#### Transcript Splines Asilomar09

Asilomar Conference November 3, 2009 Spline-based Spectrum Cartography for Cognitive Radios Gonzalo Mateos, Juan A. Bazerque and Georgios B. Giannakis ECE Department, University of Minnesota Acknowledgments: ARL/CTA grant no. DAAD19-01-2-0011 USDoD ARO grant no. W911NF-05-1-0283 1 Cooperative Spectrum Sensing Cooperation improves performance, e.g., [Ganesan-Li’07], [Ghasemi-Sousa’07] Idea: collaborate to form a spatial map of the spectrum Goal: find s.t. is the spectrum at position Specification: coarse approximation suffices Approach: factorizable model for 2 Motivation & Prior Art 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 Complemented w/ channel gain maps [Kim-Dall’Anese-Giannakis’09] RF maps boost the “cognition” capability of the network Existing approaches to spectrum cartography Spatial interpolation via Kriging [Alaya-Feki et al’08] Sparsity-aware PSD estimation [Bazerque-Giannakis‘08] Decentralized signal subspace projections [Barbarossa et al’09] 3 Frequency Basis Expansion PSD of Tx source is Basis expansion in frequency Basis functions Accommodate prior knowledge Sharp transitions (regulatory masks) Other bases possible Gaussian bells rectangular, non-overlapping 4 PSD Factorization Spatial loss function Unknown PSD model: Per sub-band factorization in space and frequency (indep. of ) 5 Problem Statement Variational regression problem with smoothing penalty (I) Available data: location of CRs Observations measured frequencies controls smoothness of Goal: estimate global PSD maps as 6 Thin-plate Splines Solution Proposition: The solutions to thin-plate splines of the form where to Problem (I) correspond is the radial basis function , and Special case: Non-overlapping bases Problem (I) decouples per sub-band Overlapping bases important for non-FDMA based CR networks 7 Parameter Estimation Plug the solution: variational problem constrained, penalized LS s.t. collecting all collecting all and collecting all Matrices (knot dependent) Unknowns s.t. s.t. Fact [Wahba’90]: but 8 Existence and Uniqueness QR decompositions: Solution given by the linear system: invertible Knots both have full-column rank. Meaning? i.e., CRs not placed on a straight line Basis functions Linearly dependent linearly independent and exhaustive Not exhaustive 9 Online PSD Tracker Real-time requirements of the sensing CRs Adapt to (slow) changes in the PSD map Online PSD tracker Exponentially-weighted moving average (EWMA) Mitigates fading and reduces periodogram variance Exponentially discards past data adaptive Criterion: Problem (I) with the substitution are recursively given by Matrices are time-invariant and computed offline 10 Simulation Setup Sub-band assignment transmitters (Tx) rectangular basis CRs (Rx) located uniformly at random -tap Rayleigh channels +path-loss+noise measured frequencies CR computes periodogram 11 Aggregate PSD Map Predicted distribution of (aggregate) power in space 0dB -30dB Five transmitters localized Smoothness enforced through the penalty 12 Selection of Spline-based PSD map estimator Selection of linear smoother via leave-one-out cross-validation OCV GCV OCV GCV 13 Global PSD Maps The estimated PSD maps reveal (un-)occupied bands across space 14 Further Numerical Tests Shadowing effects 0dB -high Tx antennas -high, -wide wall knife-edge effect on Tx power Effect of the wall identified -20dB Tracking a transmitter’s departure EWMA with Central transmitter departs at Estimator adapts to RF power levels 15 Concluding Summary Cooperative PSD map estimation Fundamental task in cognitive radio networks Factorizable model for the power map in frequency/space PSD estimation as regularized regression Thin-plate penalty enforces smoothness Bi-dimensional splines arise in the solution Conditions on the bases for existence and uniqueness Online PSD tracking Global PSD maps reveal (un-)occupied bands across space 16 Major Result on Splines Lemma: [Duchon’77] Let denote a subset of with finite cardinality, and the space of Sobolev functions where is well defined. Let argument be a functional which depends upon its only through its restriction to , i.e., Consider the variational problem Then is a thin-plate function of the form s.t. 17 Proof of Proposition Rewrite Problem (I) as Lemma Coefficients depend on Next minimization step Red terms depend on only through Lemma Likewise for subsequent minimizations 18