𝑖 2 + 𝑌𝑖 2 = 𝑑𝑖𝑁 𝑋𝑁 2 󳨐⇒ 𝑑𝑖𝑁 − 𝐸𝑁 = −2𝑋𝑁𝑋𝑖 − 2𝑌𝑁𝑌𝑖 + 𝐾,

(13)

2 2 where 𝐸𝑁 = 𝑋𝑁 + 𝑌𝑁 and 𝐾 = 𝑋𝑖2 + 𝑌𝑖2 . Set 𝑇

𝑃𝑐 = [𝑋𝑖 , 𝑌𝑖 , 𝐾] , −2𝑋1 −2𝑌1 [ −2𝑋2 −2𝑌2 [ 𝑀𝑐 = [ .. .. [ . . [−2𝑋𝑁 −2𝑌𝑁

(14) 1 1] ] .. ] , .] 1]

𝑑12 − 𝐸1 ] [ 2 [ 𝑑2 − 𝐸2 ] ]. [ ℎ𝑐 = [ .. ] ] [ . 2 − 𝐸 𝑑 𝑁] [ 𝑁

(15)

(16)

We can derive from (8) that 𝑀𝑐 𝑆𝑐 = ℎ𝑐 .

(17)

Using the least square algorithm [30], from (14), we can get −1

𝑃𝑐 = (𝑀𝑐𝑇 𝑀𝑐 ) 𝑀𝑐𝑇 ℎ𝑐 .

(18)

Then, the position of 𝑖󸀠 th UnknownNode should be (𝑃𝑐 (1) , 𝑃𝑐 (2)) .

(19)

4.2. Our Improved Weighted Centroid DV-Hop Localization Algorithm. The accuracy improvement of Hyperbolic-DVhop localization algorithm is still limited, so we have further proposed our improved weighted centroid DV-hop localization algorithm. Firstly, we simply replace the hyperbolic algorithm with centroid algorithm. Secondly, we analyse the reason leading to errors and then offer our solutions which turn out to improve the accuracy.


5

4.2.1. Refinement on the Third Step—Using Centroid Algorithm to Calculate the UnknownNodes’ Location. In the second step in original DV-hop algorithm, we can get the distance, 𝑑𝑖𝑗 , between BeaconNodes and UnknownNodes via (2). The UnknownNodes receive and restore the information transmitted by at least three different BeaconNodes. Then we set a threshold distance and all the BeaconNodes with the distance between them and the UnknownNode less than the threshold distance will be selected. At last, the centroid of all the selected BeaconNodes is calculated and is taken as the UnknownNode’s estimated location [31]. Suppose there are BeaconNodes 𝐵𝑖 (𝑋𝑖 , 𝑌𝑖 ), 𝑖 = 1, 2, . . . , 𝑛, distributed around the UnknownNode UN𝐽 (𝑋𝑗 , 𝑌𝑗 ), and the threshold distance is 𝐷threshold ⋅ 𝐷𝑖,𝐽 represents the distance between 𝐵𝑖 (𝑋𝑖 , 𝑌𝑖 ) and UN𝐽 (𝑋𝑗 , 𝑌𝑗 ). The positions of the BeaconNodes whose 𝐷𝑖,𝐽 is less than 𝐷threshold are restored in 𝐵𝑘 (𝑋𝑘 , 𝑌𝑘 ). Thus the position of UnknownNode UN𝐽 (𝑋𝑗 , 𝑌𝑗 ) can be calculated using: 𝑋𝑗 =

∑ 𝑋𝑘 , 𝑘

𝑌𝑗 =

∑ 𝑌𝑘 . 𝑘

(20)

4.2.2. Further Analyses of the Reason for Error in DV-Hop. We have discussed one of the reasons why the error exists in original DV-hop in Section 4.1; that is, the original DVhop takes the average HopSize of BeaconNode nearest to the UnknownNode. We replace it with the average of average HopSizes of all BeaconNodes, as the average HopSize of the UnknownNode. But the improvement of accuracy is limited. After further analysing the DV-hop algorithm, we found that error can be further reduced by adopting a new method. In the second step of the original DV-hop, UnknownNode uses (2) to calculate the distance between itself and BeaconNodes. The distance can also be calculated by multiplying BeaconNodes’ HopSize by HopCount from another perspective. In other words, the HopSize of UnknownNodes is not necessary anymore. That means we can cut out one little step from the second step of DV-hop. 4.2.3. Refinement on the Second Step of DV-Hop. The HopSize of UnknownNode is not needed anymore and the step to calculate the HopSize of UnknownNode is cut out. The HopSize𝑖 can be replaced by HopSize𝑗 and then the distance between UnknownNode 𝑖 and BeaconNode 𝑗 can be obtained by 𝑑𝑖𝑗 = HopSize𝑗 ∗ HopCount𝑖𝑗 .

(21)

Since we have known the HopSize of every BeaconNode and the HopCount between every node and every noode, it is easy to get the distance between every UnknownNode and each BeaconNode. The accuracy will be improved since the total number of steps for localization is decreased. From mathematical point of view, the calculation result will be less accurate in the case where an estimated value is calculated (added, subtracted, multiplied, divided, etc.) for 𝑁 + 1 times than the case where it is calculated for 𝑁 times. So the accuracy is improved when we cut out one calculating step from original DV-hop algorithm.

4.2.4. Further Analyses of the Reason for Error in Centroid Algorithm. In the centroid algorithm, the unknown node’s estimation coordinate is the centroid of the polygon formed by BeaconNodes. And centroid algorithm cannot reflect the influence of the distance between BeaconNodes and UnknownNodes. In fact, the BeaconNodes which are closer to the UnknownNode will have a greater impact on the UnknownNode. Different BeaconNodes have different effect on the UnknownNode, considering the distance between them. But the centroid algorithm treats them as the same kind of node. In order to embody the effect of BeaconNodes which have different distance on the accuracy calculation, our improved weighted centroid DV-hop (IWC-DV-hop) algorithm is proposed. 4.2.5. Refinement on the Centroid Algorithm. As what (2) and (21) have showed, distance between BeaconNodes and UnknownNodes has a direct correlation with HopCounts. So the effect of distance on accuracy can be replaced by HopCounts. Then we can calculate the location of UnknownNodes by 𝑥𝑒𝑠𝑡 =

∑𝑛𝑖=1 𝑤𝑖 𝑥𝑖 , ∑𝑛𝑖=1 𝑤𝑖

𝑦𝑒𝑠𝑡 =

∑𝑛𝑖=1 𝑤𝑖 𝑦𝑖 , ∑𝑛𝑖=1 𝑤𝑖

(22)

where (𝑥𝑒𝑠𝑡 , 𝑦𝑒𝑠𝑡 ) is the coordinates of the UnknownNode. And we suppose it has received 𝑛 BeaconNodes’ message ({(𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ), . . . , (𝑥𝑛 , 𝑦𝑛 )}). 𝑤𝑖 is the weight of BeaconNode 𝑖, and its value can be obtained by 𝑤𝑖 =

∑𝑛𝑖=1 hop𝑖 , 𝑛 hop𝑖

(23)

where hop𝑖 is the HopCount between the UnknownNode and the BeaconNode 𝑖. These weights determine the relative importance of each quantity on the average. Consequently, when the weight 𝑤𝑖 is larger than the other weights, the weights 𝑤𝑖 play a more important role on the average. This scheme ensures that the closest BeaconNode will affect the localization result mostly. Another advantage with our improvement to the original centroid algorithm is that without threshold the weighted solution can make full use of all BeaconNodes available. While only those BeaconNodes whose 𝐷𝑖,𝐽 is less than 𝐷threshold are used in the original centroid algorithm. On the other hand, how to choose a perfect threshold is also a complex issue requiring more discussion, which is avoided in the weighted solution.

5. Simulation and Performance Comparison We compare the performance of different locating methods by evaluating the average location error for each algorithm. The performance comparisons are conducted using a simulator developed with MATLAB R2013a, and all the nodes are deployed in a 500 m ∗ 500 m area. The comparison with other two researchers’ algorithms has also been made in this section. We have realized their algorithms by programming according to their description in [24, 25].

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Figure 2: Comparison with different BeaconNode ratios.

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0.45 0.4 0.35 Error ratio

5.1. Varying the BeaconNode Ratio. Firstly we compare the performance of three localization algorithms with different BeaconNode ratios. Suppose the number of all nodes is 800, and the communication radius of BeaconNode is 100 m. We calculated the average of results of 50 simulations and set the average as the error ratio. Gradually, we increased the ratio of BeaconNode and get the different results in different situations. The performance comparison with 3 different localization algorithms is shown in Figure 2. The upper line shows the error ratio of original DV-hop algorithm varies with the increase of the BeaconNode ratio, while the middle line shows the error ratio of hyperbolic-DV-hop algorithm and the lower line shows the error ratio of our IWC-DV-hop. It is obvious to observe that our IWC-DV-hop algorithm is much superior to the other two. In fact, the accuracy improvement of our hyperbolic-DV-hop over the original DV-hop is 8.2%, while our IWC-DV-hop over the original DV-hop is 63%. Figure 3 shows the comparison between our hyperbolicDV-hop algorithm, our IWC-DV-hop algorithm, Jian Li’s WDV-hop algorithm in [24], and Bingjiao Zhang’s WCL algorithm in [25]. Overall, it is obvious that our IWC-DVhop algorithm performs best with the least error ratio. We can also conclude that our proposed two algorithms are more stable, for their fluctuation range is smaller with the variety of BeaconNode ratio than that of Jian Li’s WDVhop algorithm and Binjjiao Zhang’s WCL algorithm. In addition, the environment with low density of BeaconNodes is simulated and compared. The result shows that our IWCDV-hop algorithm still has the best accuracy and is qualified for this kind of environment.

0.3 0.25 0.2 0.15 0.1 0.05 0 300 350 400 450 500 550 600 650 700 750 800 Node number Original DV-hop Hyperbolic-DV-hop IWC-DV-hop

Figure 4: Comparison with different node number.

number, we set the BeaconNode ratio to 50% and communication radius to 100 m. We calculated the average of results from 50 simulation experiments and set the average as the error ratio. Furthermore, we increased nodes number from 300 to 800 to get the different results in different situations. Figure 4 illustrates the contrast with three different algorithms with different nodes number. The upper line shows the error ratio of original DV-hop algorithm varies with the increase of the nodes number, while the middle line shows the error ratio of hyperbolic-DV-hop algorithm and the lower line shows the error ratio of our IWC-DV-hop. We observe that, to a certain extent, the hyperbolic algorithm improves the accuracy, while the IWC-DV-hop algorithm improves the

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Figure 5: Comparison with different node number.

Figure 6: Comparison with different BeaconNode communication radii.

5.3. Varying Communication Radius. In this scenario, we investigate the impact of increasing communication radius of BeaconNodes. The BeaconNode ratio is set as 50%, and node number is 800. We calculated the average of results of 50 simulations and set the average as the error ratio. We construct the performance comparison by artificially increasing the communication radius of BeaconNode with 10 m, and the result is plotted in Figures 6 and 7. As what has been introduced above, the upper line shows the error ratio of original DV-hop algorithm varies with the increase of the communication radius, while the middle line shows the error ratio of hyperbolic-DV-hop algorithm and the lower line shows the error ratio of our IWC-DV-hop. We note that the accuracy is greatly improved by IWC-DVhop, and the hyperbolic algorithm also works better than the original DV-hop. According to our records, the accuracy improvement of our hyperbolic-DV-hop over the original DV-hop is 9.3%, while our IWC-DV-hop over the original DV-hop is 62%. Figure 7 indicates that, in general, the accuracy of our IWC-DV-hop is much better than Jian Li’s WDV-hop algorithm and Bingjiao Zhang’s WCL algorithm. It is worth mentioning that Jian Li’s algorithm has better accuracy than our

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accuracy on a much larger scale. The accuracy improvement of our hyperbolic-DV-hop over the original DV-hop is 10.3%, while our IWC-DV-hop over the original DV-hop is 59%. Figure 5 shows the relationship between error ratio and node number. In this simulation, we compare the accuracy of our proposed hyperbolic-DV-hop, our proposed IWC-DVhop algorithm, Jian Li’s WDV-hop algorithm in [24], and Bingjiao Zhang’s WCL algorithm in [25]. This simulation result has proven the validity of IWC-DV-hop, since it always has the least error ratio and fluctuation.

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Figure 7: Comparison with different BeaconNode communication radii.

hyperbolic-DV-hop algorithm when the BeaconNode communication radius is larger than 60 m, but our hyperbolicDV-hop algorithm performs better in those environments, the communication radius is small. We can conclude that the change in BeaconNode communication radius had minimal impact on performance, which means our two proposed algorithms have better stability.

6. Conclusions Localization problem is one of the most important and different problems for WSNs. After a detailed study of

8 positioning problem of wireless sensor network node location, two refined distance vector-hop (DV-hop) localization algorithms, hyperbolic-DV-hop localization algorithm and IWC-DV-hop localization algorithm, are proposed in this paper. Compared to the original DV-hop algorithm, the accuracy is improved by both algorithms, between which IWC-DV-hop performs even better. Our hyperbolic-DV-hop algorithm provides better accuracy than the original DVhop algorithm because it chooses the average of average HopSizes of all BeaconNodes as the average HopSize of the UnknownNode instead of taking the average HopSize of BeaconNode nearest to the UnknownNode. It also replaces the ML localization algorithm with hyperbolic location algorithm with better performance results. Our IWC-DV-hop algorithm improves the accuracy by selecting appropriate BeaconNodes and replacing ML localization with centroid localization. A weight scheme has also been proposed to improve the accuracy by taking the influence of different BeaconNodes into consideration. Then we run simulations to compare the performance of different localization algorithms with different parameters. The results of simulations show that both our hyperbolic-DV-hop algorithm and IWC-DVhop are much better than DV-hop localization algorithm. In addition, comparing to other localization algorithms for WSNs or other refined algorithms based on DV-hop, our IWC-DV still performs excellently. According to our simulations, our IWC-DV algorithm not only has a much better precision but also performs much more steadily no matter how the parameters change. In the future, we will be interested in refining our localization algorithms even further and try to use it in the 3D situations. Some test-bed can also be constructed to allow us to compare real results with those simulation results.


Acknowledgments This work is supported by National Training Program of Innovation and Entrepreneurship for Undergraduates (201410058046), National Nature Science Foundation of China (61272006), and Tianjin Higher Education Fund for Science and Technology Development under Grant no. 20110808.

References [1] H. Dai, A. G. Chen, X. F. Gu, and L. He, “localization algorithm for large-scale and low-density wireless sensor networks,” Electronics Letters, vol. 47, no. 15, pp. 881–883, 2011. [2] G. Mao, B. Fidan, and B. D. O. Anderson, “Wireless sensor network localization techniques,” Computer Networks, vol. 51, no. 10, pp. 2529–2553, 2007. [3] F. B. Wang, L. Shi, and F. Y. Ren, “Self-localization systems and algorithms for wireless sensor networks,” Journal of Software, vol. 16, no. 5, pp. 857–868, 2005.

International Journal of Distributed Sensor Networks [4] G. Q. Gao and L. Lei, “An improved node localization algorithm based on DV-HOP in WSN,” in Proceedings of the IEEE International Conference on Advanced Computer Control (ICACC ’10), vol. 4, pp. 321–324, March 2010. [5] H. Chen, K. Sezaki, P. Deng, and C. S. Hing, “An improved DV-hop localization algorithm for wireless sensor networks,” in Proceedings of the 3rd IEEE Conference on Industrial Electronics and Applications (ICIEA ’08), pp. 1557–1561, Singapore, June 2008. [6] T. S. Rappapport, Wireless Communications: Principles and Practice, Prentice Hall, Upper Saddle River, NJ, USA, 1996. [7] L. Girod and D. Estrin, “Robust range estimation using acoustic and multimodal sensing,” in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1312– 1320, November 2001. [8] X. Cheng, A. Thaeler, G. Xue, and D. Chen, “TPS: a time-based positioning scheme for outdoor wireless sensor networks,” in Proceedings of the 23rd IEEE Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM ’04), pp. 2685–2696, Hong Kong, China, March 2004. [9] D. J. Torrieri, “Statistical theory of passive location systems,” IEEE Transactions on Aerospace and Electronic Systems, vol. 20, no. 2, pp. 183–198, 1984. [10] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-less low-cost outdoor localization for very small devices,” IEEE Personal Communications, vol. 7, no. 5, pp. 28–34, 2000. [11] D. Niculescu and B. Nath, “Ad hoc positioning system (APS),” in Proceedings of the IEEE Global Telecommunicatins Conference (GLOBECOM ’01), pp. 2926–2931, San Antonio, Tex, USA, November 2001. [12] D. Niculescu and B. Nath, “DV based positioning in Ad hoc networks,” Telecommunication Systems, vol. 22, no. 1–4, pp. 267– 280, 2003. [13] R. Nagpal, “Organgizing a glabal coordinate system from local information on an amporphous computer,” Tech. Rep. AI Memo 1666, MIT Artifical Intelligence Laboratory, 1999. [14] T. He, C. D. Huang, B. M. Blum, J. A. Stankovic, and T. Abdelzaher, “Range-free localization schemes for large scale sensor netowrks,” in Proceedings of the 9th the Annual International Conference on Mobile Computing and Networking, pp. 81–95, ACM, San Diego, Calif, USA, 2003. [15] Y. Shang, W. Ruml, Y. Zhang, and M. P. J. Fromherz, “Localization from mere connectivity,” in Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking & Computing (MobiHoc ’03), pp. 201–212, Annapolis, Md, USA, 2003. [16] Y. Shang and W. Ruml, “Improved MDS-based localization,” in Proceedings of the IEEE Conference on Computer Communications (INFOCOM ’04), pp. 2640–2651, Hong Kong, March 2004. [17] T. He, C. Huang, B. M. Blum, J. A. Stankovic, and T. Abdelzaher, “Range-free localization schemes for large scale sensor networks,” in Proceedings of the 9th Annual International Conference on Mobile Computing and Networking (MobiCom ’03), pp. 81–95, San Diego, Calif, USA, 2003. [18] S. Dulman and P. Havinga, “Statistically enhanced localization schemes for randomly deployed wireless sensor networks,” in Proceedings of the IEEE Intelligent Sensors, Sensor Networks and Information Processing Conference (ISSNIP 󸀠 04), pp. 403–410, December 2004. [19] L. Gui, A. Wei, and T. Val, “A two-level range-free localization algorithm for wireless sensor networks,” in IEEE Conference on Wireless Communications Networking and Mobile Computing, pp. 1–4, 2010.

International Journal of Distributed Sensor Networks [20] L. Gui, T. Val, and A. Wei, “Improving localization accuracy using selective 3-anchor DV-hop algorithm,” in Proceedings of the IEEE 74th Vehicular Technology Conference (VTC Fall ’11), pp. 1–5, San Francisco, Calif, USA, September 2011. [21] H. Zhang, X. H. Zhang, and A. S. Ma, “Research on the localization technology for wireless sensor network,” Applied Mechanics and Materials, vol. 513–517, pp. 1490–1493, 2014. [22] Z. G. Lin, L. Zhao, L. Li, Z. X. Chen, and X. Wang, “An improved DV-HOP on weighted and distributed calculation method,” Advanced Materials Research, vol. 787, pp. 1044–1049, 2013. [23] B. Deng, G. Huang, L. Zhang, and H. Liu, “Improved centroid localization algorithm in WSNs,” in Proceedings of the 3rd International Conference on Intelligent System and Knowledge Engineering (ISKE ’08), vol. 1, pp. 1260–1264, November 2008. [24] J. Li, J. Zhang, and L. Xiande, “A weighted DV-Hop localization scheme for wireless sensor networks,” in Proceedings of the IEEE International Conference on Scalable Computing and Communications and the 8th International Conference on Embedded Computing (SCALCOM-EMBEDDEDCOM ’09), pp. 269–272, Dalian, China, September 2009. [25] B. Zhang, M. Ji, and L. Shan, “A weighted centroid localization algorithm based on DV-hop for wireless sensor network,” in Proceedings of the 8th International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM ’12), Shanghai, China, September 2012. [26] C. Xie, “Research on improved DV-HOP localization algorithm based on weighted least square method,” in Proceedings of the IEEE International Symposium on Knowledge Acquisition and Modeling Workshop (KAM ’08), pp. 773–776, Wuhan, China, December 2008. [27] L. Sun, J. Li, and Y. Chen, Wireless Sensor Network, Tsinghua University Press, Beijing, China, 2005. [28] I. S. Jacobs and C. P. Bean, “Fine particles, thin films and exchange anisotropy,” in Magnetism, Vol. III, G. T. Rado and H. Suhl, Eds., pp. 271–350, Academic Press, New York, NY, USA, 1963. [29] D. J. Torrieri, “Statistical theory of passive location system,” IEEE Transactions on Aerospace and Electronic Systems, vol. 20, no. 2, pp. 183–198, 1984. [30] Y. T. Chan and K. C. Ho, “Simple and efficient estimator for hyperbolic location,” IEEE Transactions on Signal Processing, vol. 42, no. 8, pp. 1905–1915, 1994. [31] H.-Q. Cheng, H. Wang, and H.-K. Wang, “An improved centroid localization algorithm based on weighted average in WSN,” in Proceedings of the 3rd International Conference on Electronics Computer Technology (ICECT ’11), vol. 4, pp. 258– 262, April 2011.

9


Research Article Amorphous Localization Algorithm Based on BP Artificial Neural Network Lin-zhe Zhao,1 Xian-bin Wen,1 and Dan Li2 1

Key Laboratory of Computer Vision and System, Ministry of Education, Tianjin University of Technology, Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin 300384, China 2 School of Computer Science and Technology, Tianjin University, Tianjin, China Correspondence should be addressed to Lin-zhe Zhao; [email protected] Received 16 October 2014; Accepted 1 January 2015 Academic Editor: Chang Wu Yu Copyright © 2015 Lin-zhe Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Accurate localization of nodes is one of the key issues of wireless sensor network (WSN). Because the disadvantages of the classical Amorphous algorithm will produce large localization error in the process of localization, an improved localization algorithm is proposed in this paper to solve the problem. The improved algorithm introduces the received signal strength threshold to modify the minimum hop from the unknown node to the beacon node. Then, Back Propagation (BP) algorithm is introduced to optimize the threshold and reduce the localization error. The simulation results show that the localization accuracy of the improved algorithm is higher than that of other algorithms and the energy consumption does not increase too much.

1. Introduction Wireless sensor network (WSN) is composed of a large number of sensor nodes, which can monitor and acquire physical information in the distribution detection area in real time. WSN has gained a wide attention in recent years and has been applied to the medical field, the industry, the military, and other fields, such as remote health, smart home, battlefield reconnaissance, and environmental monitoring. The location of sensor nodes needs not be engineered or predetermined. This means that sensor nodes can be deployed randomly in inaccessible terrains or disaster relief operations. In addition, this also means that sensor network protocols and algorithms must possess self-organizing capabilities [1]. The localization problem of WSNs can be interpreted that, in a sensor network in which the own location of multiple nodes has been known, which is named by the beacon node, and the location of target node which is named by the unknown nodes is obtained by the sensor information and effective localization algorithm. According to most of the applications in WSN, the information without space and time data is unvalued. In order to better monitor and track

the target, the location of sensor node is one of the most important information [2, 3]. In recent years, localization of wireless sensor network node has become a popular and significant issue. At present, according to whether or not the network needs to measure the actual distances between network nodes, the WSN localization algorithm can be divided into two categories: one is Range-Based algorithm and the other is Range-Free algorithm. The Range-Based algorithm mainly includes the measurements of angle and distance, such as Received Signal Strength Indicator (RSSI), Time of Arrival (TOA), Time Difference on Arrival (TDOA), and Angle of Arrival (AOA) [3]. Then, the location of the unknown node can be calculated by the trilateral measurement method and triangulation method. So it needs extra hardware supporting, large computing, and communicating with high energy consumption. The Range-Free approach only depends on the connectivity of nodes such as the hops for localization without any extra hardware supporting. It includes Centroid algorithm [4], Distance Vector-Hop (DV-Hop) [5], and Amorphous algorithm. Although the localization accuracy of the range-based algorithm is usually higher than that of

2


the range-free algorithm, because of the simple hardware support, the lower consumption, and the antinoise ability, the range-free algorithm is widely used in many applications. In this case, how to improve the localization accuracy of Range-Free approach has drawn much attention. Many experts put forward the optimization algorithms based on intelligent algorithm to improve the localization accuracy. For example, BP localization algorithm based on virtual nodes (VN-BP) with lower localization error has been proposed in [6]. The advantages of BP artificial neural network in error optimization are formidable self-organizing and selflearning capacity, error back-propagation, and error minimum principle [7]. Therefore, it is a good idea to introduce BP neural networks in the localization algorithm to improve the localization accuracy. Although Amorphous algorithm is easy to implement, it will produce large error in the process of localization. In order to solve the problem, an improved localization algorithm is proposed in this paper to reduce the localization error. The proposed algorithm is clarified in two aspects: (1) modify the minimum hop from the unknown node to the beacon node by setting the received signal strength threshold; (2) BP algorithm is introduced to optimize the threshold and reduce the localization error.

2. Amorphous Localization Algorithm Amorphous algorithm is similar to DV-Hop algorithm, and the idea is to calculate the hop distance between two nodes instead of the linear distance between them. Amorphous algorithm is composed of the following three steps [8]. 2.1. Calculate the Minimum Hop from the Unknown Node to the Beacon Node. Every beacon node sends messages to the unknown nodes by flooding method. Formula (1) is used to calculate the minimum hop from the node 𝑖 to 𝑘: 𝑆(𝑖,𝑘) =

∑𝑗∈nbrs(𝑖) ℎ(𝑗,𝑘) + ℎ(𝑖,𝑘) |nbrs (𝑖)| + 1

− 0.5,

(1)

where 𝑆(𝑖,𝑘) is the minimum hop from the unknown node 𝑖 to the beacon node 𝑘; ℎ(𝑗,𝑘) is the integer hop from the unknown node 𝑗 to the beacon node 𝑘; ℎ(𝑖,𝑘) is the integer hop from the unknown node 𝑖 to the beacon node 𝑘; nbrs(𝑖) are the neighbor nodes around the unknown node 𝑖; |nbrs(𝑖)| is the number of the neighbor nodes around the unknown node 𝑖. 2.2. Calculate the Distance from the Unknown Node to the Beacon Node. Formula (2) is used to calculate the average distance of one hop: 1

√1−𝑡2 )

HopSize = 𝑟 (1 + 𝑒−𝑛local − ∫ 𝑒−(𝑛local /𝜋)(arcos 𝑡−𝑡 −1

𝑑𝑡) , (2)

where 𝑟 is the wireless range of the node and 𝑛local is the average connectivity of the network. The distance 𝑑 from the unknown node to the beacon node can be calculated on the basis of the average distance of one hop and the minimum hop from the unknown node

to the beacon node. It can be expressed by the following formula: 𝑑 = HopSize𝑖 × 𝑆(𝑖,𝑘) .

(3)

2.3. Adopt the Least Squares Method to Locate. When the estimated distances from the unknown node to three or more than three beacon nodes have been obtained, the location of the unknown node can be calculated. It is shown as 2

2

2

2

(𝑥1 − 𝑥) + (𝑦1 − 𝑦) = 𝑑12 .. . (𝑥𝑛 − 𝑥) + (𝑦𝑛 − 𝑦) =

(4)

𝑑𝑛2 .

The above formula will be solved by the least squares method; the location of the unknown node can be obtained: −1

𝑋 = (𝐴𝑇 𝐴) 𝐴𝑇 𝑏,

(5)

where (𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ), (𝑥3 , 𝑦3 ), . . . , (𝑥𝑛 , 𝑦𝑛 ) are the coordinates of 𝑛 beacon nodes, (𝑥, 𝑦) is the coordinates of the unknown node, and 𝑑1 , 𝑑2 , 𝑑3 , . . . , 𝑑𝑛 are the distances from the unknown node to the beacon nodes: 2 (𝑥1 − 𝑥𝑛 ) 2 (𝑦1 − 𝑦𝑛 ) ] [ .. .. 𝐴=[ ], . . 2 (𝑥 − 𝑥 ) 2 (𝑦 − 𝑦 ) 𝑛−1 𝑛 𝑛−1 𝑛 ] [ [ 𝑏=[

𝑥12 − 𝑥𝑛2 + 𝑦12 − 𝑦𝑛2 + 𝑑𝑛2 − 𝑑12 .. .

(6)

] ].

2 2 2 2 2 2 [𝑥𝑛−1 − 𝑥𝑛 + 𝑦𝑛−1 − 𝑦𝑛 + 𝑑𝑛 − 𝑑𝑛−1 ]

The location of the unknown node can be calculated on the basis of (5) and (6).

3. The Improved Amorphous Algorithm Based on BP Artificial Neural Network Because the disadvantages of the classical Amorphous algorithm will produce large localization error in the process of localization, an improved localization algorithm is proposed in this paper to solve the problem. The improved algorithm introduces the received signal strength threshold to modify the minimum hop from the unknown node to the beacon node. Then, BP algorithm is introduced to optimize the threshold and reduce the localization error. The improved algorithm is composed of the following two parts. Sections 3.1 and 3.2 describe the two parts, respectively. The flow chart of the improved algorithm is shown in Figure 1. 3.1. The Choice of the Hop. In the classical Amorphous localization algorithm, there exists large deviation when calculating the distance 𝑑. As shown in Figure 2, nodes 𝐴 and 𝐵 are both in the wireless range of node 𝑂. As a result, the hops of them are set as one, while the disparity of |𝑂𝐴| and |𝑂𝐵| is huge [9]. This is a common problem existing in the real

International Journal of Distributed Sensor Networks M = whi M Dissatisfy

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3.2. BP Algorithm Will Be Introduced to Optimize the Threshold to Reduce the Localization Error. The received signal strength threshold has been set to modify the minimum hop. How many the threshold should be set to minimize the localization error? In order to solve the problem, the BP neural network is introduced to optimize the threshold and reduce the localization error. BP neural network is one of the most widely used models in neural networks. BP algorithm is a neural network of three-tier or more structure. This method with the Error Minimum Principle and using recursive method to find the network weight and threshold of each node among BP algorithms is called deepest gradient descent. The advantages of BP artificial neural network in error optimization are formidable self-organizing and self-learning capacity, error back-propagation, and error minimum principle [7, 10]. A three-layer BP neural network is constructed in this paper (see Figure 3) and operated on the sensor nodes. Steps of the proposed algorithm are as follows.

Satisfy Calculate the location of the unknown node

Figure 1: The flow chart of the improved algorithm.

R A O

Step 1. Firstly, the BP neural network is operated on the sensor nodes and trained by the data of beacon nodes.

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Figure 2: Principle diagram of hops correction.

network. So the hops between two nodes cannot reflect the distance between them exactly. Then, an improved method is proposed to solve the problem in the classical algorithm. The signal strength analysis module is added to the sensor node, and the initial threshold is set as 𝑀, which is the signal strength when the transmission distance is 𝑅/2. The signal strength and the distance between two nodes are usually inversely proportional. So the hop is set as 0.5 if the received signal strength is above the threshold 𝑀; otherwise the hop is set as 1. Therefore, 1, 0.5,

(1) Hidden layer

Figure 3: Neural network model.

Construct the error function

ℎ(𝑖,𝑘) = {

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where rssi(𝑖,𝑘) is the signal strength which the unknown node 𝑖 receives from the beacon node 𝑘 and ℎ(𝑖,𝑘) is the hop from the unknown node 𝑖 to the beacon node 𝑘.

Step 2. The received signal strength threshold 𝑀 and the signal strength (rssi(𝑖,𝑚) , rssi(𝑖,𝑛) , rssi(𝑖,𝑟) ) which the unknown node 𝑖 receives from three beacon nodes are two units of the input layer. The estimated distances (𝑑𝑚 , 𝑑𝑛 , 𝑑𝑟 ) from the unknown node 𝑖 to three beacon nodes are the output layer [11]. Step 3. Calculate the input and output of each unit in the network. The input net1 of hidden layer (1) net1 = 𝑀 = 𝑤ℎ𝑖 𝑀,

(8)

(1) where 𝑤ℎ𝑖 is the weight from unit 𝑖 to unit ℎ. The output 𝑓1 (net1 ) of hidden layer

1, 𝑓1 (net1 ) = ℎ(𝑖,𝑘) = { 0.5,

rssi(𝑖,𝑘) ≤ net1 rssi(𝑖,𝑘) > net1 .

(9)

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(2) , 𝛿ℎ = 𝑓󸀠 1 (net1 ) ∑ (𝑡𝑘 − 𝑑𝑘 ) 𝑓󸀠 2 (net2 ) 𝑤𝑘ℎ

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Figure 4: Distribution of nodes (the red star points represent the anchor nodes and the little blue circles represent the unknown nodes).

𝑘=1

where 𝑡𝑘 is the desired output and 𝑑𝑘 is the actual output. Step 5. Update each weight 𝑤 output unit 𝑘: (2) (2) 𝑤𝑘ℎ = 𝑤𝑘ℎ + 𝜂𝛿𝑘 𝑓1 (net1 )

(14)

and hidden unit ℎ (1) (1) = 𝑤ℎ𝑖 + 𝜂𝛿ℎ 𝑀, 𝑤ℎ𝑖

(15)

where 𝜂 is the learning rate, 𝑖 is the input, and 𝑘 is the output. (1) 𝑀; update the threshold 𝑀 until it obtains Thus, 𝑀 = 𝑤ℎ𝑖 an appropriate value. Step 6. This training process is carried out continuously until the network error is reduced to an acceptable level or preset numbers of learning. Then, the location of the unknown node can be calculated on the basis of (5) and (6) in Section 2.

4. Simulation and Analysis 4.1. The Simulation Platform and Distribution of Nodes. MATLAB simulation software can be used to verify the feasibility of the proposed algorithm. Network deployment area is 1000 m × 1000 m, the node coordinates are generated randomly, the number is 300, there are 60 beacon nodes, the proportion of beacons is 20%, the wireless range is 300 m, and communication model is Regular Model. Distribution of nodes is shown in Figure 4. 4.2. The Definition of Localization Error. The localization error is an extremely important indicator to evaluate the localization performance. Formula (16) is used to calculate the localization error: localization error =

√(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 𝑅

,

(16)

where (𝑥1 , 𝑦1 )2 is the actual location of the unknown node and (𝑥1 , 𝑦1 )2 is the estimated location, where 𝑅 is the wireless range.

4.3. The Simulations of Amorphous Algorithm, VN-BP Algorithm, and Improved Algorithm. First of all, Amorphous algorithm, VN-BP algorithm, and improved algorithm can be simulated to get the localization error figure. It is shown in Figure 5. In Figure 5, the little blue circle represents the estimated location of the unknown node and the blue line represents the localization error of the unknown node. The localization error of Amorphous algorithm is 0.23107. BP Localization Algorithm Based on Virtual Nodes (VN-BP) is proposed in [6]. In Figure 5(b), the localization error is 0.14818, the localization accuracy of VN-BP algorithm is higher than that of the original Amorphous algorithm. In Figure 5(c), the localization error of improved algorithm is 0.11958. We can see that the localization accuracy of the improved algorithm is higher than that of the original Amorphous algorithm and VN-BP algorithm obviously from the figure. 4.4. The Simulations of Amorphous, VN-BP, and Improved Algorithm in Different Proportions of Beacons. The localization error figures of Amorphous algorithm, VN-BP algorithm, and improved algorithm in different proportions of beacons are shown in Figure 6. It is clear that the localization accuracy of the improved algorithm is superior to that of the original Amorphous algorithm and VN-BP algorithm; it improves the localization accuracy effectively. And the localization error of improved algorithm declines as the proportion of beacons increases. 4.5. The Simulations under Different Conditions Are Completed for the Five Algorithms. To find out the performance of the algorithm proposed in this paper, the simulations under different conditions are completed for Centroid, DVHOP, Amorphous, VN-BP (which selects the unknown nodes as the subbeacons based on the virtual nodes), and the improved algorithm. All the nodes in the simulation are randomly distributed in the area 1000 m × 1000 m. Each set of


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Figure 5: The localization error figures: (a) Amorphous algorithm, (b) VN-BP algorithm, and (c) Improved algorithm.

the condition is run for 1000 times so as to factually reflect the localization error of different algorithms. The average value of the localization error is used for the comparison [6]. In the condition of different proportions of beacons, there are 300 nodes in total with wireless range being set as 300 m. From Figure 7(a), it is clear that the localization error of the five algorithms decreases as the proportion of beacons increases. Under the same proportion of beacons, the localization error of the improved algorithm decreases, respectively, about 25%, 20%, and 12% on average than that of the Centroid algorithm, DV-HOP algorithm, and Amorphous algorithm. The localization accuracy of the improved algorithm is higher than that of the VN-BP algorithm, and the localization error is reduced by about 3% on average, demonstrating the effectiveness of the improved algorithm in different proportions of beacons. In the simulation of different wireless range, the proportion of beacons is set by 20% and the total number of nodes is set by 300. Figure 7(b) shows the result of the simulation.

It is clear that the localization error of the improved algorithm and VN-BP algorithm decreases as the wireless range increases. In the same condition, the localization error of the improved algorithm decreases, respectively, about 17% and 18% on average than that of the Centroid algorithm and DVHOP algorithm. The localization error of the Amorphous algorithm increases when wireless range is more than 300 m. And the improved algorithm overcomes the problem. The localization accuracy of the improved algorithm is higher than that of the VN-BP algorithm, and the localization error is reduced by about 3% on average, reflecting the effectiveness of the improved algorithm in different wireless range. Based on different numbers of nodes, wireless range is set as 300 m, and the proportion of beacons is set as 20%. The result is shown in Figure 7(c). It is clear that the localization error of the 5 algorithms decreases as the number of nodes increases. In the same number of nodes, the localization error of the improved algorithm decreases, respectively, about 23%, 22%, and 15% on average than that of the Centroid

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Figure 6: The localization error in different proportions of beacons: (a) Amorphous, (b) VN-BP, and (c) Improved algorithm.

algorithm, DV-HOP algorithm, and Amorphous algorithm. The localization accuracy of the improved algorithm is higher than that of the VN-BP algorithm, and the localization error is reduced by about 3.5% on average, showing the effectiveness of the improved algorithm in different numbers of nodes. 4.6. The Computing Time of Improved Algorithm Is Compared with Other Algorithms. According to the analysis of the paper [12], computational costs make up only a small part of all energy consumption in WSN; most of the energy consumption is communication. So decreasing the energy consumption of communication is the key to extend the lifecycle of network. In order to decrease the energy consumption of communication, all nodes cannot send the information to

a central node to calculate their location, because the energy consumption of communication is too large. The improved localization algorithm should adopt distributed computation, and the neural network is constructed and operated on the sensor nodes. Table 1 compares the average computing time needed by the five algorithms to localize a single node and localization error. All experiments are conducted on the same computer. Network deployment area is 1000 m × 1000 m, the node coordinates are generated randomly, the number is 300, the proportion of beacons is 20%, the wireless range is 300 m, and communication model is Regular Model. Due to the simple calculation, Centroid algorithm generally requires less computing time than the other four algorithms. Amorphous algorithm is similar to DV-Hop algorithm. VN-BP

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Figure 7: The localization error under different conditions: (a) proportion of beacons and localization error, (b) wireless range and localization error, and (c) number of nodes and localization error.

algorithm and improved algorithm require more computing time because of the neural network model construction. In addition, VN-BP algorithm takes much time to find virtual nodes. The algorithms need more computing time as the localization error decreases. This is an unavoidable problem. But the improved algorithm decreases the energy consumption of communication and greatly improves localization accuracy; the total energy consumption increases a little bit. The results will be described in Section 4.7. 4.7. The Energy Consumption of Improved Algorithm Is Compared with Other Algorithms. To find out the performance of the algorithm proposed in this paper, the simulations of

Table 1: Average computing time to localize single node and localization error. Centroid DV-HOP Amorphous VN-BP Improved algorithm

Computing time (s) 0.383 0.512 0.536 0.859 0.772

Localization error 0.29858 0.30776 0.23107 0.14818 0.11958

energy consumption under different conditions are completed for Centroid, DV-HOP, Amorphous, VN-BP, and


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the improved algorithm. All the nodes in the simulation are randomly distributed in the area 1000 m × 1000 m. Each set of the condition is run for 1000 times so as to factually reflect the energy consumption of different algorithms. In the condition of different proportions of beacons, there are 300 nodes in total with wireless range being set as 300 m. The result of energy consumption is shown in Figure 8(a). It is clear that energy consumption of the five algorithms increases as the proportion of the beacons increases. Under the same proportion of beacons, the energy consumption

of Centroid algorithm is lowest, because it broadcasts only once. DV-hop algorithm needs to broadcast twice, so the energy consumption of communication is large [13]. The energy consumption of improved algorithm and that of classical Amorphous algorithm are almost the same. They all adopt distributed computation, so the energy consumption of communication is decreased. But the improved algorithm takes some time to construct the neural network; energy consumption increases a little bit. A little increase of energy consumption can be ignored considering that the improved

International Journal of Distributed Sensor Networks algorithm greatly improves localization accuracy. Moreover, the energy consumption of improved algorithm is lower than that of VN-BP algorithm. This reflects the effectiveness of the improved algorithm in energy consumption. In the case of different wireless range, there are 300 nodes in total with proportion of beacons being set as 20%, and in the case of different numbers of nodes, wireless range is set as 300 m, the proportion of beacons is set as 20%. The similar result can be got in Figures 8(b) and 8(c). It is certain that accurate localization will bring more energy consumption. So localization algorithm should be designed according to different applications.

5. Conclusions The localization accuracy is an extremely important indicator to evaluate the localization performance. The higher the localization accuracy is, the better the localization performance is. Because the disadvantages of the classical Amorphous algorithm will produce large localization error in the process of localization, an improved localization algorithm is proposed in this paper to solve the problem. The improved algorithm introduces the received signal strength threshold to modify the minimum hop from the unknown node to the beacon node. Then, BP algorithm is introduced to optimize the threshold and reduce the localization error. Simulation results show that the improved algorithm overcomes the error stack effect of traditional Amorphous algorithm in the case of multiple hops and greatly improves localization accuracy. In addition, the localization accuracy of improved algorithm is higher than that of other algorithms and the energy consumption does not increase too much. Therefore, it is suitable for the actual large-scale network node localization.


Acknowledgments This work is supported by the National Natural Science Foundation of China (nos. 61472278, 60872064, and 61102125) and the Natural Science Foundation of Tianjin (no. 12JCYBJC10200).

References [1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Wireless sensor networks: a survey,” Computer Networks, vol. 38, no. 4, pp. 393–422, 2002. [2] L. Haibo, W. Yingna, and P. Bao, “A Localization method of Wireless sensor network based on two-hop focus,” Procedia Engineering, vol. 15, pp. 2021–2025, 2011. [3] J. Yao, J. Li, L. Wang, and Y. Han, “Wireless sensor network localization based on improved particle swarm optimization,” in Proceedings of the International Conference on Computing, Measurement, Control and Sensor Network (CMCSN ’12), pp. 72–75, July 2012.

9 [4] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-less low-cost outdoor localization for very small devices,” IEEE Personal Communications, vol. 7, no. 5, pp. 28–34, 2000. [5] D. Niculescu and B. Nath, “DV based positioning in Ad Hoc networks,” Telecommunication Systems, vol. 22, no. 1–4, pp. 267– 280, 2003. [6] R. Liu, K. Sun, and J. Shen, “BP localization algorithm based on virtual nodes in wireless sensor network,” in Proceedings of the 6th International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM ’10), pp. 351–355, September 2010. [7] M. Wang, D. Xiu, R. Wang, F. Du, and Y. Shi, “Data mining research in wireless sensor network based on genetic BP algorithm,” in Proceedings of the 2nd International Conference on Measurement, Information and Control (ICMIC ’13), pp. 243– 247, 2013. [8] Y. Liu, “An adaptive multi-hop distance localization algorithm in WSN,” Manufacturing Automation, vol. 33, pp. 161–163, 2011. [9] W.-X. An, J.-M. Zhao, and D.-A. Li, “Research of localization algorithm for wireless sensor networks based on Amorphous,” Transducer and Microsystem Technologies, vol. 32, pp. 33–35, 2013. [10] T. Lian, M. Xie, J. Xu, L. Chen, and H. Gao, “Modified BP neural network model is used for oddeven discrimination of integer number,” in Proceedings of the International Conference on Optoelectronics and Microelectronics (ICOM ’13), vol. 13, pp. 67–70, 2013. [11] W. Yu, W. Ying-Long, X. Xin, and L. Dong-Yue, “BP algorithm based localization for wireless sensor networks,” Shandong Science, vol. 22, pp. 66–69, 2009. [12] H. Guo, K.-S. Low, and H.-A. Nguyen, “Optimizing the localization of a wireless sensor network in real time based on a low-cost microcontroller,” IEEE Transactions on Industrial Electronics, vol. 58, no. 3, pp. 741–749, 2011. [13] Mansoor-ul-Haque, F. A. Khan, and M. Iftikhar, “Optimized energy-efficient iterative distributed localization for wireless sensor networks,” in Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, pp. 1407–1412, 2013.


Research Article Detecting the Boundary of Sensor Networks from Limited Cyclic Information Carlos Lara-Alvarez,1 Juan J. Flores,2 and Chieh-Chih Wang3 1

Centro de Investigacion en Matematicas (CIMAT), Unidad Zacatecas, 98068 Zacatecas, ZAC, Mexico Division de Estudios de Posgrado, Facultad de Ingenieria Electrica, Universidad Michoacana, 58390 Morelia, MICH, Mexico 3 Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan 2

Correspondence should be addressed to Carlos Lara-Alvarez; [email protected] Received 8 September 2014; Revised 15 January 2015; Accepted 21 January 2015 Academic Editor: S. Kami Makki Copyright © 2015 Carlos Lara-Alvarez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We address the problem of finding the boundaries of a set of points by using a limited range-cyclic order detector. This sensor, denoted by lrcod, is able to detect nearby objects and enumerate them by their cyclic order; neither distance nor the angular position of each object is provided. Boundaries are important in many applications such as detecting the breakdown of networks, insufficient coverage or connectivity, abnormal functioning sensors, and virtual coordinates for routing. We studied the information space of the lrcod sensors and established their capabilities to find inner and outer boundaries. Our proposed approach uses local information to recognize points on the boundary. To discover the complete boundary we define the Right Hand Without Crossings (RHWoC) rule. We also provide a correctness proof of this rule. The experimental evaluation confirms the effectiveness to find the boundary of large sensor networks.

1. Introduction Sensor networks can be seen as a large collection of nodes that reason about the state of the world, that is, their environment. Distributed sensing improves the signal-to-noise ratio because sensors are closer to the phenomena. Obtaining the boundary of a sensor network is important in many engineering applications. Boundaries are useful to detect the breakdown of a network, insufficient coverage or connectivity, abnormal functioning sensors, virtual coordinates for routing, and so forth. Boundary recognition is a challenging issue in wireless sensor networks [1]. The distinction between inner and boundary nodes of the network can provide valuable knowledge to a broad spectrum of algorithms. In practice, sensors are deployed randomly in a given area. Even if the initial deployment is known, the topology may vary because of sensors failure. Furthermore, many applications require dynamic sensors. Ad hoc networks are often modeled as unit disk graphs (UDG), where each point can reach other points in a given radius 𝜆. Accurately defining the boundary of a UDG without

a general position system (GPS) is a challenge that should not be underestimated. The problem we address in this paper is to find the boundaries of a set of points by using a limited range-cyclic order detector. This sensor, denoted by lrcod, can detect nearby objects and enumerate them by their cyclic order; neither distance nor the angular position of each object is provided. The lrcod sensor only measures the cyclic order, unlike bearing measurement systems, where the angle of each object is known. lrcod network can be implemented from visual sensors with miniaturized omnidirectional systems [2, 3]. Another implementation could use infrared based relative positioning sensors specially designed to enable interrobot spatial coordination [4, 5]. Although the model studied in this paper considers a sensor network as its main application, many ideas developed here could be applied to other areas such as mobile robots. For mobile robots, landmarks are like sensors in a network, and the robot provides the communication link between them. Tovar et al. [6] consider a robot that moves in the plane and that is able to sense only the cyclic order of landmarks

2 with respect to its current position. In contrast to [6], which considers an infinite range sensor model, this paper studies a sensor that can only detect objects at a maximum range of 𝜆. A limited sensing model is more convenient for real applications; for example, laser scans have a maximum range, walls in indoor environments can occlude some objects, and so forth. The rest of the paper is organized as follows: Section 2 introduces some basic definitions used along this paper. Section 3 describes the assumed network model and defines the problem. Section 4 describes how to detect a single point on a boundary and using it to recover the complete boundary. Section 5 describes the simulations and results obtained from the proposed algorithm, showing that it works under different conditions. Section 6 describes the related work. Finally, Section 7 concludes this work.


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Figure 1: Sequence of landmarks detected by the sensor. lrcod(V4 ) = [V5 , V3 , V1 , V6 ]; up to a cyclic permutation, landmark 2 cannot be detected because it is outside the reach of the sensor; analogously, lrcod(V3 ) = [V4 , V5 , V2 ]. Only the cyclic order is preserved; the sensed angular position of each landmark may be quite different from the real one.

2. Basic Definitions lrcod is a sensor that gives the counterclockwise cyclic permutations of other sensors or landmark labels. This sensor is able to see landmarks only within a given distance; for a given sensor the parameter 𝜆 is the maximum range at which the cyclic order detector can sense a landmark. Henceforth, lrcod(x) is a function that returns the readings of the sensor; those readings are permutations of the set of sensor identifiers as seen from the point x ∈ R2 . Figure 1 illustrates an example of the sequences of landmarks detected from two positions lrcod(V4 ) = [V5 , V3 , V1 , V6 ] and lrcod(V3 ) = [V4 , V5 , V2 ]. A set of landmarks is in R2 . Let 𝐺 = (𝑉, 𝐸) be a connected graph where 𝑉 is the set of landmark labels and 𝐸 = {{x, y} | x, y ∈ 𝑉, ‖x, y‖ ≤ 𝜆} with (y ≠x); here ‖⋅, ⋅‖ denotes the Euclidean distance between two points. An edge between two landmarks x, y ∈ 𝑉 means that lrcod can detect the landmark y from landmark x and vice versa. We assume that graph 𝐺 is connected; therefore every landmark is detectable from at least one other landmark. Circular Sector. Given two landmarks V𝑖 and V𝑗 seen from Vx , we define Λ󸀠 (Vx ; V𝑖 , V𝑗 ) as the points within the counterclockwise circular sector V𝑖 Vx V𝑗 . Let the output of the sensor be of the form lrcod(Vx ) = [𝐴, V𝑖 , 𝐵, V𝑗 , 𝐶], in which 𝐴, 𝐵, and 𝐶 are subsequences of lrcod(Vx ) separated by the labels corresponding to landmarks V𝑖 and V𝑗 ; then Λ󸀠 (Vx ; V𝑖 , V𝑗 ) = 𝐵 and Λ󸀠 (Vx ; V𝑗 , V𝑖 ) = 𝐶 + 𝐴, where + is the concatenation of two sequences. The sensor V𝑥 shown in Figure 2 reads lrcod(Vx ) = [V3 , V4 , V5 , V1 , V2 ]. Landmarks V3 , V5 define two sectors Λ󸀠 (Vx ; V3 , V5 ) = {V4 } and Λ󸀠 (Vx ; V5 , V3 ) = {V1 , V2 }. Walk. A walk in a graph 𝐺 is a sequence 𝑊 := V0 𝑒0 V1 ⋅ ⋅ ⋅ Vℓ−1 𝑒ℓ−1 Vℓ , whose components are alternately vertices and edges of 𝐺 (not necessarily distinct), such that V𝑖 and V𝑖+1 are the ends of edge 𝑒𝑖 , 0 ≤ 𝑖 ≤ ℓ − 1 [7]. We say that 𝑊 connects the initial point V0 to the final point Vℓ and refer to 𝑊 as a V0 Vℓ -walk or simply V0 𝑊Vℓ . A walk in a graph is closed if its initial and terminal vertices are identical. In case of no confusion we use the succession of vertices 𝑊 =

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Figure 2: Orientation and circular sectors. Circular sectors detected from lrcod: Λ󸀠 (Vx ; V3 , V5 ) = {V4 } and Λ󸀠 (Vx ; V5 , V3 ) = {V1 , V2 }.

V0 V1 ⋅ ⋅ ⋅ Vℓ−1 Vℓ to denote the walk 𝑊 = V0 𝑒0 V1 ⋅ ⋅ ⋅ Vℓ−1 𝑒ℓ−1 Vℓ . The vertices of a closed walk also define a polygon 𝑃. Polygon. A polygon 𝑃 is a finite sequence 𝑃 = V0 V1 ⋅ ⋅ ⋅ V𝑝−1 of points in the plane. The set of edges of a polygon is 𝐸(𝑃) = {𝑒𝑖 | 𝑖 = 0, . . . , 𝑝−1}, where 𝑒𝑖 is defined as an edge connecting V𝑖 and V(𝑖+1 mod 𝑝) . Simple Polygon. A simple polygon is a polygon composed of nonrepeating vertices where none of its edges 𝑒𝑖 and 𝑒𝑗 , for 𝑖 ≠𝑗, intersect at points different from their extremes. The Jordan curve theorem states that every simple closed curve divides the plane into an interior region bounded by the curve and an unbounded region containing all exterior points. Weakly Simple Polygon. A polygon 𝑃 is weakly simple if, for any 𝜖 > 0, there is a simple closed curve 𝑄 such that 𝑑F (𝑃, 𝑄) < 𝜖, where 𝑑F is the Fréchet distance [8] between two polygons 𝑃 and 𝑄.

International Journal of Distributed Sensor Networks In other words, a polygon is weakly simple if it can be made simple by an arbitrarily small perturbation of the entire curve. Informally, a weakly simple polygon is a polygon in which sides can touch each other but cannot cross. Any weakly simple polygon 𝑃 partitions R2 \ 𝑃 into two regions: interior and exterior. 𝜆-Polygon. A 𝜆-polygon 𝑃 is a weakly simple polygon where ‖𝑒𝑖 ‖ ≤ 𝜆 for all 𝑒𝑖 ∈ 𝐸(𝑃). 𝜆-Boundary. A 𝜆-boundary or outer boundary 𝑃 of a set of points 𝑉 is a 𝜆-polygon such that 𝑉 \ 𝑃 lies in the interior of 𝑃. Diagonal. A diagonal is a line that connects two nonconsecutive vertices of a polygon; an inner diagonal lies totally in the interior of the polygon. Hole. A hole or inner boundary is an empty 𝜆-polygon with vertices 𝐻 ⊂ 𝑉 such that every inner diagonal is larger than 𝜆. A hole with 𝑛 = |𝐻| vertices will be called 𝑛-hole.

3. Network Model Throughout this paper, we consider sensor networks that satisfy the following assumptions. (i) The system is composed of a set of sensors 𝑉, located in a region 𝑅 ⊂ R2 ; each sensor V ∈ 𝑉 has a distinctive identity. (ii) Every sensor V ∈ 𝑉 is able to detect lrcod(V). (iii) All wireless sensors have the same maximum transmission range 𝜆. Assumptions (i) and (iii) involve that each sensor can gather information from other sensors within its transmission range. Therefore, we can build a network graph whose vertices correspond to the labeled nodes and whose edges correspond to communication links. Henceforth, we will indistinctly call them sensors, nodes, or landmarks. Assumption (i) admits that sensors can move arbitrarily in 𝑅 but they are static for a period of time 𝜏 long enough to recover the boundaries. Furthermore, we assume that landmarks are in general position (i.e., no three landmarks are collinear). Assumption (iii) is usual in ad hoc networks modeled as unit disk graphs (UDG), where each point can reach other points in a given radius 𝜆. Problem Definition. Given a network 𝑉 of lrcod sensors, find the boundary polygon of 𝑉 and its holes by using only the local information provided by the lrcod sensors. The task we are solving is to find the perimeter of a set of points; this task can be done by simply following the sequence of extreme points on the convex hull. Classical algorithms such as the Graham Scan [9] can be used to find the convex hull (CH) of the point set. Unfortunately, following it by using the limited rangecyclic order detector is in certain cases impossible, because the distance between two consecutive extreme points could be greater than the maximum range of the sensor. Hence, when

3 following the CH of the set of lrcod sensors, the algorithm can reach a position where the sensor cannot see the next point in the CH, so it has to share information through other points inside the CH (points at sight) until the next point on the CH (not at sight before) is reached. Once it has done so, it can continue finding points on the CH. This approach is time consuming and sometimes unnecessary.

4. Proposed Algorithm The proposed approach finds points on the boundary following the procedure described in Section 4.1. Those points can be used as seeds to find the complete sequence of points on the boundary, as described in Section 4.2. 4.1. Boundary Points. This subsection presents a series of definitions and lemmas that are the basis for discerning boundary from inner points. According to our definition of holes a 𝜆-polygon 𝑃 formed by less than four points cannot be a boundary; hence, three connected points on the vertices of a triangle cannot form a hole; therefore, a hole must have at least four landmarks. 𝜆-Point. A 𝜆-point is a point V𝑗 for which ∃V𝑖 , V𝑘 ∈ lrcod(V𝑗 ) such that Λ󸀠 (V𝑗 ; V𝑖 , V𝑘 ) = {} and V𝑖 ∉ lrcod(V𝑘 ). Points V𝑖 , V𝑘 are called adjacent boundary points of V𝑗 . 4.2. Detecting a Boundary from 𝜆-Points. This subsection addresses how to use 𝜆-points as seeds to find the boundary sequence. Let ℎ𝑗 and ℎ𝑗+1 be two consecutive points on the convex hull and let (ℎ𝑗 , ℎ𝑗+1 ) be the line segment joining them. Since the graph 𝐺 is connected, there exists at least one ℎ𝑗 ℎ𝑗+1 -walk. We are looking for the walk such that the area comprised by 𝑊󸀠 := ℎ𝑗 𝑊ℎ𝑗+1 (ℎ𝑗+1 , ℎ𝑗 )ℎ𝑗 does not contain any landmark V ∈ 𝑉; we refer to the walk that satisfies this condition as a pocket. The right-hand rule (RH) is a heuristic rule that has been used extensively to deliver packages in ad hoc networks [10]. It has also been used to find the convex hull for an unlimited cyclic sensor [6]. The right-hand rule states that if the last two previously visited vertices of a walk are V𝑗−1 , V𝑗 then the next vertex is the first vertex that appears immediately by turning right from the edge (V𝑗−1 , V𝑗 ). When the greedy multi-hop forwarding algorithm gets stuck, the right hand rule can be used to go around holes. To complete a boundary starting from a border walk, we define the Right Hand Without Crossings (RHWoC)rule. Given a partial walk 𝑊󸀠 = [V0 ⋅ ⋅ ⋅ V𝑗−1 𝑒𝑗−1 V𝑗 ] such that its last edge is (V𝑗−1 , V𝑗 ), the RHWoC rule selects the first landmark V𝑘 in lrcod(V𝑗 ) that follows V𝑗−1 in the permutation such that the edge 𝑒𝑗 := (V𝑗 , V𝑘 ) does not intersect at a single point to any previous edge in the walk. To illustrate the RHWoC rule, Figure 4 shows an example to find the next point of the walk 𝑊 = V1 𝑒12 V2 𝑒24 V4 𝑒46 V6 . When the point V6 is analyzed, V3 appears in the permutation immediately after point V4 but it is not selected because the edge (V6 , V3 ) intersects the boundary; then the correct point V5 is selected.

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Figure 3: (a) Detectable 4-hole, the sensor V𝑗 reads an empty sector. (b) Nondetectable hole, all sensors on the hole read nonempty sectors.

4.3. Discovering Intersections. The RHWoC rule selects an edge that does not cross any previous edge in the border walk, but the border walk 𝑊 could be a long sequence of vertices and edges. The following lemma states that when two edges intersect at a single point, at least one vertex of an edge must see at least one vertex of the other edge. Lemma 1. Let 𝑒𝑖𝑗 = (V𝑖 , V𝑗 ) and 𝑒𝑘𝑛 = (V𝑘 , V𝑛 ) be two edges, and let lrcod𝑘,𝑛 := lrcod(V𝑘 ) ∪ lrcod(V𝑛 ) be the landmarks seen from V𝑘 or V𝑛 . If (V𝑖 ∉ lrcod𝑘,𝑛 ) ∧ (V𝑗 ∉ lrcod𝑘,𝑛 ) then 𝑒𝑖𝑗 does not intersect 𝑒𝑘𝑛 . Proof. When 𝑒𝑖𝑗 and 𝑒𝑘𝑛 intersect at a single point 𝑞, there exist two points V𝑎 ∈ {V𝑖 , V𝑗 } and V𝑏 ∈ {V𝑘 , V𝑛 } such that ‖V𝑎 𝑞‖ ≤ 𝜆/2 and ‖V𝑏 𝑞‖ ≤ 𝜆/2. From the law of cosines it can be shown that ‖V𝑎 , V𝑏 ‖ ≤ 𝜆; in other words, V𝑏 ∈ lrcod(V𝑎 ). We are interested in determining if two different edges intersect in a single point. When discovering crossing edges, the endpoints of two edges could be the same; hence, there is only one rather than two distinct edges. This condition is easily detectable and can be used to discover connectivity issues of the network such as bridges and articulation points [11]. Lemma 2 states how to use the information provided by lrcod to detect that two edges in a walk intersect in a single point. Lemma 2. Let Vℎ 𝑒ℎ𝑖 V𝑖 𝑒𝑖𝑗 V𝑗 𝑒𝑗𝑘 V𝑘 ⋅ ⋅ ⋅ V𝑚 be a border walk, and let V𝑛 ∈ lrcod(V𝑚 ) be a point seen from V𝑚 ; edge 𝑒𝑛 := (V𝑚 , V𝑛 ) intersects edge 𝑒𝑖𝑗 in a single point when (V𝑛 ∈ Λ󸀠𝑗 (V𝑘 , V𝑖 ))∧(V𝑛 ∈ Λ󸀠𝑖 (V𝑗 , Vℎ )) and lrcod(V𝑛 ) is arranged as [V𝑖 , 𝐴, V𝑘 , 𝐵, V𝑗 , 𝐶] with 𝐴, 𝐵, and 𝐶 subsequences of lrcod(V𝑛 ). Proof. Figure 5(a) illustrates this lemma for V𝑚 = V𝑘 . Every point satisfying (V𝑛 ∈ Λ󸀠𝑗 (V𝑘 , V𝑖 )) ∧ (V𝑛 ∈ Λ󸀠𝑖 (V𝑗 , Vℎ )) is in the red shadow. When lrcod(V𝑛 ) = [V𝑖 , 𝐴, V𝑘 , 𝐵, V𝑗 , 𝐶], meaning that V𝑛 is in the circular sector (of a large circle) V𝑗 V𝑘 V𝑖 , then 𝑒𝑘𝑛 intersects edge 𝑒𝑖𝑗 . Corollary 3. Let V𝑗 be a 𝜆-point with adjacent boundary points V𝑖 , V𝑘 . If V𝑛 ∉ Λ󸀠𝑗 (V𝑘 , V𝑖 ), 𝑒𝑘𝑛 := (V𝑘 , V𝑛 ) does not intersect 𝑒𝑖𝑗 . In other words, a walk of four consecutive boundary points can be found from two consecutive 𝜆-points.

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Figure 4: Finding the boundary from a partial walk. The boundary is recovered from the walk V1 V2 V4 V6 ; point V5 ∈ lrcod(V6 ) is selected because 𝑒65 := (V6 , V5 ) does not intersect any previous edge.

Proof. Corollary 3 follows directly from Lemma 2. In Figure 5(b), the edge (V𝑘 , V𝑛 ) does not cross any edge when V𝑛 ∉ Λ 𝑗 (V𝑘 , V𝑖 ). Point V𝑘 is also a 𝜆-point because Λ󸀠𝑗 (V𝑖 , V𝑘 ) is empty. Observation 1. It is always possible to define intersections for a walk with four or more vertices by using Lemma 2. Although it is useful to find inner diagonals of a hole, 𝜆-points are better to find points on the boundary when two consecutive points on the convex hull cannot see each other. Sometimes small holes do not have detectable 𝜆points; for instance, the 4-hole shown in Figure 3(a) can be detected because Λ󸀠𝑗 (V𝑘 , V𝑖 ) is empty, but Λ󸀠𝑗 (V𝑘 , V𝑖 ) = {V} in Figure 3(b). Those undetectable cases never occur for points on the convex hull. Observation 2. If there is only a single 𝜆-point (a walk with three points instead of four) then Corollary 3 can be used. We can define some other cases such as that shown in Figure 5(c) where V𝑛 ∈ Λ 𝑗 (V𝑘 , V𝑖 ) and lrcod(V𝑛 ) reads [V𝑖 , 𝐴, V𝑘 , 𝐵, V𝑗 , 𝐶]. Unfortunately, the limited range of the sensor prevents us from using this case for all situations because sometimes V𝑖 ∉ lrcod(V𝑛 ). We say that the intersection is undetermined for the cases where we cannot determine if an edge intersects or not with another edge of a walk. In such cases, the boundary could be completed by using other 𝜆-points.


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Data: A finite set of connected points 𝑉 ⊂ 𝑅2 . Result: A list L of the boundary and detectable holes of 𝑉. (1) Exchange sensor information between neighbors; (2) L ← {}; (3) foreach V𝑖 ∈ 𝑉 do (4) if V𝑖 𝑖𝑠 𝑎 𝜆-𝑝𝑜𝑖𝑛𝑡 𝑤𝑖𝑡ℎ 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝑠 𝑝𝑜𝑖𝑛𝑡𝑠 V𝑎 , V𝑏 then (5) 𝑊󸀠 ← V𝑎 (V𝑎 , V𝑖 )V𝑖 (V𝑖 , V𝑏 )V𝑏 ; (6) if 𝑊󸀠 𝑖𝑠 𝑛𝑜𝑡 𝑢𝑠𝑒𝑑 𝑖𝑛 L 𝑎𝑛𝑑 V𝑏 𝑖𝑠 𝑎 𝜆-𝑝𝑜𝑖𝑛𝑡 𝑤𝑖𝑡ℎ 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑝𝑜𝑖𝑛𝑡𝑠 V𝑖 , V𝑐 then (7) 𝑊 ← RHWoC (V𝑎 , V𝑖 , V𝑏 , V𝑐 ); (8) L ← L ∪ 𝑊; (9) end (10) end (11) end (12) return L Algorithm 1: Boundary (𝑉).

(1) 𝑒𝑗 ← (V𝑖−1 , V𝑖 ); (2) V𝑗−1 ← V𝑖−1 ; (3) V𝑗 ← V𝑖 ; (4) 𝑊 ← V𝑖−3 V𝑖−2 V𝑖−1 V𝑖 ; (5) repeat (6) 𝐿 ← lrcod(V𝑗 ); (7) Circular-rotate 𝐿 until V𝑗−1 is the last element of 𝐿; (8) foreach V𝑘 ∈ 𝐿 do (9) if (𝑘 = 𝑗 − 1) 𝑜𝑟 (V𝑗 , V𝑘 ) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑎𝑛𝑦 𝑒𝑑𝑔𝑒 𝑜𝑓 𝑊 then (10) 𝑒𝑗 ← (V𝑗 , V𝑘 ); (11) V𝑗−1 ← V𝑗 ; (12) V𝑗 ← V𝑘 ; (13) 𝑊 ← 𝑊𝑒𝑗 V𝑗 ; (14) break; (15) end (16) end (17) until (𝑒𝑗 ∉ 𝑊); (18) return 𝑊 Algorithm 2: Function RHWoC (V𝑖−3 , V𝑖−2 , V𝑖−1 , V𝑖 ).

4.4. Algorithm Overview. To find the boundaries of a set of points 𝑉 we propose Algorithms 1 and 2. In Algorithm 1 each sensor first obtains local information (line 1), then it applies the definition of Section 4.1 to recognize two consecutive 𝜆-points (lines 4–6). If the 𝜆-point and its corresponding adjacent points are not already used, then a boundary (or a hole) has been detected and the function RHWoC is called (line 7). Function alg:closed recovers the complete border until it finds an edge that already is on the boundary. When no more borders can be found then the list L is returned (line 12). Correctness of Function RHWoC. Let W be the outer area reachable with the lrcod from vertices of 𝑊. At each step, the next landmark V𝑘 in the walk is selected according to the RHWoC rule. The rule’s precondition asserts that V1 , . . . , V𝑖−1 are 𝜆-convex hole points; then W does not contain any landmark. For the following steps, RHWoC rule ensures that

𝑆𝑗 = Λ󸀠 (V𝑗 ; V𝑗−1 , V𝑘 ) ∩ W does not contain any landmark. As ⋃∀V𝑗 ∈𝑊 𝑆𝑗 = W and the walk never crosses itself, the Jordan theorem asserts that 𝑊 is a boundary and the set of landmarks 𝑉 \ 𝑊 are in the same side of 𝑊. Finally, based on our connectivity assumption, the walk reaches a previously detected edge and the cycle of lines 5–17 terminates correctly.

5. Simulations The proposed algorithm has been implemented for randomly generated sets in a quadrangle of 30 × 30. The number of nodes and the value of 𝜆 are user-defined parameters. Examples of networks for different values of 𝑛 are illustrated in Figure 6. The proposed algorithm finds correctly the outer boundaries of networks which are relatively sparse (Figures 6(a) and 6(b)) and dense (Figures 6(c) and 6(d)). Figures 6(a) and 6(b) also show the detectable inner boundaries.

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6. Related Work A well-known technique to find the boundary of a set of points is the 𝛼-shape [12]. Alpha shapes are closely related to alpha complexes, subcomplexes of the Delaunay triangulation of the point set. Some approaches to detect boundaries in networks are (a) distribution of nodes in the nonhole regions [13], (b) length of the shortest path between two nodes to detect cycles [14, 15], and (c) patterns that the nodes produce [1]. In robotics, landmarks are useful for navigation, place recognition, localization, and mapping. The proposed approach deals with a more challenging setting compared with [6], on which a robot that is only able to detect the cyclic angular order of landmarks in an unlimited range can accomplish simple tasks, such as navigation and patrolling. One of the advantages of this approach is that the memory of the robot does not save the complete state of the environment. To compensate, it considers an infinite range in the landmark order detector.

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7. Conclusions The problem we address in this paper is to find the boundaries of a sensor network by using a limited range-cyclic order detector. By using local information we show how to detect 𝜆-points; these points are on the boundary or on a hole. The Right Hand Without Crossings (RHWoC) rule can be used to discover other landmarks from an incomplete border walk. We provide a correctness proof of the RHWoC rule. Some holes could not be detected by using local information. We also show the configurations where the boundary is detectable; the most remarkable case is when there are two consecutive 𝜆-points. Therefore, the proposed technique is appropriate for large sensor networks. The experimental evaluation confirms the effectiveness to find the boundary of large sensor networks.


References [1] O. Saukh, R. Sauter, M. Gauger, and P. J. Marrón, “On Boundary recognition without location information in wireless sensor networks,” ACM Transactions on Sensor Networks, vol. 6, no. 3, article 20, 2010. [2] P. Ferrat, C. Gimkiewicz, S. Neukom, Y. Zha, A. Brenzikofer, and T. Baechler, “Ultra-miniature omni-directional camera for an autonomous flying micro-robot,” in Conference on Optical and Digital Image Processing, vol. 7000 of Proceedings of SPIE, 2008. [3] B. Tavli, K. Bicakci, R. Zilan, and J. M. Barceló-Ordinas, “A survey of visual sensor network platforms,” Multimedia Tools and Applications, vol. 60, no. 3, pp. 689–726, 2012. [4] J. Pugh, X. Raemy, C. Favre, R. Falconi, and A. Martinoli, “A fast onboard relative positioning module for multirobot systems,”

[12]

[13]

[14]

[15]

IEEE/ASME Transactions on Mechatronics, vol. 14, no. 2, pp. 151– 162, 2009. J. F. Roberts, T. Stirling, J.-C. Zufferey, and D. Floreano, “3-D relative positioning sensor for indoor flying robots,” Autonomous Robots, vol. 33, no. 1-2, pp. 5–20, 2012. B. Tovar, L. Freda, and S. M. Lavalle, “Learning combinatorial map information from permutations of landmarks,” International Journal of Robotics Research, vol. 30, no. 9, pp. 1143–1156, 2011. A. Bondy and U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, Springer, 2008. A. Ribó Mor, Realization and counting problems for planar structures: trees and linkages, polytopes and polyominoes [Ph.D. thesis], Freie Universität Berlin, Berlin, Germany, 2006. R. L. Graham and F. F. Yao, “Finding the convex hull of a simple polygon,” Journal of Algorithms, vol. 4, no. 4, pp. 324–331, 1983. R. B. Muhammad, “A distributed graph algorithm for geometric routing in ad hoc wireless networks,” Journal of Networks, vol. 2, no. 6, pp. 50–57, 2007. G. Ausiello, D. Firmani, and L. Laura, “Real-time monitoring of undirected networks: articulation points, bridges, and connected and biconnected components,” Networks, vol. 59, no. 3, pp. 275–288, 2012. H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel, “On the shape of a set of points in the plane,” IEEE Transactions on Information Theory, vol. 29, no. 4, pp. 551–559, 1983. S. Funke and C. Klein, “Hole detection or: how much geometry hides in connectivity?”,” in Proceedings of the 22nd Annual Symposium on Computational Geometry (SCG ’06), pp. 377–385, ACM, New York, NY, USA, 2006. A. Kröller, S. P. Fekete, D. Pfisterer, and S. Fischer, “Deterministic boundary recognition and topology extraction for large sensor networks,” in Proceedings of the 17th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA ’06), pp. 1000–1009, ACM, New York, NY, USA, 2006. Y. Wang, J. Gao, and J. S. B. Mitchell, “Boundary recognition in sensor networks by topological methods,” in Proceedings of the 12th Annual International Conference on Mobile Computing and Networking (MOBICOM ’06), pp. 122–133, ACm, New York, NY, USA, September 2006.


Research Article Optimal Energy Allocation Scheme in Distributed Estimation for Wireless Sensor Networks over Rayleigh Fading Channels Eni D. Wardihani,1,2 Wirawan,1 and Gamantyo Hendrantoro1 1

Department of Electrical Engineering, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia Department of Electrical Engineering, Politeknik Negeri Semarang, Semarang 50275, Indonesia

2

Correspondence should be addressed to Eni D. Wardihani; [email protected] Received 28 August 2014; Revised 24 November 2014; Accepted 9 December 2014 Academic Editor: S. Kami Makki Copyright © 2015 Eni D. Wardihani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We focus on the issue of energy and bandwidth limitations in Wireless Sensor Networks (WSN). To tackle this issue, we propose the optimal energy allocation scheme and the optimal number of quantization bits by using alternating optimization method. Firstly, we determine the optimal energy allocation scheme which minimizes the reconstruction error on fusion center by keeping the number of quantization bits per sensor fixed. Secondly, we determine the optimal number of quantization bits per sensor such that the energy allocation scheme achieves the minimum reconstruction error. To find the optimal energy allocation scheme and the optimal number of quantization bits jointly, we propose an iterative algorithm. The results show that the proposed algorithms do not only achieve better results than equal energy allocation scheme but also produce reconstruction error close to that of unquantized estimation. This paper also investigates the effects of sensor noise observation and propagation losses on the design of optimal energy allocation scheme. The optimal energy allocation scheme suggests the allocation of more energy to sensors with small variance noise of sensor observation.

1. Introduction Wireless Sensor Networks (WSN) is one of the new technologies growing very rapidly in the last two decades. It happens because WSN is one of the solutions to challenging problems in various fields, such as environment, health, military, industrial, and residential applications. However, many technical problems have to be faced in developing WSN technology to a broader and better technology. The main challenges of WSN are the energy and bandwidth limitations, because in many instances sensor is powered by battery which has limited lifetime. In addition, communication equipments which are used to transmit data over the wireless networks have to consider the bandwidth availability and range of sensor [1]. Energy and bandwidth efficiency algorithms on WSN have been widely investigated. Various signal processing methods have been proposed [2]. Prior study of energy and bandwidth efficiency on WSN assumes that the distribution of sensor noise observation is known [3]. Unfortunately, characterizing the exact distribution of sensor noise from

observation is impractical, especially for applications in a dynamic sensing environment [4]. The authors of [5] proposed a Universal Decentralized Estimation (UDES) that does not require the knowledge of sensor noise distributions. Based on this system, several algorithms to overcome energy and bandwidth limitations have been developed, such as [6, 7]. However, all of these works assume that each sensor transmits the quantized data to the fusion center without error. This is not very realistic because the links between the sensors and the FC are affected by attenuation and fading that can degrade estimation performance. In [8–11], the authors consider an imperfect channel modelled as a Binary Symmetric Channel (BSC) and an Additive White Gaussian Noise (AWGN) channel. However, the channels are not fading in either of the cases. In contrast to the works which have been mentioned above, the main purpose of this paper is to optimize energy and bandwidth in distributed estimation system for WSN over a flat fading Rayleigh channel with path loss effects without prior knowledge of sensor noise distribution. Some

2

International Journal of Distributed Sensor Networks Flat fading Rayleigh channel n

𝜃

Sensor

A= 𝜃+n

Quantizer

̂ ̂ Fusion b1 , b2 , . . . ̂ A center

A Q b1 , b2 , . . .

Figure 1: Single sensor to fusion center.

additional details regarding the contributions of the paper are listed below: (i) we propose the optimal energy allocation scheme and the optimal number of quantization bits by using alternating optimization method. Firstly, we determine the optimal energy allocation scheme per sensor which minimizes the reconstruction error at fusion center by keeping the number of quantization bits fixed. Secondly, we determine the optimal number of quantization bits at sensor such that the optimal energy allocation scheme achieves the minimum reconstruction error at fusion center; (ii) we present an iterative algorithm to find the optimal energy allocation scheme and the optimal number of quantization bits jointly; (iii) this paper investigates the effect of the sensor noise observation, which has been not considered in previous efforts [6–11]. We formulate the optimal energy allocation scheme and the optimal number of quantization bits that can minimize the mean absolute reconstruction error at the fusion center as a convex optimization which results in a close form of each of them. The analytical form of the optimal number of quantization bits in a single sensor to a fusion center suggests that the optimal number can be found when the energy allocation for each bit is equal and produces the reconstruction error close to unquantized estimation. In the distributed estimation system, the optimal energy allocation scheme suggests the allocation of more energy to sensors with small variance noise of sensor observation. This paper is organized as follows. Section 2 discusses the optimal energy allocation scheme and the optimal number of quantization bits on a simple network consisting of a sensor and fusion center. The optimal energy allocation scheme and the optimal number of quantization bits on distributed estimation system consists of a set of sensors and a fusion center which are analyzed in Section 3. Simulation results are then presented in Section 4 and our conclusions can be found in Section 5.

𝐿

𝐴 𝑄 = ∑𝑏𝑖 2−𝑖 ,

𝑖 = 1, . . . , 𝐿.

(2)

𝑖=1

The quantization bits {𝑏𝑖 }𝐿𝑖=1 are transmitted to fusion center over Rayleigh flat fading channel with path loss effect. At the fusion center the sensor observations are reconstructed as 𝐿

̂ = ∑̂𝑏𝑖 2−𝑖 , 𝐴

𝑖 = 1, . . . , 𝐿.

(3)

𝑖=1

In the following, we firstly focus to derive the optimal number of quantization bits if the energy allocation of sensor is fixed. Afterwards, we use the optimal number of quantization bits to determine the optimal energy allocation scheme in sensor by minimizing the reconstruction error. 2.1. Optimal Number of Quantization Bits for Equal Energy Allocation in Each Bit. A simple scenario is adopted where all quantization bits have equal energy allocation 𝐸𝑏𝑖 = 𝐸𝑏 , ∀𝑖. The total transmission energy of all bits is fixed to 𝐸. When using an 𝐿-bit quantizer, the energy per bit (𝐸𝑏 ) depends on 𝐿 since 𝐸𝑏 = 𝐸/𝐿. The channel between sensor and fusion center is flat fading Rayleigh channel with the error probability in high SNR (𝛾 ≫ 1) for BPSK modulation [12]: 𝑃𝑒 (𝛾) =

1 , 4𝛾

(4)

where 𝛾 = 𝐸𝑏 /𝑁0 = 𝐸/𝐿𝑁0 and 𝑁0 denotes noise power spectral density. The reconstruction error in fusion center is defined as 𝐴− ̂ written as follows: 𝐴, ̂ = 𝐴 − 𝐴𝑄 + 𝐴𝑄 − 𝐴 ̂ 𝐴−𝐴

2. Single Sensor to Fusion Center We analyze the system as depicted in Figure 1, where single sensor making observation on an unknown parameter 𝜃. The observation is corrupted by noise and is described by 𝐴 = 𝜃 + 𝑛,

where the noise 𝑛 has zero mean and variance 𝜎2 . Suppose 𝐴 ∈ [−𝑊, 𝑊] is the signal range that sensor can observe, where 𝑊 is a known parameter decided by the sensors dynamic range. The sensor observations are normalized in the range [0, 1] using linear transformation. The sensor quantizes the real valued measurement 𝐴 to 𝐴 𝑄. The signal is quantized into 𝐿 bits with uniform quantization so that

(1)

∞

𝐿

𝐿

𝐿

𝑖=1

𝑖=1

𝑖=1

𝑖=1

= ∑𝑏𝑖 2−𝑖 + ∑𝑏𝑖 2−𝑖 − ∑𝑏𝑖 2−𝑖 + ∑̂𝑏𝑖 2−𝑖 ∞

𝐿

𝑖=𝐿+1

𝑖=1

= ∑ 𝑏𝑖 2−𝑖 + ∑ (𝑏𝑖 − ̂𝑏𝑖 ) 2−𝑖 .

(5)


3

By using triangle inequality, the limit absolute value of reconstruction error can be written as follows: ∞ 𝐿 󵄨󵄨 ̂󵄨󵄨󵄨󵄨 ≤ ∑ 2−𝑖 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖 − ̂𝑏𝑖 󵄨󵄨󵄨󵄨 2−𝑖 󵄨󵄨𝐴 − 𝐴 󵄨 󵄨 󵄨 󵄨 𝑖=𝐿+1 −𝐿

≤2

𝑖=1

𝐿

(6)

𝐿

In order to determine the mean absolute reconstruction error with respect to 𝑝 := [𝑝1 , . . . , 𝑝𝐿 ]𝑇 , we can formulate the optimization problem as follows:

󵄨 󵄨 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖 − ̂𝑏𝑖 󵄨󵄨󵄨󵄨 2−𝑖 . 𝑖=1

min𝑓𝑜 (𝑝; 𝐿) := 2−𝐿 + ∑𝑃𝑒 (

𝑝𝑖 𝐸 −𝑖 )2 𝑁0

subject to 𝑓𝑖 (𝑝) := 𝑝𝑖 ≥ 0,

𝑖 = 1, . . . , 𝐿,

𝑝

𝑖=1

(11)

𝐿

𝑔 (𝑝) := ∑𝑝𝑖 = 1.

If we take the expectation of both sides of (6), we can obtain

𝑖=1

𝐿

󵄨 ̂󵄨󵄨󵄨󵄨] ≤ 2−𝐿 + ∑𝐸 [󵄨󵄨󵄨󵄨𝑏𝑖 − ̂𝑏𝑖 󵄨󵄨󵄨󵄨] 2−𝑖 , 𝐸 [󵄨󵄨󵄨󵄨𝐴 − 𝐴 󵄨 󵄨 󵄨

(7)

𝑖=1

where 𝐸[|𝑏𝑖 − ̂𝑏𝑖 |] = 𝑃𝑒 (𝛾), so that 𝐿 󵄨 ̂󵄨󵄨󵄨󵄨] ≤ 2−𝐿 + 𝑃𝑒 (𝛾) ∑2−𝑖 𝐸 [󵄨󵄨󵄨󵄨𝐴 − 𝐴 󵄨

𝑝𝑖∗ = (√

𝑖=1

≤ 2−𝐿 + 𝑃𝑒 (𝛾) (1 − 2−𝐿 ) ≤ 2−𝐿 + (1 − 2−𝐿 ) 𝑃𝑒 (

(8)

𝐸 ) := 𝑓 (𝐿) . 𝐿𝑁0

𝐿 opt = arg min𝑓 (𝐿) 𝐿

= arg min 2−𝐿 + (1 − 2−𝐿 ) 𝑃𝑒 (

𝐸 ). 𝐿𝑁0

(9)

In the first term of (9), 2−𝐿 decreases as 𝐿 increases. The second term, (1 − 2−𝐿 )𝑃𝑒 (𝐸/𝐿𝑁0 ), increases with increasing 𝐿. It shows that there is an optimal value of 𝐿. Therefore, the function 𝑓(𝐿) which is the sum of two terms, will reach a minimum value. The values of 𝐿 which minimize the function 𝑓(𝐿) can be found by numerical method. Section 4 gives some examples of the optimal number of quantization bits 𝐿 opt . 2.2. Optimal Energy Allocation in Each Bit. In the previous subsection, the energy allocation for each bit is equal. However, each bit in (8) has a different weight. Therefore, we can optimize the energy allocation for each bit. When the total number of quantization bits 𝐿 is fixed, the energy of each bit is 𝐸𝑏 = 𝑝𝑖 𝐸, where 𝑝𝑖 is the proportion of energy of the 𝑖th bit from the total energy of sensor 𝐸. We can optimize 𝐸𝑏 by optimizing 𝑝𝑖 , 𝑖 = 1, . . . , 𝐿 with ∑𝐿𝑖=1 𝑝𝑖 = 1, 𝑝𝑖 ≥ 0. The error probability in (4) becomes 𝑃𝑒 (𝛾) = 𝑃𝑒 (𝐸𝑏𝑖 /𝑁0 ) = 𝑃𝑒 (𝑝𝑖 𝐸/𝑁0 ) and (9) becomes 󵄨 ̂󵄨󵄨󵄨󵄨] = 2−𝐿 + (1 − 2−𝐿 ) 𝑃𝑒 ( 𝑝𝑖 𝐸 ) . 𝐸 [󵄨󵄨󵄨󵄨𝐴 − 𝐴 󵄨 𝐿𝑁0

1 ), 4 (𝐸/𝑁0 ) 2𝑖 𝜐∗

𝑖 = 1, . . . , 𝐿,

(12)

where 𝜐∗ is a constant chosen such that ∑𝐿𝑖=1 𝑝𝑖∗ = 1 can be fulfilled.

Based on (8), reconstruction error on fusion center can be minimized by determining the value of 𝐿 which minimizes 𝑓(𝐿) function. Hence, the optimal number of quantization bits is

𝐿

Proposition 1. (1) The objective function in (11) is a convex function. ∗ (2) For a given 𝐿, (11) admits a unique solution 𝑝𝐿 := [𝑝1∗ , . . . , 𝑝𝐿∗ ]𝑇 that can minimize the objective function ∗ 𝑓𝑜 (𝑝𝐿 ; 𝐿):

Proof. See Appendix A.

3. Distributed Estimation System: Multisensor to Fusion Center The distributed estimation system assumed herein consists of a set of sensors and a fusion center with a star topology. This system is used to observe and estimate an unknown deterministic parameter 𝜃 as shown in Figure 2. Each sensor makes an observation which is corrupted by additive noise and described by 𝑥𝑘 = 𝜃 + 𝑛𝑘 ,

𝑘 = 1, 2, . . . , 𝐾,

(13)

where 𝑥𝑘 ∈ [−𝑊, 𝑊] and 𝑊 is sensor observation range that depends on sensor specification. The sensor noises {𝑛𝑘 , 𝑘 = 1, 2, . . . , 𝐾} are zero mean, spatially uncorrelated with variance 𝜎𝑘2 . The sensor observations are normalized in the range [0, 1] by using linear transformation. Sensor 𝑘 quantizes its observation 𝑥𝑘 to the 𝐿 𝑘 quantization bits and the quantization results are 𝐿𝑘

(𝑥𝑘 )𝑄 = ∑𝑏𝑖(𝑘) 2−𝑖 .

(14)

𝑖=1

𝐿

𝑘 Bits {𝑏𝑖(𝑘) }𝑖=1 are sent to fusion center over the wireless channel, which is again modeled as a flat fading Rayleigh channel with error probability 𝑃𝑒𝑘 (𝛾). The signal reconstruction at the fusion center is

𝐿𝑘

(10)

𝑥̂𝑘 = ∑̂𝑏𝑖(𝑘) 2−𝑖 . 𝑖=1

(15)

4

International Journal of Distributed Sensor Networks n1 x1 = 𝜃 + n1 Sensor1

Quantizer1

(x1 )Q

n2 x2 = 𝜃 + n2 𝜃

Sensor2

Quantizer2

nK

.. .

xK = 𝜃 + nK

(x2 )Q

.. .

Fusion center

𝜃̂

(xK )Q

QuantizerK

SensorK

Figure 2: Distributed estimation system: multisensor to fusion center.

The problem of interest is to design the optimal energy allocation scheme which minimizes the mean absolute reconstruction error 𝐸|𝜃̂ − 𝜃|2 when the number of quantization bits per sensor is fixed 𝐿 𝑘 . Then, we determine the optimal number of quantization bits per sensor such that the energy allocation scheme achieves the minimum reconstruction error 𝐸|𝜃̂ − 𝜃|2 . If the estimation error in fusion center is [13] −1 𝐾

𝐾 1 𝜃̂ − 𝜃 = ( ∑ 2 ) 𝜎 𝑘=1 𝑘

𝑥̂𝑘 − 𝜃 𝜎𝑘2 𝑘=1 ∑

(16)

𝐾

1 = (∑ 2 ) 𝜎 𝑘=1 𝑘

𝐾

𝜎𝑘2

󵄨󵄨 ∞ 󵄨 󵄨 󵄨 𝑘 󵄨󵄨∑𝑖=𝐿 +1 2−𝑖 + ∑𝐿𝑖=1 𝐸󵄨󵄨󵄨󵄨 (̂𝑏𝑖(𝑘) − 𝑏𝑖(𝑘) ) 󵄨󵄨󵄨󵄨2−𝑖 󵄨󵄨󵄨󵄨 𝑘 󵄨 ≤∑ , 𝜎𝑘2 𝑘=1

𝑥̂𝑘 − (𝑥𝑘 − 𝑛𝑘 ) 𝜎𝑘2 𝑘=1 ∑

−1 𝐾

(17)

𝑥̂ − 𝑥𝑘 + 𝑛𝑘 . ∑ 𝑘 𝜎𝑘2 𝑘=1

(18)

2

2 (∑𝐾 𝑘=1 (1/𝜎𝑘 ))

2

2 (∑𝐾 𝑘=1 (1/𝜎𝑘 ))

where 𝐸|̂𝑏𝑖(𝑘) − 𝑏𝑖(𝑘) | = 𝑃𝑒 (𝛾) is error probability of BPSK scheme in Rayleigh channels. Then, the reconstruction error can be written as −𝐿 𝑘

󵄨2 𝐾 2 󵄨 𝐸 󵄨󵄨󵄨󵄨𝜃̂ − 𝜃󵄨󵄨󵄨󵄨 = ∑ 𝑘=1

2 ̃𝑘 /𝜎𝑘2 )) (∑𝐾 𝐸 [(∑𝐾 𝑘=1 (𝑥 𝑘=1 (𝑛𝑘 /𝜎𝑘 ))]

2 ∑𝐾 𝑘=1 (1/𝜎𝑘 )

(19)

󵄨 −𝑖 󵄨󵄨󵄨 𝐸 󵄨󵄨󵄨󵄨(̃𝑏𝑖(𝑘) − 𝑏𝑖(𝑘) ) 2−𝑖 + ∑∞ 𝑖=𝐿 𝑘 +1 2 󵄨󵄨

𝐾

−1 𝐾

−2 󵄨 𝐾 𝐾 󵄨󵄨 𝑥̃ + 𝑛 󵄨󵄨󵄨2 1 󵄨2 󵄨 󵄨 󵄨 𝐸 󵄨󵄨󵄨󵄨𝜃̂ − 𝜃󵄨󵄨󵄨󵄨 = ( ∑ 2 ) 𝐸 󵄨󵄨󵄨 ∑ 𝑘 2 𝑘 󵄨󵄨󵄨 󵄨 󵄨󵄨 𝜎 𝜎 󵄨 𝑘 𝑘=1 𝑘 󵄨𝑘=1 󵄨 󵄨󵄨󵄨∑𝐾 (𝑥̃ /𝜎2 )󵄨󵄨󵄨2 󵄨 𝑘=1 𝑘 𝑘 󵄨󵄨 =󵄨 2 2 (∑𝐾 𝑘=1 (1/𝜎𝑘 ))

+

󵄨 󵄨󵄨 󵄨 󵄨 𝐾 𝐾 󵄨󵄨𝑥 𝐸 󵄨󵄨𝑥̃ 󵄨󵄨 󵄨 ̃𝑘 − (𝑥𝑘 )𝑄 + (𝑥𝑘 )𝑄 + 𝑥𝑘 󵄨󵄨󵄨 ≤ ∑ 󵄨 2𝑘 󵄨 ≤ ∑ 󵄨 𝜎𝑘 𝜎𝑘2 𝑘=1 𝑘=1

𝑘=1

If reconstruction error is defined as 𝑥̂𝑘 − 𝑥𝑘 = 𝑥̃𝑘 , then

+

󵄨󵄨 𝐾 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ ( 𝑥̃𝑘 )󵄨󵄨󵄨 󵄨󵄨 2 󵄨󵄨 𝜎 󵄨󵄨𝑘=1 𝑘 󵄨󵄨

≤∑

and we have 𝜃 = 𝑥𝑘 − 𝑛𝑘 , then 𝐾 1 𝜃̂ − 𝜃 = ( ∑ 2 ) 𝜎 𝑘=1 𝑘

̃𝑘 /𝜎𝑘2 ) is bounded, we can instead minimize that ∑𝐾 𝑘=1 (𝑥 𝐾 | ∑𝑘=1 (𝑥̃𝑘 /𝜎𝑘2 )|2 , where the upper bound is written as

.

We can minimize 𝐸|𝜃̂ − 𝜃|2 by minimizing ̃𝑘 /𝜎𝑘2 )|2 . This is because for any bounded 𝐸| ∑𝐾 𝑘=1 (𝑥 random variable 𝑍 ∈ [−𝑈, 𝑈] with the pdf 𝑝(𝑧), we 𝑈 have 𝐸|𝑍|2 = ∫−𝑈 |𝑧|2 𝑝(𝑧)𝑑𝑧 = 𝑈𝐸|𝑍|. By noticing

+ (1 − 2−𝐿 𝑘 ) 𝑃𝑒 (𝛾) 𝜎𝑘2

.

(20)

3.1. Energy Optimization for Equal Number of Bits per Sensor. We first consider here a simple situation where each sensor transmits an equal number of quantization bits (𝐿 𝑘 = 𝐿) ∀𝑘. The total energy of all sensors is 𝐸𝑇 . Energy allocation for the 𝑘th sensor is 𝐸𝑘 = 𝑧𝑘 𝐸𝑇 , where 𝑧𝑘 (𝑘 = 1, . . . , 𝐾) is the energy proportion of 𝑘th sensor and the energy bit of 𝑘th sensor is 𝐸𝑏(𝑘) = 𝑧𝑘 𝐸𝑇 /𝐿. The error probability of Rayleigh channel with BPSK modulation (4) can be expressed as 𝑃𝑒𝑘 (𝛾) = 𝑃𝑒𝑘 (

𝐸𝑏(𝑘) 𝑧𝐸 ) = 𝑃𝑒𝑘 ( 𝑘 𝑇 ) . 𝑁0 𝐿𝑁0

(21)

Substituting (21) into (20), it can be obtained that −𝐿

󵄨2 𝐾 2 󵄨 𝐸 󵄨󵄨󵄨󵄨𝜃̂ − 𝜃󵄨󵄨󵄨󵄨 = ∑ 𝑘=1

+ (1 − 2−𝐿 ) 𝑃𝑒𝑘 (𝑧𝑘 𝐸𝑇 /𝐿𝑁0 ) 𝜎𝑘2

.

(22)


5

The optimal energy allocation scheme is the solution for optimization problem of 𝑧 := [𝑧1 , . . . , 𝑧𝐾 ]𝑇 : −𝐿

2

𝐾

min𝑓𝑜 (𝑧; 𝐿) := ∑ 𝑧

−𝐿

+ (1 − 2 ) 𝑃𝑒𝑘 (𝑧𝑘 𝐸𝑇 /𝐿𝑁0 ) 𝜎𝑘2

𝑘=1

subject to 𝑓𝑘 (𝑧) := −𝑧𝑘 ≤ 0,

𝑘 = 1, . . . , 𝐾,

Algorithm 3 (iterative algorithm). Consider the following. (1) The step 𝑙, with 𝐿 𝑘 , 𝑘 = 1, . . . , 𝐾 finds 𝑧𝑙 [𝑧1(𝑙) , . . . , 𝑧𝑘(𝑙) ] as the optimal solution of (26).

=

(2) Update 𝐿𝑙𝑘 ke 𝐿(𝑙+1) based on iteration 𝑘 (23)

𝐿𝑙+1 𝑘 = arg min [ 𝐿𝑘

𝐾

𝑔 (𝑧) := ∑ 𝑧𝑘 = 1.

𝑧𝐸 1 + (1 − 2−𝐿 𝑘 ) 𝑃𝑒𝑘 ( 𝑘 𝑇 )] . 2𝐿 𝑘 𝐿𝑁0

(27)

(3) Back to step (𝑙 + 1).

𝑘=1

Proposition 2. (1) Equation (23) is a convex function. (2) Each sensor transmits the same fixed number of bits 𝐿. The channels between sensor and fusion center are flat fading Rayleigh channels. These are influenced by path loss 𝑎𝑘 = 𝑑𝑘𝛼 , where 𝑑𝑘 is the distance between the 𝑘th sensor and fusion center, and 𝛼 is path loss exponent of the wireless channel [14]. Supposing BPSK modulation the optimization problem in (23) admits a unique solution 𝑧∗𝐿 := [𝑧1∗ , . . . , 𝑧𝐿∗ ]𝑇 that can minimize the objective function 𝑓𝑜 (𝑧; 𝐿) 𝑎𝐿 𝑧𝑘∗ = (√ 2 ∗ 𝑘 ), 4𝜎𝑘 𝜐 (𝐸𝑇 /𝑁0 )

𝑘 = 1, . . . , 𝐾,

(24)

where 𝜐∗ is a constant chosen to enforce the constraint ∗ ∑𝐾 𝑘=1 𝑧𝑘 = 1.

If 𝑃𝑒𝑘 (𝛾) is convex and (𝐿 𝑘 )𝐾 𝑘=1 are fixed, then the problem in (26) is definitely convex. Therefore, the energy allocation 𝑧𝑘 can be found by numerical calculation. Step (1) of Algorithm 3 can be easily resolved. It can be proved that the objective function is always decreasing from one iteration to the next: 𝑓𝑜 (𝑧(𝑙+1) ; 𝐿𝑙+1 𝑘 𝑘 = 1, . . . , 𝐾) 𝑘 = 1, . . . , 𝐾) ≤ 𝑓𝑜 (𝑧(𝑙) ; 𝐿(𝑙+1) 𝑘

(28)

≤ 𝑓𝑜 (𝑧(𝑙) ; 𝐿(𝑙) 𝑘 𝑘 = 1, . . . , 𝐾) . Simulation result shows that Algorithm 3 converges after 3-4 iterations. In Section 4, the approach to Algorithm 3 will be used to obtain the optimal values of 𝐿 𝑘 and 𝑧𝑘 , 𝑘 = 1, . . . , 𝐾.

Proof. See Appendix B. Optimal number of quantization bits can be found by numerical calculations, where the value is the solution of the following problem 𝐿 opt = arg min𝑓opt (𝑧∗ ; 𝐿) , 𝐿

3.2. Optimal Energy Allocation Scheme for Different Number of Bits per Sensor. The 𝑘th sensor transmits 𝐿 𝑘 quantization bits, 𝑘 = 1, . . . , 𝐾. The optimal energy allocation scheme that can minimize the reconstruction error at the fusion center is the solution of the following optimization problem: 𝐾

𝑧𝐸 1 1 1 [ + (1 − 𝐿 ) 𝑃𝑒𝑘 ( 𝑘 𝑇 )] 2 2𝐿 𝑘 𝑘 2 𝐿 𝜎 𝑘 𝑁0 𝑘=1 𝑘

min𝑓𝑜 (𝑧; 𝐿 𝑘 ) := ∑

subject to 𝑓𝑘 (𝑧) := −𝑧𝑘 ≤ 0, 𝐾

𝑔 (𝑧) := ∑ 𝑧𝑘 = 1,

This section provides numerical examples to corroborate the analytical result derived in previous sections. Parameters that will be used in simulations are summarized in Table 1.

(25)

where 𝑓opt (𝑧∗ ; 𝐿) is the optimal value of the objective function (23), if the number of quantization bits per sensor is 𝐿.

𝑧

4. Numerical Result

𝑘 = 1, . . . , 𝐾.

𝑘=1

(26) Given a set of 𝐿 𝑘 , 𝑘 = 1, . . . , 𝐾, the optimal solution of (26) is equal to (23). The problem of interest is to obtain the minimum value of 𝑓𝑜 (𝑧; 𝐿 𝑘 , 𝑘 = 1, . . . , 𝐾) with respect to 𝐿 𝑘 and 𝑧𝑘 , 𝑘 = 1, . . . , 𝐾. Joint optimization of 𝐿 𝑘 and 𝑧𝑘 can be done by the following iterative algorithm.

4.1. Single Sensor to Fusion Center. In the first system, a single sensor transmits its quantized observation to the fusion center over flat fading Rayleigh channels. In the equal energy allocation scheme, along with the increasing number of quantization bits, the reconstruction error will increase. This does not happen in the optimal scheme as shown in Figure 3. The explanation for this behaviour is that as 𝐿 increases, the equal scheme increases the probability of error for all transmitted bits. The optimal scheme can be found by solving the convex optimization problem in (11) using the interior point method [15]. The result shows that the optimal scheme provides greater energy on a smaller bit index (Most Significant Bit) as shown in Figure 4. This is in agreement with that of [5]. 4.2. Distributed Estimation System: Multisensor to Fusion Center. In the second system, suppose that 𝐾 = 10 sensors are deployed with variance noise of sensor observation denoted by (𝜎1 , 𝜎2 , . . . , 𝜎10 ). The distances of the sensors to the fusion center (𝑑𝑘 , 𝑘 = 1, . . . , 𝐾) are varied. The path loss exponent of wireless channel is assumed 𝛼 = 3.5. The quantized observations are transmitted over AWGN and Rayleigh channel. The total energy budget was set to 𝐸𝑇 /𝑁0 = 200.

6

International Journal of Distributed Sensor Networks Table 1: Parameters in simulation.

Parameters 𝐾 𝛼 𝐸𝑇 /𝑁0 𝐿 = 𝐿 opt 𝜎𝑘2 𝑑𝑘 1 6, ∀𝑘 1, ∀𝑘 0.01 ∗ 𝑘 = 1, ∀𝑘 = 0.01 ∗ 𝑘

𝐿 opt 𝑑𝑘 𝜎𝑘2 𝑑𝑘 𝜎𝑘2

Specifications 10 3.5 200 7, ∀𝑘, 𝑘 = 1, . . . , 𝐾 0.01 × 𝑘 𝑑𝑘 ∈ [1, 10] Case 2 6, ∀𝑘 𝑘/4 0.01, ∀𝑘 𝑘/4 0.01, ∀𝑘

Corresponding figure Figures 3–8 Figures 3–8 Figures 3–8 Figure 4 Figures 5 and 6

3 6, ∀𝑘 𝑑𝑘 ∈ [1, 10] 0.01 ∗ 𝑘 𝑑𝑘 ∈ [1, 10] = 0.01 ∗ 𝑘

100

Figure 7 Figure 8

2.5

10−1 1.5 pi

Reconstruction error

2

1 10−2 0.5

10−3

0

5

10 15 20 Number of quantization bits (L)

25

30

Equal energy allocation Optimal energy allocation

Figure 3: Reconstruction error with equal energy allocation scheme among bits as in (8) and optimal energy allocation scheme among bits as in (11) for single sensor to FC over Rayleigh channel.

By using the parameters in Table 1, Figure 5 shows the comparison between equal energy allocation scheme and optimal energy allocation scheme in (23) for distributed estimation system over AWGN and Rayleigh channels. This comparison is done for a variable number of bits 𝐿 by defining 2 10 a specific set of values for {𝑑𝑘 }10 𝑘=1 and {𝜎𝑘 }𝑘=1 . Both types of channels show the benefits of optimal energy allocation scheme. In Rayleigh channel, the optimal energy allocation scheme is effective to decrease the reconstruction error significantly. It is observed that the increasing number of quantization bits for each sensor decreases the reconstruction error to the floor. The optimal number of quantization bits in (25) ranges from 5 to 10. After the optimal number of quantization bits, the increasing number of quantization bits will not influence the reconstruction error at the fusion center.

0

1

2

3

4

5 6 7 Bit index: i

8

9

10

Figure 4: Optimal energy allocation scheme for single sensor to FC with 𝐿 = 𝐿 opt = 7.

Figure 6 compares equal energy allocation scheme, optimal energy allocation scheme in (23), and joint optimization by using iterative algorithm method in (26). It shows that Algorithm 3 has the best performance. Simulation is done assuming the total number of quantization bits is equal in every sensor. As explained in Section 3.2, we can find the optimal value of 𝑧𝑘 and 𝐿 𝑘 , 𝑘 = 1, . . . , 𝐾, jointly by using Algorithm 3. The resulting optimal energy allocation scheme and the optimal number of quantization bits per sensor are depicted in Figures 7 and 8. Figure 7 shows the optimal energy allocation scheme among sensors if we choose the optimal value of 𝐿 in (25), 𝐿 𝑘 = 𝐿 opt . The appropriate optimal energy allocation scheme is the numerical solutions of the convex problem in (23). We also investigated the effect of variance noise of sensor toward the optimal energy allocation scheme. In case the distance between sensors and fusion center is equal, the optimal energy allocation scheme will allocate low energy to sensors which have high variance noise of sensor observation.


7 100

0.45 0.4

(a) Reconstruction error

Reconstruction error

0.35 0.3 0.25 0.2 0.15

10−1 (b)

10−2

0.1 (c)

0.05 10−3

0 5 10 15 Number of quantization bits (L k )

0

20

AWGN with equal energy allocation AWGN with optimal energy allocation Rayleigh with optimal energy allocation Rayleigh with equal energy allocation

Figure 5: Reconstruction error of objective function in (23) with equal and optimal energy allocation schemes in distributed estimation system over AWGN and Rayleigh fading channel, for 𝐾 = 10, 𝜎𝑘2 = 0.01 × 𝑘, 𝑘 = 1, 2, . . . , 𝐾, and 𝑑𝑘 = 1, ∀𝑘.

8

10

12

14

16

18

20

Optimal energy allocation Equal energy allocation Joint optimization

Figure 6: (a) Reconstruction error of objective function in (23) with equal energy allocation scheme among sensors. (b) Reconstruction error of objective function in (23) with optimal energy allocation scheme among sensors. (c) Reconstruction error of objective function in (26) with joint optimization using Algorithm 3. The link between sensors and fusion center are modelled as flat fading Rayleigh channel, for 𝐾 = 10, 𝜎𝑘2 = 0.01 × 𝑘, 𝑘 = 1, 2, . . . , 𝐾, and 𝑑𝑘 ∈ [1, 10]. 0.2

5. Conclusion

0.1 0

1

2

3

4

5 6 7 8 Sensor index (k)

9

10

9

10

9

10

Case 1 (a)

zk

0.4 0.2 0

1

2

3

4


Case 2 (b)

0.4 zk

Energy and bandwidth are serious limitations for the wide development of WSN. This has motivated us to propose the optimal energy allocation scheme and the optimal number of quantization bits for WSN over Rayleigh fading channels with path loss effects. We introduce a concept of alternating optimization and iterative algorithm to facilitate the solution. We formulate the optimal energy allocation scheme and the optimal number of quantization bits as a convex optimization and present these closed form. We also have found the optimal number of quantization bits that can minimize reconstruction error when the energy allocation for each bit is equal. The optimal energy allocation scheme has allocated more energy for most significant bit. We observe that the optimal energy allocation scheme decreases the reconstruction error to the floor when the number of quantization bits increases. It is in contrast with the equal energy allocation scheme, because in this scheme, the increasing number of quantization bits increases the probability of error per bit transmitted. In distributed estimation system, we have shown that the optimal energy allocation scheme per sensor depends on the number of quantization bits per sensor, path loss, and variance noise of sensor observation. The optimal number of quantization bits per sensor can be found with the help of our convex optimization formula. In Rayleigh channel,

6

Number of quantization bits (L k )

zk

Figure 8 shows the optimal number of quantization bits per sensor based on iteration in (27). It shows that the sensors with high variance noise of sensor observation will allocate less number of quantization bits.

4

2

0.2 0

1

2

3

4

5

6

7

8

Sensor index (k) Case 3 (c)

Figure 7: Optimal energy allocation scheme for 𝐿 𝑘 = 𝐿 opt = 6, ∀𝑘. Case 1: 𝑑𝑘 = 1, ∀𝑘, 𝜎𝑘2 = 0.01 ∗ 𝑘, 𝑘 = 1, . . . , 𝐾, case 2: 𝑑𝑘 = 𝑘/4, 𝜎𝑘2 = 0.01, ∀𝑘, and case 3: 𝑑𝑘 ∈ [1, 10], 𝜎𝑘2 = 0.01 ∗ 𝑘, 𝑘 = 1, . . . , 𝐾.

8

International Journal of Distributed Sensor Networks ∗

(2) 𝑝𝐿 := [𝑝1∗ , . . . , 𝑝𝐿∗ ]𝑇 is the optimal solution of ∗ ∗ 𝑓𝑜 (𝑝𝐿 ; 𝐿) when transmitting 𝐿 bits and 𝑝𝐿̃ := [𝑝̃1∗ , 𝑝̃2∗ , ∗ 𝑇 ̃ when transmit. . . , 𝑝̃𝐿 ] is the optimal solution of 𝑓𝑜 (𝑝𝐿 ; 𝐿) ̃ bits. ting 𝐿 ̃ we can construct the following 𝐿 Because 𝐿 > 𝐿, dimensional vector

Lk

10 5 0

1

2

3

4


9

10

Case 1

𝑇

(a)

𝑇

𝑝𝐿 := [𝑝1 , . . . , 𝑝𝐿 ] =

Lk

10 5 0

1

2

3

4

5

6

7

8

9

10

𝑝𝑖 ≥ 0,

Case 2 (b)

(A.2)

]

𝑖 = 1, . . . , 𝐿,

𝐿

̃ 𝐿

𝑖=1

𝑖=1

∑𝑝𝑖 = ∑𝑝̃𝑖∗ = 1.

∗

2

∗

𝑓𝑜 (𝑝𝐿 ; 𝐿) ≥ 𝑓𝑜 (𝑝𝐿 ; 𝐿) . 1

2

3

4

5

(A.3)

As 𝑝𝐿 is the optimal solution when transmitting 𝐿 bits, we have

4 Lk

[

.

By construction, 𝑝𝐿 is a feasible point for the optimization problem in (11) when transmitting 𝐿 bits because

Sensor index (k)

0

[𝑝∗𝑇 0, . . . , 0] ̃ , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝐿 ̃ 𝐿−𝐿

6

7

8

9

(A.4)

We can write down 𝑓𝑜 (𝑝𝐿 ; 𝐿) explicitly as [8]

10

Sensor index (k)

𝐿

𝑓𝑜 (𝑝𝐿 ; 𝐿) = 2−𝐿 + ∑𝑃𝑒 (

Case 3

𝑖=1

(c)

̃ 𝐿

Figure 8: Optimal number of quantization bits per sensor: 𝐿 𝑘 , 𝑘 = 1, . . . , 𝐾. Case 1: 𝑑𝑘 = 1, ∀𝑘, 𝜎𝑘2 = 0.01 ∗ 𝑘, 𝑘 = 1, . . . , 𝐾, case 2: 𝑑𝑘 = 𝑘/4, 𝜎𝑘2 = 0.01, ∀𝑘, and case 3: 𝑑𝑘 ∈ [1, 10], 𝜎𝑘2 = 0.01 ∗ 𝑘, 𝑘 = 1, . . . , 𝐾.

the optimal energy allocation scheme is effective to decrease the reconstruction error significantly. It is observed that the increasing number of quantization bits for each sensor decreases the reconstruction error to the floor. After the optimal number of quantization bits, the increasing number of quantization bits does not influence the reconstruction error at the fusion center. The optimal number of quantization bits produces reconstruction error close to that of unquantized estimation. The optimal energy allocation scheme suggests the allocation of more energy to sensors with small variance noise observation. Joint optimization using our iterative algorithm has the best estimation performance compared to equal and optimal energy allocation schemes, individually.

= 2−𝐿 + ∑𝑃𝑒 ( 𝑖=1

𝑝𝑖 𝐸 −𝑖 )2 𝑁0 𝐿 𝑝̃𝑖 𝐸 −𝑖 ) 2 + ∑ 𝑃𝑒 (0) 2−𝑖 𝑁0 ̃ 𝑖=𝐿+1

(A.5)

𝐿 ̃ ̃ + 1 ∑ 2−𝑖 = 2−𝐿 − 2−𝐿 + 𝑓𝑜 (𝑝𝐿∗̃ , 𝐿) 2 ̃ 𝑖=𝐿+1

̃

∗

∗

̃ < 𝑓𝑜 (𝑝̃ , 𝐿) ̃ . = 2−𝐿−1 − 2−𝐿−1 + 𝑓𝑜 (𝑝𝐿̃ , 𝐿) 𝐿 If 𝑃𝑒 (0) = 1/2 from (A.4) and (A.5), it follows readily that ∗ ∗ ̃ 𝑓𝑜 (𝑝𝐿 ; 𝐿) < 𝑓𝑜 (𝑝𝐿̃ ; 𝐿). ∗ The objective function (11) is convex and 𝑝𝐿 := [𝑝1∗ , ∗ . . . , 𝑝𝐿∗ ]𝑇 is the optimal solution that can minimize 𝑓𝑜 (𝑝𝐿 ; 𝐿). Then, we can use Karush Kuhn Tucker (KKT) condition, which shows that there are {𝜆∗𝑖 }𝐿𝑖=1 and 𝜐∗ until [15] 𝑝𝑖∗ ≤ 0,

𝜆∗𝑖 ≥ 0,

𝑝𝑖∗ 𝜆∗𝑖 = 0,

𝑖 = 1, . . . , 𝐿,

(A.6)

𝐿

∑𝑝𝑖∗ = 1,

(A.7)

𝑖=1

Appendices

𝐿

∇𝑓𝑜 (𝑝𝑖∗ ; 𝐿) + ∑𝜆∗𝑖 ∇𝑓𝑖 (𝑝𝑖∗ ) + 𝜐∗ ∇𝑔𝑖 (𝑝𝑖∗ ) = 0,

A. Proof of Proposition 1 (1) Problem (11) is convex if and only if 𝑃𝑒 (𝛾) is convex. In special cases, for example, BPSK over flat fading Rayleigh channel, error probability in high SNR is as in (4). 𝑃𝑒 (𝛾) is convex if and only if ∇2 𝑃𝑒 (𝛾) ⪰ 0, which reduces to the simple condition 𝑃𝑒󸀠󸀠 (𝛾) ≥ 0. The second derivative of 𝑃𝑒 (𝛾) to 𝛾 is 𝑃𝑒󸀠󸀠 (𝛾) =

1 . 2𝛾3

where ∇ is the gradient of (A.8). It follows from (A.6)–(A.8) that the {𝑝𝑖∗ }𝐿𝑖=1 must meet 2−𝑖

󵄨 𝐸 𝑑𝑃𝑒 (𝛾) 󵄨󵄨󵄨 󵄨󵄨 − 𝜆∗𝑖 + 𝜐∗ = 0, 𝑁0 𝑑𝛾 󵄨󵄨󵄨𝛾=(𝐸/𝑁0 )𝑝∗ 𝑖

(A.1)

(A.8)

𝑖=1

𝑖 = 1, . . . , 𝐿.

(A.9)


9

We take the special case of BPSK with error probability as in (4). Derivative of 𝑃𝑒 (𝛾) to 𝛾 can be calculated as follows: 𝑑𝑃𝑒 (𝛾) 1 = − 2. 𝑑𝛾 4𝛾

(A.10)

Substitution of (A.10) into (A.9) results in the form of optimal energy allocation scheme as follows: 𝑝𝑖∗ = (√

1 ), 4 (𝐸/𝑁0 ) 2𝑖 𝜐∗

𝑖 = 1, . . . , 𝐿.

(A.11)

B. Proof of Proposition 2 (1) Problem (23) is convex if and only if 𝑃𝑒 (𝛾) is convex. The proof is like that of Proposition 1. (2) If the objective function (23) is convex and 𝑧∗𝐿 := ∗ ∗ 𝑇 ] is the optimal solution. We can apply Karush [𝑧1 , . . . , 𝑧𝐾 Kuhn Tucker (KKT) condition, which shows that there must exist {𝜆∗𝑘 }𝐾 𝑘=1 and 𝜐∗ until [15] 𝑧𝑘∗ ≥ 0,

𝜆∗𝑘 ≥ 0,

𝜆∗𝑘 𝑧𝑘∗ = 0,

𝑘 = 1, . . . , 𝐾,

(B.1)

𝐾

∑ 𝑧𝑘∗ = 1,

(B.2)

𝑘=1

𝐾

∇𝑓𝑜 (𝑧∗ ; 𝐿) + ∑ 𝜆∗𝑘 ∇𝑓𝑘 (𝑧∗ ) + 𝜐∗ ∇𝑔𝑘 (𝑧∗ ) = 0,

(B.3)

𝑘=1

where ∇ is the gradient of (B.3). From (B.1)–(B.3), the following must be satisfied 󵄨 1 𝐸𝑇 𝑑𝑃𝑒 (𝛾) 󵄨󵄨󵄨 󵄨󵄨 𝜎𝑘2 𝐿𝑁0 𝑑𝛾 󵄨󵄨󵄨𝛾=(𝐸

𝑇 /𝐿𝑁0 )𝑧𝑘

− 𝜆∗𝑘 + 𝜐∗ = 0,

(B.4)

𝑘 = 1, . . . , 𝐾. If the channel between 𝑘th sensor with FC is influenced by path loss 𝑎𝑘 = 𝑑𝑘𝛼 , where 𝑑𝑘 is the distance between the 𝑘th sensor and FC and 𝛼 is path loss exponent of the wireless channel [14], then (A.1) becomes 𝑃𝑒 (𝛾) =

𝑎𝑘 . 4𝛾

(B.5)

𝑎𝑘 𝐿 ), 4𝜎𝑘2 𝜐∗ (𝐸𝑇 /𝑁0 )

𝑘 = 1, . . . , 𝐾.

Sensor observation in single sensor to fusion center system 𝜃: The deterministic parameter to be estimated 𝑛: Sensor noise observation The quantization result of 𝐴 𝐴 𝑄: ̂ 𝐴: The estimation of 𝐴 𝐿: The number of quantization bits 𝑖: Index of bit The 𝑖th quantization bits 𝑏𝑖 : The energy per bit 𝐸𝑏 : The energy allocation for 𝑖th bit 𝐸𝑏𝑖 : 𝐸: The total energy of all bits 𝛾: Signal-to-noise ratio Noise power spectral density 𝑁0 : The error probability 𝑃𝑒 : The energy proportion of 𝑖th bit 𝑝𝑖 : The optimal energy proportion of 𝑖th bit 𝑝𝑖∗ : 𝐾: Total number of sensor 𝑘: Index sensor 𝑊: Sensor observation range The observation of 𝑘th sensor 𝑥𝑘 : The quantization result of 𝑥𝑘 (𝑥𝑘 )𝑄: 𝑃𝑒𝑘 : Error probability of 𝑘th sensor 𝐸‖𝜃̂ − 𝜃‖2 : The absolute reconstruction error The number of quantization bit of 𝑘th 𝐿 𝑘: sensor The total energy of all sensors 𝐸𝑇 : The energy of 𝑘th sensor 𝐸𝑘 : The energy proportion of 𝑘th sensor 𝑧𝑘 : The optimal energy proportion of 𝑘th 𝑧𝑘∗ : sensor The energy per bit of 𝑘th sensor 𝐸𝑏(𝑘) : 𝑑𝑘 : The distance between 𝑘th sensor and fusion center 𝛼: Path loss exponent Path loss. 𝑎𝑘 :


Acknowledgment

(B.6)

References

By substituting (B.6) into (B.4), we can obtain the optimal energy allocation scheme as follows: 𝑧𝑘∗ = (√

𝐴:

This work is part of research program funded by Graduate Research Grant no. 11588/PL4/PPK/LK/2013 from the Indonesian Ministry of National Education.

Its derivative to 𝛾 can be calculated as follows: 𝑑𝑃𝑒 (𝛾) 𝑎 = − 𝑘2 . 𝑑𝛾 4𝛾

Notations

(B.7)

[1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Wireless sensor networks: a survey,” Computer Networks, vol. 38, no. 4, pp. 393–422, 2002. [2] H. Liu, G. Liu, Y. Liu, L. Mo, and H. Chen, “Adaptive quantization for distributed estimation in energy-harvesting wireless sensor networks: a game-theoretic approach,” International

10

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12] [13] [14] [15]

International Journal of Distributed Sensor Networks Journal of Distributed Sensor Networks, vol. 2014, Article ID 217918, 9 pages, 2014. J. Li and G. AlRegib, “Distributed estimation in energyconstrained wireless sensor networks,” IEEE Transactions on Signal Processing, vol. 57, no. 10, pp. 3746–3758, 2009. R. Nowak, U. Mitra, and R. Willett, “Estimating inhomogeneous fields using wireless sensor networks,” IEEE Journal on Selected Areas in Communications, vol. 22, no. 6, pp. 999–1006, 2004. Z.-Q. Luo and J.-J. Xiao, “Universal decentralized estimation in a bandwidth constrained sensor network,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 4, pp. iv/829–iv/832, 2005. J.-J. Xiao, S. Cui, Z.-Q. Luo, and A. J. Goldsmith, “Linear coherent decentralized estimation,” IEEE Transactions on Signal Processing, vol. 56, no. 2, pp. 757–770, 2008. M. Noori and M. Ardakani, “Energy efficiency of universal decentralized estimation in random sensor networks,” IEEE Transactions on Wireless Communications, vol. 10, no. 12, pp. 4023–4028, 2011. X. Luo and G. B. Giannakis, “Energy-constrained optimal quantization for wireless sensor networks,” Eurasip Journal on Advances in Signal Processing, vol. 2008, Article ID 462930, 2008. G. Liu, B. Xu, and H. Chen, “Decentralized estimation over noisy channels in cluster-based wireless sensor networks,” International Journal of Communication Systems, vol. 25, no. 10, pp. 1313–1329, 2012. S. Talarico, N. A. Schmid, M. Alkhweldi, and M. C. Valenti, “Distributed estimation of a parametric field: algorithms and performance analysis,” IEEE Transactions on Signal Processing, vol. 62, no. 5, pp. 1041–1053, 2014. M. Tang, J. Bai, J. Li, and Y. Xin, “Distributed optimal power and rate control in wireless sensor networks,” The Scientific World Journal, vol. 2014, Article ID 580854, 8 pages, 2014. J. G. Proakis and M. Salehi, Digital Communications, McGrawHill, Boston, Mass, USA, 2008. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, New York, NY, USA, 2010. T. S. Rappaport, Wireless Communications: Principles and Practice, Prentice Hall, 2002. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, NY, USA, 2004.


Research Article Decentralized Kalman Filtering with Multilevel Quantized Innovation in Wireless Sensor Networks Zhi Zhang and Jianxun Li Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China Correspondence should be addressed to Zhi Zhang; [email protected] Received 19 September 2014; Accepted 25 January 2015 Academic Editor: Chang Wu Yu Copyright © 2015 Z. Zhang and J. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Because of low power consumption and limited power supply significance in wireless sensor networks (WSNs), this paper studies the multilevel quantized innovation Kalman filtering (MQI-KF) for decentralized state estimation in WSNs since the MQI-KF can help to save power. In the first place, the common features of the practical low energy consumption WSNs are explored. On this basis, the new quantization scheme is presented. Besides, this paper explores the quantization state estimation by adopting the Bayesian method rather than the traditional iterated conditional expectation method. After that, this paper proposes a new decentralized state estimation algorithm (MQI-KF) for WSNs. Information entropy is analyzed to evaluate the performance of the quantization scheme. Performance analysis and simulations show that the MQI-KF is more efficient than the other decentralized Kalman filtering (KF) algorithms, and the accuracy of its estimation is close to that of the standard KF based on nonquantized measurements. Since the new quantization scheme and algorithm take into consideration the features of real WSNs which are based on the universal network protocol IEEE 802.15.4 standard, they can almost be applied into all practical WSNs with low energy consumption.

1. Introduction Decentralized state estimation is of great importance in studying WSNs. Against this background, this paper mainly discusses the decentralized state estimation of a Gaussian Markov stochastic process in WSNs. In general, a WSN is composed of a fusion center (FC) and lots of sensors. The computation capability and energy of these sensors are quite limited. Each one of them firstly acquires noisy measurements within its sensing range. Then the measurements will be transmitted to the FC which processes them with the KF to get the state estimation with minimum mean square error (MMSE). To save energy and the communication bandwidth, the measurements should be minimized through quantizing before the transmission. This is because the quantization helps to reduce the amount of bits and the energy consumption in transmission. Hence, the focus of the research is the decentralized state estimation with quantized measurements. 1.1. Related Work. To save energy and reduce bandwidth in WSNs, several decentralized state estimation approaches with

quantized measurements were put forward in [1–5]. However, one disadvantage of these algorithms is that quantizing measurements directly may lead to different levels of accuracy. This means that when the measurement value is large, there appears large quantization noise. In order to solve this problem, the KF for quantizing innovation at multiple levels was developed in [6]; an efficient quantization scheme was presented in [7, 8]; on the basis of the quantization scheme in [7, 8], the transmission strategy and the multilevel quantization state estimation were combined together in [9], leading to better performance and less bandwidth consumption. However, on the one hand, these quantization schemes are theoretical ones which were put forward without considering the features of the practical low power consumption WSNs. So in this paper, a new quantization scheme is proposed on the basis of analyzing these features. On the other hand, these decentralized state estimations in [6, 9] were derived by adopting the iterated conditional expectation method directly on the condition that the prior probability density function (pdf) of the state vector is

2


Gaussian. However, these researches did not analyze the case of the posterior pdf of the state based on the quantized innovation. By contrast, this paper studies the situation of the posterior pdf explicitly to prepare for the further study of the state estimation with quantized innovation. It innovatively utilizes a new approach for the state estimation which is a new perspective for the decentralized state estimation. By taking all these factors into consideration, a novel multilevel quantized innovation KF algorithm is proposed on the basis of the new quantization scheme and the decentralized state estimation. 1.2. Contributions of This Paper. The main contributions of this paper are listed as follows: (1) It puts forward an efficient quantization scheme by flexibly utilizing different communication modes. (2) This paper adopts the Bayesian method for the MQIKF instead of the iterated conditional expectation method used in [6, 9]. (3) It puts forward a decentralized state estimation algorithm based on the new quantization scheme and innovative KF. Model and preliminary in WSNs are presented in Section 2. Section 3 introduces the details of MQI-KF. Performance analysis and experiments are explored in Sections 4 and 5, respectively. Finally, Section 6 concludes the research in this paper. Notation. For a discrete random variable 𝜓, the probability mass function is presented as Pr{𝜓}. When a random variable 𝑥 is conditioned on 𝑑, the pdf is denoted by 𝑝(𝑥 | 𝑑). This paper utilizes 𝑝(𝑥) = N[𝑥; 𝜇, 𝑃] to denote the Gaussian pdf with mean 𝜇 and covariance 𝑃. Φ(⋅) is the standard normal 𝑧 distribution function; namely, Φ(𝑧) = ∫−∞ N[𝑥; 0, 1]𝑑𝑥.

2. Model and Preliminary This paper analyzes a discrete-time linear stochastic system in the WSN which consists of FC and 𝑁 sensors. The system can be denoted by 𝑋𝑘 = 𝐹𝑘−1 𝑋𝑘−1 + 𝑤𝑘−1 ,

(1)

𝑦𝑘𝑖 = ℎ𝑘𝑖 𝑋𝑘 + V𝑘𝑖 ,

(2)

where 𝑘 = 1, 2, . . . and 𝑖 = 1, 2, . . . , 𝑁. 𝐹𝑘−1 ∈ R𝑛×𝑛 refers to the dynamic model of the system; 𝑋𝑘 , 𝑤𝑘−1 ∈ R𝑛 , 𝑦𝑘𝑖 , V𝑘𝑖 ∈ R, and ℎ𝑘𝑖 ∈ R1×𝑛 is the measurement model of the 𝑖th sensor which is activated at time step 𝑘. Both 𝑤𝑘−1 and V𝑘𝑖 are zero mean, white, and Gaussian noises independent of each other; their covariance matrixes are 𝑄𝑘−1 ∈ R𝑛×𝑛 and 𝑅𝑘𝑖 ∈ R𝑛 , respectively. In order to save communication bandwidth and energy, the 𝑖th activated sensor obtains the measurement 𝑦𝑘𝑖 and then gets the innovation 𝑦̃𝑘𝑖 through computation within the sensor. This can be computed by 𝑖 , 𝑦̃𝑘𝑖 = 𝑦𝑘𝑖 − 𝑦̂𝑘|𝑘−1

(3)

Figure 1: The WSN for target tracking.

𝑖 where 𝑦̂𝑘|𝑘−1 is the one-step predicted measurement. It is transmitted from FC, which is the same with the situations in [6–9]. The 𝑦̃𝑘𝑖 is quantized in the 𝑖th activated sensor before it is sent to the FC. The quantization model is like this:

𝑏𝑘𝑖 = 𝑞𝑖 [𝑦̃𝑘𝑖 ] ,

(4)

where 𝑞𝑖 [⋅] is the quantization scheme; 𝑏𝑘𝑖 is the quantized innovation of the 𝑖th activated sensor at time step 𝑘. The 𝑦̃𝑘𝑖 can be quantized and encoded into binary data. The quantization schemes and transmission strategies in [6–9] are only applicable for the quantized innovation 𝑏𝑘𝑖 . However, the real situation is that, except for the quantized innovation 𝑏𝑘𝑖 , the wireless data packet transmitted from the activated sensors to the FC also contains destination address, source address, Frame Check Sequence (FCS), and so forth. Thus, by considering the features in the low energy consumption WSNs in practice, this paper designs an energysaving, efficient decentralized state estimation algorithm. The low energy consumption WSN designed for target tracking experiments is demonstrated in Figures 1 and 2. It is constructed on the basis of the universal network protocol IEEE 802.15.4 standard [10]. This protocol is widely acknowledged as a standard technology in the WSNs with low energy consumption [11]. Figure 3 shows the format of the data packet defined by this protocol. Two remarks are listed here for introducing the decentralized state estimation algorithm in Section 3. Remark 1. This paper utilizes the round-robin mode in the protocol to avoid jamming of FC’s receiver, since this mode only allows the activated sensor to transmit data packet per time step. This means that the sensor index corresponds to the time step. For instance, at time step 𝑘, ℎ𝑘𝑖 = ℎ𝑘 , 𝑏𝑘𝑖 = 𝑏𝑘 , 𝑦𝑘𝑖 = 𝑦𝑘 , and 𝑦̃𝑘𝑖 = 𝑦̃𝑘 . Remark 2. Two different modes can be adopted in the transmission of the data packet from the sensors to the FC, namely, broadcast communication and peer to peer communication. Which one to choose depends on the parameter in the MHR. With the same energy management mechanism in the PHR [10], the amounts of energy they consume are the same.


3

Control, management, and FC

Wireless sensor network

Mobile targets

Figure 2: The components of the WSN.

3. Kalman Filtering Based on Multilevel Quantized Innovation

The quantization scheme is demonstrated as follows: 𝑏

In this section, firstly a more efficient quantization scheme is designed for normalized innovation; secondly, an innovative quantization KF is derived by adopting the Bayesian approach. Finally, a decentralized state estimation algorithm is developed based on this quantization scheme and the new quantization KF. 3.1. Quantization Scheme Design. To quantize the scalar innovation 𝛾𝑘 at time step 𝑘, the FC transmits the one-step prediction 𝑦̂𝑘|𝑘−1 and the square root of the covariance matrix of innovation Ω𝑘 to the 𝑖th activated sensor ̂𝑘|𝑘−1 , 𝑦̂𝑘|𝑘−1 = ℎ𝑘 𝑋 Ω𝑘 =

√ℎ𝑘 𝑃𝑘|𝑘−1 ℎ𝑘𝑇

(5) + 𝑅𝑘 .

(6)

For the 𝑖th activated sensor, the normalized innovation 𝛾𝑘 can be computed by 𝛾𝑘 =

(𝑦𝑘 − 𝑦̂𝑘|𝑘−1 ) . Ω𝑘

(7)

−𝜂𝑙 { { { { { {−𝜂𝑙𝑝 { { { { { .. { { { . { { { { { { {−𝜂1𝑏 { { { { 𝑝 { { −𝜂1 { { { { 𝑏𝑘 = 𝑞 [𝛾𝑘 ] = {0 { { { 𝑝 { { 𝜂1 { { { { { { 𝜂𝑏 { 1 { { { { { . { .. { { { { { { 𝑝 { { 𝜂𝑙 { { { { 𝑏 {𝜂𝑙

if 𝛾𝑘 ∈ (𝜏1 (𝑘) , 𝜏2 (𝑘)] if 𝛾𝑘 ∈ (𝜏2 (𝑘) , 𝜏3 (𝑘)] .. . if 𝛾𝑘 ∈ (𝜏2𝑙−1 (𝑘) , 𝜏2𝑙 (𝑘)] if 𝛾𝑘 ∈ (𝜏2𝑙 (𝑘) , 𝜏2𝑙+1 (𝑘)] if 𝛾𝑘 ∈ (𝜏2𝑙+1 (𝑘) , 𝜏2𝑙+2 (𝑘)]

(8)

if 𝛾𝑘 ∈ (𝜏2𝑙+2 (𝑘) , 𝜏2𝑙+3 (𝑘)] if 𝛾𝑘 ∈ (𝜏2𝑙+3 (𝑘) , 𝜏2𝑙+4 (𝑘)] .. . if 𝛾𝑘 ∈ (𝜏4𝑙 (𝑘) , 𝜏4𝑙+1 (𝑘)] if 𝛾𝑘 ∈ (𝜏4𝑙+1 (𝑘) , 𝜏4𝑙+2 (𝑘)) ,

where {𝜏1 (𝑘), 𝜏2 (𝑘), . . . , 𝜏4𝑙+1 (𝑘), 𝜏4𝑙+2 (𝑘)} are the thresholds of the scheme; 𝜏4𝑙+2 (𝑘) = +∞, 𝜏1 (𝑘) = −∞; 𝐿 is the number 𝑝 of the quantization levels, and 𝐿 = 4𝑙 + 1. When 𝑏𝑘 = ±𝜂𝑚 (1 ≤ 𝑚 ≤ 𝑙), the data packet will be transmitted to FC in 𝑏 (1 ≤ 𝑚 ≤ 𝑙), the the peer to peer manner. And if 𝑏𝑘 = ±𝜂𝑚 broadcast manner will be chosen. In the case of 𝑏𝑘 = 0, there

4


MAC layer

MAC header (MHR)

Data payload (MAC payload)

MAC footer (MFR)

PHY layer Synchronizaton header (SHR)

PHY header (PHR)

MAC protocol data unit (PHY payload)

Figure 3: The structure of data packet in IEEE 802.15.4.

will be no transmission of the data packet. After receiving the data packet from the activated sensor, the FC can determine the level of the 𝛾𝑘 by reading data payload of the packet and the transmission mode parameter in the MHR. For example, when 𝑏𝑘 ∈ {𝜂1𝑏 , −𝜂1𝑏 }, 𝑏𝑘 will be transmitted 𝑝 𝑝 by one bit through broadcast mode, and if 𝑏𝑘 ∈ {𝜂1 , −𝜂1 }, it can be transmitted by one bit through peer to peer mode. 𝑝 Thus, when 𝑏𝑘 ∈ {0, ±𝜂1 , ±𝜂1𝑏 }, 𝑏𝑘 will be transmitted using binary data no more than one bit. Therefore, quantizing the normalized innovation 𝑏𝑘 using one bit in the 𝑖th activated sensor, there are five quantization levels using the new quantization scheme. On the other hand, there are three levels in [7–9] and two levels in [6]. It is obvious that more quantization levels contribute to more estimation accuracy. Thus, the performance of the quantization scheme proposed in this paper is better than the others, which will be seen in the performance analysis. 3.2. State Estimation by Adopting the Bayesian Approach. The state estimations in [6, 9] are based on the method of iterated conditional expectation, which can be expressed as follows: ̂𝑘|𝑘 = 𝐸 [𝑋𝑘 | 𝑏𝑘 ] = 𝐸 [𝐸 [𝑋𝑘 | 𝑏𝑘−1 , 𝑦𝑘 ] | 𝑏𝑘 ] . 𝑋 1 1

(9)

Different from this method, this subsection studies the posterior probability density of the state 𝑋𝑘 conditioned on the quantized innovation 𝑏1𝑘 . Once the posterior density 𝑝(𝑋𝑘 | 𝑏1𝑘 ) is described explicitly, the MMSE of the state ̂𝑘|𝑘 can be determined. estimation 𝑋 Naturally, adopting the Bayesian method, the 𝑝(𝑋𝑘 | 𝑏1𝑘 ) can be derived through two steps.

(1) Prediction Step. From 𝑝(𝑋𝑘−1 | 𝑏1𝑘−1 ) to 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ), we can obtain 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ) by

(2) Correction Step. From 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ) to 𝑝(𝑋𝑘 | 𝑏1𝑘 ), 𝑝(𝑋𝑘 | 𝑏1𝑘 ) can be calculated as follows: 𝑝 (𝑋𝑘 | 𝑏1𝑘 ) (𝑛+1)/2

= [(2𝜋) 𝜏

⋅

󵄨󵄨 𝑃 󵄨1/2 −1 󵄨󵄨 𝑘|𝑘−1 𝑅𝑘 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 Ω2 󵄨󵄨󵄨 ] 󵄨 󵄨 𝑘 𝑋 𝑇

∫𝜏 𝑗+1 𝑒−(𝛾−̂𝛾

) 𝑃(̂ 𝛾𝑋 )−1 (𝛾−̂ 𝛾𝑋 )/2

𝑗

(11) ̃ ̃𝑇 𝑃−1 𝑋/2 𝑘|𝑘−1

𝑑𝛾 × 𝑒−𝑋

,

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

̃ = 𝑋𝑘 − 𝑋 ̂𝑘|𝑘−1 , 𝛾̂𝑋 = ℎ𝑋/Ω ̃ 𝑘 , and 𝑃(̂ where 𝑋 𝛾𝑋 ) = 𝑅𝑘 /Ω2𝑘 . 𝛾 and Ω𝑘 are defined in (6) and (7), respectively. On the basis ̂𝑘|𝑘 can be computed by of 𝑝(𝑋𝑘 | 𝑏1𝑘 ), the conditional mean 𝑋 ̂𝑘|𝑘 = 𝑋 ̂𝑘|𝑘−1 + 𝜔𝑘 𝑋

𝑃𝑘|𝑘−1 ℎ𝑘𝑇 √ℎ𝑘 𝑃𝑘|𝑘−1 ℎ𝑘𝑇 + 𝑅𝑘

,

(12)

where 𝜔𝑘 = [(2𝜋)1/2 ]

−1

2 exp [−𝜏𝑗2 (𝑘) /2] − exp [−𝜏𝑗+1 (𝑘) /2]

Φ [𝜏𝑗+1 (𝑘)] − Φ [𝜏𝑗 (𝑘)]

.

(13)

Proof. See Appendix B. Beside, its conditional covariance matrix 𝑃𝑘|𝑘 is demonstrated as follows: 𝑃𝑘|𝑘 = 𝑃𝑘|𝑘−1 − 𝜉𝑘

𝑃𝑘|𝑘−1 ℎ𝑘𝑇 ℎ𝑘 𝑃𝑘|𝑘−1 ℎ𝑘 𝑃𝑘|𝑘−1 ℎ𝑘𝑇 + 𝑅𝑘

,

(14)

where 𝜉𝑘 = 𝜔𝑘2 − [(2𝜋)1/2 ] ⋅

−1

2 𝜏𝑗 (𝑘) exp [−𝜏𝑗2 (𝑘) /2] − 𝜏𝑗+1 (𝑘) exp [−𝜏𝑗+1 (𝑘) /2]

Φ [𝜏𝑗+1 (𝑘)] − Φ [𝜏𝑗 (𝑘)]

(15) .

Proof. See Appendix C. 𝑝 (𝑋𝑘 |

𝑏1𝑘−1 )

= [(2𝜋)

𝑛/2

󵄨󵄨 󵄨1/2 −1 󵄨󵄨𝑃𝑘|𝑘−1 󵄨󵄨󵄨 ]

̂𝑘|𝑘−1 )𝑇 𝑃−1 (𝑋𝑘 − 𝑋 ̂𝑘|𝑘−1 ) } { (𝑋𝑘 − 𝑋 𝑘|𝑘−1 ⋅ exp {− }. 2 } { Proof. See Appendix A.

(10)

The multilevel quantized innovation KF is demonstrated in (12), (13), (14), and (15). In this subsection, the posterior probability density 𝑝(𝑋𝑘 | 𝑏1𝑘 ) is studied explicitly. On the basis of this, the multilevel quantized innovation KF is derived. The approach adopted here is different from the iterated conditional expectation method in [6, 9], and this provides a new method for the future study of the decentralized state estimation.


5

3.3. Algorithm. This subsection focuses on a new decentralized state estimation algorithm. It is proposed on the basis of the new quantization scheme, the multilevel quantized innovation KF, and transmission strategy in [9] (see the following). Decentralized state estimation algorithm is as follows. ̂0|0 , 𝑃0|0 , 𝐿 and {𝜏1 (𝑘), 𝜏2 (𝑘), . . . , 𝜏4𝑙+1 (𝑘), (1) Initialization. 𝑋 𝜏4𝑙+2 (𝑘)} are set in the FC. For 𝑖 = 1, . . . , 𝑁, 𝐿 and {𝜏1 (𝑘), 𝜏2 (𝑘), . . . , 𝜏4𝑙+1 (𝑘), 𝜏4𝑙+2 (𝑘)} are set in the 𝑖th sensor. ̂𝑘|𝑘−1 (2) Estimation. When 𝑘 ≥ 1 one-step state prediction 𝑋 and 𝑃𝑘|𝑘−1 can be computed in the FC. (2.1) Prediction Step. According to (A.3) and (A.4) ̂𝑘|𝑘−1 = 𝐹𝑘−1 𝑋 ̂𝑘−1|𝑘−1 , 𝑋

(16)

𝑇 𝑃𝑘|𝑘−1 = 𝐹𝑘−1 𝑃𝑘−1|𝑘−1 𝐹𝑘−1 + 𝑄𝑘−1 .

(2.2) Correction Step. The computations of 𝑦̂𝑘|𝑘−1 and Ω𝑘 are conducted in the FC which transmits them to the activated 0 sensor. Then in this sensor 𝛾𝑘 can be computed. If 𝑏𝑘 ≠ after quantization, the data packet will be transmitted from the sensor to the FC by applying the transmission strategy in [9]; if 𝑏𝑘 = 0, there will be no transmission of the data packet. At last, by utilizing (12), (13), (14), and (15) the decentralized state estimation can be computed ̂𝑘|𝑘−1 + 𝜔𝑘 ̂𝑘|𝑘 = 𝑋 𝑋

𝑃𝑘|𝑘 = 𝑃𝑘|𝑘−1 − 𝜉𝑘

𝑃𝑘|𝑘−1 ℎ𝑘𝑇 √ℎ𝑘 𝑃𝑘|𝑘−1 ℎ𝑘𝑇 + 𝑅𝑘


, (17)

,

Δ𝑃𝑘 = 𝑃𝑘|𝑘 − 𝑃𝑘|𝑘−1 = 𝜌𝑘

2 exp [−𝜏𝑗2 (𝑘) /2] − exp [−𝜏𝑗+1 (𝑘) /2]

Φ [𝜏𝑗+1 (𝑘)] − Φ [𝜏𝑗 (𝑘)]

2 𝜏𝑗 (𝑘) exp [−𝜏𝑗2 (𝑘) /2] − 𝜏𝑗+1 (𝑘) exp [−𝜏𝑗+1 (𝑘) /2]

Φ [𝜏𝑗+1 (𝑘)] − Φ [𝜏𝑗 (𝑘)]

(18) .

Both the broadcast and peer to peer communication modes are characteristics of the WSNs which are based on the IEEE 802.15.4. Therefore, the quantization scheme and the algorithm put forward on the basis of these modes are applicable in the majority of the actual WSNs which consume less energy.

4. Performance Analysis In this section, the focus is the performance of the MQI-KF.


𝐿

𝐻 = ∑ Pr (𝑗) log2 [

,

(19)

1 ]. Pr (𝑗)

(20)

Since the normalized innovation 𝛾𝑘 is a standard normal distribution (𝑝(𝛾 | 𝑏1𝑘−1 ) = N(𝛾; 0, 1) in Appendix B) and the quantization scheme in (8), its information entropy can be obtained by 𝜏𝑖+1 (𝑘)

⋅ log2 [(∫

−1

Msechu 1

4.2. Performance Analysis of the Quantization Scheme. The comparison of the performances of different quantization schemes are made on the condition that the transmission bandwidth is one bit. In this way, the comparison will not be affected by different transmission strategies. The quantization scheme is regarded as an information source and quantized innovation 𝑏𝑘 is its output. Hence, the value of the information entropy can be the standard to evaluate different quantization schemes. The bigger the value is, the more information the 𝑏𝑘 contains. The information entropy is defined in [13]

𝑖=1 𝜏𝑖 (𝑘)

,

You 1.536

where 𝜌𝑘 is the coefficient of Δ𝑃𝑘 . When the value of Δ𝑃𝑘 is maximum, the thresholds are optimal. There is a famous solution for obtaining the optimal thresholds and it is proposed in the Lloyd-Max quantization scheme [12].

𝐿

−1

Xu 1.536

4.1. Optimal Thresholds of the Quantization Scheme. The error covariance matrix correction is computed by

𝐻 = ∑∫

𝜉𝑘 = 𝜔𝑘2 − [(2𝜋)1/2 ] ⋅

𝐻 (bit)

Zhang 2.203

𝑗=1

where 𝜔𝑘 = [(2𝜋)1/2 ]

Table 1: Information entropies.

1 −𝑥2 /2 𝑑𝑥 𝑒 √2𝜋

𝜏𝑖+1 (𝑘)

𝜏𝑖 (𝑘)

−1

(21)

1 −𝑥2 /2 𝑑𝑥) ] . 𝑒 √2𝜋

Therefore in the case of one-bit transmission bandwidth, the information entropies of MQI-KF and [6, 7, 9] can be seen in Table 1. From Table 1 we can see that, compared with quantization schemes in [6, 7, 9], the quantization scheme in this paper has greater information entropy. In other words, the quantized innovation of this quantization scheme contains more information. Thus, this scheme is more efficient because the number of its quantization levels is the biggest when the bandwidth is one bit. 4.3. Performance Analysis of the State Estimation Algorithm. The coefficient of the error covariance matrix correction 𝜌𝑘 is adopted as the criterion to evaluate different quantization

6

International Journal of Distributed Sensor Networks Table 2: Values of 𝜌𝑘 of different algorithms.

10 9

𝜌𝑘 Zhang 0.9201 0.9859

1 bit 2 bits

Xu 0.8098 0.9560

You 0.8098 0.9201

Msechu 0.6366 0.8825

state estimation algorithms. For standard KF, 𝜌𝑘 = 1; for the MQI-KF, 𝜌𝑘 can be obtained by referring to [6]

7 6 5 4 3

𝑗

𝜌𝑘 = 𝐸 [𝜌𝑘 | 𝑏1𝑘 ]

2 2

=

8

RMSE (m)

𝑐

2 2 1 𝐿 {exp [−𝜏𝑗 (𝑘) /2] − exp [−𝜏𝑗+1 (𝑘) /2]} . ∑ 2𝜋 𝑗=1 Φ [𝜏𝑗+1 (𝑘)] − Φ [𝜏𝑗 (𝑘)]

(22)

Therefore, when the 𝑐 (the amount of transmission bandwidth’s bytes) are the same, the values of 𝜌𝑘 of different algorithms can be seen in Table 2. According to Table 2, the algorithm put forward in this paper is the most accurate one.

5. Results of Simulation Experiments In order to demonstrate the performance of the MQI-KF, two simulation experiments are performed. To evaluate the performances of the quantization schemes of MQI-KF and those in [6, 7, 9], the experiment is conducted with onebit transmission bandwidth. The second experiment aims to evaluate the efficiency of MQI-KF, and the comparison is made on the basis of the same transmission bandwidth. A simulation scenario for target tracking in the WSN is designed. The following items are detailed information about it: (1) The WSN consists of a FC and 𝑁 (𝑁 = 50) sensors which are randomly arranged over a region of 100 m × 100 m. The coordinate of each sensor is known. (2) The single target in this region is regarded as a point object. The state of the target at time 𝑘 is defined as 𝑇 𝑦𝑘 𝑦𝑘̇ 𝑥𝑘̈ 𝑦𝑘̈ ] . In this formula, 𝑥𝑘 𝑋𝑘 = [𝑥𝑘 𝑥𝑘̇ and 𝑦𝑘 are the positions at the 𝑋 and 𝑌 coordinates, and 𝑦𝑘̇ are the velocities at the two respectively; 𝑥𝑘̇ and 𝑦𝑘̈ are the acceleration of the target at the axes; 𝑥𝑘̈ 𝑋 and 𝑌 coordinates, respectively. (3) The time step of the FC is a constant (𝑇 = 1 s). During this time step, the three sensors which are the nearest of the moving target will be activated. Each activated sensor measures the distance between itself and the moving target and quantizes the innovation locally. Then the data packet will be transmitted to the FC. Finally, the algorithms under this scenario are over Monte-Carlo 1000 runs in the experiments. This paper assumes that the target moves in the 2D Cartesian coordinate system, and the dynamic model is assumed as the constant acceleration (CA) model for the state vector.

1 0

0

50

100

150

200

250

300

350

400

t (step)

EKF Zhang-5 levels Xu-3 levels

You-3 levels Msechu-2 levels

Figure 4: RMSE of the positions along the 𝑥-axis for the estimators.

For the 𝑖th activated sensor, its coordinate is (𝑥𝑖 , 𝑦𝑖 ), and the measurements are modeled as 2

2

𝑍𝑘𝑖 = ℎ𝑘𝑖 (𝑋𝑘 ) + V𝑘 = √(𝑥𝑘 − 𝑥𝑖 ) + (𝑦𝑘 − 𝑦𝑖 ) + V𝑘 ,

(23)

where the white Gaussian noise sequences 𝑤𝑘−1 and V𝑘 are independent of each other and their variances are var(𝑤𝑘−1 ) = 0.012 and var(V𝑘 ) = 102 . The measurement equation is nonlinear and is linearized in the simulation experiments. A comparison is made among the MQI-KF in this paper, the other algorithms in [6, 7, 9], and the standard KF (EKF) based on nonquantized innovation. This paper assumes that no communication loss appears in this process. 5.1. Simulation without Considering the Transmission Strategy. Figures 4 and 5 show the root mean square error (RMSE) of the positions at the 𝑋 and 𝑌 coordinates, respectively. It can be seen that with the one-bit transmission bandwidth, the performance of the MQI-KF is obviously better than the algorithms in [6, 7, 9]. The reason is that the quantization scheme in this paper has more quantization levels. Therefore, it is more efficient. 5.2. Simulation with Transmission Strategy. Figures 6 and 7 show that the performance of the MQI-KF is obviously more accurate than those in [6, 7, 9], on the basis of the same twobit transmission bandwidth. Besides, it is close to the standard KF (EKF) which is based on nonquantized measurements. This means that, using the same bandwidth, the MQI-KF has higher accuracy since it is based on efficient quantization scheme.

6. Conclusion This paper studies an innovative decentralized state estimation algorithm MQI-KF. It is based on the quantization

7

10

10

9

9

8

8

7

7

6

6

RMSE (m)

RMSE (m)


5 4

4

3

3

2

2

1

1

0

0

50

100

150

200 250 t (step)

EKF Zhang-5 levels Xu-3 levels

300

350

400

0

0

50

100

150

200

250

300

350

400

t (step)

You-3 levels Msechu-2 levels

EKF Zhang-2 bits Xu-2 bits

Figure 5: RMSE of the positions along the 𝑦-axis for the estimators.

10

You-2 bits Msechu-2 bits

Figure 7: RMSE of the positions along the 𝑦-axis for the estimators.

Appendices

9

A. Prediction Step

8

Similar to the cases in [6, 9], 𝑝(𝑋𝑘−1 | 𝑏1𝑘−1 ) is assumed to be a Gaussian conditional density:

7 RMSE (m)

5

6 5 4

󵄨 󵄨1/2 𝑝 (𝑋𝑘−1 | 𝑏1𝑘−1 ) = [(2𝜋)𝑛/2 󵄨󵄨󵄨𝑃𝑘−1|𝑘−1 󵄨󵄨󵄨 ]

3

(A.1) ̂𝑘−1|𝑘−1 )𝑇 𝑃−1 ̂ { (𝑋𝑘−1 − 𝑋 𝑘−1|𝑘−1 (𝑋𝑘−1 − 𝑋𝑘−1|𝑘−1 ) } ⋅ exp {− , } 2 } {

2 1 0

−1

0

50

100

150

200

250

300

350

400

t (step)

EKF Zhang-2 bits Xu-2 bits

You-2 bits Msechu-2 bits

Figure 6: RMSE of the positions along the 𝑥-axis for the estimators.

scheme proposed by considering the characteristics of the practical WSNs with low energy consumption. Through studying the posterior pdf, the decentralized KF is derived by adopting the Bayesian approach. On this basis, a comprehensive decentralized state estimation algorithm is put forward. The performance analysis and simulation experiments show that the MQI-KF is better than those in [6, 7, 9]. Moreover, the quantization scheme and algorithm proposed in this paper can be applied into most of the WSNs which cost less energy in practice.

̂𝑘−1|𝑘−1 is the conditional mean and 𝑃𝑘−1|𝑘−1 is its where 𝑋 conditional covariance matrix. The 𝑋𝑘 is a linear combination of 𝑋𝑘−1 and 𝑤𝑘−1 in (1). If the joint pdf 𝑝(𝑋𝑘−1 , 𝑤𝑘−1 | 𝑏1𝑘−1 ) is Gaussian, so is the 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ). Since 𝑤𝑘−1 is independent of 𝑏1𝑘−1 and 𝑋𝑘−1 , it can be expressed by

𝑝 (𝑋𝑘−1 , 𝑤𝑘−1 | 𝑏1𝑘−1 ) = 𝑝 (𝑋𝑘−1 | 𝑏1𝑘−1 ) 𝑝 (𝑤𝑘−1 ) .

(A.2)

According to the system equation (1), the pdf 𝑝(𝑤𝑘−1 ) is Gaussian, and 𝑝(𝑋𝑘−1 | 𝑏1𝑘−1 ) has been assumed to be Gaussian in (A.1). So their product 𝑝(𝑋𝑘−1 , 𝑤𝑘−1 | 𝑏1𝑘−1 ) is also Gaussian. Thus, the conditional density 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ) is Gaussian.

8


For the conditional density 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ), its conditional ̂𝑘|𝑘−1 can be derived by mean 𝑋 ̂𝑘|𝑘−1 = 𝐸 [𝑋𝑘 | 𝑏𝑘−1 ] 𝑋 1

𝛾=

= 𝐸 [𝐹𝑘−1 𝑋𝑘−1 + 𝑤𝑘−1 | 𝑏1𝑘−1 ] = 𝐹𝑘−1 𝐸 [𝑋𝑘−1 | 𝑏1𝑘−1 ] + 𝐸 [𝑤𝑘−1 | 𝑏1𝑘−1 ]

(A.3)

̂𝑘−1|𝑘−1 . = 𝐹𝑘−1 𝑋 And the conditional covariance matrix 𝑃𝑘|𝑘−1 can be obtained by 𝑇

̂𝑘|𝑘−1 ) (𝑋𝑘 − 𝑋 ̂𝑘|𝑘−1 ) | 𝑏𝑘−1 ] 𝑃𝑘|𝑘−1 = 𝐸 [(𝑋𝑘 − 𝑋 1 (A.4)

̃𝑘−1 + 𝑤𝑘−1 )𝑇 | 𝑏𝑘−1 ] = 𝐹𝑘−1 𝑃𝑘−1|𝑘−1 𝐹𝑇 ⋅ (𝐹𝑘−1 𝑋 1 𝑘−1 + 𝑄𝑘−1 , ̂𝑘−1|𝑘−1 . Thus, the pdf can be presented ̃𝑘−1 = 𝑋𝑘−1 − 𝑋 where 𝑋 explicitly as follows: −1

󵄨 󵄨1/2 𝑝 (𝑋𝑘 | 𝑏1𝑘−1 ) = [(2𝜋)𝑛/2 󵄨󵄨󵄨𝑃𝑘|𝑘−1 󵄨󵄨󵄨 ] 𝑇

̂𝑘|𝑘−1 ) 𝑃−1 (𝑋𝑘 − 𝑋 ̂𝑘|𝑘−1 ) } { (𝑋𝑘 − 𝑋 𝑘|𝑘−1 ⋅ exp {− }. 2 { }

=

𝑝 (𝑏1𝑘−1 , 𝑏) 𝑝 (𝑏 | 𝑋𝑘 , 𝑏1𝑘−1 ) 𝑝 (𝑏 | 𝑏1𝑘−1 )

× 𝑝 (𝑋𝑘 | 𝑏1𝑘−1 )

× 𝑝 (𝑋𝑘 | 𝑏1𝑘−1 ) 𝜏𝑗+1

=

∫𝜏

𝑗

𝑝 (𝛾 | 𝑋𝑘 , 𝑏1𝑘−1 ) 𝑑𝛾

𝜏𝑗+1

∫𝜏

𝑗

𝑝 (𝛾 | 𝑏1𝑘−1 ) 𝑑𝛾

× 𝑝 (𝑋𝑘 | 𝑏1𝑘−1 ) .

ℎ𝐸 [𝑋𝑘 | 𝑏1𝑘−1 ] + 𝐸 [V𝑘 | 𝑏1𝑘−1 ] − 𝑦̂𝑘|𝑘−1

(B.1)

(B.4)

Ω𝑘 ̂𝑘|𝑘−1 − 𝑦̂𝑘|𝑘−1 ℎ𝑋 = 0. Ω𝑘

The conditional covariance can be obtained by 𝑇

𝑃 (̂ 𝛾) = 𝐸 {[𝛾 − 𝛾̂] [𝛾 − 𝛾̂] | 𝑏1𝑘−1 } =

ℎ𝑃ℎ𝑇 + 𝑅 = 1. (B.5) Ω2𝑘

Thus, 𝑝(𝛾 | 𝑏1𝑘−1 ) = N(𝛾; 0, 1). For the numerator 𝑝(𝛾 | 𝑋𝑘 , 𝑏1𝑘−1 ) in (B.1), the 𝛾 conditioning on 𝑋𝑘 means that 𝑋𝑘 is regarded as a known vector in this conditional density. Besides, ℎ, 𝑦̂𝑘|𝑘−1 , and Ω𝑘 in (B.2) are constants at this time step. According to (B.2), 𝛾 is a linear combination of a Gaussian random variable V𝑘 and a known vector, so 𝑝(𝛾 | 𝑋𝑘 , 𝑏1𝑘−1 ) is also a Gaussian pdf. Thus, the conditional mean 𝛾̂𝑋 can be obtained by 𝛾̂𝑋 = 𝐸 [𝛾 | 𝑋𝑘 , 𝑏1𝑘−1 ] =

Pr {𝛾 ∈ (𝜏𝑗 , 𝜏𝑗+1 ] | 𝑋𝑘 , 𝑏1𝑘−1 } Pr {𝛾 ∈ (𝜏𝑗 , 𝜏𝑗+1 ] | 𝑏1𝑘−1 }

𝛾̂ = 𝐸 [𝛾 | 𝑏1𝑘−1 ]

=

𝑝 (𝑋𝑘 , 𝑏1𝑘−1 , 𝑏)

(B.3)

Considering that 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ) in (A.5) and 𝑝(V𝑘 ) in (2) are Gaussian densities, the joint pdf 𝑝(𝑋𝑘 , V𝑘 | 𝑏1𝑘−1 ) is also Gaussian. So 𝑝(𝛾 | 𝑏1𝑘−1 ) is a Gaussian density. The conditional mean 𝛾̂ can be derived by

(A.5)

At time step 𝑘, we assume 𝛾𝑘 ∈ (𝜏𝑗 (𝑘), 𝜏𝑗+1 (𝑘)] (1 ≤ 𝑗 ≤ 4𝑙+2). To be brief, let 𝑏 = 𝑏𝑘 , 𝛾 = 𝛾𝑘 , ℎ = ℎ𝑘 , 𝑅 = 𝑅𝑘 , 𝜏𝑗 = 𝜏𝑗 (𝑘), ̃ = 𝑋𝑘 − 𝑋 ̂𝑘|𝑘−1 , and 𝑃 = 𝑃𝑘|𝑘−1 at this time 𝜏𝑗+1 = 𝜏𝑗+1 (𝑘), 𝑋 step. By referring to (8) and 𝛾 ∈ (𝜏𝑗 , 𝜏𝑗+1 ], the posterior pdf 𝑝(𝑋𝑘 | 𝑏1𝑘−1 , 𝑏) can be obtained using the Bayesian rule:

(B.2)

where Ω2𝑘 = ℎ𝑃ℎ𝑇 + 𝑅 in (6). The conditional density 𝑝(𝑋𝑘 | 𝑏1𝑘−1 ) has already been established at Appendix A. For the denominator 𝑝(𝛾 | 𝑏1𝑘−1 ) in (B.1), it will be Gaussian if the joint pdf 𝑝(𝑋𝑘 , V𝑘 | 𝑏1𝑘−1 ) is Gaussian, since 𝛾 in (B.2) is a linear combination of 𝑋𝑘 and V𝑘 . Since V𝑘 is independent from 𝑏1𝑘−1 and 𝑋𝑘−1 , there is this formula:

=

B. Correction Step for the Conditional Mean

=

(ℎ𝑋𝑘 + V𝑘 − 𝑦̂𝑘|𝑘−1 ) , Ω𝑘

𝑝 (𝑋𝑘 , V𝑘 | 𝑏1𝑘−1 ) = 𝑝 (𝑋𝑘 | 𝑏1𝑘−1 ) 𝑝 (V𝑘 ) .

̃𝑘−1 + 𝑤𝑘−1 ) = 𝐸 [(𝐹𝑘−1 𝑋

𝑝 (𝑋𝑘 | 𝑏1𝑘−1 , 𝑏) =

Currently, the aim is to describe the posterior conditional density 𝑝(𝑋𝑘 | 𝑏1𝑘−1 , 𝑏) explicitly. By referring to (2) and (7), there can be this formula:

̃ + V𝑘 | 𝑋𝑘 , 𝑏𝑘−1 ] 𝐸 [ℎ𝑋 1 Ω𝑘

̃ ℎ𝑋 = . Ω𝑘

(B.6)

And its conditional covariance 𝑃(̂ 𝛾𝑋 ) is 𝑇

𝑃 (̂ 𝛾𝑋 ) = 𝐸 {[𝛾 − 𝛾̂𝑋 ] [𝛾 − 𝛾̂𝑋 ] | 𝑋𝑘 , 𝑏1𝑘−1 } =

𝐸 [V𝑘 V𝑘𝑇 | 𝑋𝑘 , 𝑏1𝑘−1 ] Ω2𝑘

𝑅 = 2. Ω𝑘

(B.7)


9

So 𝑝(𝛾 | 𝑋𝑘 , 𝑏1𝑘−1 ) = N[𝛾; 𝛾̂𝑋 , 𝑃(̂ 𝛾𝑋 )]. 𝑘−1 Considering 𝑝(𝛾 | 𝑏1 ) = N(𝛾; 0, 1), 𝑝(𝛾 | 𝑋𝑘 , 𝑏1𝑘−1 ) = N[𝛾; 𝛾̂𝑋, 𝑃(̂ 𝛾𝑋 )], and (A.5), the posterior density 𝑝(𝑋𝑘 | 𝑘−1 𝑏1 , 𝑏) in (B.1) can be rewritten as

̃ 𝑘) × Define 𝐼𝑛 ∈ R𝑛 , 𝐼𝑛 = [1, 0, . . . , 0]𝑇 , 𝑙𝑗+1 = (𝜏𝑗+1 − ℎ𝑋/Ω 𝑛 ̃ 𝑘 ) × 𝐼𝑛 , 𝛼 ∈ R , 𝛼 = 𝛽 × 𝐼𝑛 , 𝐴 ∈ R𝑛×𝑛 , and 𝐼𝑛 , 𝑙𝑗 = (𝜏𝑗 − ℎ𝑋/Ω 𝐴 = 𝑅/Ω2𝑘 ×𝐼𝑛×𝑛 , and 𝐼𝑛×𝑛 is an 𝑁-order identity matrix. Since 𝛽, 𝑅, and Ω𝑘 are scalar values, we have that

𝑝 (𝑋𝑘 | 𝑏1𝑘−1 , 𝑏) ∫

󵄨󵄨1/2 −1

󵄨󵄨 󵄨 𝑃𝑅 󵄨 = [(2𝜋)(𝑛+1)/2 󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨 󵄨󵄨 Ω𝑘 󵄨󵄨 𝜏

×

𝑙𝑗

]

exp (−

𝛼𝑇𝐴−1 𝛼 ) 𝑑𝛼 2

(B.8)

𝑋 𝑇

∫𝜏 𝑗+1 𝑒−(𝛾−̂𝛾

𝑙𝑗+1

) 𝑃(̂ 𝛾𝑋 )−1 (𝛾−̂ 𝛾𝑋 )/2

𝑗

̃ ̃𝑇 𝑃−1 𝑋/2

𝑑𝛾 × 𝑒−𝑋

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

=∫

̃ 𝑘 𝜏𝑗 −ℎ𝑋/Ω

.

Thus, the posterior density 𝑝(𝑋𝑘 | 𝑏1𝑘 ) is described explicitly. ̃ = 𝑋𝑘 − 𝑋 ̂𝑘|𝑘−1 , there is the formula: Since 𝑋

(B.13)

−1

exp [− [

𝛽𝑇 (𝑅/Ω2𝑘 ) 𝛽 2

] 𝑑𝛽. ]

Thus, (B.12) can be rewritten as

{⋅} = − 𝑃 ∫

̂𝑘|𝑘−1 + 𝑋 ̃ | 𝑏𝑘−1 , 𝑏] 𝐸 [𝑋𝑘 | 𝑏1𝑘−1 , 𝑏] = 𝐸 [𝑋 1 ̃ | 𝑏𝑘−1 , 𝑏] . ̂𝑘|𝑘−1 + 𝐸 [𝑋 =𝑋 1

̃ 𝑘 𝜏𝑗+1 −ℎ𝑋/Ω

R𝑛

̃𝑇 𝑃−1 𝑋 ̃ 𝑋 𝜕 [exp (− )] ̃ 2 𝜕𝑋 (B.14)

(B.9) ⋅∫

𝑙𝑗+1

𝑙𝑗

𝛼𝑇 𝐴−1 𝛼 ̃ exp (− ) 𝑑𝛼 𝑑𝑋. 2

By referring to (B.8), there will be this:

̃ | 𝑏𝑘−1 , 𝑏] = 𝐸 [𝑋 1

󵄨 󵄨1/2 −1 [(2𝜋)(𝑛+1)/2 󵄨󵄨󵄨󵄨𝑃𝑅/Δ2𝑘 󵄨󵄨󵄨󵄨 ] Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

By interchanging the integrals in (B.14), there will be × {⋅}

̃𝑇 −1 ̃

̃ exp (− 𝑋 𝑃 𝑋 ) {⋅} = ∫ 𝑋 2 R𝑛 ⋅∫

𝜏𝑗+1

𝜏𝑗

(B.10)

̃ ̃ 𝑇 𝑅 −1 1 ℎ𝑋 ℎ𝑋 ̃ exp [− (𝛾 − ) ( 2 ) (𝛾 − )] 𝑑𝛾 𝑑𝑋. 2 Ω𝑘 Ω𝑘 Ω𝑘

̃ | 𝑏𝑘−1 , 𝑏]. The purpose here is to obtain the value of 𝐸[𝑋 1 ̃ 𝑘 , {⋅} can be expressed as follows: Defining 𝛽 = 𝛾 − ℎ𝑋/Ω ̃ exp [− 𝑋 𝑃 𝑋 ] {⋅} = ∫ 𝑋 𝑛 2 R ⋅∫


(B.11)

𝛽𝑇 (𝑅/Ω2𝑘 ) 𝛽

] 𝑑𝛽 𝑑𝑋. ̃

2

]

𝛼𝑇𝐴−1 𝛼 ) 2

+ ̃𝑇𝑃−1 𝑋 ̃ 󵄨󵄨󵄨ℎ (𝜏𝑗+1 −𝛽)Ω𝑘 𝑋 󵄨 𝑑𝛼 ⋅ exp (− )󵄨󵄨󵄨 2 󵄨󵄨 +

(B.15)

ℎ (𝜏𝑗 −𝛽)Ω𝑘

R𝑛

−1

exp [− [

R𝑛

= − 𝑃 ∫ exp (−

̃𝑇 −1 ̃


{⋅} = − 𝑃 ∫ exp (−

𝜎1 𝜎 ) 𝑑𝛼 + 𝑃 ∫ exp (− 2 ) 𝑑𝛼, 𝑛 2 2 R

where ℎ+ is a Moore-Penrose pseudoinverse matrix [14], and 𝜎1 = 𝛼𝑇 𝐴−1 𝛼

Furthermore, 𝑇

+ [ℎ+ (𝜏𝑗+1 − 𝛽) Ω𝑘 ] 𝑃−1 [ℎ+ (𝜏𝑗+1 − 𝛽) Ω𝑘 ]

̃𝑇 𝑃−1 𝑋 ̃ 𝑋 𝜕 [exp (− )] {⋅} = − 𝑃 ∫ ̃ 2 R 𝑛 𝜕𝑋 ⋅∫



𝑇

exp (−

𝛽

−1 (𝑅/Ω2𝑘 )

2

𝛽

(B.16) (B.12)

̃ ) 𝑑𝛽 𝑑𝑋.

𝑇

−1

𝜎2 = 𝛼 𝐴 𝛼 𝑇

+ [ℎ+ (𝜏𝑗 − 𝛽) Ω𝑘 ] 𝑃−1 [ℎ+ (𝜏𝑗 − 𝛽) Ω𝑘 ] .

10

International Journal of Distributed Sensor Networks Because 𝑑𝛼 = 𝑑𝑏𝑗+1 and 𝑑𝛼 = 𝑑𝑏𝑗 , the formula can be changed into

Then 𝜎1 =

Ω2𝑘

[𝐼𝑛𝑇 𝛽𝑇 𝑅−1 𝐼𝑛×𝑛 𝛽𝐼𝑛 𝑇

{⋅} = − 𝑃 exp (−

+ 𝑇

+ (𝜏𝑗+1 − 𝛽) (ℎ ) 𝑃−1 ℎ+ (𝜏𝑗+1 − 𝛽)] 𝑇

= Ω2𝑘 {𝛽𝑇 [𝑅−1 + (ℎ+ ) 𝑃−1 ℎ+ ] 𝛽 + 𝑇

⋅ ∫ exp (−

=

⋅ exp (−

−1 𝑇

𝑇

{[𝛽 − 𝜏𝑗+1 𝑅 (ℎ𝑃ℎ + 𝑅) ] 𝑇

× [ℎ𝑃ℎ 𝑅 (ℎ𝑃ℎ + 𝑅) ]

−1

𝑇

𝜏𝑗2 2

−1

2 (ℎ𝑃ℎ𝑇 + 𝑅) } = [𝛽 + 𝜏𝑗+1

(B.17)

−2 −1

=

× [ℎ𝑃ℎ𝑇 𝑅 (ℎ𝑃ℎ𝑇 + 𝑅) ] × [𝛽

𝜏𝑗2 2

) − exp (−

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

⋅ 𝑃 [exp (−

𝑇

− 𝜏𝑗+1 𝑅 (ℎ𝑃ℎ𝑇 + 𝑅) 𝐼𝑛 ]

= [(2𝜋)1/2 ]

−1

× [ℎ𝑃ℎ𝑇 𝑅 (ℎ𝑃ℎ𝑇 + 𝑅) 𝐼𝑛×𝑛 ] × [𝛼 − 𝜏𝑗+1 𝑅 (ℎ𝑃ℎ + 𝑅)

2

R𝑛

󵄨 󵄨1/2 −1 [(2𝜋)(𝑛+1)/2 󵄨󵄨󵄨󵄨𝑃𝑅/Ω2𝑘 󵄨󵄨󵄨󵄨 ]

−1

2 = [𝛼 − 𝜏𝑗+1 𝑅 (ℎ𝑃ℎ𝑇 + 𝑅) ] + 𝜏𝑗+1

−1

𝑏𝑗𝑇 𝐵−1 𝑏𝑗

(B.20)

) 𝑑𝑏𝑗

2 𝜏𝑗+1

2

)] .

Thus, (B.10) can be rewritten as

− 𝜏𝑗+1 𝑅 (ℎ𝑃ℎ𝑇 + 𝑅) ]

𝑇

) 𝑑𝑏𝑗+1 + 𝑃

2

̃ | 𝑏𝑘−1 , 𝑏] 𝐸 [𝑋 1

−1 𝑇

−2

𝑇 𝑏𝑗+1 𝐵−1 𝑏𝑗+1

) ∫ exp (−

⋅ 𝑃 [exp (−

× [𝛽 − 𝜏𝑗+1 𝑅 (ℎ𝑃ℎ + 𝑅) ]

−1

)

= [(2𝜋)𝑛/2 |𝐵|1/2 ]

−1 −1

𝑇

2

R𝑛

+ 𝑇

2 − 𝜏𝑗+1 (ℎ ) 𝑃−1 ℎ+ (𝛽 + 𝛽𝑇 )} + (ℎ ) 𝑃−1 ℎ+ 𝜏𝑗+1

Ω2𝑘

2 𝜏𝑗+1

2 𝐼𝑛 ] + 𝜏𝑗+1

=

×


2 + 𝜏𝑗+1 ,

𝜏𝑗2 2

) − exp (−

[(2𝜋)𝑛/2 |𝐵|1/2 ]

2 𝜏𝑗+1

2

)]

(B.21)

𝑃ℎ𝑇

−1

√ℎ𝑃ℎ𝑇 + 𝑅

2 exp (−𝜏𝑗2 /2) − exp (−𝜏𝑗+1 /2)

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

.

By referring to (B.9), there will be

where 𝑏𝑗+1 = 𝛼−𝜏𝑗+1 𝑅(ℎ𝑃ℎ𝑇 +𝑅)−1 𝐼𝑛 and 𝐵 = ℎ𝑃ℎ𝑇 𝑅(ℎ𝑃ℎ𝑇 + 𝑅)−2 𝐼𝑛×𝑛 . Similarly,

× [ℎ𝑃ℎ𝑇 𝑅 (ℎ𝑃ℎ𝑇 + 𝑅) 𝐼𝑛×𝑛 ] −1

𝑇

× [𝛼 − 𝜏𝑗 𝑅 (ℎ𝑃ℎ + 𝑅)

(B.18)

𝐼𝑛 ] + 𝜏𝑗2

where 𝑏𝑗 = 𝛼 − 𝜏𝑗 𝑅(ℎ𝑃ℎ𝑇 + 𝑅)−1 𝐼𝑛 . Substituting (B.17) and (B.18) into (B.15), there will be

+ 𝑃 exp (−

𝜏𝑗2 2

) ∫ exp (−


2

R𝑛

) ∫ exp (− R𝑛

√ℎ𝑃ℎ𝑇 + 𝑅

(B.22) ,

1/2 −1

𝜔𝑘 = [(2𝜋)

]

2 exp (−𝜏𝑗2 /2) − exp (−𝜏𝑗+1 /2)

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

.

(B.23)

C. Correction Step for the Conditional Covariance Matrix

= 𝑏𝑗𝑇 𝐵−1 𝑏𝑗 + 𝜏𝑗2 ,

2

𝑃ℎ𝑇

where −1

−2

{⋅} = − 𝑃 exp (−

̂𝑘|𝑘−1 + 𝜔𝑘 =𝑋

𝑇

−1

𝜎2 = [𝛼 − 𝜏𝑗 𝑅 (ℎ𝑃ℎ𝑇 + 𝑅) 𝐼𝑛 ]

2 𝜏𝑗+1

̂𝑘|𝑘 = 𝐸 [𝑋𝑘 | 𝑏𝑘−1 , 𝑏] = 𝑋 ̂𝑘|𝑘−1 + 𝐸 [𝑋 ̃ | 𝑏𝑘−1 , 𝑏] 𝑋 1 1

𝑏𝑗𝑇 𝐵−1 𝑏𝑗 2

̂𝑘|𝑘 ) (𝑋𝑘 − 𝑋 ̂𝑘|𝑘 )𝑇 | 𝑏𝑘 ] 𝑃𝑘|𝑘 = 𝐸 [(𝑋𝑘 − 𝑋 1

) 𝑑𝛼 (B.19)

) 𝑑𝛼.

For the covariance matrix 𝑃𝑘|𝑘 ,

̃𝑎 + 𝑋 ̃𝑎 − 𝑋 ̂𝑘|𝑘 ) = 𝐸 [(𝑋𝑘 − 𝑋 𝑘 𝑘 𝑇

̃𝑎 + 𝑋 ̃𝑎 − 𝑋 ̂𝑘|𝑘 ) | 𝑏𝑘 ] , ⋅ (𝑋𝑘 − 𝑋 𝑘 𝑘 1

(C.1)


11

where

Similarly,

̃𝑎 = 𝐸 [𝑋 ̃ | 𝑏𝑘−1 , 𝛾] = 𝐸 [𝑋𝑘 | 𝑏𝑘−1 , 𝛾] − 𝑋 ̂𝑘|𝑘−1 , 𝑋 𝑘 1 1 ̂𝑘|𝑘 = 𝐸 [𝑋𝑘 | 𝑏𝑘−1 , 𝑏] = 𝑋 ̂𝑘|𝑘−1 + 𝐸 [𝑋 ̃ | 𝑏𝑘−1 , 𝑏] . 𝑋 1 1

(C.2)

̂𝑘|𝑘 ) (𝑋𝑘 − 𝑋 ̂𝑘|𝑘 ) | 𝑃𝑘|𝑘 = 𝐸 [(𝑋𝑘 − 𝑋

𝑏1𝑘 ]

̃ | 𝑏𝑘−1 , 𝑏)] ̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 𝐸 {[𝐸 (𝑋 1 1

= 𝐸 {[𝑋𝑘

̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋𝑘 | 𝑏1𝑘−1 , 𝛾) + 𝐸 (𝑋 1

𝑇

̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 ̃ | 𝑏𝑘−1 , 𝑏)] | 𝑏𝑘 } = 𝐾 ⋅ [𝐸 (𝑋 1 1 1 (C.3)

̃ | 𝑏𝑘−1 , 𝑏)] × [𝑋𝑘 − 𝐸 (𝑋𝑘 | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 1 1

× 𝐸 {[𝛾 − 𝐸 (𝛾 | 𝑏1𝑘−1 , 𝑏)] 𝑇

Besides, since 𝐸{[𝑋𝑘 − 𝐸(𝑋𝑘 | 𝑏1𝑘−1 , 𝛾)] | 𝑏1𝑘 } = 0, (C.3) will be 𝑏1𝑘−1 , 𝛾)]

× Var [𝛾 | 𝑏1𝑘 ] × 𝐾𝑇 , where Var {𝛾 | 𝑏1𝑘 } = 𝐸 {𝛾2 | 𝑏1𝑘 } − 𝐸2 {𝛾 | 𝑏1𝑘 } .

𝑇

⋅ [𝑋𝑘 − 𝐸 (𝑋𝑘 | 𝑏1𝑘−1 , 𝛾)] | 𝑏1𝑘 } (C.4) ̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 ̃ | 𝑏𝑘−1 , 𝑏)] + 𝐸 {[𝐸 (𝑋 1 1

Var {𝛾 | 𝑏1𝑘−1 , 𝑏} = ∫

𝑇

(C.10)

− 𝐸 {𝛾 |

𝑏1𝑘−1 , 𝑏} .

For the second term in (C.10), according to (B.21) and (C.7), we can obtain 𝐸 {𝛾 | 𝑏1𝑘−1 , 𝑏}

𝑇 𝑏1𝑘−1 , 𝛾)]

|

𝑏1𝑘 }

𝑃ℎ𝑇 ℎ𝑃 . = 𝑃− ℎ𝑃ℎ𝑇 + 𝑅

(C.5)

1/2 −1

= [(2𝜋)

For the second term in (C.4), according to (6), (7), and ̃ | 𝑏𝑘−1 , 𝛾] is given by the KF in [15], 𝐸[𝑋 1 ̃ | 𝑏𝑘−1 , 𝛾] = 𝐸 [𝑋𝑘 | 𝑏𝑘−1 , 𝛾] 𝐸 [𝑋 1 1 − 𝐸 [𝑋𝑘|𝑘−1 | 𝑏1𝑘−1 , 𝛾] ̂𝑘|𝑘−1 𝑏1𝑘−1 , 𝛾] − 𝑋

̂𝑘|𝑘−1 + =𝑋

𝛾2 𝑝 (𝛾 | 𝑏1𝑘−1 , 𝑏) 𝑑𝛾

2

𝐸 {[𝑋𝑘 − 𝐸 (𝑋𝑘 | 𝑏1𝑘−1 , 𝛾)]

= 𝐸 [𝑋𝑘 |

𝜏𝑗+1

𝜏𝑗

Here, the purpose is to obtain the 𝑃𝑘|𝑘 in (C.4). For the first term in (C.4), using the KF in [15], there will be

(C.9)

Since we assume 𝛾 ∈ (𝜏𝑗 , 𝜏𝑗+1 ] at time step 𝑘 and 𝑝(𝛾 | 𝑏1𝑘−1 ) = N(𝛾; 0, 1), (C.9) can be rewritten as

̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 ̃ | 𝑏𝑘−1 , 𝑏)] | 𝑏𝑘 } . ⋅ [𝐸 (𝑋 1 1 1

⋅ [𝑋𝑘 − 𝐸 (𝑋𝑘 |

(C.8)

⋅ [𝛾 − 𝐸 (𝛾 | 𝑏1𝑘−1 , 𝑏)] | 𝑏1𝑘 } × 𝐾𝑇 = 𝐾

𝑇

̃ | 𝑏𝑘−1 , 𝑏)] | 𝑏𝑘 } . ̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 + 𝐸 (𝑋 1 1 1

𝑃𝑘|𝑘 = 𝐸 {[𝑋𝑘 − 𝐸 (𝑋𝑘 |

(C.7)

Substituting (C.6) and (C.7) into the second term of (C.4), we can obtain

Therefore, (C.1) can be rewritten as 𝑇

̃ | 𝑏𝑘−1 , 𝑏] = 𝐾 × 𝐸 [𝛾 | 𝑏𝑘−1 , 𝑏] . 𝐸 [𝑋 1 1

]

(C.11)

And, for the 𝑝(𝛾 | 𝑏1𝑘−1 , 𝑏) in the first term of (C.9), since the quantized normalized innovation 𝑏 is a discrete value, using the Bayesian rule 𝑏1𝑘−1 , 𝑏)

𝑃ℎ𝑇 × (𝑦𝑘 − 𝑦̂𝑘|𝑘−1 ) (C.6) ℎ𝑃ℎ𝑇 + 𝑅

=

=

󵄨 󵄨 𝑝 (𝛾 | 𝑏1𝑘−1 ) 𝑝 (𝑏 󵄨󵄨󵄨𝛾󵄨󵄨󵄨 𝑏1𝑘−1 ) 𝑝 (𝑏 | 𝑏1𝑘−1 )

󵄨 󵄨 𝑝 (𝛾 | 𝑏1𝑘−1 ) Pr (𝑏 󵄨󵄨󵄨𝛾󵄨󵄨󵄨 𝑏1𝑘−1 ) Pr (𝑏 | 𝑏1𝑘−1 )

(C.12) .

By referring to (8) and this paper assuming that the system is a Gaussian Markov stochastic process in the WSN, we can obtain the conditional probabilities Pr(𝑏 | 𝑏1𝑘−1 ) = Pr(𝑏) and Pr(𝑏 | 𝛾, 𝑏1𝑘−1 ) = 1. Thus, we have

𝑃ℎ𝑇

𝑦𝑘 − 𝑦̂𝑘|𝑘−1 = × 𝑇 √ℎ𝑃ℎ + 𝑅 √ℎ𝑃ℎ𝑇 + 𝑅 = 𝐾 × 𝛾, where 𝐾 = 𝑃ℎ𝑇 /√ℎ𝑃ℎ𝑇 + 𝑅 and 𝛾 𝑦̂𝑘|𝑘−1 )/√ℎ𝑃ℎ𝑇 + 𝑅 is defined in (7).

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

= 𝜔𝑘 .

𝑝 (𝛾 |

̂𝑘|𝑘−1 −𝑋

2 exp (−𝜏𝑗2 /2) − exp (−𝜏𝑗+1 /2)

=

(𝑦𝑘 −

𝑝 (𝛾 | 𝑏1𝑘−1 , 𝑏) =

𝑝 (𝛾 | 𝑏1𝑘−1 ) Pr (𝑏)

=

𝑝 (𝛾 | 𝑏1𝑘−1 ) Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

. (C.13)

12

International Journal of Distributed Sensor Networks For the first term in (C.9), it can be obtained as follows: ∫

𝜏𝑗+1

𝜏𝑗

̃ | 𝑏𝑘−1 , 𝑏)] ̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 𝐸 {[𝐸 (𝑋 1 1

𝛾2 𝑝 (𝛾 | 𝑏1𝑘−1 , 𝑏) 𝑑𝛾

𝑇

̃ | 𝑏𝑘−1 , 𝛾) − 𝐸 (𝑋 ̃ | 𝑏𝑘−1 , 𝑏)] | 𝑏𝑘 } = 𝐾 ⋅ [𝐸 (𝑋 1 1 1

𝜏

∫𝜏 𝑗+1 𝛾2 𝑝 (𝛾 | 𝑏1𝑘−1 ) 𝑑𝛾 𝑗

=

(C.14)

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 ) [(2𝜋)1/2 ]

=

Substituting (C.18) into (C.8), we have

× (1 − 𝜉𝑘 ) × 𝐾𝑇 . Further, Substituting (C.5) and (C.20) into (C.4), we can obtain

−1

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

× {⋅} ,

𝑃𝑘|𝑘 = 𝑃 −

𝑃ℎ𝑇 ℎ𝑃 + 𝐾 × (1 − 𝜉𝑘 ) × 𝐾𝑇 ℎ𝑃ℎ𝑇 + 𝑅

= 𝑃−

𝑃ℎ𝑇 ℎ𝑃 𝑃ℎ𝑇 ℎ𝑃 + (1 − 𝜉 ) 𝑘 ℎ𝑃ℎ𝑇 + 𝑅 ℎ𝑃ℎ𝑇 + 𝑅

where 𝜏𝑗+1

2

𝛾 𝛾 exp (− ) 𝑑𝛾. 2 2

{⋅} = ∫

𝜏𝑗

(C.15)

= 𝑃 − 𝜉𝑘

Furthermore, {⋅} = − ∫

𝜏𝑗+1

𝜏𝑗

𝛾𝑑 exp (−

= − 𝛾 exp (−

𝛾2 ) 2

󵄨𝜏𝑗+1

𝛾2 󵄨󵄨󵄨 )󵄨󵄨 2 󵄨󵄨󵄨𝜏

𝑗

= 𝜏𝑗+1 exp (−

2 𝜏𝑗+1

2

∫

𝜏𝑗

𝛾 𝑝 (𝛾 |

𝜏𝑗+1

+∫

𝜏𝑗

exp (−

) − 𝜏𝑗 exp (−

𝜏𝑗2 2

𝛾2 ) 𝑑𝛾 2

(C.16)

= [(2𝜋)

]

References

2 𝜏𝑗 exp (−𝜏𝑗2 /2) − 𝜏𝑗+1 exp (−𝜏𝑗+1 /2) (C.17)

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

+ 1. Substituting (C.11) and (C.17) into (C.10), there will be Var {𝛾 | 𝑏1𝑘−1 , 𝑏} = [(2𝜋)1/2 ]

−1

2 𝜏𝑗 exp (−𝜏𝑗2 /2) − 𝜏𝑗+1 exp (−𝜏𝑗+1 /2)

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

(C.18)

+ 1 − 𝜔𝑘2 = 1 − 𝜉𝑘 , where 𝜉𝑘 = 𝜔𝑘2 − [(2𝜋)1/2 ] ⋅

−1

2 𝜏𝑗 exp (−𝜏𝑗2 /2) − 𝜏𝑗+1 exp (−𝜏𝑗+1 /2)

Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )

Acknowledgments This work was jointly supported by National Natural Science Foundation (61175008); Shanghai Aerospace Science and Technology Innovation Fund (SAST201448); Aerospace Science and Technology Innovation Fund and Aeronautical Science Foundation of China (20140157001).

)

𝑏1𝑘−1 , 𝑏) 𝑑𝛾

1/2 −1

𝑃ℎ𝑇 ℎ𝑃 . ℎ𝑃ℎ𝑇 + 𝑅

The authors declare that there is no conflict of interests regarding the publication of this paper.

Thus, (C.14) can be rewritten as 2

(C.21)

Conflict of Interests

+ √2𝜋 [Φ (𝜏𝑗+1 ) − Φ (𝜏𝑗 )] .

𝜏𝑗+1

(C.20)

(C.19) .

[1] K. A. Clements and R. A. Haddad, “Approximate estimation for systems with quantized data,” IEEE Transactions on Automatic Control, vol. 17, no. 2, pp. 235–239, 1972. [2] K. Zhang and X. R. Li, “Optimal sensor data quantization for best linear unbiased estimation fusion,” in Proceedings of the 43rd IEEE Conference on Decision and Control (CDC ’04), pp. 2656–2661, December 2004. [3] E. Sviestins and T. Wigren, “Optimal recursive state estimation with quantized measurements,” IEEE Transactions on Automatic Control, vol. 45, no. 4, pp. 762–767, 2000. [4] G. Ing and M. J. Coates, “Parallel particle filters for tracking in wireless sensor networks,” in Proceedings of the IEEE 6th Workshop on Signal Processing Advances in Wireless Communications (SPAWC ’05), pp. 935–939, June 2005. [5] Z. Duan, V. P. Jilkov, and X. R. Li, “State estimation with quantized measurements: approximate MMSE approach,” in Proceedings of the 11th International Conference on Information Fusion (FUSION ’08), Cologne, Germany, July 2008. [6] E. J. Msechu, S. I. Roumeliotis, A. Ribeiro, and G. B. Giannakis, “Decentralized quantized Kalman filtering with scalable communication cost,” IEEE Transactions on Signal Processing, vol. 56, no. 8, pp. 3727–3741, 2008. [7] K. You, L. Xie, S. Sun, and W. Xiao, “Multiple-level quantized innovation Kalman filter,” in Proceedings of the 17th World Congress, International Federation of Automatic Control (IFAC ’08), pp. 1420–1425, July 2008.

International Journal of Distributed Sensor Networks [8] K. You, L. Xie, S. Sun, and W. Xiao, “Quantized filtering of linear stochastic systems,” Transactions of the Institute of Measurement and Control, vol. 33, no. 6, pp. 683–698, 2011. [9] J. Xu and J. X. Li, “State estimation with quantised sensor information in wireless sensor networks,” IET Signal Processing, vol. 5, no. 1, pp. 16–26, 2011. [10] Institute of Electrical and Electronics Engineers, IEEE Std. 802.15.4-2003 Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low Rate Wireless Personal Area Networks (LR-WPANs), IEEE Press, New York, NY, USA, 2003. [11] M. R. Palattella, N. Accettura, X. Vilajosana et al., “Standardized protocol stack for the internet of (important) things,” IEEE Communications Surveys and Tutorials, vol. 15, no. 3, pp. 1389– 1406, 2013. [12] S. P. Lloyd, “Least squares quantization in PCM,” IEEE Transactions on Information Theory, vol. 28, no. 2, pp. 129–137, 1982. [13] S. Haykin, Communication Systems, John Wiley & Sons, New York, NY, USA, 4th edition, 2001. [14] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and applications, Springer, New York, NY, USA, 2nd edition, 2003. [15] Y. Bar-Shalom, X. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation, John Wiley & Sons, New York, NY, USA, 2001.

13


Research Article A Hierarchical Scheduling Scheme in WSNs Based on Node-Failure Pretreatment Hai-yuan Liu,1,2 Yi-nan Guo,1 Mei-rong Chen,1,2 and Yuan-shun Zhu1 1

School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China College of Sciences, China University of Mining and Technology, Xuzhou 221116, China

2

Correspondence should be addressed to Yi-nan Guo; [email protected] Received 19 September 2014; Revised 9 February 2015; Accepted 9 February 2015 Academic Editor: S. Kami Makki Copyright © 2015 Hai-yuan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. One of the most important challenges in designing wireless sensor networks is how to recover a broken network within a very short time as the active nodes are failed or out-of-energy. Focusing on this problem, a hierarchical scheduling scheme based on node-failure pretreatment is proposed, in which the global optimization method is used to find the minimum connected tree and the local multilayer recovery algorithm is used to find a candidate sensor node instead of the failed one. Three highlights of this scheme are as follows: (1) The importance of sensor nodes is defined in terms of their locations in minimum connected tree and coverage acreage. (2) The neighborhood radius of failed sensor node varies with its importance, and then its candidate-node set is dynamically constructed. (3) A novel multilayer recovery strategy including node recovery and regional recovery is presented. Simulation results show that hierarchical scheduling scheme finds the optimal candidate sensor node in less time to make the repaired network with lowest energy consumption. Though the less sensor nodes are activated, the network lifetime is slightly shorter. Moreover, this scheme can be applied in the problem that the communication radius of sensor node is less than two times of its sensing radius.

1. Introduction Wireless sensor networks (WSNs) are ad hoc networks composed by a large number of tiny and autonomous wireless sensor nodes [1]. These battery-powered nodes endowing with sensing, processing, and wireless communication capabilities are deployed in the detected area to collect monitoring data from the environment and transform them to sink nodes [2]. Now WSNs have been widely applied to environmental monitoring, space exploration, military, and other fields. Each wireless sensor node has compact size and finite power supply. In particular, sensor nodes placed in dangerous environments are difficult to recharge or replace batteries. So they are cheap and their lifetimes are limited. In order to effectively monitor the detected area as long as possible and against probably exhausted sensor nodes, the robust and fully connected network is composed of the least nodes. We call this network structure as the minimum connected tree (MCT). How to rationally schedule the randomly distributed wireless sensor nodes so as to minimize energy consumption

and prolong the operational lifetime of whole network is the key issue in WSNs. A promising approach for solving this problem is sleep scheduling strategy. That is, a part of sensor nodes keeps running or being active while others called redundant nodes put into low-power sleeping mode. All active nodes form the minimum connected tree, which satisfies the requirement of energy-efficiency coverage and has least communication consumption. Once one of the active nodes runs out of the power, the communication linkages corresponding to this node are broken. In order to recover network as soon as possible, a neighbor sleeping node, which can reform the new minimum connected tree with other active nodes, will wake up instead of the out-of-energy node [3]. This scheduling strategy can effectively decrease the redundant monitoring data and prolong the network lifespan. Many researches had been done on the sleep scheduling strategy. Some schemes met the needs of coverage, which made the full use of sensor nodes in large-scale network environment by maximizing the amount of alternate-node

2 sets [4–6]. An adaptive sleep scheduling for probing environment was proposed in [7]. The states of sensor nodes were decided in terms of the probing and monitoring mechanism. In these algorithms, with the increasing number of wireless sensor nodes, fewer nodes are activated to reduce energy consumption and ensure a linear increase of network lifetime. A distributed scheduling algorithm was presented to prolong the network lifetime according to the overlapping relationship of coverage acreage between the neighbor nodes [8]. Another distributed algorithm took maximum network lifetime and the coverage rate as the optimization objectives [9]. In the above methods, the obtained wireless sensor networks had the optimal coverage rate. But the active nodes may not be fully connected. In order to monitor target area effectively and ensure the full connection among the active nodes, Zhang and Hou [10] proved that if the communication radius of sensor node is larger than twice that of its sensing radius, the network satisfying complete coverage consists of fully connected sensor nodes. However, once the above constraint condition cannot be satisfied, how to construct fully connected network with optimal coverage rate is the key issue. Nakamura et al. [11] transformed the full-connection coverage problem into an ILP model and solved it by the commercial optimization package CPLEX [12]. By comparing WSNs’ performances gotten from an ILP model and evolutionary algorithms, Quintão et al. [13] indicated that population-based method could achieve reasonable solutions for WSNs with a considerably lower computational cost. Nevertheless, it only considered the coverage rate of the network as a relevant criterion. The papers [14–16] presented hybrid algorithms based on the density control. They took the relationship among coverage, connectivity, and energy consumption of WSNs into account and focused on finding a sensor node set, which not only met the coverage requirements, but also maximized the network lifetime. A distributed protocol [17], which provided a transition mechanism of sensor nodes’ states, was presented. A sensor node has four states: sleeping, passive, test, and active. Each passive node obtains information and switches to sleeping in test state. Thus, according to the situation of WSNs, sensor nodes can change their states dynamically and reduce the possibility of overlapping coverage with other sensor nodes. However, the rational requirement of the network coverage was not guaranteed. It is worth noting that the real-time performances of WSNs are highlighted in practice. When an active node is failed, to reconstruct the minimum connected tree by global optimization methods or repair the network by locally calculating all the paths between the failed node and its neighbor nodes are both time-consuming and have the direct influence on the real-time performances of WSNs. To solve the above problem, Martins et al. [15] presented a dynamic hybrid approach combining the global optimization method with a local online algorithm so as to correct the failures caused by the out-of-energy nodes. The global optimization method was employed for solving the coverage problem by genetic algorithm (GA). The local online algorithm (LOA) was intended to restore the network in terms of the failures of active nodes. However, a new node was

International Journal of Distributed Sensor Networks chosen from the neighborhood of a failed node to replace the invalid one based on the least sum of the distances from the parent nodes to the sink node and the distances between the son nodes and sink node in this algorithm. Each son node or parent node was derived from the rooted sink node. It is easy to see that the computational cost of this algorithm is greatly affected by the size of the neighborhood radius. When the scale of wireless sensor network is small, the neighborhood of failed node is usually small. It is possible to evaluate all neighbor sensor nodes and choose the optimal one having the least energy consumption and less computation. On the contrary, when the scale of the wireless sensor network is large, there are a lot of sleeping nodes to be evaluated. It will be too expensive and time-consuming to calculate the distances from these sleeping nodes to the parent or son nodes of the failed nodes. In this paper, a hierarchical scheduling method based on node-failure pretreatment strategy (FPS) is proposed. Its goal is to construct an active-sensor-node set to make the repaired network having the lowest energy consumption and maintaining the better performances, such as the connectivity and the coverage rate, as well. The FPS-based hierarchical scheduling method combines the global optimization method with the local multilayer recovery strategy. The global optimization method rebuilds the network so as to find the minimum connected tree satisfying the requirements of network performances. In FPS-based local multilayer recovery strategy, the neighborhood radius of each sensor node varies with its importance, which is determined by its location in minimum connected tree and coverage acreage. Subsequently, the candidate-node set of this node is gotten by certain rules before the failure of this node. Considering the energy consumption and connectivity of the network, an optimal candidate node is chosen from its candidate-node set by certain rules when an active sensor node is failed. And then this candidate node replaces the failed one by node recovery strategy. Once no rational candidate node can be found, regional recovery strategy is done to repair the invalid region affected by the failed node. The above node recovery strategy and regional recovery strategy are composed of local multilayered recovery strategy. The remainder of this paper is organized as follows. In Section 2, a mathematical model describing minimum energy consumption based on network connectivity and coverage is formulated. The detailed hierarchical scheduling scheme is illustrated in Section 3. Then in Section 4, the comparative experiments are done and the experimental results are further analyzed. Finally, the conclusions and future work are drawn in Section 5.

2. Problem Analysis 2.1. Problem Description and Assumptions. To obtain a WSN having the longest network lifetime and the best qualities of service, such as connectivity and coverage, by rationally switching the “sleeping” and “active” modes of wireless sensor nodes in the target area is the goal of the scheduling optimization methods. It actually means that only a part of sensor nodes keeps being active, which construct the network


3

with the acceptable performances, and others are put into the low-powered sleeping mode. Once a sensor node cannot be connected to any sink node, which is called the failed node, or the coverage rate is less than a preset threshold, we call it the network failure. In the case of the network failure caused by some invalid sensor nodes, how to make full use of the sleeping nodes nearby the invalid nodes to recover the WSN so as to effectively prolong the network lifespan is a dynamic scheduling problem. Suppose there are 𝑁 sensor nodes randomly distributed in the target region. The set of all sensor nodes is expressed as 𝑆 = {𝑠1 , 𝑠2 , . . . , 𝑠𝑁}. Some assumptions are considered to construct the mathematical model of scheduling in WSNs. (i) Each sensor node has a unique ID and its position is known. (ii) All sensor nodes are homogeneous and have the equal energy. (iii) All sensor nodes cannot move after being deployed. (iv) The detecting area of each sensor node is assumed to be a circle with the sensing radius 𝑟𝑠 . And the communication area of each sensor node is also a circle with the communication radius 𝑟𝑐 . Typically, 𝑟𝑠 is less than 𝑟𝑐 . (v) Each sensor node has three states: active, sleeping, and dead. The working nodes keep being active, while the failed nodes are in the dead state. And inactive nodes are in sleeping state, which can be activated as the candidate nodes in some cases and transform to the active state. (vi) The energy consumption of a sleeping node is 0. The switching time among different modes of a sensor node is negligible.

In the above formula, 𝐴 active is the acreage of effectively detecting area covered by the minimum connected tree. 𝐴 stands for the acreage of the target region. 2.2.4. The Energy Consumption. According to the communication structure of minimum connected tree in WSNs, when a packet with 𝑘 bits is transmitted from 𝑠𝑖 to 𝑠𝑗 , the energy consuming by 𝑠𝑖 is defined as follows: 𝐸𝑖𝑗 (𝑘) = 𝑘 (2𝐸elec + 𝐸amp × 𝑑𝑖𝑗2 ) .

𝐸elec is the energy consumption of communication for a transmitter or receiver. 𝐸amp is the energy consumption of communication for an amplifier. 𝑑𝑖𝑗 is the quadratic distance between 𝑠𝑖 and 𝑠𝑗 . If the distance between two active sensor nodes is larger than the communication radius of the sensor node, the whole wireless sensor network is not fully connected. Because the sensor nodes are randomly deployed, it is difficult to ensure WSNs fully connected by arbitrarily selecting the active nodes. Considering the above situations, the energy consumption of 𝑠𝑖 is modified as follows: 2 {𝑘 × (2𝐸elec + 𝐸amp × 𝑑𝑖𝑗 ) , 𝐸𝑖𝑗 (𝑘) = { 𝑘 × (2𝐸elec + 𝐸amp × 𝑑𝑖𝑗10 ) , {

2.2.1. The Networks’ Lifetime. The network’s operational lifetime is defined as the duration from the beginning of the network operation to the network failure: 𝑇 (𝑆) = 𝑇failure − 𝑇beginning .

(1)

2.2.2. The Networks’ Connectivity. The wireless sensor network, in which there is a feasible path between any two sensor nodes in the target area, is called fully connected WSN. 2.2.3. The Coverage Rate. The coverage rate of WSNs is one of the important qualities of service. Based on the binary sensing model of sensor node, the coverage rate of WSNs for the certain target area is defined as follows: 𝑓𝑐 =

𝐴 active ∩ 𝐴 × 100%. 𝐴

(2)

𝑑𝑖𝑗 ≤ 𝑟𝑐 , 𝑑𝑖𝑗 > 𝑟𝑐 .

(4)

According to the above formula, if the distance between two sensor nodes is larger than the communication radius of sensor node, the required energy consumption of 𝑠𝑖 is far more than others. The energy consumption of communication for the WSN is defined as the sum energy consumption that all active nodes send a data packet with 𝑘 bits to the sink nodes [18]: 𝐸= ∑(

2.2. The Measurements of WSNs’ Performances. Four commonly used metrics including the network’s lifetime, connectivity, coverage rate, and the communication energy consumption are defined as follows.

(3)

𝑖∈𝑆󸀠

∑ arc(𝑗,ℎ)∈𝑝𝑖

2 𝑘 × (2𝐸elec + 𝐸amp × 𝑑𝑗ℎ ) − 𝐸elec ) . (5)

In the above formula, 𝑆󸀠 is the active-node set. 𝑝𝑖 is the shortest path from 𝑠𝑖 to sink nodes. 𝑑𝑖𝑘 is the quadratic distance between 𝑠𝑗 and 𝑠ℎ . The goal of the scheduling method is to keep the least sensor nodes active, which satisfies the monitoring and communication requirements with the minimum energy consumption, and put others into low-powered sleeping state. Hence the objective function of the scheduling optimization problem is shown as follows: min

𝐸 + 𝑝𝑐 × (1 − 𝑓𝑐 )

s.t.

𝑓𝑐 > 𝑓𝜃 ,

(6)

where 𝑓𝜃 is the preset threshold of the network’s coverage rate. 𝑝𝑐 is the penalty factor, which is used to penalize the acreage of the target region not covered by WSNs. From the above formula, we know that aforesaid model is in essence to find the fully connected minimum connected tree having minimum energy consumption and maximum coverage rate at the same time.

4


input: A set of sensor nodes, sensing radius and communication radius of sensor node, size of the target area, preset threshold of coverage rate and other parameters (1) Initialize the population; (2) Evaluate initial population; (3) while NOT stop condition do (4) Single point crossover operation in terms of probability 𝑝𝑐 ; (5) Flip mutation operation with mutation probability 𝑝𝑚 ; (6) Roulette wheel select operation; (7) Create the minimum spanning tree by Dijkstra algorithm; (8) Evaluate new population; (9) end while Algorithm 1: Global optimization.

3. The Hierarchical Scheduling Scheme The hierarchical scheduling scheme is in essence a multilayer dynamic scheduling strategy based on the node-failure pretreated strategy (FPS). At first, the global optimization method is adopted to determine which sensor nodes are chosen from the nodes randomly deployed in the target area and activated so as to construct the minimum connected tree, in which all sensor nodes are fully connected and the coverage rate of corresponding network is larger than the preset threshold. Secondly, when an active node is out-of-energy or failed, the full connectivity of the wireless sensor network is broken and its coverage rate for the target area is decreased. This invalid WSN must be restored in a very short time. For the sake of prolonging the network lifetime, the sleeping nodes need to be evaluated so as to find one being activated instead of the failed node rapidly. This process is called the recovery process of the WSN. In this paper, a dynamic multilayer recovery strategy is proposed to restore the broken WSN and ensure that the recovered network is fully connected, which keeps larger coverage rate than the preset threshold. In a word, this dynamic hierarchical scheduling scheme combines the global optimization method with local multilayer recover strategy based on the node-failure pretreatment. The highlight of this scheme is the local multilayer recover strategy. 3.1. Finding Minimum Connected Tree by Global Optimization Method. It is difficult to find a fully connected active-node set, which has the least energy consumption and satisfies the requirement of coverage rate by the random selection method. In particular, it is time-consuming and no optimal solutions can be found for the WSN containing enormous sensor nodes. In order to efficiently obtain the minimum connected tree, genetic algorithm (GA) is adopted as the global optimization method. Each individual is encoded as a binary vector representing a set of active nodes. The objective function expressed by (6) is defined as the fitness function of GA. The single point crossover operation and flip mutation operators are adopted

during the evolution process. The pseudocode of the global optimization method is listed in Algorithm 1. From (4), we know that when the distance between two active nodes is larger than the communication radius, its energy consumption for sending data is set a large value. Subsequently, in the evolution of GA, the individuals containing disconnected sensor nodes will have the higher energy consumption and then cannot be selected into the offspring. Through the above definition and evolutionary operation, this algorithm can guarantee the full connectivity of active nodes in the WSN. The optimal solution found by GA is the full-connected active-node set, which is having the lowest energy consumption and larger coverage rate than the preset threshold. If a sink node is regarded as the root of a minimum connected tree, these active nodes are connected by the minimum spanning tree and shortest path algorithms. Here, the shortest path between the sink node and each active node is calculated so as to build the minimum connected tree. 3.2. Local Multilayer Recovery Strategy. By the above global optimization method, the minimum connected tree of the WSN is established based on the shortest path between each sensor node and sink node. When an active sensor node is out-of-energy or failed, a local multilayer recovery strategy is proposed so as to find an optimal substitution node to repair the invalid network. The key issue of this strategy is how to measure the candidate nodes lying in the neighborhood of failed node according to some specific criteria and how to recover the invalid network as soon as possible. Consequently, the node’s importance is defined to evaluate the sensor node and decide the size of neighborhood. And then a multilayer recovery strategy is proposed to repair the network step by step with less time-consumption in terms of certain rules. 3.2.1. The Definition of the Sensor Nodes’ Importance. The loss of the wireless sensor network caused by the failed sensor node is quite different. Several sensor nodes lying in the communication linkage of minimum connected tree are the articulation points. The failure of these nodes may result in


5

the invalid linkage or disconnection from one or more other nodes to the sink node [19]. In order to rationally measure the destroyed degree of a failed sensor node to the whole WSN, the importance of sensor node is defined as a weighted sum of various losses caused by the failed node. Suppose 𝑠𝑖 is the failed node; CR𝑖 denotes the variation of the network’s coverage rate caused by 𝑠𝑖 . AR𝑖 stands for the number of input and output paths associated with 𝑠𝑖 , which reflects the variation of the network’s connectivity. 𝛼 and 𝛽 represent the weights of CR𝑖 and AR𝑖 , respectively. Consider 𝑊𝑖 = 𝛼CR𝑖 + 𝛽AR𝑖 s.t. 𝛼 + 𝛽 = 1.

(7)

(A) Accumulative the Loss Function of the Coverage Rate. When a sensor node is failed, one or more nodes may be disconnected from the sink node. That means both the failed node and subsequently disconnected nodes cannot provide the valid monitoring information. Their coverage for the detecting area is blank [20]. So the variation of the network’s coverage acreage caused by 𝑠𝑖 , expressed by CAR𝑖 , is defined as the sum of the acreage covered by the failed node and the cumulative acreage detected by all disconnected sensor nodes. Consider |𝑁𝑆𝑖 |

𝑗

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