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Cramér-Rao-Type Bounds for Localization

Abstract

The localization problem is fundamentally important for sensor networks. This paper, based on "Estimation bounds for localization" by the authors (2004 © IEEE), studies the Cramér-Rao lower bound (CRB) for two kinds of localization based on noisy range measurements. The first is anchored localization in which the estimated positions of at least nodes are known in global coordinates. We show some basic invariances of the CRB in this case and derive lower and upper bounds on the CRB which can be computed using only local information. The second is anchor-free localization where no absolute positions are known. Although the Fisher information matrix is singular, a CRB-like bound exists on the total estimation variance. Finally, for both cases we discuss how the bounds scale to large networks under different models of wireless signal propagation.

References

  1. Chang C, Sahai A: Object tracking in a 2D UWB sensor network. Proceedings of the 38th Conference on Signals, Systems and Computers (Asilomar '04), November 2004, Pacific Grove, Calif, USA 1: 1252–1256.

    Google Scholar 

  2. Seada K, Helmy A, Govindan R: On the effect of localization errors on geographic face routing in sensor networks. Proceedings of 3rd International Symposium on Information Processing in Sensor Networks (IPSN '04), April 2004, Berkeley, Calif, USA 71–80.

    Google Scholar 

  3. Doherty L, Pister KSJ, El Ghaoui L: Convex position estimation in wireless sensor networks. Proceedings of IEEE 20th Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE INFOCOM '01), April 2001, Anchorage, Alaska, USA 3: 1655–1663.

    Google Scholar 

  4. Savarese C, Rabaey J, Langendoen K: Robust positioning algorithms for distributed ad-hoc wireless sensor networks. Proceedings of the General Track Annual Technical Conference (USENIX '02), June 2002, Monterey, Calif, USA 317–327.

    Google Scholar 

  5. Bulusu N, Heidemann J, Estrin D: GPS-less low-cost outdoor localization for very small devices. IEEE Personal Communications 2000, 7(5):28–34. 10.1109/98.878533

    Article  Google Scholar 

  6. Niculescu D, Nath B: Ad hoc positioning system (APS). Proceedings of IEEE Global Telecommunications Conference (Globecom '01), November 2001, San Antonio, Tex, USA 5: 2926–2931.

    Article  Google Scholar 

  7. Nagpal R, Shrobe HE, Bachrach J: Organizing a global coordinate system from local information on an Ad Hoc sensor network. Proceedings of 2nd International Workshop Information Processing in Sensor Networks (IPSN '03), April 2003, Palo Alto, Calif, USA, Lecture Notes in Computer Science 2634: 333–348.

    Article  Google Scholar 

  8. Chintalapudi KK, Dhariwal A, Govindan R, Sukhatme G: Ad-hoc localization using ranging and sectoring. Proceedings of 23th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM '04), March 2004, Hong Kong 4: 2662–2672.

    Google Scholar 

  9. Moses RL, Patterson R: Self-calibration of sensor networks. Unattended Ground Sensor Technologies and Applications IV, April 2002, Orlando, Fla, USA, Proceedings of the SPIE 4743: 108–119.

    Article  Google Scholar 

  10. Moses RL, Krishnamurthy D, Patterson R: A self-localization method for wireless sensor networks. EURASIP Journal on Applied Signal Processing 2003, (4):348–358.

    MATH  Google Scholar 

  11. Hightower J, Want R, Boriello G: SpotON: an indoor 3d location sensing technology based on RF signal strength. In Tech. Rep. UW CSE 00-02-02. University of Washington, Seattle, Wash, USA; February 2000.

    Google Scholar 

  12. Girod L, Estrin D: Robust range estimation using acoustic and multimodal sensing. Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS '01), November 2001, Maui, Hawaii, USA 3: 1312–1320.

    Google Scholar 

  13. Savvides A, Han C-C, Strivastava MB: Dynamic fine-grained localization in Ad-Hoc networks of sensors. Proceedings of the 7th Annual International Conference on Mobile Computing and Networking (MOBICOM '01), July 2001, Rome, Italy 166–179.

    Chapter  Google Scholar 

  14. Savarese C, Rabaey JM, Beutel J: Location in distributed ad-hoc wireless sensor networks. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '01), May 2001, Salt Lake City, Utah, USA 4: 2037–2040.

    Google Scholar 

  15. Capkun S, Hamdi M, Hubaux J-: GPS-free positioning in mobile ad-hoc networks. Proceedings of the 34th Annual Hawaii International Conference on System Sciences (HICSS '01), January 2001, Maui, Hawaii, USA 3481–3490.

    Google Scholar 

  16. Ji X, Zha H: Sensor positioning in wireless ad-hoc sensor networks using multidimensional scaling. Proceedings of 23th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM '04), March 2004, Hong Kong 4: 2652–2661.

    Google Scholar 

  17. Smith ST: Intrinsic Cramér-Rao bounds and subspace estimation accuracy. Proceedings of the IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM '00), March 2000, Cambridge, Mass, USA 489–493.

    Google Scholar 

  18. Patwari N, Hero AO III, Perkins M, Correal NS, O'Dea RJ: Relative location estimation in wireless sensor networks. IEEE Transactions on Signal Processing 2003, 51(8):2137–2148. 10.1109/TSP.2003.814469

    Article  Google Scholar 

  19. Savvides A, Garber W, Adlakha S, Moses RL, Strivastava MB: On the error characteristics of multihop node localization in ad-hoc sensor networks. Proceedings of 2nd International Workshop Information Processing in Sensor Networks (IPSN '03), April 2003, Palo Alto, Calif, USA, Lecture Notes in Computer Science 2634: 317–332.

    Article  Google Scholar 

  20. Wang H, Yip L, Yao K, Estrin D: Lower bounds of localization uncertainty in sensor networks. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '04), May 2004, Montreal, Quebec, Canada

    Google Scholar 

  21. Stoica P, Marzetta TL: Parameter estimation problems with singular information matrices. IEEE Transactions on Signal Processing 2001, 49(1):87–90. 10.1109/78.890346

    Article  MathSciNet  Google Scholar 

  22. Dersan A, Tanik Y: Passive radar localization by time difference of arrival. Proceedings of Military Communications Conference (MILCOM '02), October 2002, Anaheim, Calif, USA 2: 1251–1257.

    Article  Google Scholar 

  23. Chernyak VS: Fundamentals of Multisite Radar Systems. Gordon and Breach Science, Australia; 1998.

    Google Scholar 

  24. Chang C: Localization and object-tracking in an ultrawideband sensor network, M.S. thesis. EECS Department, University of California Berkeley, Berkeley, Calif, USA; 2004.

    Google Scholar 

  25. Hoffman K, Kunze RA: Linear Algebra. Prentice-Hall, Englewood Cliffs, NJ, USA; 1992.

    MATH  Google Scholar 

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Correspondence to Cheng Chang.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Chang, C., Sahai, A. Cramér-Rao-Type Bounds for Localization. EURASIP J. Adv. Signal Process. 2006, 094287 (2006). https://doi.org/10.1155/ASP/2006/94287

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