Transcript Document 7477741

Optimum Passive
Beamforming in Relation to
Active-Passive Data Fusion
Bryan A. Yocom
Final Project Report
EE381K-14 – MDDSP
The University of Texas at Austin
May 01, 2008
What is Data Fusion?
Combining information from multiple
sensors to better perform signal processing
Active-Passive Data Fusion:
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
Active Sonar – gives good range estimates
Passive Sonar – gives good bearing estimates
and information about spectral content
Image from http://www.atlantic.drdc-rddc.gc.ca/factsheets/22_UDF_e.shtml
Passive Beamforming
A form of spatial filtering
Narrowband delay-and-sum beamformer
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Planar wavefront, linear array
Suppose 2N+1 elements
Sampled array output: xn = a(θ)sn + vn
Steering vector: w(θ) = a(θ) (aka array pattern)
Beamformer output: yn = wH(θ)xn
Direction of arrival estimation: precision limited
by length of array
 e  jNkd sin 
  j ( N 1) kd sin 
e






 jkd sin
 e



w ( )  
1

 e jkd sin 





 j ( N 1) kd sin 
e

 e jNkd sin 


The Goal
Given that we have prior information about
the location of contact:

Design a passive sonar beamformer to
provide minimum error in direction of arrival
(DOA) estimation while additionally providing
a low entropy measurement (accurate and
precise)
How? Use the prior information.
Passive Beamforming
& Data Fusion
Assume a data fusion framework has collected prior
information about the state of a contact via

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Active sonar measurements
Previous passive sonar measurements
Prior information is represented in the form of a onedimensional continuous random variable, Φ, with
probability density function (PDF):
p( ) where   [0,  ]
The information provided by a passive horizontal line
array measurement can be represented in terms of a
likelihood function [Bell, et al, 2000]:

L( |  )  exp K (a( ) R a( ))
H
1
x
1

Bayesian Updates
Posterior PDF:
p ( |  ) 
L( |  ) p ( )

 L( |  ' ) p( ' )d '
0
Differential entropy:

H   p( ) log 10 ( p( )) d
0
Entropy improvement:
H ( )  H prior  H posterior( )

Expected entropy improvement:  H   p( )H ( )d
0
Expected error in DOA estimate:

    p( ) arg max p( | , )  d
0

Adaptive Beamforming
Most common form is Minimum Variance Distortionless Response
(MVDR) beamformer (aka Capon beamformer) [Capon, 1969]
Given cross-spectral matrix Rx
and replica vector a(θ)
Minimize
wHR
xw
subject to
wHa(θ)=1:
1
Rx 
K
K
x x
i 1
i
H
i
R x1a( )
w
a( ) H R x1a( )
Direction of arrival estimation: much more precise,
but sensitive to mismatch (especially at high SNR)
Rx is commonly “diagonally-loaded” to make MVDR more robust:
R x  R x  I
Sensitivity to mismatch
Mismatch of 2 degrees
[Li, et al, 2003]
With limited computational resources how
can we solve this problem?
Cued Beams [Yudichak, et al, 2007]
Steer (adaptive) beams more densely in areas of high
prior probability
Previously cued beams were steered within a certain
number of standard deviations from the mean of an
assumed Gaussian prior PDF
Improvements were seen, but a need still exists to fully
cover bearing and generalize to any type of prior PDF
Generalized Cued Beams
0.01
p()
Goal: generalize cued beams for any
type of prior pdf, i.e., non-gaussian
0.005
Given prior pdf, p(Φ), the cumulative
distribution function (CDF) is given by:
F ( )   p(t )dt ,

  0, 180
60
80
100
120
140
160
180
100
120
140
160
180
0.5
0
20
40
60
80
(F)
180

If it assumed that Φ(F) can be solved for
(which is always the case for a discrete
pdf) we can define the steered angle of
the nth beam according to:
 n 
n  

N

1


where n  0, 1, 2, ..., N - 1is the beam number
when there are N beams to be steered
40

By a change of variables, (switch the
abscissa and ordinate), we obtain:
 ( F )   p(t )dt
20
1
0
F
0

F()

0
90
0
0
0.2
0.4
0.6
F
0.8
1
Robust Capon Beamformer
[Li, et al, 2003]
Use a Robust Capon Beamformer (RCB) instead of the standard,
diagonally loaded, MVDR beamformer.
The RCB is essentially a more robust derivation of the MVDR
beamformer for cases when the look direction is not precisely known.
Assign an uncertainty set (matrix B) to the look direction:
min (Bu  a ) H R 1 (Bu  a )
u
B is an N x L matrix:
subject to u  1
B  a (0 )  a(1 ) a( 2 ) ... a( L )
Solution to the optimization problem is somewhat involved
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Uses Lagrange multiplier methodology
Eigendecomposition of (BHR-1B) – slightly more complex then MVDR
Find the root of a non-trivial equation (e.g. via the Newton-Rhapson method)
Robust Capon Beamformer (RCB)
Assign a different uncertainty set to each beam
based on its distance from the two adjacent
beams. Essentially, vary the beamwidth of each
beam.
Goal: Full azimuthal coverage.
Although finely spaced beams will not cover
every bearing, all directions will be covered by at
least one beam. If a contact is detected the data
fusion framework will trigger the cued beams to
be steered in that direction.
Cued Beams with RCB
p()
0.01
0.005
0
0
20
40
60
80
100
120
140
160
180

Prior probability
Maximum Response Axes
Wide beams in areas of low probability
Narrow beams in areas of high prior probability
Results – Entropy Improvement
ABF Surveillance Beams
ABF Cued Beams
1.5
<entropy improvement>
<entropy improvement>
1.5
1
0.5
0
40
1
0.5
0
40
30
-40
30
-20
20
Priori pdf width
10
(degrees in one standard deviation)
40
-40
20
0
20 Diagonal Loading
(dB rel. to noise)
Priori pdf width 10
(degrees in one standard deviation)
RCB Surveillance Beams
20
40
0
Diagonal Loading
(dB rel. to noise)
RCB Cued Beams
1.5
<entropy improvement>
1.5
<entropy improvement>
-20
1
0.5
0
40
1
0.5
0
40
30
-40
20
Priori pdf width 10
(degrees in one standard deviation) 40
-20
0
20 Diagonal Loading
(dB rel. to noise)
30
-40
-20
20
Priori pdf width 10
(degrees in one standard deviation) 40
0
20
Diagonal Loading
(dB rel. to noise)
Results – Expected DOA Error
ABF Surveillance Beams
ABF Cued Beams
3
<error in DOA estimate>
<error in DOA estimate>
3
2
1
0
40
30
2
1
0
40
30
-40
-20
20
Priori pdf width
10
(degrees in one standard deviation)
20
40
-40
0
Diagonal Loading
(dB rel. to noise)
Priori pdf width
10
(degrees in one standard deviation)
RCB Surveillance Beams
20
40
0
Diagonal Loading
(dB rel. to noise)
RCB Cued Beams
3
<error in DOA estimate>
3
<error in DOA estimate>
-20
20
2
1
0
40
30
-40
-20
20
Priori pdf width 10
(degrees in one standard deviation) 40
20
0
Diagonal Loading
(dB rel. to noise)
2
1
0
40
30
-40
-20
20
Priori pdf width
10
(degrees in one standard deviation)
20
40
0
Diagonal Loading
(dB rel. to noise)
Challenges
Different amounts of noise are present in each beam
of RCB because the beamwidths differ
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This needs to be accounted for by somehow weighting the
beams
Wider beams also lessen the ability for the beamformer to adapt
to interferers
γ term in likelihood function is SNR dependent

L( |  )  exp K (a( ) H R x1a( )) 1
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
The value of γ basically controls how much peaks in the
beamformer output are emphasized.
RCB seems to be especially sensitive to this term
With proper choice of beam weightings and γ RCB could
outperform ABF
Beamformers used in a Bayesian
Tracker (time permitting)
Questions?