Cooperating to Build a Radio Map to Support Spectrum Agility Song - PowerPoint PPT Presentation

Cooperating to Build a Radio Map to Support Spectrum Agility Song Liu, Wade Trappe, Larry J. Greenstein December 3rd, 2007 WINLAB Fall 2007 Research Review

Overview � Motivation and Background � Optimal Random Field Reconstruction � Balanced Spectrum Sampling � Field Estimation by Hierarchical Interpolation � Summary

Background Motivation � A Typical Spatial Distribution of Spectral Intensity Challenges: � Sources with unknown locations � Random Variations • Correlated • Non-stationary Building Radio Maps is harder than it looks.

Background Physical Facts � Radio Propagation Model (log-normal) ⎛ ⎞ − x x = − γ + 0 x x ⎜ ⎟ ( ) 10 log ( ) (dB) P P s 0 10 ⎝ d ⎠ 0 � Spatially Correlated � : path loss exponent � � s ( x ) : shadow fading, normally distributed with zero mean and ⎛ ⎞ − x x ⎜ ⎟ = σ − i j x x 2 Cov( ( ) ( )) exp s s ⎜ ⎟ i j dB X ⎝ ⎠ C

Background Spectrum Reconstruction � Reconstruction Criterion for a Random Field: � Mean Square Error (MSE) + ) N M 1 ∑ = − x x 2 x x [( ( ) ( )) |{ ,..., }] MSE E P P 1 m m N M = + 1 m N � N : the number of sensors (CRs) � M : the number of positions of interest ) : radio power estimate (in dB) � P

Background Spectrum Reconstruction (cont’d) + ) N M 1 ∑ = − x x 2 x x [( ( ) ( )) |{ ,..., }] MSE E P P 1 m m N M = + m N 1 � Sampling � Given N sensors, what are the best locations to place them? � Estimation � Given measured data at known locations, how to estimate spectrum level at an unknown location? A joint process of sampling and estimation.

Optimal Reconstruction Optimal Random Field Mapping � Optimal estimation in a stationary environment ˆ( = x x x x � MMSE (unbiased): ) [ ( ) | ( ),..., ( )] P E P P P m m 1 N � Gaussian process: [ P ( x m ), P ( x 1 ), … , P ( x N )] T ~ N ( � , C ) μ ⎡ x ⎤ ( ) m ⎡ ⎤ σ ⎢ ⎥ 2 C μ μ ⎡ ⎤ x x ( ) ( ) ⎢ ⎥ = ⎢ C = = ⎢ 12 1 dB ⎥ μ m ⎥ ⎢ ⎥ μ M ⎣ ⎦ C C ⎣ ⎦ N ⎢ ⎥ 21 22 μ x ⎣ ( ) ⎦ N ˆ( = μ + − − x x C C 1 P μ � Optimal estimate: ) ( ) ( ) P 12 22 m m N N − σ = σ − 2 2 C C C 1 � Minimum variance: x x |{ } 12 22 21 dB m N

Optimal Reconstruction Optimal Sensor Placement � A set of sensor locations: A N = { x 1 , …, x N } + ) N M 1 ∑ − x x 2 arg min [( ( ) ( )) | ] E P P A m m N M A = + 1 m N N � Given the optimal estimate, MMSE is equivalent to the maximum entropy criterion ⇔ = A A A arg min ( | ) arg max ( ) H H N N N A A N N + + + x x A x A L arg max ( ) arg max ( | ) arg max ( | ) H H H − 1 2 1 1 N N x x x 1 2 N 1 = π σ x A 2 ( | ) log(2 ) H e + x A 1 | n n + 2 1 n n

Optimal Reconstruction � Optimal � Impractical � Optimal estimation has to know � and C in prior: � � , P 0 ( d 0 ), x 0 , X C � Only work in stationary environments � Implicit solution: sub-optimal implementation by discretizing the node position N = 64 100 100 � Optimal sampling only 90 90 depends on C 80 80 70 70 60 60 y (meters) y (meters) Tend to place sensors uniformly; 50 50 40 40 slightly favor boundaries 30 30 20 20 10 10 0 0 0 0 20 20 40 40 60 60 80 80 100 100 x (meters) x (meters)

Optimal Reconstruction Case study: N = 25, � = 2, � dB = 4 dB, X C = 200 m � Linear Interpolation (N=25) MMSE Estimation 14 2 Random Equal 12 Root mean square error (dB) Root mean square error (dB) MinVar 1.5 10 8 1 6 4 0.5 2 0 0 [0,20] [20,40] [40,60] [60,80] [80,100] [0,20] [20,40] [40,60] [60,80] [80,100] Distance from the source (meters) Distance from the source (meters) Sub-optimal placement (MinVar) does not show performance improvement � by either the optimal estimation or linear approximation (interpolation) In regions close to a radio source, all three placement schemes have high � reconstruction errors using the approximation approach

Balanced Spectrum Sampling Balanced Spectrum Sampling � Balance between the uncertainty and the rapid decreases of spectrum intensity � On average, the spectrum intensity changes logarithmically along the direction from the source � Keep a uniform spectrum resolution across all power levels P (dB) 0 Received Power (dB) -20 50 -40 50 0 0 x (m) -50 log(d) -50 y (m)

Balanced Spectrum Sampling � Case study: N = 64, � = 2, � dB = 4 dB, X C = 125 m Linear Interpolation 12 100 MinVar 90 Balanced 10 Root mean square error (dB) 80 70 8 60 y (meters) 6 50 40 4 30 20 2 10 0 0 [0,20] [20,40] [40,60] [60,80] [80,100] 0 10 20 30 40 50 60 70 80 90 100 x (meters) Distance from the source (meters) � Reconstruction error can be significantly reduced in regions close to the source without compromising the performance of “outer” regions

Field Estimation by Interpolation Approximate Spectrum Mapping � Nonparametric Estimation – Interpolation � Nearest Neighbor � Linear � Spline � Hierarchical Interpolation using Compact Supported Functions � Radial basis functions [Wendland’95] � B-splines (Cubic)

Field Estimation by Interpolation Hierarchical Interpolation by 2-D Radial Basis Functions � 2-D Radial Basis Functions ( ) N ∑ = c φ − α x x x ( ) || || s k j j k = 1 j

Field Estimation by Interpolation Distance Matrix by Compactly Support Functions � Sparse Matrix � Unstructured: unpredictable complexity depends on the support radius � Worst case: non-sparse

Field Estimation by Interpolation Reduce Computational Complexity by Segmentation � Data are usually spatially clustered in reality � Segment the area of interest based on the knowledge of scene and/or machine learning techniques → 3 3 ( ) ( ( / ) ) O n O k n k

Summary � Building a radio map in a random field is a joint optimization of sensor placement and reconstruction accuracy � By balancing the uncertainty and estimation error, the reconstruction accuracy in regions close to radio sources can be significantly improved without impairing the overall performance � The spectrum over the area of interest is approximated by interpolation. If sensors are locally clustered, the complexity of interpolation methods can be greatly reduced by segmentation

Thank You! Questions?

m 10 = C X , db 4 = dB σ , 2 = γ

Application of Spectrum Map in Localization � Weighted Centroid � Radio Map “Plug- Localization in”

Enhanced Centroid Localization γ = σ dB = 2 , 1 dB � Random placement 100m x 100m, 7 10 Uniform Weighted 9 6 Interpolate Ideal interpolate RMSE of the location estimates (m) RMSE of the location estimates (m) 8 5 7 4 6 5 3 4 2 3 Uniform Weighted 1 2 Interpolate Ideal interpolate 1 0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 80 Number of anchor nodes Number of anchor nodes N = 25 N = 100