基于多维变换的无线传感器网络定位算法研究
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摘要
无线传感器网络技术以其独特的优势和潜力被确定为人类21世纪中最重要的技术之一。随着无线通信、微加工集成以及嵌入式微处理器技术等支撑技术的发展,无线传感器网络在生活中的应用更是深入到各个角落。在相关技术中,传感器节点的定位技术是传感器网络工作和管理中的一个最为基本和关键的问题。离开了位置信息,节点所收集到的其他监测信息往往毫无意义。因此,确定事件发生的位置或获取消息的节点位置是无线传感器网络最基本的功能之一,对传感器网络应用的有效性起着关键的作用。
本文首先简要的介绍了无线传感器网络的特征,例如规模大、一次性部署、能量有限等。这些特征都是研究传感器网络以及针对其设计配套的软硬件时所必须考虑的因素。随后,本文提出了传感器网络的节点定位问题,深入研究了目前的各种节点定位算法并总结了现有的各种基于距离的和距离无关的定位算法。接下来本文指出了基于距离的传统多维变换定位算法的不足并提出了两种新的改进中心式定位算法。其中,改进算法一以相合分解代替特征值分解,能够简化原有定位算法中矩阵分解的步骤并提高定位精度,但时间复杂度保持为O(n3),适用于对定位精度要求高的应用场合;改进算法二采用强迫正定Cholesky分解,虽然定位精度比前者稍差,但是时间复杂度可以降为O(n~2),更适用于对实时性要求比较高的应用场合。此外,本文还提出了新颖的拼图分割算法对传感器网络进行有效的分割,再结合中心式定位算法实现分布式的传感器网络节点定位。MATLAB的仿真结果表明,这两种改进的中心式定位算法以及提出的分布式定位算法的性能符合理论推导,它们能够在不同的应用场合实现较高精度的节点定位。
With its unique advantages and potential, the technology of wireless sensor networks has been identified as one of the most important technologies in the 21st century. As its supporting technologies such as wireless communication, micro-fabrication and integration, and embedded microprocessor have greatly advanced, the application of wireless sensor networks has penetrated into every aspect of human lives. Among all the related technologies, node localization is the most fundamental and key issue in the implementation and management of sensor networks. The detected information by a node is often meaningless without its knowledge of position. Therefore, to determine the location of an event or a node is one of the underlying functions of wireless sensor networks, which plays an essential role in the effectiveness of the whole network.
This paper first briefly introduces the features of wireless sensor networks, such as its large scale, one-time deployment and limited power, which must be taken into account when doing relevant research or design. Then, the paper poses the problem of node localization, carefully studies current localization algorithms and summarizes existing range-based and range-free localization algorithms. Moreover, this paper analyzes the drawbacks of the traditional multidimensional scaling localization algorithm and proposes two improved MDS-based centralized localization algorithms. The first improved algorithm replaces the previous eigen-decomposition with congruent-decomposition, simplifying the steps of matrix decomposition and enhancing localization accuracy while maintaining a computational complexity of O(n3). This algorithm is suitable in scenarios where high localization accuracy is required. The second improved algorithm, using forced positive-definite Cholesky decomposition, can reduce the computational complexity to O(n2) with limited error, which fits well where real-time response is the primary concern. This paper also proposes a distributed localization algorithm which applies the jigsaw puzzle segmentation. Simulations using MATLAB show that all the proposed algorithms exhibit good performances and can realize node localization in corresponding scenarios.
本文首先简要的介绍了无线传感器网络的特征,例如规模大、一次性部署、能量有限等。这些特征都是研究传感器网络以及针对其设计配套的软硬件时所必须考虑的因素。随后,本文提出了传感器网络的节点定位问题,深入研究了目前的各种节点定位算法并总结了现有的各种基于距离的和距离无关的定位算法。接下来本文指出了基于距离的传统多维变换定位算法的不足并提出了两种新的改进中心式定位算法。其中,改进算法一以相合分解代替特征值分解,能够简化原有定位算法中矩阵分解的步骤并提高定位精度,但时间复杂度保持为O(n3),适用于对定位精度要求高的应用场合;改进算法二采用强迫正定Cholesky分解,虽然定位精度比前者稍差,但是时间复杂度可以降为O(n~2),更适用于对实时性要求比较高的应用场合。此外,本文还提出了新颖的拼图分割算法对传感器网络进行有效的分割,再结合中心式定位算法实现分布式的传感器网络节点定位。MATLAB的仿真结果表明,这两种改进的中心式定位算法以及提出的分布式定位算法的性能符合理论推导,它们能够在不同的应用场合实现较高精度的节点定位。
With its unique advantages and potential, the technology of wireless sensor networks has been identified as one of the most important technologies in the 21st century. As its supporting technologies such as wireless communication, micro-fabrication and integration, and embedded microprocessor have greatly advanced, the application of wireless sensor networks has penetrated into every aspect of human lives. Among all the related technologies, node localization is the most fundamental and key issue in the implementation and management of sensor networks. The detected information by a node is often meaningless without its knowledge of position. Therefore, to determine the location of an event or a node is one of the underlying functions of wireless sensor networks, which plays an essential role in the effectiveness of the whole network.
This paper first briefly introduces the features of wireless sensor networks, such as its large scale, one-time deployment and limited power, which must be taken into account when doing relevant research or design. Then, the paper poses the problem of node localization, carefully studies current localization algorithms and summarizes existing range-based and range-free localization algorithms. Moreover, this paper analyzes the drawbacks of the traditional multidimensional scaling localization algorithm and proposes two improved MDS-based centralized localization algorithms. The first improved algorithm replaces the previous eigen-decomposition with congruent-decomposition, simplifying the steps of matrix decomposition and enhancing localization accuracy while maintaining a computational complexity of O(n3). This algorithm is suitable in scenarios where high localization accuracy is required. The second improved algorithm, using forced positive-definite Cholesky decomposition, can reduce the computational complexity to O(n2) with limited error, which fits well where real-time response is the primary concern. This paper also proposes a distributed localization algorithm which applies the jigsaw puzzle segmentation. Simulations using MATLAB show that all the proposed algorithms exhibit good performances and can realize node localization in corresponding scenarios.
引文
[1] Feng Zhao, Leonidas J. Guibas, Wireless Sensor Networks: An Information Processing Approach, Morgan Kaufmann Publishers, 2004
[2] Chee-Yee Chong, Srikanta P Kumar, “Sensor networks: evolution, opportunities, and challenges,” in Proc. IEEE, vol. 91, no. 8. August 2003, pp. 1247-1255
[3] Deborah Estrin, Ramesh Govindan, John Heidemann, and Satish Kumar, “Next century challenges: scalable coordination in sensor networks,” in Proc. 5th Annual International Conference on Mobile Computing and Networking (MobiCom 1999), Seattle, WA, ACM Press, 1999, pp. 263-270
[4] Philippe Bonnet, Johannes Gehrke, Praveen Seshadri, “Querying the physical world,” in IEEE Personal Communication, vol. 5, no. 5, 2000, pp. 10-15
[5] 21 ideas for the 21st century, Business Week, August 30, 1999, pp. 78-167
[6] Mark Weiser, “The computer for the twenty-first century,” in Scientific American, September 1991, pp. 94-100
[7] National Research Council, Embedded, Everywhere: A Research Agenda for Networked Systems of Embedded Computers, National Academy Press, 2001
[8] Deborah Estrin, David Culler, Kris Pister, Gaurav Sukhatme, “Connecting the physical world with pervasive networks,” in IEEE Pervasive Computeing, January 2002, pp. 59-69
[9] Ian F Akyildiz, Weilian Su, Yogesh Sankarasubramaniam, Erdal Cayirci, “A survey on sensor networks,” in IEEE Communications Magazine, vol. 40, no. 8, August 2002, pp. 102-114
[10] 孙利明等,无线传感器网络,北京:清华大学出版社,2005 年 5 月
[11] Haitham Hamza, “Security in wireless sensor networks,” [Online]. Available: http://cse.unl. edu/~rohini/csce990/slides/hhamza.ppt
[12] Tian He, Chengdu Huang, Brian M. Blum, John A. Stankovic, and Tarek A, “Range-free localization schemes for large scale sensor networks,” in Proc. of the 9th Annual International Conference on Mobile Computing and Networking, (MOBICOM), 2003, pp. 81-95
[13] Neal Patwari, Joshua N. Ash, Spyros Kyperountas, Alfred O. Hero III, Randolph L. Moses, and Neiyer S. Correal, “Locating the nodes: cooperative localization in wireless sensor networks,” in IEEE Signal Processing Magazine, July, 2005, pp. 54-69
[14] Guolin Sun, Jie Chen, Wei Guo, and K.J. Ray Liu, “Signal processing techniques in network-aided positioning,” in IEEE Singal Processing Magazine, July 2005
[15] Priyantha N B, Balakrishnam H, Demaine E, and Teller S, “Anchor-free distributedlocalization in sensor networks,” in Technical Report MIT-LCS-TR-892, MIT Lab for Computer Science, April 2003
[16] N. Patwari, A.O. Hero, III, M. Perkins, N.S. Correal, and R.J. O’Dea, “Relative location estimation in wireless sensor networks,” in IEEE Trans. Signal Processing, (Special Issue on Signal Processing in Networks), November 2002, pp. 2137-2148
[17] N. Bulusu, J. Heidemann, D. Estrin, and T. Tran, “Self-configuration localization systems: Design and experimental evaluation,” in ACM Transaction Embedded Computer Systems, vol. 3, no. 1, February 2004, pp. 24-60
[18] P. Biswas and Y. Ye, “A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization,’’ Technical Report, Department of Computer Science, Stanford University, Stanford, CA, October 2003
[19] L. Doherty, K.S.J. Pister, and L.E. Ghaoui, “Convex position estimation in wireless sensor networks,’’ in Proc. IEEE INFOCOM 2001, vol. 3, 2001, pp. 1655-1663
[20] R.L. Moses, D. Krishnamurthy, and R. Patterson, “A self-localization method for wireless sensor networks,’’ in EURASIP J. Applied Signal Processing, no. 4, March 2003, pp. 348-358
[21] J. Albowicz, A. Chen, and L. Zhang, “Recursive position estimation in sensor networks,’’ in Proc. IEEE Int. Conf. on Network Protocols, November 2001, pp. 35-41
[22] A. Savvides, C.C. Han, and M.B. Srivastava, “Dynamic fine-grained localization in ad-hoc networks of sensors,’’ in Proc. IEEE Mobicom, July 2001, pp. 166-179
[23] S. Capkun, M. Hamdi, and J.P. Hubaux, “GPS-free positioning in mobile ad hoc networks,’’ in Proc. 34th IEEE Hawaii Int. Conf. System Sciences (HICSS-34), January. 2001, pages 9008-9022
[24] Peter H. Dana. Global Positioning System Overview, [Online]. Available: http://www.colora do.edu/geography/gcraft/notes/gps/gps.html
[25] Savarese C, Rabaey J, Beutel J, “Locationing in distributed ad-hoc wireless sensor networks,” in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Salt Lake City, UT, May 2001, pp. 2037-2040
[26] J. Hightower, Borriello G, “Location systems for ubiquitous computing,” in IEEE Computer, vol. 34, August 2001, pp. 57-66
[27] J. Hightower, Borriello G, “Location sensing techniques,” Aug, 2001 [Online]. Available: www.cs.washington.edu/research/portolano/papers/UW-CSE-01-07-01.pdf
[28] G. Carter, Coherence and Time Delay Estimation, Piscataway, NJ: IEEE Press, 1993
[29] C. Knapp and G. Carter, “The generalized correlation method for estimation oftime delay,’’ in IEEE Trans. Acoust., Speech, Signal Processing, vol. 24, no. 4, 1976, pp. 320-327
[30] E. Robinson and A. Quazi, “Effect of sound-speed profile on differential time delayestimation,’’ in Journal of Acoustic Society America, vol. 77, no. 3, 1985, pp. 1086-1090
[31] K. Pahlavan, P. Krishnamurthy, and J. Beneat, “Wideband radio propagation modeling for indoor geolocation applications,’’ in IEEE Commun. Magazine, vol. 36, no. 4, April 1998, pp. 60-65
[32] F. Sivrikaya and B. Yener, “Time synchronization in sensor networks: A survey,’’ in IEEE Network, vol. 18, no. 4, July-August 2004, pp. 45-50
[33] J.Y. Lee and R.A. Scholtz, “Ranging in a dense multipath environment using an UWB radio link,’’ in IEEE J. Sel. Areas Commun., vol. 20, no. 9, December 2002, pp. 1677-1683
[34] Priyantha N. B, Chakraborty A, and Padmanabhan H, “The cricket location support system,” in Proc. of 6th ACM International Conference on Mobile Computing and Networking (MOBICOM), Augest 2000, pp. 32-43
[35] Chen Hongyang, Deng Ping, Xu Yongjun, Li Xiaowei, “A robust location algorithm with biased extended Kalman filtering of TDOA data for wireless sensor network,” in Proc. Of 2005 International Conference, vol. 2, September 2005, pp. 883-886
[36] N. Patwari, R.J. O’Dea, and Y. Wang, “Relative location in wireless networks,’’ in Proc. IEEE VTC, vol. 2, May 2001, pp. 1149-1153
[37] A.J. Coulson, A.G. Williamson, and R.G. Vaughan, “A statistical basis for log normal shadowing effects in multipath fading channels,’’ in IEEE Trans. Vehicle Technology, vol. 46, no. 4, April 1998, pp. 494-502
[38] T.S. Rappaport, Wireless Communications: Principles and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1996.
[39] P. Bahl, and V. Padmanabhan, “RADAR: an in-building RF-based user location and tracking system,” in Proc. of INFOCOM, March 2000, pp.775-784
[40] Jeffery Hightower, Roy Want, and Gaetano Borriello, “SpotON: An indoor 3d location sensing technology based on RF signal strength”, UW CSE 2000-02-02, University of Washington, Seattle, WA, February 2000
[41] D. Niculescu and B. Nath, “Ad hoc positioning system (APS) using AoA,” in Proc. IEEE INFOCOM 2003, San Francisco, CA, vol. 3, pp. 1734-1743
[42] H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 1968.
[43] Ken Hinckley, Mike Sinclair, “Touch-sensing input devices,” in Proc. of the 1999 Conference on Human Factors in Computing Systems (CHI 1999), ACM, 1999
[44] Kurt Partridge, Larry Arnstein, Gaetano Borriello, Turner Whitted, “Fast intrabody signaling,” in Demonstration at Wireless and Mobile Computer Systems and Applications (WMCSA), December 2000
[45] Roy Want, Andy Hopper, Veronica Falcao, Jon Gibbons, “The active badge location system,” in ACM Transactions on Information Systems, vol. 10, no. 1, January 1992, pp. 91-102
[46] Ji X, Zha H. Sensor positioning in wireless ad-hoc sensor networks using multidimensional scaling, in INFOCOM 2004. Twenty-third Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 4, March 2004, pp. 2652-2661
[47] Bulusu N, Heidemann J, Estrin D, “GPS-less low cost outdoor localization for very small devices,” in IEEE Personal Communications, vol. 7, no. 5, 2000, pp. 28-34
[48] Patwari N, Alfred O. Hero, “Using proximity and quantized RSS for sensor localization in wireless networks,” in Proc. of 2nd ACM International Conference on Wireless Sensor Networks and Applications (WSNA), ACM Press, September 2003, pp. 20-29
[49] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A survey on sensor networks,” in IEEE Communications Magazine, vol. 40, no. 8, August 2002, pp. 102-114
[50] Forrest W. Young, Multidimensional Scaling. [Online]. Available: http://forrest.psych.unc.edu /teaching/p208a/mds/mds.html
[51] Metric Multidimensional Scaling. [Online]. Available: www.quantlet.com/mdstat/scripts/mva /htmlbook/mvahtmlnode99.html
[52] Golub G H, Van Loan C F, Matrix Computations, 3rd Edition. Baltimore: The John Hopkins University Press, 1996
[53] 北京大学数学系几何与代数教研室代数小组,高等代数(第二版),北京:高等教育出版社,1988 年
[54] 粟塔山等,最优化计算原理与算法程序设计,长沙:国防科技大学出版社,2001 年
[55] 邓乃扬,无约束最优化计算方法,北京:科学出版社,1988 年
[2] Chee-Yee Chong, Srikanta P Kumar, “Sensor networks: evolution, opportunities, and challenges,” in Proc. IEEE, vol. 91, no. 8. August 2003, pp. 1247-1255
[3] Deborah Estrin, Ramesh Govindan, John Heidemann, and Satish Kumar, “Next century challenges: scalable coordination in sensor networks,” in Proc. 5th Annual International Conference on Mobile Computing and Networking (MobiCom 1999), Seattle, WA, ACM Press, 1999, pp. 263-270
[4] Philippe Bonnet, Johannes Gehrke, Praveen Seshadri, “Querying the physical world,” in IEEE Personal Communication, vol. 5, no. 5, 2000, pp. 10-15
[5] 21 ideas for the 21st century, Business Week, August 30, 1999, pp. 78-167
[6] Mark Weiser, “The computer for the twenty-first century,” in Scientific American, September 1991, pp. 94-100
[7] National Research Council, Embedded, Everywhere: A Research Agenda for Networked Systems of Embedded Computers, National Academy Press, 2001
[8] Deborah Estrin, David Culler, Kris Pister, Gaurav Sukhatme, “Connecting the physical world with pervasive networks,” in IEEE Pervasive Computeing, January 2002, pp. 59-69
[9] Ian F Akyildiz, Weilian Su, Yogesh Sankarasubramaniam, Erdal Cayirci, “A survey on sensor networks,” in IEEE Communications Magazine, vol. 40, no. 8, August 2002, pp. 102-114
[10] 孙利明等,无线传感器网络,北京:清华大学出版社,2005 年 5 月
[11] Haitham Hamza, “Security in wireless sensor networks,” [Online]. Available: http://cse.unl. edu/~rohini/csce990/slides/hhamza.ppt
[12] Tian He, Chengdu Huang, Brian M. Blum, John A. Stankovic, and Tarek A, “Range-free localization schemes for large scale sensor networks,” in Proc. of the 9th Annual International Conference on Mobile Computing and Networking, (MOBICOM), 2003, pp. 81-95
[13] Neal Patwari, Joshua N. Ash, Spyros Kyperountas, Alfred O. Hero III, Randolph L. Moses, and Neiyer S. Correal, “Locating the nodes: cooperative localization in wireless sensor networks,” in IEEE Signal Processing Magazine, July, 2005, pp. 54-69
[14] Guolin Sun, Jie Chen, Wei Guo, and K.J. Ray Liu, “Signal processing techniques in network-aided positioning,” in IEEE Singal Processing Magazine, July 2005
[15] Priyantha N B, Balakrishnam H, Demaine E, and Teller S, “Anchor-free distributedlocalization in sensor networks,” in Technical Report MIT-LCS-TR-892, MIT Lab for Computer Science, April 2003
[16] N. Patwari, A.O. Hero, III, M. Perkins, N.S. Correal, and R.J. O’Dea, “Relative location estimation in wireless sensor networks,” in IEEE Trans. Signal Processing, (Special Issue on Signal Processing in Networks), November 2002, pp. 2137-2148
[17] N. Bulusu, J. Heidemann, D. Estrin, and T. Tran, “Self-configuration localization systems: Design and experimental evaluation,” in ACM Transaction Embedded Computer Systems, vol. 3, no. 1, February 2004, pp. 24-60
[18] P. Biswas and Y. Ye, “A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization,’’ Technical Report, Department of Computer Science, Stanford University, Stanford, CA, October 2003
[19] L. Doherty, K.S.J. Pister, and L.E. Ghaoui, “Convex position estimation in wireless sensor networks,’’ in Proc. IEEE INFOCOM 2001, vol. 3, 2001, pp. 1655-1663
[20] R.L. Moses, D. Krishnamurthy, and R. Patterson, “A self-localization method for wireless sensor networks,’’ in EURASIP J. Applied Signal Processing, no. 4, March 2003, pp. 348-358
[21] J. Albowicz, A. Chen, and L. Zhang, “Recursive position estimation in sensor networks,’’ in Proc. IEEE Int. Conf. on Network Protocols, November 2001, pp. 35-41
[22] A. Savvides, C.C. Han, and M.B. Srivastava, “Dynamic fine-grained localization in ad-hoc networks of sensors,’’ in Proc. IEEE Mobicom, July 2001, pp. 166-179
[23] S. Capkun, M. Hamdi, and J.P. Hubaux, “GPS-free positioning in mobile ad hoc networks,’’ in Proc. 34th IEEE Hawaii Int. Conf. System Sciences (HICSS-34), January. 2001, pages 9008-9022
[24] Peter H. Dana. Global Positioning System Overview, [Online]. Available: http://www.colora do.edu/geography/gcraft/notes/gps/gps.html
[25] Savarese C, Rabaey J, Beutel J, “Locationing in distributed ad-hoc wireless sensor networks,” in IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Salt Lake City, UT, May 2001, pp. 2037-2040
[26] J. Hightower, Borriello G, “Location systems for ubiquitous computing,” in IEEE Computer, vol. 34, August 2001, pp. 57-66
[27] J. Hightower, Borriello G, “Location sensing techniques,” Aug, 2001 [Online]. Available: www.cs.washington.edu/research/portolano/papers/UW-CSE-01-07-01.pdf
[28] G. Carter, Coherence and Time Delay Estimation, Piscataway, NJ: IEEE Press, 1993
[29] C. Knapp and G. Carter, “The generalized correlation method for estimation oftime delay,’’ in IEEE Trans. Acoust., Speech, Signal Processing, vol. 24, no. 4, 1976, pp. 320-327
[30] E. Robinson and A. Quazi, “Effect of sound-speed profile on differential time delayestimation,’’ in Journal of Acoustic Society America, vol. 77, no. 3, 1985, pp. 1086-1090
[31] K. Pahlavan, P. Krishnamurthy, and J. Beneat, “Wideband radio propagation modeling for indoor geolocation applications,’’ in IEEE Commun. Magazine, vol. 36, no. 4, April 1998, pp. 60-65
[32] F. Sivrikaya and B. Yener, “Time synchronization in sensor networks: A survey,’’ in IEEE Network, vol. 18, no. 4, July-August 2004, pp. 45-50
[33] J.Y. Lee and R.A. Scholtz, “Ranging in a dense multipath environment using an UWB radio link,’’ in IEEE J. Sel. Areas Commun., vol. 20, no. 9, December 2002, pp. 1677-1683
[34] Priyantha N. B, Chakraborty A, and Padmanabhan H, “The cricket location support system,” in Proc. of 6th ACM International Conference on Mobile Computing and Networking (MOBICOM), Augest 2000, pp. 32-43
[35] Chen Hongyang, Deng Ping, Xu Yongjun, Li Xiaowei, “A robust location algorithm with biased extended Kalman filtering of TDOA data for wireless sensor network,” in Proc. Of 2005 International Conference, vol. 2, September 2005, pp. 883-886
[36] N. Patwari, R.J. O’Dea, and Y. Wang, “Relative location in wireless networks,’’ in Proc. IEEE VTC, vol. 2, May 2001, pp. 1149-1153
[37] A.J. Coulson, A.G. Williamson, and R.G. Vaughan, “A statistical basis for log normal shadowing effects in multipath fading channels,’’ in IEEE Trans. Vehicle Technology, vol. 46, no. 4, April 1998, pp. 494-502
[38] T.S. Rappaport, Wireless Communications: Principles and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1996.
[39] P. Bahl, and V. Padmanabhan, “RADAR: an in-building RF-based user location and tracking system,” in Proc. of INFOCOM, March 2000, pp.775-784
[40] Jeffery Hightower, Roy Want, and Gaetano Borriello, “SpotON: An indoor 3d location sensing technology based on RF signal strength”, UW CSE 2000-02-02, University of Washington, Seattle, WA, February 2000
[41] D. Niculescu and B. Nath, “Ad hoc positioning system (APS) using AoA,” in Proc. IEEE INFOCOM 2003, San Francisco, CA, vol. 3, pp. 1734-1743
[42] H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 1968.
[43] Ken Hinckley, Mike Sinclair, “Touch-sensing input devices,” in Proc. of the 1999 Conference on Human Factors in Computing Systems (CHI 1999), ACM, 1999
[44] Kurt Partridge, Larry Arnstein, Gaetano Borriello, Turner Whitted, “Fast intrabody signaling,” in Demonstration at Wireless and Mobile Computer Systems and Applications (WMCSA), December 2000
[45] Roy Want, Andy Hopper, Veronica Falcao, Jon Gibbons, “The active badge location system,” in ACM Transactions on Information Systems, vol. 10, no. 1, January 1992, pp. 91-102
[46] Ji X, Zha H. Sensor positioning in wireless ad-hoc sensor networks using multidimensional scaling, in INFOCOM 2004. Twenty-third Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 4, March 2004, pp. 2652-2661
[47] Bulusu N, Heidemann J, Estrin D, “GPS-less low cost outdoor localization for very small devices,” in IEEE Personal Communications, vol. 7, no. 5, 2000, pp. 28-34
[48] Patwari N, Alfred O. Hero, “Using proximity and quantized RSS for sensor localization in wireless networks,” in Proc. of 2nd ACM International Conference on Wireless Sensor Networks and Applications (WSNA), ACM Press, September 2003, pp. 20-29
[49] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A survey on sensor networks,” in IEEE Communications Magazine, vol. 40, no. 8, August 2002, pp. 102-114
[50] Forrest W. Young, Multidimensional Scaling. [Online]. Available: http://forrest.psych.unc.edu /teaching/p208a/mds/mds.html
[51] Metric Multidimensional Scaling. [Online]. Available: www.quantlet.com/mdstat/scripts/mva /htmlbook/mvahtmlnode99.html
[52] Golub G H, Van Loan C F, Matrix Computations, 3rd Edition. Baltimore: The John Hopkins University Press, 1996
[53] 北京大学数学系几何与代数教研室代数小组,高等代数(第二版),北京:高等教育出版社,1988 年
[54] 粟塔山等,最优化计算原理与算法程序设计,长沙:国防科技大学出版社,2001 年
[55] 邓乃扬,无约束最优化计算方法,北京:科学出版社,1988 年
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