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© THALES UK LTD AND/OR ITS SUPPLIERS. THIS INFORMATION CARRIER CONTAINS PROPRIETARY INFORMATION WHICH SHALL NOT BE USED, REPRODUCED OR DISCLOSED TO THIRD PARTIES WITHOUT PRIOR WRITTEN AUTHORIZATION BY THALES UK LTD AND/OR ITS SUPPLIERS, AS APPLICABLE. www.thalesgroup.com Degradation of Covariance ReconstructionBased Robust Adaptive Beamformers SSPD 2014 Samuel D. Somasundaram Maritime Mission Systems, Thales UK Andreas Jakobsson Department of Mathematical Statistics, Lund University, Sweden THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN Overview of Presentation 2 /11 / Adaptive beamforming background Covariance matrix reconstruction Results Conclusions © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN Background 3 /11 / Beamformer (spatial filter)– combines sensor outputs to steer a receive beam in a specified direction Array measurement model xn sna0 n n 2 R E xn xnH 02a0a0H Q Idea is to recover signal waveform w H xn sn w H a0 1 © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page MVDR w H nn 0 w n n 0 min E w n n H 02 E sn Q EnnnnH H 2 minw Q w H SNR w a0 2 0 H 2 wHQ w ˆ 1a Q H ˆ H w MVDR H 1 min w Qw subject to w a 1 ˆ a a Q MPDR or Capon beamformer - does not require signal-free snapshots ˆ 1a 2 R H H H w n n 0 min E w x n min w Rw w MPDR H 1 ˆ a a R H ˆ H min w Rw subject to w a 1 1 K ˆ R SCM THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN x x K k 1 k H k Background 4 /11 / MPDR sensitive to errors in steering vector model and R estimate a a0 Pointing errors, calibration errors, multipath propagation Motivated diagonally loaded beamformers Include worst-case optimisation, robust Capon beamformer ˆ R ˆ R DL SCM DL I w H a 1 | w H a | 1, a a 2 2 © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page More recently, covariance matrix reconstruction based approaches have been proposed Reconstruct, IAA Reconstruct Reconstructs Q and inserts into MVDR equation Rationale is that MVDR is less sensitive to SOI steering vector errors IAA Can be interpreted as reconstructing R and inserting into MPDR equation THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN Covariance matrix reconstruction 5 /11 / Vector of angles sampling SOI region SOI Region θ 0 Noise-plus-interference region θ Vector of angles sampling NPI region Integrate spatial response over some angular © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page 1800 ˆ C region ˆ ( )a( )a H ( )d P Region Reconstruct forms NPI covariance using Capon estimator H H ˆ P Q Capon ( )a( )a ( )d A(θ )Pˆ Capon (θ )A (θ ) w MVDR ˆ 1a Q H 1 ˆ a a Q IAA can be viewed as reconstructing data covariance θ θ θ ˆ A (θ)Pˆ (θ) A (θ) R IAA H Pˆ IAA ( )a( )a ( )d H THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN w MPDR ˆ 1a R H 1 ˆ a a R Algorithms Evaluated 6 /11 / Reconstruct Q using Capon estimator and insert into MVDR equation-> MVDR-Q-Capon Reconstruct R using IAA estimator and insert into MPDR equation -> MPDR-R-IAA, IAA Reconstruct Q using IAA estimator and insert into MVDR equation> MVDR-Q-IAA Reconstruct R using Capon estimator and insert into MPDR © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page equation -> MPDR-R-Capon Recon-Est - MVDR-Q-Capon with additional robustness to SOI steering vector error Sample covariance based estimators MPDR-SCM and RCB-SCM THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN Results – No steering vector errors 7 /11 / © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page 20 element ULA, K = 60 snapshots, 4 sources embedded in white Gaussian noise SOI is source nominally at 900 Covariance matrix reconstruction works well when there are no steering vector errors THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN Results – AOA Error Only 8 /11 / Intf AOA Error Only SOI now at 90-1.220 © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page SOI + Intf AOA Errors Reconstruction based on Capon estimator degrades significantly Reconstruction based on IAA estimator better THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN Results – Arbitrary Errors 9 /11 / © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page Intf Arbitrary Error Only All covariance matrix reconstruction highly sensitive to arbitrary steering vector errors THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN Conclusions 10 /11 / Covariance matrix reconstruction based approaches highly sensitive to the structure of the noise-plus-interference Previous results had not shown this sensitivity Noise plus-interference can take many forms and we often don’t really know its structure Interference not necessarily point sources, could be near-field, platform etc. © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page SCM-based approaches insensitive to noise and interference structure MPDR sensitive to SOI steering vector errors Diagonal loading (e.g, in RCB) fixes the sensitivity to SOI steering vector errors In many realistic scenarios, diagonally loaded SCM based adaptive beamforming preferable to covariance matrix reconstruction THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN 11 /11 / Thank you for your time © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page Any questions? THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN 12 /11 / Adaptive beamforming Theory – Frequency Domain S0 1 2 M S0(t-t1) S0(t-t2) S0(t-tM) © THALES UK LTD. AND/OR ITS SUPPLIERS Subject to restrictive legend on title page FFT S0(w)exp(-jwtM) S0(w)exp(-jwt2) S0(w)exp(-jwt1) Signal of interest can be written as Frequency-domain measurement can be written as THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014 OPEN exp( jwt1 ) exp( jwt ) 2 . S 0 (w )a( , w ) S 0 (w ) . . exp( jwt M ) x(w) S0 (w)a( , w) n(w)