Optimum Passive Beamforming in Relation to Active-Passive Data Fusion

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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

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What is Data Fusion?

• Combining information from multiple sensors to
  better perform signal processing
• Active-Passive Data Fusion
• 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/fac
tsheets/22_UDF_e.shtml
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Passive Beamforming

• A form of spatial filtering
• Narrowband delay-and-sum beamformer
• Planar wavefront, linear array
• Suppose 2N1 elements
• Sampled array output xn a(?)sn vn
• Steering vector w(?) a(?) (aka array pattern)
• Beamformer output yn wH(?)xn
• Direction of arrival estimation precision
  limited by length of array

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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.

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Passive Beamforming Data Fusion

• Assume a data fusion framework has collected
  prior information about the state of a contact
  via
• Active sonar measurements
• Previous passive sonar measurements
• Prior information is represented in the form of a
  one-dimensional continuous random variable, F,
  with probability density function (PDF)
• The information provided by a passive horizontal
  line array measurement can be represented in
  terms of a likelihood function Bell, et al,
  2000

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Bayesian Updates

• Posterior PDF
• Differential entropy
• Entropy improvement
• Expected entropy improvement
• Expected error in DOA estimate

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Adaptive Beamforming

• Most common form is Minimum Variance
  Distortionless Response (MVDR) beamformer (aka
  Capon beamformer) Capon, 1969
• Given cross-spectral matrix Rxand replica vector
  a(?)
• Minimize wHRxw subject to wHa(?)1
• 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

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Sensitivity to mismatch
Li, et al, 2003
Mismatch of 2 degrees

• With limited computational resources how can we
  solve this problem?

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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

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Generalized Cued Beams

• Goal generalize cued beams for any type of prior
  pdf, i.e., non-gaussian
• Given prior pdf, p(F), the cumulative
  distribution function (CDF) is given by
• By a change of variables, (switch the abscissa
  and ordinate), we obtain
• If it assumed that F(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

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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
• B is an N x L matrix
• Solution to the optimization problem is somewhat
  involved
• 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)

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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.

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Cued Beams with RCB
Prior probability
Maximum Response Axes
Wide beams in areas of low probability
Narrow beams in areas of high prior probability
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Results Entropy Improvement
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Results Expected DOA Error
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Challenges

• Different amounts of noise are present in each
  beam of RCB because the beamwidths differ
• This needs to be accounted for by somehow
  weighting the beams
• Wider beams also lessen the ability for the
  beamformer to adapt to interferers
• ? term in likelihood function is SNR dependent
• The value of ? 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 ? RCB
  could outperform ABF

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Beamformers used in a Bayesian Tracker (time
permitting)
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Questions?