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Observation Uncertainty in Gaussian Sensor Networks Anand D. Sarwate Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2006-3 http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-3.html January 23, 2006 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED 23 JAN 2006 2. REPORT TYPE 00-00-2006 to 00-00-2006 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Observation Uncertainty in Gaussian Sensor Networks 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION University of California at Berkeley,Electrical Engineering and REPORT NUMBER Computer Sciences,Berkeley,CA,94720 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE 86 unclassified unclassified unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 Copyright © 2006, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Acknowledgement I'd like to thank my advisor, Professor Michael Gastpar, for guiding this work, as well as Professor Anant Sahai for useful feedback on the draft. I had helpful discussions about this work with Bobak Nazer, Dan Hazen, and Krishnan Eswaran. This work was supported by an NDSEG Fellowship from the United States Department of Defense, and the National Science Foundation under award CCF-0347298. Finally, thanks to Elizabeth Foster-Shaner for her infinite patience. Observation Uncertainty in Gaussian Sensor Networks by Anand Dilip Sarwate S.B. Electrical Engineering (Massachusetts Institute of Technology), 2002 S.B. Mathematics (Massachusetts Institute of Technology), 2002 A thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in Engineering - Electrical Engineering and Computer Sciences in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Michael Gastpar, Chair Professor Anant Sahai Fall 2005 The thesis of Anand Dilip Sarwate is approved. Chair Date Date University of California, Berkeley Fall 2005 Observation Uncertainty in Gaussian Sensor Networks Copyright c 2005 (cid:13) by Anand Dilip Sarwate Abstract Observation Uncertainty in Gaussian Sensor Networks by Anand Dilip Sarwate Master of Science in Engineering - Electrical Engineering and Computer Sciences University of California, Berkeley Professor Michael Gastpar, Chair The term \sensor network" encompasses a wide range of engineering systems with dramatically di(cid:11)erent characteristics. We consider a speci(cid:12)c class of sensor networks whose objective is to reconstructa sourceat acentral terminal. Ourobjective inthisthesisisto quantifythe asymptotic error in reconstructing the source as the number of data sources, sensors, and model complexity increases. We consider three types of estimation systems { unconstrained estimators for vector Gaussian sources that are allowed direct access to the sensor observations, estimators for discrete sources that receive information via rate constrained links from the sensors, and estimators for scalar Gaussians whose input is the output of a multiple-access channel. We (cid:12)rst establish bounds on the optimal estimator performance of these networks using a centralized estimator with access to all of the sensor observations. We assume the observations are noisy linear functions of the source and are thus speci(cid:12)ed by a matrix. Because the asymptotic error depends only on the spectral properties of this matrix, we can use tools from matrix analysis to give bounds on the spectrum and error in terms of the entries of the matrix for a number of di(cid:11)erent scenarios. Finally, we look at the case where the matrix is partially unknown. In some 1 cases we can estimate the matrix directly from the data and in others we must minimize the worst mismatch distortion. These problems can also be looked at in a more information-theoretic framework. We look at a lossless distributed source coding problem in which the joint distribution of the sources is partially unknown. Although for any (cid:12)nite number of sensors standard multi-terminal source codes can easily be adapted to handle the model uncertainty across time, we show a rate penalty is incurred if the number of sensors and blocklength go to simultaneously. This represents one kind of 1 tradeo(cid:11) between delay and complexity for the scaling behavior of these systems. Finally, we look at the case where the sensors must communicate their observations across an additive white Gaussian noise multiple-access channel. With a known correlation structure, the optimalerrorconvergesto0as1=M, whereM isthenumberofsensors. However, asimplefeedback schemeusingK bitsbroadcasttoallsensorscanprovideadistortionthatscalesto0asM K=(K+2). (cid:0) We conjecture that providing similar feedback to an optimal source code will not improve the performance beyond that of our protocol. Professor Michael Gastpar Thesis Committee Chair 2 I dedicate this thesis to my parents, Dilip V. Sarwate and Sandhya D. Sarwate, and to my brother, Sanjiv D. Sarwate. Without their love and support I would never have gotten this far. i