Location Estimation in Sensor Networks Moshe Mishali

(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. wireless networkautonomous sensors temperaturesoundvibrationpressure Wikipedia

CodeBlue

Model Fusion Center Sensors

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

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

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

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

References 1. Z.-Q. Luo, "Universal decentralized estimation in a bandwidth constrained sensor network," IEEE Trans. on Inf. Th., June A. Ribeiro and G. B. Giannakis, "Bandwidth-constrained distributed estimation for wireless sensor Networks-part I: Gaussian case," IEEE Trans. on Sig. Proc., March 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 J.-J. Xiao and Z.-Q. Luo, “Universal decentralized detection in a bandwidth- constrained sensor network”, IEEE Trans. on Sig. Proc., August 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 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

Known Noise PDF – Case 1 Design:

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

Known Noise PDF – Case 2 Design:

Generalizing Case 2 Known Noise PDF

Example: Known Noise PDF with Unknown Variance

Unknown Noise PDF Setup Binary observations: Linear estimator:

1. Develop a universal linear -unbiased estimator for 2. Given such an estimator design the sensor network to achieve Method

A Universal Linear -Unbiased Estimator A necessary and sufficient condition

Construction (1)

Construction (2)

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 1 st bit ¼ of the sensors for the 2 nd bit, and so on… Exact expressions can be found in [1] For small, it requires

Simulations

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

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

Steps 1. Introduce “confidence” (i.e. prior) on 2. Derive lower-bound for the CRLB 3. Derive upper-bound for the CRLB 4. Implementation

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

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

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

Step 4/4 - Implementation 1. Formulate an optimization problem for, which are the “closest” pair to the one of the condition of step Discretize the objective.

Advanced II – Detection Fusion Center Constraints: 1.Each is a bit, 1 or 0. 2.The noise PDF is unknown. It is assumed that

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

Advanced III – Energy Constraint Fusion Center The BLUE estimator: Setup

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

Probabilistic Quantization Signal range Quant. Step Bernoulli The Quasi-BLUE estimator:

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

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

Simulations

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