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

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Standard Capon Beamforming (SCB)

Signal power estimate

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Norm Constrained Capon Beamforming (NCCB)

Loading level determined by norm constraint.

Diagonal loading:

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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!

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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!

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Covariance Fitting

Same signal power estimate as SCB!

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Our Robust Capon Beamformer(RCB)

?Incorporate ellipsoidal uncertainty set into covariance fitting

is of full column rank.

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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)

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Scaling Ambiguityo Uncertainty in SOI steering vector cause

scaling ambiguity

and yield same

o Add constraint to eliminateambiguity

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Main Steps of Our RCB

o Step 1:

o Step 2: Obtain Lagrange multiplier

o Step 3:

o Step 4:

o Step 5:

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?Obtain weight vector based on

?Diagonal loading (spherical constraint)!

?Waveform estimate

Waveform Estimation

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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)

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Numerical Examples?M = 10 sensors

?Uniform linear array with half-wavelength spacing

?Array calibration error exists (independent complex Gaussian random variables added)

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Power Estimate vs. AngleTrue powers denoted by circles.

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Making NCCB Have Same Diagonal Loading Level As RCB

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NCCB and RCB Having Same Diagonal Loading Level

NCCB

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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

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Steps of CRCBo Following similar steps in RCB

o Concentrating out

with

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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

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Numerical Examples

?M = 10 sensors, 40 snapshots

?Uniform linear array with half-wavelength spacing

?100 Monte-Carlo trials for average output SINR

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Power Estimate vs. Angle

SOI (-10 deg, 30 dB)

Coherent multipath(9 deg, 27 dB)

Non-coherent signal(-40 deg, 40 dB)

Assume

CRCB

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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:

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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!

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Array Calibration Errors

Array steering vector with calibration errors

where

Random amplitude errorRandom phase error

For small calibration errors