Robust Adaptive Beamforming - University of California, …dsp.ucsd.edu/~yuzhe/My Talks/Robust...
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Robust Adaptive Beamforming with application to Matched Field Processing Yong Han Goh & Y. Jin
Transcript of Robust Adaptive Beamforming - University of California, …dsp.ucsd.edu/~yuzhe/My Talks/Robust...
Robust Adaptive Beamformingwith application to Matched Field Processing
Yong Han Goh & Y. Jin
Outline
Background4 Robust Adaptive Beamforming Methods
LCMPWhite Noise Gain Constraint Gershman et al.Stoica et al.
SimulationsIntro to Matched Field Processing
MFP simulationsMFP results on the actual data
Conclusion
To Start with…
Shortcoming:Does not provide sufficient robustness against mismatch between presumed and actual signal steering vectorTends to suppress the SOI by adaptive nulling
0 50 100 150-40
-30
-20
-10
0
10
θ -space
Bea
mpa
ttern
B( θ)
in d
BMPDR
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-20
-10
0
10
θ -space
Bea
mpa
ttern
B( θ)
in d
BMPDR
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0
10
θ -space
Bea
mpa
ttern
B( θ)
in d
BMPDR
Effect of mismatch in beamscan
No mismatch case
Mismatch case
15-elt ULANL = 50 dBSNRin = 63 dB
SourceSL = 140 dBR = 7 km
Steering Vector Mismatch Due to…
look direction mismatcharray perturbationarray manifold mismodelingwavefront distortionssource local scattering…
Approaches to Robust Adaptive BeamformingLinear Constraint Minimum Power Beamformer
Directional Constraints
0.891 0.891u uN N
⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦C 1 v v
H w =C g
111
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
g
70 80 90 100 110-40
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0
10
θ -space
Bea
mpa
ttern
B( θ)
in d
B
MPDR
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0
10
θ -space
Bea
mpa
ttern
B( θ)
in d
B
LCMP
White Noise Gain Constraint
Array Gain:
Signal perturbed by uncorrelated random errors:
Sensitivity:
Sensitivity increases as white noise gain decreasesUWLA: low sensitivity; MVDR: high sensitivity
2
21
H
Hn
w vG
w S w w= ⎯⎯⎯⎯→distortionless
white noise
2 2Hs s s n nR v v Sσ σ= +
2/ 1distortionlessdG dS wG G
ξ⎛ ⎞= ⎯⎯⎯⎯⎯→ =⎜ ⎟⎝ ⎠
( )2 2Hs s s n nR v v I Sσ ξ σ= + +
White Noise Gain ConstraintGoal: reduce sensitivityImpose quadratic constraint on white noise gain to increase robustness
Solution is
Adjust ε until white noise gain constraint is satisfied
No simple relation
1
H 1
( ) ( )WNGCR I dw
d R I dεε
−
−
+=
+
Hw Rw w dδ
= ≤ 2w
1min subject to 1, and .H S Hw Rw w d w δ−= ≥2 2
wmin subject to 1, and .H
Shortcoming of WNGC
Relationship between ε and δ2 is not simple
Need approach to compute the ε based on the uncertainty of the steering vector
Iterative procedure is required to adjust the diagonal loading factor
In practice, can use ad hoc method to determine it
Looking for methods that…
Have sufficient robustness against arbitrary steering vector mismatch
Have sound mathematical framework
Are computationally easy to implement
One approach:2003: Sergiy A. Vorobyov, Alex B. Gershman, and Zhi-Quan Luo
Gershman et al. :Worst-Case Performance Optimization
Steering vector distortion: ∆
The actual steering vector belongs to
εΔ ≤
( ) { }| ,A ε ε= + ≤c c v e e
presumed s.v.
Any vector satisfies constraintSet of vectors
Worst-Case optimizationFor all vector in A(ε), the array response should NOT be smaller than 1
( ) { }| ,A ε ε= + ≤c c v e e
v
c2
c1
• Uncertainty Set• Search space
1 ( )Hw A ε≥ ∈c c for all
Formulation of Robust BF
Looks good, butNon-linearNon-convexInfinite # of constraints
1 ( )H H
www R Aw ε≥ ∈c cmin for al subject to l
1HH
ww wRw wε− ≥vmin subject to
1 { } 0H
w
H Hw w ww Rw ε≥ + =v vmin subject Im to &
( ) 120 2 1 1( )Hw R I
R Iλ λε
λ λε−
− −= ++
vv v
Another Method: Stoica et al. 2003:Directly Estimate Signal Power σs
2
Using w0HRw0 as σs
2 estimate, MPDR gives
It can be shown that it’s the solution to
2 2H Hs s s i i i nR Sσ σ= + +∑v v v v
2 1s H
s sRσ =
v v
2
2 2 0Hs sR
σσ σ− ≥v vmax subject to
Covariance fitting problem
Adding in uncertainty
Given an uncertainty “ellipsoid”
Estimate the power of SOI by
2
2 2 0Hs sR
σσ σ− ≥v vmax subject to
( ) ( )1 1p p
HC−− − ≤v vvv
( ) ( )2
2 2
,
1
0
& 1
H
H
p p
R
C
σσ σ
−
− ≥
− − ≤
vvv
v v v v
max subject to
By derivation, we get
Assume C = εI, we have
( ) ( )1 1 1HH
p pR C− −− − ≤v
v v v v v vmin subject to
21 HpR ε− − ≤
vv v v vmin subject to
11
0 pR Iλ
−−⎛ ⎞= +⎜ ⎟⎝ ⎠
v vSolution:
Direct estimation of the actual steering vector
10
0 10 0
H
RwR
−
−=v
v v
( ) ( )2
2 2
,
1
0
& 1
H
H
p p
R
C
σσ σ
−
− ≥
− − ≤
vvv
v v v v
m ax subject to
Relationship Between Stoica’s and Gershman’s method
It can be shown that,
21 HpR ε− − ≤
vv v v vmin subject to
10
0 10 0
H
RwR
−
−=v
v vLet v0 denote the optimal solution, and
Then w0 is the optimal solution to
1 { } 0H H H
ww Rw w w wε≥ + =v vmin subject to & Im
StociaG
ershm
an
Simulation 1: Beampatterns of LCMP, WNGC, Stoica and Gershman
ε0=0.217, εSoitca=0.3, εGershman=√0.3
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MPDRLCMP
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Bea
mpa
ttern
B( θ)
in d
B
MPDRStoica
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B( θ)
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B
Compare the 4 methods using sample covariance matrix
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B( θ)
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B MPDRLCMP
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B( θ)
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B MPDRWNGC
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B( θ)
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B MPDRStoica
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B MPDRGershman
Comparison of algorithms in beamscanspace in presence of mismatch
MFP overview
……
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.
.
.
.
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.
.
.X
True source position (atrue)
Array
c(z)
csed(z), ρsed(z)
Test source positions for replica (a)
B(a) = wH(a)S(atrue)w(a)Peak of the output of the beamformer B(a) is at atrue
w depends on the beamformer used
SwellEx’ 96 experiment
source
VLA
60 m
21-elt94.125 to212.25 md = 5.63m
4 knots
Sound speed profile
Processing of received array dataVert.Array
ReceivedTime series
1
2
L
L-1
chunk k
chunk’s FFTs
49 Hz
64 Hz
snapshots
X00
.
.
.X01
.
.
.
(...)
Spectrogram of data
∑=
=K
k
Hkikii K
S1
XX1ˆ estimate, CSDM
Fi = 49, 64, 79, 94, 112, 130, 148, 166, 201, 235, 283, 338, 388 Hz
Nfft = 213
Lk = 213
Fs = 1500 HzKaiser Window
(β=7.85)
Simulation results at F = 148 HzSNRin = 10 dB, 21 element arraySource at r = 3000 m, z = 60 m
Simulation results at F = 148 Hz
WNGC = 0.5N
Simulation results at F = 148 Hzsource
Experimental Results at F = 49 HzSource at r = 3000 m and z = 60 m
Experimental Results at F = 49 Hz
WNGC = 0.5N
Experimental Results at F = 49 Hz
Results averaged over first 5 frequencies
Results averaged over first 5 frequencies
WNGC = 0.5N
Results averaged over first 5 frequencies
Results averaged over all 13 frequencies
Results averaged over all 13 frequencies
WNGC = 0.5N
Results averaged over all 13 frequencies
Conclusions
Investigated various robust ABF algorithmsMFP results improved as we average over frequencies (except for MPDR)MUSIC best localized the source for this particular set of MFP data
ReferencesH. Cox, R. M. Zeskind, M. M. Owen, “Robust Adaptive Beamforming,” lEEE Trans. Acoust., Speech, Signal Processing, vol. 35, pp. 1365-1376, Oct. 1987.J. Li, P. Stoica, Z. S. Wang, “On Robust Capon Beamforming and Diagonal Loading,” lEEE Trans. Signal Processing , vol. 51, pp. 1702-1715, Oct. 2003.S. A. Vorobyov, A. B. Gershman, and Z. Q. Luo, “Robust adaptive beamformingusing worst case performance optimization,” lEEE Trans. Signal Processing , vol. 51, pp. 313-323, Feb. 2003.A. B. Baggeroer, W. A. Kuperman, P. N. Mikhalevsky “An Overview of Matched Field Methods in Ocean Acoustics,” lEEE Journal Oceanic Engineering , vol. 18, pp. 401-424, Oct. 1993.H. L. V. Trees, Optimum Array Processing, John Wiley & Sons, Inc, 2002.F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics, Springer-Verlag New York, Inc, 2000.SwellEx-96 Experiment data (http://www.mpl.ucsd.edu/swellex96/)..H. Schmidt, A. B. Baggeroer, W. A. Kuperman, E. K. Scheer, “Environmentally tolerant beamforming for high-resolution matched field processing: Deterministic mismatch,” J. Acoust. Soc. Am. 88, pp. 1851-1862, 1990.
Normal mode representation of pressure field
The normal mode representation of the field p(r,z) at a range r and depth z from the source is given by
where ρ(zs) is the density at the source depth zs, kn is the mode propagation constant for mode n, and Un are normalized eigenvectors of the following eigenvalue problem,
. The eigenvectors Un are zero at z = 0, and satisfy the local boundary conditions atthe ocean bottom.
( )( )
( ) ( ) rik
n n
nsn
s
i
nek
zUzUziezrp ∑
−
=πρ
π
8,
4
( ) ( ) 0)( 222
2
=−+ zUkzKdz
Udnn
n
