Collaborative Localization Using Weighted Centroid Localization (WCL) Algorithm in CR Networks

Transcription

1 Collaborative Localization Using Weighted Centroid Localization (WCL) Algorithm in CR Networks Simulation and Theoretical Results Jun Wang Paulo Urriza Prof. Danijela Čabrić UCLA CORES Lab March 12, 2010

2 Outline 1 Background of the Problem 2 System Model 3 Theoretical Results 4 Simulation Results 5 Summary

3 Outline 1 Background of the Problem 2 System Model 3 Theoretical Results 4 Simulation Results 5 Summary

4 Motivation Location is a vital piece of information for dynamic spectrum access networks!

5 Existing Techniques General Categories Range-based estimates link distance between unknown and anchor, requires channel model, sensitive to errors, less reliable Range-free less hardware, less depend on channel conditions, coarse but more reliable RSS Fingerprint Matching indoor environment, requires off-line phase to build fingerprint map for a set of known locations, expensive

6 This Work 1000 Sample Node Distribution 900 y coordinate Focuses on range-free techniques Non-interactive Localization In particular, WCL WCL x coordinate ˆL p = M i=1 w il i M i=1 w i (1) Figure: Collaborative Localization of a Primary User

7 Outline 1 Background of the Problem 2 System Model 3 Theoretical Results 4 Simulation Results 5 Summary

8 Node Locations Location of the ith node: L i = [ xi Nodes are placed in a grid and L i are known. Location of the primary user: L p = y i ] i = 1, 2,..., N (2) [ xp y p ] (3) x p, y p U(0, α) Cov(x p, y p ) = 0 (4)

9 Signal Model Received power of the ith user ( ) Li L p P i = P 0 10γ lg + s i db (5) d 0 s = [s 1, s 2,..., s N ] N(0, Ω s ) (6) Two Cases for investigation { σ 2 Ω s = db I N, i.i.d case {Ω s } ij = σdb 2 e L i L j /X c, correlated case (7)

10 WCL Scheme Threshold Node Selection Select a subset of M out of N nodes with highest reveived power. Number of possible subsets: T = C M N = N! M!(N M)! Equivalent to use the ratio of top M/N nodes to perform localization. (8)

11 WCL Scheme (Cont.) Relative Span Weighted Centroid 1 Weighting Factor w i = P i P min P (9) P min minimum received power, P = P max P min span Estimated Location ˆL p = M i=1 w il i M i=1 w i = M i=1 [(P i P min )L i ] M i=1 (P. (10) i P min ) 1 Laurendeau C. and Barbeau M.

12 Outline 1 Background of the Problem 2 System Model 3 Theoretical Results 4 Simulation Results 5 Summary

13 Analysis of the Localization Error of WCL Average Localization Error of Subset t e t = etx 2 + ety, 2 (11) e tx, e ty - 1D localization error for subset t Average Localization Error for Fixed L p e avg = β t - Prob(tth subset selected) T β t e t (12) t=1

14 Analysis of the Localization Error of WCL(Cont.) Overall Localization Error e loc = α α 0 0 e avg (x p, y p )f (x p, y p )dx p dy p (13) Based on our assumption on primary user location: e loc = 1 α 2 α α 0 0 e avg (x p, y p )dx p dy p (14) If nodes are place uniformly, requires 2N + 2 times integration! Grid-like nodes placement fits the indoor localization scenario.

15 1-D Localization Error (I.I.D. Case) Define Then µ i = P 0 10γ lg( L i L p d 0 ) P min (15) P i P min = µ i + s(l i ) N(µ i, σ 2 db ) (16) 1-D Location Estimate M i=1 ˆx p = [(P i P min )x i ] M i=1 (P = a i P min ) b. (17)

16 1-D Localization Error (I.I.D. Case) (Cont.) Distribution of a and b given by M M a = [(P i P min )x i ] N( µ i x i, σdb 2 b = i=1 i=1 M xi 2 ) = N(m a, σa) 2 i=1 M M (P i P min ) N( µ i, MσdB 2 ) = N(m b, σb 2 ) (18) i=1 i=1 Correlation Coefficient M ρ ab = σ2 db i=1 x i σ a σ b x = M x 2 1, (19)

17 1-D Localization Error (I.I.D. Case) (Cont.) Exact distribution of the ratio of two Gaussian RVs p X (x) = σ 1 σ 2 π(σ1 2x2 2ρσ 1 σ 2 x + σ2 2) [ ( 2 1 X1 exp 2(1 ρ 2 ) σ1 2 2ρ X 1 X2 + X )] 2 2 σ 1 σ 2 σ2 2 + X 1 σ2 2 X 2 ρσ 1 σ 2 + ( X 2 σ1 2 X 1 ρσ 1 σ 2 )x 2π(σ 2 1 x 2 2ρσ 1 σ 2 x + σ2 2 ( )3/2 ( exp X 2 X 1 x) 2 ) 2(σ1 2x2 2ρσ 1 σ 2 x + σ2 2 [ ( ) X1 σ2 2 X 2 ρσ 1 σ 2 + ( X 2 σ1 2 X 1 ρσ 1 σ 2 )x 1 2Q σ 1 σ 2 (1 ρ 2 ) 1/2 (σ 2 1 x2 2ρσ 1 σ 2 x + σ 2 2 )1/2 )]

18 1-D Localization Error (I.I.D. Case) (Cont.) Alternatively, Gaussian Approximation Finally, m ˆxp (m a /m b ) + σ 2 b m a/m 3 b ρ abσ a σ b /m 2 b, (20) σ 2ˆx p σ 2 b m2 a/m 4 b + σ2 a/m 2 b 2ρ abσ a σ b m a /m 3 b (21) Mean and Variance of 1-D Localization Error e tx = m ˆxp x p (22) σ 2 e tx = σ 2ˆx p (23)

19 Accuracy of the Gaussian Approximation Figure: Exact v.s. Approximated Mean and Variance

20 1-D Localization Error (Correlated Case) Correlation Model E[s(L i )s(l j )] = σ 2 db e L i L j /X c = σ 2 db λ ij, (24) Mean and Variance of a and b m a = σ 2 a = σ 2 b = M x i µ i, M m b = µ i i=1 i=1 M M M M R ij = x i x j σdb 2 λ ij i=1 j=1 i=1 j=1 M M M M R ij = σdb 2 λ ij (25) i=1 j=1 i=1 j=1

21 1-D Localization Error (Correlated Case) (Cont.) Correlation Coefficient ρ ab = σ2 db M i=1 M j=1 x iλ ij σ a σ b = 1 T Λx (x T Λx1 T, (26) Λ1) 1/2 Exact PDF and Gaussian approximation still applicable. Apply the same token for localization error in y-axis.

22 2-D Localization Error Expression of 2-D Error Denote where 2-D Error is given by e t = [e tx, e ty ] T N(ē t, Ω t ) (27) ē t = e tx, e ty ] T [ ] σ 2 Ω t = etx ρ etx e ty σ etx σ ety ρ etx e ty σ etx σ ety σe 2 ty e t = (28) e 2 tx + e 2 ty = e t 2. (29)

23 Methods to Evaluate the 2-D Error 1. De-correlation e t = Ω 1/2 t (e t ē t ) = [e tx, e ty] N(0, I) et 2 = (ω ω21)e 2 2 tx + (ω ω22)e 2 2 ty +2( e tx ω 11 + e ty ω 21 )e tx + 2( e tx ω 12 + e ty ω 22 )e ty +2(ω 11 ω 12 + ω 21 ω 22 )e txe ty. (30) 2. Characteristic Function of Gaussian Quadratic Forms ψ e 2 t (jω) = I jωω t 1 exp{ ē t T Ω 1 t [I (I jωω t ) 1 ]ē t }, (31)

24 Bottleneck of 2-D Error Evaluation Closed-form expression of ρ etx e ty ρ etx e ty σ etx σ ety = E [(e tx e tx )(e ty e ty )] = E {[ ( ˆx p x p ) (m ˆxp x p ) ] [ (ŷ p y p ) (mŷp y p ) ]} = E [ ( ˆx p m ˆxp )(ŷ p mŷp ) ] = E ˆx p ŷ p ] m ˆxp mŷp [ M i=1 = E [(P ] M i P min )x i ] i=1 [(P i P min )y i ] M i=1 (P M i P min ) i=1 (P i P min ) m ˆxp mŷp (32)

25 Outline 1 Background of the Problem 2 System Model 3 Theoretical Results 4 Simulation Results 5 Summary

26 Simulation Environment 1 Similar to Relative Span Weighted Centroid 2 2 Random transmitter position - uniformly distributed within a m 2 simulation grid 3 Sensor nodes are distributed in a grid 4 Node Densities: (0.25 to 10) per m 2 2 C. Laurendeau

27 Localization Error of WCL Algorithm 1000 Estimated Location Using All Nodes y coordinate x coordinate Figure: WCL Using all of the nodes in centroid

28 Optimal Node Participation - Uncorrelated Case Optimal Ratio Density Figure: Optimal number of node that participate (minimum mean error)

29 Normalized Mean Error - Uncorrelated Case 2.5 Mean Localization Error (Normalized to Nodespacing) using optimal number of nodes Node Spacing = 0 2 = 2 = 4 = 6 = 8 Mean Error (m) = 10 = 12 = 14 = 16 = 18 = Density (Nodes / 100mx100m) Figure: Mean error normalized to node spacing

30 Normalized Std. Dev. of Error - Uncorrelated Case Normalized Std. Dev. of Error (m) Node Spacing = 0 = 2 = 4 = 6 = 8 = 10 = 12 = 14 = 16 = 18 = 20 Std. Dev. of Error (Normalized to Nodespacing) using optimal number of nodes Density (Nodes / 100mx100m) Figure: Std. Dev. of error normalized to node spacing

31 500m 1000m WCL with Confined PU Position 1000m 500m Total Area for PU Total Area for Sensor Nodes Confine PU position to the center of the map ( 1 4 of the area) Reduces boundary problems Benefits of high cooperation ratio can be exploited Main Drawback: reduces the effective area that can be localized.

32 Optimal Node Participation - Confined PU Position Optimal Ratio of Participation (Transmitter Confined to 500m x 500m area) Optimal Ratio Density Figure: Optimal ratio of nodes that participate (confined PU position)

33 Normalized Mean Error - Confined PU Position Mean Error (m) Mean Localization Error (Normalized to Nodespacing) for confined area using optimal ratio of nodes Node Spacing = 0 = 2 = 4 = 6 = 8 = 10 = 12 = 14 = 16 = 18 = Density (Nodes / 100mx100m) Figure: Mean error normalized to node spacing (confined PU position)

34 Normalized Std. Dev. of Error - Confined PU position Std. Dev. of Error (m) Std. Dev. of Localization Error (Normalized to Nodespacing) using optimal ratio of nodes for confined area Node Spacing = 0 = 2 = 4 = 6 = 8 = 10 = 12 = 14 = 16 = 18 = Density (Nodes / 100mx100m) Figure: Normalized std. dev. of error (confined PU position)

35 1000m Proposal: Adaptive Participation WCL (APWCL) 1000m Total Area for Participating Nodes Coarse Node Position (via WCL on 4-7 nodes) Two-stage WCL Coarse WCL with 5 nodes (accuracy is within 1 node space) Fine WCL using subset of node area with higher participation X and Y localization can also be decoupled Total Area for Sensor Nodes

36 1000m APWCL - Calculating Participation 1000m R Total Area for Participating Nodes Coarse Node Position (via WCL on 4-7 nodes) Take R to be the distance to the closest map edge Get area of circle with radius R Participation = Area * Node Density Reduces errors due to boundary problem Total Area for Sensor Nodes

37 APWCL - Normalized Mean Error Comparison Normalized Mean Error (m) Normalized Mean Error Comaparison of Centroid Techniques Strongest Node Top 5 APWCL Ver. 2 Normalized Mean Error (m) Normalized Mean Error Comaparison of Centroid Techniques (Correlated Case) Strongest Node Top 5 APWCL Ver Figure: Uncorrelated [left], Correlated [right]

38 APWCL - Normalized Std. Dev. Comparison Normalized Std. Dev. of Error (m) Normalized Std. Dev. of Error Comaparison of Centroid Techniques 1 Strongest Node 0.9 Top APWCL Ver Normalized Std. Dev. of Error (m) Normalized Std. Dev. of Error Comaparison of Centroid Techniques (Correlated Case) 1 Strongest Node 0.9 Top APWCL Ver Figure: Uncorrelated [left], Correlated [right]

39 Outline 1 Background of the Problem 2 System Model 3 Theoretical Results 4 Simulation Results 5 Summary

40 Conclusion 1 Performance of WCL was analyzed 2 Theoretical framework of the WCL with node selection and relative span weighting was established 3 Proposed an improvement to WCL which solves the boundary problem and potentially improves its accuracy

41 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

42 Bibliography I S. Liu et al., Non-interactive localization of cognitive radios based on dynamic signal strength mapping, Proc. of the Sixth international conference on Wireless On-Demand Network Systems and Services, pp , Utah, USA, 2009 C. Laurendeau et al., Centroid localization of uncooperative nodes in wireless networks using a relative span weighting method, EURASIP J. on Wireless Commun. and Networking, vol. 2010, id , pp10, L. Liechty et al., Developing the best 2.4 GHz propagation model from active network measurements, in Proceedings of the 66th IEEE Vehicular Technology Conference (VTC 07), pp , Sept, M. K. Simon, Probability distributions involving Gaussian random variables, Boston, Kluwer Academic Publishers, 2002.

43 Bibliography II J. Hayya et al., A note on the ratio of two normaly distributed variables, Management Science, vol.21, no.11, pp , Jul A. Laub, Matrix analysis for science and engineers,siam, 2005.

44 Thank you very much Questions?