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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
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Overview of Presentation
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Adaptive beamforming background
Covariance matrix reconstruction
Results
Conclusions
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Background
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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
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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
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x x
K
k 1
k
H
k
Background
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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
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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
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Covariance matrix reconstruction
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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
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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
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w MPDR
ˆ 1a
R
H 1
ˆ a
a R
Algorithms Evaluated
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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
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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
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Results – No steering vector errors
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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
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Results – AOA Error Only
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Intf AOA Error Only
SOI now at 90-1.220
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SOI + Intf AOA Errors
Reconstruction based on Capon estimator
degrades significantly
Reconstruction based on IAA estimator better
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Results – Arbitrary Errors
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Intf Arbitrary Error Only
All covariance matrix reconstruction highly sensitive to arbitrary steering vector errors
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Conclusions
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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.
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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
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Thank you for your time
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Any questions?
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Adaptive beamforming Theory – Frequency Domain
S0
1
2
M
S0(t-t1) S0(t-t2)
S0(t-tM)
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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
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exp( jwt1 )
exp( jwt )
2
.
S 0 (w )a( , w ) S 0 (w )
.
.
exp( jwt M )
x(w) S0 (w)a( , w) n(w)