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Location Estimation in Sensor Networks

Location Estimation in Sensor Networks. Moshe Mishali. (Wireless) Sensor Network.

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Location Estimation in Sensor Networks

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  1. Location Estimation in Sensor Networks Moshe Mishali

  2. (Wireless) Sensor Network • A wireless sensor network (WSN) is a wireless network consisting of spatially distributed autonomous devices using sensors to cooperatively monitor physical or environmental conditions, such as temperature, sound, vibration, pressure, motion or pollutants, at different locations. Wikipedia

  3. CodeBlue

  4. Model Sensors Fusion Center

  5. Maximum Likelihood Estimator • Given: • are Gaussian i.i.d. • Then, the MLE is

  6. Constrained Distributed Estimation • The communication to the fusion center is bandwidth-constrained.e.g. each sensor can send only 1 bit,

  7. Variations • Deterministic or Bayesian • Knowledge of noise structure • Known PDF (explicit) • Known PDF with unknown parameters • Unknown PDF (bounded or not) • Scalar or vector

  8. Outline • Known noise PDF • Known noise PDF, but unknown parameters • Unknown noise PDF (universal estimator) • Advanced • Dynamic range considerations • Detection in WSN • Estimation under energy constraint • (Compressive WSN) • Discussion

  9. References • Z.-Q. Luo, "Universal decentralized estimation in a bandwidth constrained sensor network," IEEE Trans. on Inf. Th., June 2005 • A. Ribeiro and G. B. Giannakis, "Bandwidth-constrained distributed estimation for wireless sensor Networks-part I: Gaussian case," IEEE Trans. on Sig. Proc., March 2006 • A. Ribeiro and G. B. Giannakis, "Bandwidth-constrained distributed estimation for wireless sensor networks-part II: unknown probability density function," IEEE Trans. on Sig. Proc., July 2006 • J.-J. Xiao and Z.-Q. Luo, “Universal decentralized detection in a bandwidth-constrained sensor network”, IEEE Trans. on Sig. Proc., August 2005 • J.-J. Xiao, S. Cui, Z.-Q. Luo and A. J. Goldsmith, “Joint estimation in sensor networks under energy constraint”, IEEE Trans. on Sig. Proc., June 2005 • W. U. Bajwa, J. D. Haupt, A. M. Sayyed and R. D. Nowak, “Joint source-channel communication for distributed estimation in sensor networks”, IEEE Trans. on Inf. Th., October 2007

  10. Known Noise PDF – Case 1 • Design:

  11. Known Noise PDF – Case 1 • CRLB for unbiased estimator based on the binary observations min

  12. Known Noise PDF – Case 2 • Design:

  13. Known Noise PDF • Generalizing Case 2

  14. Known Noise PDF withUnknown Variance • Example:

  15. Unknown Noise PDF • Setup Binary observations: Linear estimator:

  16. Method • Develop a universallinear -unbiased estimator for • Given such an estimator design the sensor network to achieve

  17. A Universal Linear -Unbiased Estimator A necessary and sufficient condition

  18. Construction (1)

  19. Construction (2)

  20. Fusion Center Estimator • To reduce MSE: • Duplicate the whole system and average, OR • Allocate sensor according to bit significance: • ½ of the sensors for the 1st bit • ¼ of the sensors for the 2nd bit, and so on… • Exact expressions can be found in [1] • For small , it requires

  21. Simulations

  22. Simulations

  23. Advanced I – Dynamic Range • Setup – Gaussian Noise PDF • The dynamic range of is large relativeto • Idea: Let each sensor use different quantization, so that some of the thresholds will be close to the real

  24. Non-Identical Thresholds

  25. Non-Identical Thresholds • There is no close form for the log-likelihood. • However, there is a closed form for the CRLB (for unbiased estimator): • Goal: minimize the CRLB instead of the MSE

  26. Steps • Introduce “confidence” (i.e. prior) on • Derive lower-bound for the CRLB • Derive upper-bound for the CRLB • Implementation

  27. Step 1/4 – “Confidence” • is the “confidence” (or prior) of • The weighted Variance/CRLB: • The optimum:

  28. Step 2/4 – Lower Bound • Derive: • + necessary and sufficient condition for achievability • Numerically:

  29. Step 3/4 – Upper Bound • For a uniform thresholds grid. • Select according [2, Th. 2] • Then,

  30. Step 4/4 - Implementation • Formulate an optimization problem for , which are the “closest” pair to the one of the condition of step 2. • Discretize the objective.

  31. Advanced II – Detection • Constraints: • Each is a bit, 1 or 0. • The noise PDF is unknown.It is assumed that Fusion Center

  32. Decentralized Detection • Suppose bounded noise • Define • Sensor decodes the th bit of , where • The decision rule at the fusion center is

  33. Advanced III – Energy Constraint FusionCenter Setup The BLUE estimator:

  34. Advanced III – Energy Constraint FusionCenter Goal: Meet target MSE under quantization + total power constraints.

  35. Probabilistic Quantization Bernoulli Quant. Step Signalrange The Quasi-BLUE estimator:

  36. Power Scheduling Const Const MSE due to BER: only a constant factor

  37. Solution 2. Non-Convex  Transformation (Hidden convexity) 1. Integer variable 3. Analytic expression (KKT conditions) • Threshold strategy: • The FC sends = threshold to all nodes (high power link). • Each sensor observes his SNR (scaled by the path loss). • If SNR> , send bits (otherwise inactive).

  38. Simulations

  39. Summary • Model • Bandwidth-constrained estimation • Known Noise PDF • Unknown Noise PDF • Extensions • Detection • Energy-constraint

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