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.

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 2005

2. 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

3. 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

4. J.-J. Xiao and Z.-Q. Luo, “Universal decentralized detection in a bandwidth-constrained sensor network”, IEEE Trans. on Sig. Proc., August 2005

5. 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

6. 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 withUnknown 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 1st bit ¼ of the sensors for the 2nd bit, and so on…

Exact expressions can be found in [1] For small , it requires

Simulations

Simulations

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

Advanced I – Dynamic Range

Non-Identical Thresholds

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 CRLB3. Derive upper-bound for the CRLB4. 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 2.

2. Discretize the objective.

Advanced II – Detection

Fusion CenterConstraints: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

FusionCenter

The BLUE estimator:

Setup

Advanced III – Energy Constraint

FusionCenter

Goal: Meet target MSE under quantization + total power constraints.

Probabilistic Quantization

Signalrange

Quant. Step

Bernoulli

The Quasi-BLUE estimator:

Power Scheduling

ConstConst

MSE due to BER: only a constant factor

Solution

1. Integer variable2. 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