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OPEN

THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014

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Degradation of Covariance Reconstruction-Based Robust Adaptive

BeamformersSSPD 2014

Samuel D. SomasundaramMaritime Mission Systems, Thales UK

Andreas JakobssonDepartment of Mathematical Statistics, Lund University,

Sweden

2 /2 /11


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Overview of Presentation

Adaptive beamforming background

Covariance matrix reconstruction

Results

Conclusions

3 /3 /11


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Background

Beamformer (spatial filter)– combines sensor outputs to steer a receive beam in a specified direction

nnn s nax 0

Array measurement model

QaaxxR HHnnE 00

20 HnnE nnQ

220 nsE

10 awH 0nHnw

MVDRQww

awH

H 2

020

SNR

Rwwxwnw Hn

Hn

H E minmin02

MPDR or Capon beamformer - does not require signal-free snapshots

1tosubjectˆmin awwQw HH

aQa

aQw

1

1

MVDR ˆ

ˆ

H

Qwwnwnw Hn

Hn

H E minmin02

1tosubjectˆmin awwRw HHaRa

aRw

1

1

MPDR ˆ

ˆ

H

Idea is to recover signal waveform nnH sxw

Hk

K

kkKxxR

1

SCM

1ˆ

4 /4 /11


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Background

MPDR sensitive to errors in steering vector model and R estimate

Pointing errors, calibration errors, multipath propagation

2

2,1||1 aaawaw HH

0aa

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

Motivated diagonally loaded beamformers

Include worst-case optimisation, robust Capon beamformer

IRR DLSCMDL ˆˆ

5 /5 /11


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Covariance matrix reconstruction

0 1800

Integrate spatial response over some angular region

SOI Region

Noise-plus-interference region

Reconstruct forms NPI covariance using Capon estimator

)()(ˆ)()()()(ˆCaponCapon

θAθPθAaaQ HH dP

θVector of angles sampling SOI region

Vector of angles sampling NPI regionθ

dP H )()()(ˆˆRegion

aaC

aQa

aQw

1

1

MVDR ˆ

ˆ

H

IAA can be viewed as reconstructing data covariance θθθ dP H

IH )()()(ˆ)()(ˆ)(ˆ

AAIAA aaθAθPθAR

aRa

aRw

1

1

MPDR ˆ

ˆ

H

6 /6 /11


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Algorithms Evaluated

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

Sample covariance based estimators MPDR-SCM and RCB-SCM

Recon-Est - MVDR-Q-Capon with additional robustness to SOI steering vector error

Reconstruct Q using IAA estimator and insert into MVDR equation-> MVDR-Q-IAA

Reconstruct R using Capon estimator and insert into MPDR equation -> MPDR-R-Capon

7 /7 /11


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Results – No steering vector errors

20 element ULA, K = 60 snapshots, 4 sources embedded in white Gaussian noiseSOI is source nominally at 900

Covariance matrix reconstruction works well when there are no steering vector errors

8 /8 /11


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Results – AOA Error Only

SOI now at 90-1.220

Reconstruction based on Capon estimator degrades significantlyReconstruction based on IAA estimator better

Intf AOA Error Only

SOI + Intf AOA Errors

9 /9 /11


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Results – Arbitrary Errors

Intf Arbitrary Error Only

All covariance matrix reconstruction highly sensitive to arbitrary steering vector errors

10 /10 /11


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Conclusions

Covariance matrix reconstruction based approaches highly sensitive to the structure of the noise-plus-interference

Previous results had not shown this sensitivity

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

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.

In many realistic scenarios, diagonally loaded SCM based adaptive beamforming preferable to covariance matrix reconstruction

11 /11 /11


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Thank you for your time

Any questions?

12 /12 /11


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Adaptive beamforming Theory – Frequency Domain

S0

1 2 M

S0(t-t1) S0(t-t2) S0(t-tM)

FFT

S0(w)exp(-jwt1)

S0(w)exp(-jwt2) S0(w)exp(-jwtM)

)exp(

.

.

.

)exp(

)exp(

)(),()(

2

1

00

Mj

j

j

SS

wt

wtwt

www aSignal of interest can be written as

)(),()()( 0 wwww nax SFrequency-domain measurement can be written as

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