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March 11, 2003 ASAP Workshop 2003 1
Robust Capon Beamforming
Uppsala UniversityPetre Stoica
University of FloridaYi Jiang
University of FloridaJian Li
University of FloridaZhisong Wang
March 11, 2003 ASAP Workshop 2003 2
Outline
? Standard Capon Beamforming (SCB)? Norm Constrained Capon Beamforming (NCCB)? Robust Capon Beamforming (RCB)? Coherent RCB (CRCB)? Simulation Results? Conclusions
March 11, 2003 ASAP Workshop 2003 3
Standard Capon Beamforming (SCB)
Signal power estimate
March 11, 2003 ASAP Workshop 2003 4
Norm Constrained Capon Beamforming (NCCB)
Loading level determined by norm constraint.
Diagonal loading:
March 11, 2003 ASAP Workshop 2003 5
Recent Robust Beamformers
?Based on original SCB formulationo Robust adaptive beamforming based on worst-case
performance optimization [Vorobyov, Gershman, Luo, 2001]
o Robust minimum variance beamforming [Lorenz, Boyd, 2001]
Directly Address Steering Vector Uncertainties!
March 11, 2003 ASAP Workshop 2003 6
Our RCB
?Based on Covariance Fittingo Robust Capon Beamforming [Stoica, Wang, Li, 2002]o On Robust Capon Beamforming and Diagonal Loading
[Li, Stoica, Wang, 2002]
?New featureso Steering vector within an uncertainty seto Incorporate uncertainty set into formulation directlyo Computationally most efficiento Conceptually simple o Scaling ambiguity eliminated
Directly Address Steering Vector Uncertainties!
March 11, 2003 ASAP Workshop 2003 7
Covariance Fitting
Same signal power estimate as SCB!
March 11, 2003 ASAP Workshop 2003 8
Our Robust Capon Beamformer(RCB)
?Incorporate ellipsoidal uncertainty set into covariance fitting
is of full column rank.
March 11, 2003 ASAP Workshop 2003 9
Our RCBo Without loss of generality, consider spherical
uncertainty set:
o Solution at boundary of uncertainty set
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Our RCBo Use Lagrange multiplier method
o Obtain Lagrange multiplier by solving
via Newton’s method (monotonic polynomial --computationally efficient)
March 11, 2003 ASAP Workshop 2003 11
Scaling Ambiguityo Uncertainty in SOI steering vector cause
scaling ambiguity
and yield same
o Add constraint to eliminateambiguity
March 11, 2003 ASAP Workshop 2003 12
Main Steps of Our RCB
o Step 1:
o Step 2: Obtain Lagrange multiplier
o Step 3:
o Step 4:
o Step 5:
March 11, 2003 ASAP Workshop 2003 13
?Obtain weight vector based on
?Diagonal loading (spherical constraint)!
?Waveform estimate
Waveform Estimation
March 11, 2003 ASAP Workshop 2003 14
Advantages of Our RCB
? Computationo Our RCB requires flops
while flops for [Vorobyov, Gershman, Luo, 2001]
o More computations needed to determine Lagrange multiplier and polynomial not monotonic for [Lorenz, Boyd, 2001] -- also flops
? Ambiguity elimination obvious for our RCB(not considered by others)
March 11, 2003 ASAP Workshop 2003 15
Numerical Examples?M = 10 sensors
?Uniform linear array with half-wavelength spacing
?Array calibration error exists (independent complex Gaussian random variables added)
March 11, 2003 ASAP Workshop 2003 16
Power Estimate vs. AngleTrue powers denoted by circles.
March 11, 2003 ASAP Workshop 2003 17
Making NCCB Have Same Diagonal Loading Level As RCB
March 11, 2003 ASAP Workshop 2003 18
NCCB and RCB Having Same Diagonal Loading Level
NCCB
March 11, 2003 ASAP Workshop 2003 19
Coherent RCB (CRCB)
o Coherent multipaths existo DOAs of multipaths known relative to DOA of SOI
? Motivation – GPS applications etc.
From Multipath Mitigation Performance of Planar GPS Adaptive Antenna Arrays for Precision Landing Ground Stations by J.H. Williams, et al, the MITRE Corporation
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CRCB
? Robust against coherent multipaths as well as steering vector errors.
Steering vector:
o Covariance fitting
Steering vector of SOISteering vectors of coherent multipaths
March 11, 2003 ASAP Workshop 2003 21
Steps of CRCBo Following similar steps in RCB
o Concentrating out
with
March 11, 2003 ASAP Workshop 2003 22
Insight of CRCB
Let
• Project data to orthogonal subspace of V• Apply RCB to projected data
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If , it is combined with
o Doubly RCB is robust against error of Vo Columns in V should be as independent as possible
o More columns in V meanso Better multipath eliminationo Loss of DOF for interference suppression.
o Error of V causes error of SOI steering vector
Choice of Multipath Subspace
March 11, 2003 ASAP Workshop 2003 24
Numerical Examples
?M = 10 sensors, 40 snapshots
?Uniform linear array with half-wavelength spacing
?100 Monte-Carlo trials for average output SINR
March 11, 2003 ASAP Workshop 2003 25
Power Estimate vs. Angle
SOI (-10 deg, 30 dB)
Coherent multipath(9 deg, 27 dB)
Non-coherent signal(-40 deg, 40 dB)
Assume
CRCB
March 11, 2003 ASAP Workshop 2003 26
Output SINR vs.
SOI (-20 deg, 30 dB)
Coherent multipath(19 deg, 27 dB)
Non-coherent signal(-40 deg, 40 dB)
1-D null space: assume
2-D null space:
March 11, 2003 ASAP Workshop 2003 27
Summary?Our RCB robust against steering vector errors.
o Much more accurate SOI power estimateo Directly related to uncertainty of steering vectoro Belongs to (extended) class of diagonal loading approaches
?Much better resolution and interference rejection capability than data-independent beamformers.
? Computationally efficient.
? Can be made robust against coherent interferences (CRCB).
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THANK YOU!THANK YOU!
March 11, 2003 ASAP Workshop 2003 29
Array Calibration Errors
Array steering vector with calibration errors
where
Random amplitude errorRandom phase error
For small calibration errors
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