Effects of Correlated Shadowing: Connectivity ...npatwari/pubs/ipsn08-talk.pdfSummary Effects of...

MotivationOur Results/Contribution

Summary

Effects of Correlated Shadowing: Connectivity,Localization, and RF Tomography

Neal Patwari Piyush Agrawal

Sensing and Processing Across Networks LabDepartment of Electrical and Computer Engineering

University of Utah, USA

International Conference on Information Processing inSensor Networks, 2008

Patwari, Agrawal Effects of Correlated Shadowing


Summary

Outline

1 MotivationCurrent Fading Models for WSN are LackingNetwork Shadowing (NeSh) Model

2 Our Results/ContributionConnectivity of Multi-hop NetworksSensor Cooperative LocalizationRF Tomographic Imaging



Summary

Current Fading Models for WSN are LackingNetwork Shadowing (NeSh) Model

Outline





Summary


Intuition: Fading Models Defy Physical Reality

Measured (a) from [Newport07], compared to three models

NeSh ModelMeasuredLevelSignal

Circular CoverageModel

(a) (c)(b) (d)i.i.d. Fading

Model

1 Coverage is not circular: Random with environment2 Fading is not i.i.d.: Spatial correlations exist



Summary


Models all Fit Generic Path Loss and Fading Model

Link (i, j) is disconnected if Pi,j < Pthr.Pi,j: measured received power at node j transmitted bynode i (dBm),

Pi,j = P̄(di,j) − Xi,j − Yi,j

P̄(di,j): ensemble mean dBm received power at distancedi,j between i and j,Xi,j: shadowing loss in dB,Yi,j: other fading losses in dB.



Summary


Fading in Circular Coverage and i.i.d. Fading Models

Circular Coverage Model:

Zero fading: Xi,j = Yi,j = 0.At distance dthr, Pi,j < Pthr, link disconnected.

i.i.d. Fading Model:

Total fading Zi,j = Xi,j + Yi,j, where {Zi,j}i,j aremutually independent.Problem: Links to neighboring points arecompletely different.



Summary


Related Research on Correlated Shadowing Model

1 Temporal correlations for MANETs [Newport07] [Wang06]2 Exponentially-decaying covariance across space

[Gudmundson91]3 Comparison: NeSh allows links without a common node:

(a)

C

D

A

(b)

C

D

A

B



Summary


Outline





Summary


Shadowing Field is the Random Environment

Start w/ random shadowing field p(x)

xi

xj

xk

xl

link a link b

shadowing field ( )p x

Assumptions about p(x):Isotropic, zero-mean GaussianCovariance:E

[p(xi)p(xj)

]=

σ2X

δ e−‖xj−xi‖

δ

δ: Distance constant

Assumption: Xa = normalized line integral of p(x),

Xa ,1

‖xj − xi‖1/2

∫ xj

xi

p(y)dy.



Summary


NeSh is 2nd-Order Version of Existing Models

The model agrees in distribution with existing models[Rappaport]:

1 Gaussian (in dB),2 Zero-mean,3 Variance σ2

X constant vs. path length.

NeSh also causes E [XaXb] > 0 for ‘close’ links a, b.Assumes no correlation in small-scale fading {Yi,j}.



Summary


Experimental Evidence

NeSh model verified in [Agrawal08]Experiment-based indoor measurements of 15 WSNsA random environment was created for each WSNBetter agreement than model of [Gudmundson91]Estimated parameters: σ2

X/σ2dB = 0.29 and δ = 21 cm



Summary


NeSh Model Has Limitations

Statistical: Not site-specific, not solution to Maxwell’s EqnsLimited tests: Not tested for multi-floor, outdoor (terrain)Not needed when environments are homogenous



Summary

Connectivity of Multi-hop NetworksSensor Cooperative LocalizationRF Tomographic Imaging

Outline





Summary


Connectivity Research: Circular and i.i.d. Models

Standard assumption: Circular coverage modelUsing i.i.d. fading model [Hekmat06], [Bettstetter05]:Increasing fading → Higher graph connectivity!Intuition: Partially, artifact of i.i.d. model



Summary


Simulation Examples

(a)

1 m

1m

1 3

1

3

(b)

1.5 m

1.5

m

Simulate two networks with same density: 1 node / m2



Summary


Using NeSh to Generate Network Connectivity

For a given geometry of sensors, determine connectivity:1 Generate vector of [Pi,j]i,j as multivariate Gaussian with

NeSh model2 Any link (i, j) is connected if Pi,j > Pthr

3 Determine whether or not the graph is connected



Summary


Results Show i.i.d. Fading Model is Optimistic

For correlated shadowing vs. i.i.d. model:Const. range: P[disconnected] up 230% (4x4), 380% (8x2)Const. connectivity: Need higher transmit powerModel disconnect highest in narrow deployments



Summary


Outline





Summary


Correlation Changes Common Assumption

Lower bounds on location variance, MSETo date: Assume link measurements are i.i.d.Pro: simplification. Con: inaccurate for RSSQuestion: Will correlations increase or decrease errors?



Summary


Localization Formulation Details

Estimate: Coordinates of unknown-location nodes,

θ = [x1, . . . , xn, y1, . . . , yn]T ,

Given: Some nodes n + 1, . . . , N have known coordinatesGiven: Measurements Pi,j between pairs (i, j) ∈ E



Summary


Lower Bound on Variance: Analysis Setup

Cramér-Rao lower bound (CRLB) on covariance matrix Cθ

of unbiased estimators of θ,

Cθ > F−1 = [Fµ + FC]−1

Total Fisher information F is sum of Fisher information1 from the mean loss on individual links, Fµ2 from the shadowing covariance between links, FC

Also consider bound using only mean term: C∗θ > F−1

µ



Summary


Lower Bound on Variance: Numerical Results

16 node (·) random network, 4 with known coordinate (x)Bound on 1-σ covariance ellipse – smaller is better

(a) (b)

(a) i.i.d.: (- - - -),NeSh: (—–)(b) i.i.d.: (- - - -),NeSh mean termonly: (—–)

Average std. dev. bound: (a) Down 4%, (b) Up 14%.



Summary


Intuition Gained: Use pair-wise data

In NeSh model:Diminished localization from individual Pi,j meas’tsIncreased localization from pairs (Pi,j, Pk,l)

Overall, location errors could be slightly reduced (4%).Algorithms should use Pi,∗ and Pj,∗ to infer distance di,j

(e.g.: RF fingerprinting, statistical learning methods)



Summary


Outline





Summary


Correlation Enables Inference About Environment

Shadowing Field � Link Losses

5 7

18 19 20 2114 15 16 17

1

2

3

4

5 6 7 8 9 10 11 12 13

Imaging would not be possible in i.i.d. fadingThis work: imaging of environment change



Summary


Method Overview

Change in shadowing field: M-pixel image vector p

xik

xjk

y

y

m

n

link k

Change in link loss, ν, vector from each link (ik, jk)

A linear combo of pixels m in between xik and xjk



Summary


Imaging is Inverse Problem in Linear Model

Overall linear model with noise n:

ν = Ap + n

Measure ν, solve for p. Least squares:

p̂ = Πν, where Π = RAT (ARAT + σ2

KIK)−1

R: Covariance in shadowing field (NeSh)σ2

KIK : Diagonal regularization termΠ: Projection matrix, calculated once



Summary


Experiment: Person Moving in Room

N

SW

Sensor

StoppingPosition

Door

WalkingPath

Key SENE

NW

SW

Environment: Empty 5m x 5m roomSensors: Crossbow Mica2, 915 MHzProtocol: Each broadcasts packet every 0.5 secMovement: Person walks in, 1 min. in each corner



Summary


Imaging Algorithm: Three Steps

1 Calculate change in path loss from past average2 Calculate tomographic image via LS3 Increase image contrast



Summary


Experiment Results: Location Seen in Image19:02:23, NE corner

19:03:23, NW corner

19:01:23, SE corner

19:00:23, SW corner

19:04:23, empty room



Summary

Summary

Shadowing is correlated: Use correlated model (NeSh)Effects: Degrade connectivity, impact localizationalgorithms, and permit RTI

Future WorkMore experiment-based modeling and RTI developmentDevelopment of link correlation-based loc. algorithms



Summary

Questions and Comments

More info on http://span.ece.utah.edu


http://span.ece.utah.edu

Appendix For Further Reading

For Further Reading I

M. Gudmundson.Correlation model for shadow fading in mobile radiosystems.IEE Electronics Letters, 27(23):2145 – 2146, 7 Nov. 1991.

M. Youssef, M. Mah, A. Agrawala.Challenges: device-free passive localization for wirelessenvironments.ACM MobiCom, Sept. 2007.

R. Hekmat and P. V. Mieghem.Connectivity in wireless ad-hoc networks with a log-normalradio model.Springer Mobile Networks and Applications, 11:351–360,April 2006.



For Further Reading II

C. Bettstetter and C. Hartmann.Connectivity of wireless multihop networks in a shadowfading environment.Wirel. Netw., 11(5):571–579, 2005.

C. Newport, D. Kotz, Y. Yuan, R.S. Gray, J. Liu, C. Elliott.“Experimental Evaluation of Wireless SimulationAssumptions”.Simulation, vol. 83, no. 9, pp. 643-661, 2007.

I P. Agrawal, N. Patwari.Correlated link shadow fading in multi-hop wirelessnetworks.Tech Report arXiv:0804.2708v2, 18 Apr 2008.



For Further Reading III

I Z. Wang, E. K. Tameh, A. Nix.Simulating correlated shadowing in mobile multihoprelay/ad-hoc networks.Tech. Rep. IEEE C802.16j-06/060, IEEE 802.16 BroadbandWireless Access Working Group. July 2006.


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