Basis Pursuit for Spectrum Cartography

1. 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

2. Goal: find s.t. is the spectrum at position Cooperative spectrum sensing • Idea: collaborate to form a spatial map of the spectrum • Cooperation improves performance, e.g., [Quan et al’08] • Approach: • Basis expansion model (BEM) for • Nonparametric basis pursuit

3. 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 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] • 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]

4. 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 Frequency basis expansion

5. Spatial loss function Unknown • Per sub-band factorization in space and frequency (indep. of ) • Goal: estimate PSD atlas as Spatial PSD model • BEM:

6. Available data: location of CRsmeasured frequencies (I) Observations • Nonparametric basis selection ( not selected) Nonparametric basis pursuit • Twofold regularization of variational LS estimator • Avoid overfitting by promoting smoothness

7. Q1: How to estimate based on ? Thin-plate splines solution Proposition 1: Estimates in (I) are thin-plate splines [Duchon’77] where is the radial basis function , and • Unique, closed-form, finitely-parameterized minimizers! • Q2: How does (I) perform basis selection?

8. Matrices ( and dependent) i) ii) iii) Proposition 2: Minimizers of (I) are fully determined by w/ as • Remark: group Lasso encourages sparse factors • Full-rank mapping: Lassoing bases • (I) equivalent to group Lasso estimator [Yuan-Lin’06]

9. Simulated test • sensing CRs, sampling frequencies • sources; raised cosine pulses • bases; (roll off x center frequency x bandwidth) frequency (Mhz) basis index Original Estimated S P E C T R U M M A P

10. Real RF data CRs -60 -50 -40 -20 (dBi) -30 -10 1 2 3 • IEEE 802.11 WLAN activity sensed 4 5 6 7 8 9 10 11 12 13 14 • Maps recovered and extrapolated Frequency bases identified 10

11. 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 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