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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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Overview of Presentation
Adaptive beamforming background
Covariance matrix reconstruction
Results
Conclusions
3 /3 /11
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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Thank you for your time
Any questions?
12 /12 /11
THALES UK Ltd. SSPD 2014, Edinburgh, Sep. 2014
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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