Splines Asilomar09

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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
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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
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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]
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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
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PSD Factorization
 Spatial loss function
Unknown
 PSD model:
 Per sub-band factorization in space and frequency (indep. of
)
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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
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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
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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
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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
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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
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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
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Aggregate PSD Map
 Predicted distribution of (aggregate) power in space
0dB
-30dB
 Five transmitters localized
 Smoothness enforced through the penalty
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Selection of
 Spline-based PSD map estimator
 Selection of
linear smoother
via leave-one-out cross-validation
OCV
GCV
 OCV
 GCV
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Global PSD Maps
 The estimated PSD maps reveal (un-)occupied bands across space
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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
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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
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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.
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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
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