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Basis Pursuit for Spectrum Cartography

Juan A. Bazerque, Gonzalo Mateos, and Georgios B. Giannakis

ECE Department, University of Minnesota

Acknowledgments: NSF grants no. CCF-0830480, 1016605 EECS-0824007, 1002180

May 25, 2011

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Cooperative spectrum sensing Cooperation improves performance, e.g., [Quan et al’08]

Goal: find s.t. is the spectrum at position

Approach: Basis expansion model (BEM) for

Nonparametric basis pursuit

Idea: collaborate to form a spatial map of the spectrum

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Motivation & prior art

Approaches to spectrum cartography Spatial interpolation via Kriging [Alaya-Feki et al’08][Kim et al’09] Sparsity-aware PSD estimation [Bazerque-Giannakis‘08] Decentralized signal subspace projections [Barbarossa et al’09]

Power spectrum density (PSD) maps envisioned for: Identification of idle bands reuse and handoff operation Localization and tracking of primary user (PU) activity Cross-layer design of CR networks

Basis pursuit [Chen et al’98], LASSO [Tibshirani’94] Scalar vs. functional coefficient selection in overcomplete BEM Specific models: COSSO [Lin-Zhang’06], SpAM [Ravikumar’09]

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Frequency basis expansion PSD of Tx source is

Basis expansion in frequency

Basis functions Accommodate prior knowledge raised-cosine Sharp transitions (regulatory masks) rectangular, non-overlapping Overcomplete basis set (large ) robustness

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Spatial PSD model Spatial loss function Unknown

Per sub-band factorization in space and frequency (indep. of )

BEM:

Goal: estimate PSD atlas as

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Twofold regularization of variational LS estimator

(I)

Nonparametric basis pursuit Available data:

location of CRs

measured frequencies

Observations

Avoid overfitting by promoting smoothness

Nonparametric basis selection ( not selected)

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Thin-plate splines solution

Unique, closed-form, finitely-parameterized minimizers!

Proposition 1: Estimates in (I) are thin-plate splines [Duchon’77]

where is the radial basis function , and

Q2: How does (I) perform basis selection?

Q1: How to estimate based on ?

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Lassoing bases (I) equivalent to group Lasso estimator [Yuan-Lin’06]

Matrices ( and dependent)

i) ii) iii)

Remark: group Lasso encourages sparse factors Full-rank mapping:

Proposition 2:

as

w/

Minimizers of (I) are fully determined by

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Simulated test

SPECTRUM MAP

basis index frequency (Mhz)

sources; raised cosine pulses sensing CRs, sampling frequencies bases; (roll off x center frequency x bandwidth)

Original Estimated

1010

Real RF data

Frequency bases identified Maps recovered and extrapolated

IEEE 802.11 WLAN activity sensed

CRs

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-50-60 -40 -30 -20 -10 (dBi)

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Concluding Summary Cooperative PSD map estimation

Fundamental task in cognitive radio networks

(Overcomplete) BEM for the power map in frequency/space

Computer simulations and real RF data for testing PSD atlas reveal (un-)occupied bands across space Source localization and identification of Tx parameters

PSD estimation as regularized nonparametric regression Thin-plate regularization effects smoothness Bi-dimensional splines arise in the solution Sparsity-encouraging penalty basis selection via group Lasso