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

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) Spatially Correlated

• : path loss exponent

s(x): shadow fading, normally distributed with zero

mean and

10

( ) 10 log ( ) (dB) P P s d γ ⎛ ⎞ − = − + ⎜ ⎟ ⎝ ⎠ x x x x

2

Cov( ( ) ( )) exp

i j i j dB C

s s X σ ⎛ ⎞ − ⎜ ⎟ = − ⎜ ⎟ ⎝ ⎠ x x x x

Background

Reconstruction Criterion for a Random Field:

Mean Square Error (MSE) N : the number of sensors (CRs) M : the number of positions of interest

• : radio power estimate (in dB)

Spectrum Reconstruction

2 1 1

1 [( ( ) ( )) |{ ,..., }]

N M m m N m N

MSE E P P M

+ = +

= −

∑

x x x x )

P )

Background

Spectrum Reconstruction (cont’d)

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.

2 1 1

1 [( ( ) ( )) |{ ,..., }]

N M m m N m N

MSE E P P M

+ = +

= −

∑

x x x x )

Background

Optimal Random Field Mapping

Optimal estimation in a stationary environment

MMSE (unbiased): Gaussian process:

[P(xm ), P(x1 ), …, P(xN )]T ~ N(, C)

Optimal estimate: Minimum variance:

1

ˆ( ) [ ( ) | ( ),..., ( )]

m m N

P E P P P = x x x x

1 12 22

ˆ( ) ( ) ( )

m m N N

P μ

−

= + − x x C C P μ

1

( ) ( ) ( ) ( )

m m N N

μ μ μ μ ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ x x x μ μ x M

2 12 21 22 dB

σ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ C C C C

2 2 1 |{ } 12 22 21

m N

dB

σ σ

−

= −

x x

C C C

Optimal Reconstruction

Optimal Sensor Placement

A set of sensor locations: AN = {x1, …, xN} Given the optimal estimate, MMSE is equivalent to the

maximum entropy criterion

1 2

1 2 1 1

arg min ( | ) arg max ( ) arg max ( ) arg max ( | ) arg max ( | )

N N N

N N N N N

H H H H H

−

⇔ = + + +

A A x x x

A A A x x A x A L

2 1

1 arg min [( ( ) ( )) | ]

N

N M m m N A m N

E P P A M

+ = +

−

∑

x x )

1

2 1 |

1 ( | ) log(2 ) 2

n n

n n

H e π σ

+

+

=

x A

x A

Optimal Reconstruction

20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 x (meters) y (meters) 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 x (meters) y (meters)

Optimal Impractical

Optimal estimation has to know and C in prior:

, P0(d0), x0, XC

Only work in stationary environments Implicit solution: sub-optimal implementation by

discretizing the node position

Optimal sampling only

depends on C Tend to place sensors uniformly; slightly favor boundaries

N = 64 Optimal Reconstruction

• 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

[0,20] [20,40] [40,60] [60,80] [80,100] 0.5 1 1.5 2 MMSE Estimation Distance from the source (meters) Root mean square error (dB)

• Case study: N = 25, = 2, dB = 4 dB, XC = 200 m

[0,20] [20,40] [40,60] [60,80] [80,100] 2 4 6 8 10 12 14 Linear Interpolation (N=25) Distance from the source (meters) Root mean square error (dB) Random Equal MinVar

Optimal Reconstruction

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) log(d)

• 50

50

• 50

50

• 40
• 20

x (m) y (m) Received Power (dB)

Balanced Spectrum Sampling

Case study: N = 64, = 2, dB = 4 dB, XC = 125 m

Reconstruction error can be significantly reduced in

regions close to the source without compromising the performance of “outer” regions

10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 x (meters) y (meters)

[0,20] [20,40] [40,60] [60,80] [80,100] 2 4 6 8 10 12 Linear Interpolation Distance from the source (meters) Root mean square error (dB) MinVar Balanced

Balanced Spectrum Sampling

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

( )

1

( ) || ||

N k j j k j

s c φ α

=

= −

∑

x x x

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

• f scene and/or machine learning techniques

) ) / ( ( ) (

3 3

k n k O n O →

Field Estimation by Interpolation

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 , db 4 , 2

dB

= = =

C

X σ γ

Application of Spectrum Map in Localization

Weighted Centroid

Localization

Radio Map “Plug-

in”

Enhanced Centroid Localization

Random placement 100m x 100m,

dB 1 , 2

dB =

= σ γ

10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 Number of anchor nodes RMSE of the location estimates (m) Uniform Weighted Interpolate Ideal interpolate 2 4 6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 9 10 Number of anchor nodes RMSE of the location estimates (m) Uniform Weighted Interpolate Ideal interpolate

N = 25 N = 100