Robust Capon Beamforming

Transcription

1 Robust Capon Beamforming Yi Jiang Petre Stoica Zhisong Wang Jian Li University of Florida Uppsala University University of Florida University of Florida March 11, 2003 ASAP Workshop

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

3 Standard Capon Beamforming (SCB) Signal power estimate March 11, 2003 ASAP Workshop

4 Norm Constrained Capon Beamforming (NCCB) Diagonal loading: Loading level determined by norm constraint. March 11, 2003 ASAP Workshop

5 Recent Robust Beamformers Directly Address Steering Vector Uncertainties! Based on original SCB formulation o Robust adaptive beamforming based on worst-case performance optimization [Vorobyov, Gershman, Luo, 2001] o Robust minimum variance beamforming [Lorenz, Boyd, 2001] March 11, 2003 ASAP Workshop

6 Our RCB Directly Address Steering Vector Uncertainties! Based on Covariance Fitting o Robust Capon Beamforming [Stoica, Wang, Li, 2002] o On Robust Capon Beamforming and Diagonal Loading [Li, Stoica, Wang, 2002] New features o Steering vector within an uncertainty set o Incorporate uncertainty set into formulation directly o Computationally most efficient o Conceptually simple o Scaling ambiguity eliminated March 11, 2003 ASAP Workshop

7 Covariance Fitting Same signal power estimate as SCB! March 11, 2003 ASAP Workshop

8 Our Robust Capon Beamformer (RCB) Incorporate ellipsoidal uncertainty set into covariance fitting is of full column rank. March 11, 2003 ASAP Workshop

9 Our RCB o Without loss of generality, consider spherical uncertainty set: o Solution at boundary of uncertainty set March 11, 2003 ASAP Workshop

10 Our RCB o Use Lagrange multiplier method o Obtain Lagrange multiplier by solving via Newton s method (monotonic polynomial -- computationally efficient) March 11, 2003 ASAP Workshop

11 Scaling Ambiguity o Uncertainty in SOI steering vector cause scaling ambiguity and yield same o Add constraint ambiguity to eliminate March 11, 2003 ASAP Workshop

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

13 Waveform Estimation Obtain weight vector based on Diagonal loading (spherical constraint)! Waveform estimate March 11, 2003 ASAP Workshop

14 Advantages of Our RCB Ambiguity elimination obvious for our RCB (not considered by others) Computation o 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 March 11, 2003 ASAP Workshop

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

16 Power Estimate vs. Angle True powers denoted by circles. March 11, 2003 ASAP Workshop

17 Making NCCB Have Same Diagonal Loading Level As RCB March 11, 2003 ASAP Workshop

18 NCCB and RCB Having Same Diagonal Loading Level NCCB March 11, 2003 ASAP Workshop

19 Coherent RCB (CRCB) 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 o o Coherent multipaths exist DOAs of multipaths known relative to DOA of SOI March 11, 2003 ASAP Workshop

20 CRCB Robust against coherent multipaths as well as steering vector errors. Steering vector: Steering vector of SOI Steering vectors of coherent multipaths o Covariance fitting March 11, 2003 ASAP Workshop

21 Steps of CRCB o Following similar steps in RCB o Concentrating out with March 11, 2003 ASAP Workshop

22 Insight of CRCB Let Project data to orthogonal subspace of V Apply RCB to projected data March 11, 2003 ASAP Workshop

23 Choice of Multipath Subspace o Error of V causes error of SOI steering vector If, it is combined with o More columns in V means o Better multipath elimination o Loss of DOF for interference suppression. o Doubly RCB is robust against error of V o Columns in V should be as independent as possible March 11, 2003 ASAP Workshop

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

25 Power Estimate vs. Angle CRCB SOI (-10 deg, 30 db) Coherent multipath (9 deg, 27 db) Non-coherent signal (-40 deg, 40 db) Assume March 11, 2003 ASAP Workshop

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

27 Summary Our RCB robust against steering vector errors. o Much more accurate SOI power estimate o Directly related to uncertainty of steering vector o 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). March 11, 2003 ASAP Workshop

28 THANK YOU! March 11, 2003 ASAP Workshop

29 Array Calibration Errors For small calibration errors Random amplitude error Random phase error Array steering vector with calibration errors where March 11, 2003 ASAP Workshop