collaborative localization using weighted centroid...

Background of the Problem System Model Theoretical Results Simulation Results Summary

Collaborative Localization Using WeightedCentroid Localization (WCL) Algorithm

in CR NetworksSimulation and Theoretical Results

Jun WangPaulo Urriza

Prof. Danijela Cabric

UCLA CORES Lab

March 12, 2010


Outline

1 Background of the Problem

2 System Model

3 Theoretical Results

4 Simulation Results

5 Summary


Motivation

Location is a vital piece ofinformation for dynamicspectrum access networks!


Existing Techniques

General Categories

Range-based – estimates link distance between unknown andanchor, requires channel model, sensitive to errors, less reliable

Range-free – less hardware, less depend on channelconditions, coarse but more reliable

RSS Fingerprint Matching – indoor environment, requiresoff-line phase to build fingerprint map for a set of knownlocations, expensive


This Work

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

x−coordinate

y−co

ordi

nate

Sample Node Distribution

Figure: Collaborative Localization of aPrimary User

Focuses on range-freetechniques

Non-interactiveLocalization

In particular, WCL

WCL

Lp =

∑Mi=1 wiLi∑Mi=1 wi

(1)


Outline


2 System Model



5 Summary


Node Locations

Location of the ith node:

Li =

[xiyi

]i = 1, 2, . . . ,N (2)

Nodes are placed in a grid and Li are known.

Location of the primary user:

Lp =

[xpyp

](3)

xp, yp ∼ U(0, α)

Cov(xp, yp) = 0 (4)


Signal Model

Received power of the ith user

Pi = P0 − 10γ lg

(‖Li − Lp‖

d0

)+ si dB (5)

s = [s1, s2, . . . , sN ] ∼ N(0,Ωs) (6)

Two Cases for investigation

Ωs =

σ2dB IN, i.i.d case

Ωsij = σ2dBe−‖Li−Lj‖/Xc , correlated case

(7)


WCL Scheme

Threshold Node Selection

Select a subset of M out of N nodes with highest reveived power.Number of possible subsets:

T = CMN =

N!

M!(N −M)!(8)

Equivalent to use the ratio of top M/N nodes to performlocalization.


WCL Scheme (Cont.)

Relative Span Weighted Centroid1

Weighting Factor

wi =Pi − Pmin

P4(9)

Pmin – minimum received power, P4 = Pmax − Pmin – span

Estimated Location

Lp =

∑Mi=1 wiLi∑Mi=1 wi

=

∑Mi=1[(Pi − Pmin)Li ]∑Mi=1(Pi − Pmin)

. (10)

1Laurendeau C. and Barbeau M.


Outline


2 System Model



5 Summary


Analysis of the Localization Error of WCL

Average Localization Error of Subset t

et =√e2tx + e2

ty , (11)

etx , ety - 1D localization error for subset t

Average Localization Error for Fixed Lp

eavg =T∑t=1

βtet (12)

βt - Prob(tth subset selected)


Analysis of the Localization Error of WCL(Cont.)

Overall Localization Error

eloc =

∫ α

0

∫ α

0eavg (xp, yp)f (xp, yp)dxpdyp (13)

Based on our assumption on primary user location:

eloc =1

α2

∫ α

0

∫ α

0eavg (xp, yp)dxpdyp (14)

If nodes are place uniformly, requires 2N + 2 times integration!

Grid-like nodes placement fits the indoor localization scenario.


1-D Localization Error (I.I.D. Case)

Define

µi = P0 − 10γ lg(‖Li − Lp‖

d0)− Pmin (15)

ThenPi − Pmin = µi + s(Li ) ∼ N(µi , σ

2dB) (16)

1-D Location Estimate

xp =

∑Mi=1[(Pi − Pmin)xi ]∑Mi=1(Pi − Pmin)

=a

b. (17)


1-D Localization Error (I.I.D. Case) (Cont.)

Distribution of a and b given by

a =M∑i=1

[(Pi − Pmin)xi ] ∼ N(M∑i=1

µixi , σ2dB

M∑i=1

x2i ) = N(ma, σ

2a)

b =M∑i=1

(Pi − Pmin) ∼ N(M∑i=1

µi ,Mσ2dB) = N(mb, σ

2b) (18)

Correlation Coefficient

ρab =σ2dB

∑Mi=1 xi

σaσb

=‖x‖1√M‖x‖2

, (19)



Exact distribution of the ratio of two Gaussian RVs

pX (x) =σ1σ2

π(σ21x

2 − 2ρσ1σ2x + σ22)×

exp

[−

1

2(1− ρ2)

(X1

2

σ21

− 2ρX1

σ1

X2

σ2+

X22

σ22

)]

+X1σ

22 − X2ρσ1σ2 + (X2σ

21 − X1ρσ1σ2)x

√2π(σ2

1x2 − 2ρσ1σ2x + σ2

2)3/2

×exp

(−

(X2 − X1x)2

2(σ21x

2 − 2ρσ1σ2x + σ22)

)[

1− 2Q

(X1σ

22 − X2ρσ1σ2 + (X2σ

21 − X1ρσ1σ2)x

σ1σ2(1− ρ2)1/2(σ21x

2 − 2ρσ1σ2x + σ22)1/2

)]



Alternatively,

Gaussian Approximation

mxp ' (ma/mb) + σ2bma/m

3b − ρabσaσb/m2

b, (20)

σ2xp ' σ

2bm

2a/m

4b + σ2

a/m2b − 2ρabσaσbma/m

3b (21)

Finally,

Mean and Variance of 1-D Localization Error

etx = mxp − xp (22)

σ2etx = σ2

xp (23)


Accuracy of the Gaussian Approximation

Figure: Exact v.s. Approximated Mean and Variance


1-D Localization Error (Correlated Case)

Correlation Model

E[s(Li )s(Lj)] = σ2dBe−‖Li−Lj‖/Xc = σ2

dBλij , (24)

Mean and Variance of a and b

ma =M∑i=1

xiµi , mb =M∑i=1

µi

σ2a =

M∑i=1

M∑j=1

R′ij =

M∑i=1

M∑j=1

xixjσ2dBλij

σ2b =

M∑i=1

M∑j=1

Rij =M∑i=1

M∑j=1

σ2dBλij (25)


1-D Localization Error (Correlated Case) (Cont.)

Correlation Coefficient

ρab =σ2dB

∑Mi=1

∑Mj=1 xiλij

σaσb=

1TΛx

(xTΛx1TΛ1)1/2, (26)

Exact PDF and Gaussian approximation still applicable.

Apply the same token for localization error in y-axis.


2-D Localization Error

Expression of 2-D Error

Denoteet = [etx , ety ]T ∼ N(et ,Ωt) (27)

whereet = [etx , ety ]T

Ωt =

[σ2etx ρetxetyσetxσetyρetxetyσetxσety σ2

ety

](28)

2-D Error is given by

et =√e2tx + e2

ty = ‖et‖2. (29)


Methods to Evaluate the 2-D Error

1. De-correlation

e′t = Ω

−1/2t (et − et)

= [e′tx , e

′ty ] ∼ N(0, I)

e2t = (ω2

11 + ω221)e

′tx

2+ (ω2

12 + ω222)e

′ty

2

+2(etxω11 + etyω21)e′tx + 2(etxω12 + etyω22)e

′ty

+2(ω11ω12 + ω21ω22)e′txe

′ty . (30)

2. Characteristic Function of Gaussian Quadratic Forms

ψe2t(jω) = |I− jωΩt|−1exp−et

TΩ−1t [I− (I− jωΩt)

−1]et, (31)


Bottleneck of 2-D Error Evaluation

Closed-form expression of ρetxety

ρetxetyσetxσety = E [(etx − etx)(ety − ety )]

= E[

(xp − xp)− (mxp − xp)] [

(yp − yp)− (myp − yp)]

= E[(xp −mxp )(yp −myp )

]= E [xp yp]−mxpmyp

= E

[∑Mi=1 [(Pi − Pmin)xi ]∑M

i=1(Pi − Pmin)

∑Mi=1 [(Pi − Pmin)yi ]∑M

i=1(Pi − Pmin)

]−mxpmyp (32)


Outline


2 System Model



5 Summary


Simulation Environment

1 Similar to Relative Span Weighted Centroid2

2 Random transmitter position - uniformly distributed within a1000× 1000m2 simulation grid

3 Sensor nodes are distributed in a grid

4 Node Densities: (0.25 to 10) per 100× 100m2

2C. Laurendeau


Localization Error of WCL Algorithm

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

x−coordinate

y−co

ordi

nate

Estimated Location Using All Nodes

Figure: WCL Using all of the nodes in centroid


Optimal Node Participation - Uncorrelated Case

02

46

810

0

5

10

15

203

3.5

4

4.5

5

5.5

6

6.5

7

DensityσdB

Opt

imal

Rat

io

Figure: Optimal number of node that participate (minimum mean error)


Normalized Mean Error - Uncorrelated Case

1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5Mean Localization Error (Normalized to Nodespacing) using optimal number of nodes

Density (Nodes / 100mx100m)

Mea

n E

rror

(m

)

Node Spacingσ

dB = 0

σdB

= 2

σdB

= 4

σdB

= 6

σdB

= 8

σdB

= 10

σdB

= 12

σdB

= 14

σdB

= 16

σdB

= 18

σdB

= 20

Figure: Mean error normalized to node spacing


Normalized Std. Dev. of Error - Uncorrelated Case

1 2 3 4 5 6 7 8 90.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ized

Std

. Dev

. of E

rror

(m

)

Std. Dev. of Error (Normalized to Nodespacing) using optimal number of nodes

Node Spacingσ

dB = 0

σdB

= 2

σdB

= 4

σdB

= 6

σdB

= 8

σdB

= 10

σdB

= 12

σdB

= 14

σdB

= 16

σdB

= 18

σdB

= 20

Figure: Std. Dev. of error normalized to node spacing


WCL with Confined PU Position

1000m

1000m

500m

500m

Total Area for Sensor Nodes

Total Area for PU

Confine PU positionto the center of themap ( 1

4 of the area)

Reduces boundaryproblems

Benefits of highcooperation ratio canbe exploited

Main Drawback:reduces the effectivearea that can belocalized.


Optimal Node Participation - Confined PU Position

02

46

810

5

10

15

20

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Density

Optimal Ratio of Participation (Transmitter Confined to 500m x 500m area)

σdB

Opt

imal

Rat

io

Figure: Optimal ratio of nodes that participate (confined PU position)


Normalized Mean Error - Confined PU Position

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Mean Localization Error (Normalized to Nodespacing) for confined area using optimal ratio of nodes


Mea

n E

rror

(m

)

Node Spacingσ

dB = 0

σdB

= 2

σdB

= 4

σdB

= 6

σdB

= 8

σdB

= 10

σdB

= 12

σdB

= 14

σdB

= 16

σdB

= 18

σdB

= 20

Figure: Mean error normalized to node spacing (confined PU position)


Normalized Std. Dev. of Error - Confined PU position

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Std

. Dev

. of E

rror

(m

)

Std. Dev. of Localization Error (Normalized to Nodespacing) using optimal ratio of nodes for confined area

Node Spacingσ

dB = 0

σdB

= 2

σdB

= 4

σdB

= 6

σdB

= 8

σdB

= 10

σdB

= 12

σdB

= 14

σdB

= 16

σdB

= 18

σdB

= 20

Figure: Normalized std. dev. of error (confined PU position)


Proposal: Adaptive Participation WCL (APWCL)

1000m

1000m

Coarse Node Position (via WCL on 4-7 nodes)

Total Area for Participating Nodes


Two-stage WCL

Coarse WCL with 5nodes (accuracy is within1 node space)

Fine WCL using subsetof node area with higherparticipation

X and Y localization canalso be decoupled


APWCL - Calculating Participation

1000m

1000mCoarse Node Position

(via WCL on 4-7 nodes)

Total Area for Participating Nodes


R Take R to be thedistance to theclosest map edge

Get area of circle withradius R

Participation =Area * Node Density

Reduces errors due toboundary problem


APWCL - Normalized Mean Error Comparison

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

σdB

Nor

mal

ized

Mea

n E

rror

(m

)

Normalized Mean Error Comaparison of Centroid Techniques

Strongest NodeTop 5APWCL − Ver. 2

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

σdB

Nor

mal

ized

Mea

n E

rror

(m

)

Normalized Mean Error Comaparison of Centroid Techniques(Correlated Case)


Figure: Uncorrelated [left], Correlated [right]


APWCL - Normalized Std. Dev. Comparison

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σdB

Nor

mal

ized

Std

. Dev

. of E

rror

(m

)

Normalized Std. Dev. of Error Comaparison of Centroid Techniques


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σdB

Nor

mal

ized

Std

. Dev

. of E

rror

(m

)

Normalized Std. Dev. of Error Comaparison of Centroid Techniques(Correlated Case)


Figure: Uncorrelated [left], Correlated [right]


Outline


2 System Model



5 Summary


Conclusion

1 Performance of WCL was analyzed

2 Theoretical framework of the WCL with node selection andrelative span weighting was established

3 Proposed an improvement to WCL which solves the boundaryproblem and potentially improves its accuracy


Future Work

1 Complete the theoretical analysis on 2-D localization error

2 Verify the theoretical framework through simulations

3 Further study of the correlated case in simulations

4 Algorithm to find the optimal threshold for APWCL


Bibliography I

S. Liu et al., ”Non-interactive localization of cognitive radiosbased on dynamic signal strength mapping”, Proc. of the Sixthinternational conference on Wireless On-Demand NetworkSystems and Services, pp. 77-84, Utah, USA, 2009

C. Laurendeau et al., ”Centroid localization of uncooperativenodes in wireless networks using a relative span weightingmethod”, EURASIP J. on Wireless Commun. and Networking,vol. 2010, id. 567040, pp10, 2010.

L. Liechty et al., ”Developing the best 2.4 GHz propagationmodel from active network measurements”, in Proceedings ofthe 66th IEEE Vehicular Technology Conference (VTC ’07),pp. 894-896, Sept, 2007.

M. K. Simon, Probability distributions involving Gaussianrandom variables, Boston, Kluwer Academic Publishers, 2002.


Bibliography II

J. Hayya et al., ”A note on the ratio of two normalydistributed variables”, Management Science, vol.21, no.11,pp.1338-1341, Jul.1975.

A. Laub, Matrix analysis for science and engineers,Siam, 2005.


Thank you very much

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collaborative localization using weighted centroid...

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