Optimum Passive Beamforming in Relation to Active-Passive Data

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					     Optimum Passive
Beamforming in Relation to
Active-Passive Data Fusion
              Bryan A. Yocom
         Literature Survey Report
          EE381K-14 – MDDSP
     The University of Texas at Austin
              March 04, 2008
       What is Data Fusion?
Combining information from multiple
sensors to better perform signal processing
Active-Passive Data Fusion:
   Active Sonar – good range estimates
   Passive Sonar – good bearing estimates

               Image from http://www.atlantic.drdc-rddc.gc.ca/factsheets/22_UDF_e.shtml
         Passive Beamforming
A form of spatial filtering                                        e  jNkd sin  
Narrowband delay-and-sum beamformer                                 j ( N 1) kd sin  
   Planar wavefront, linear array                                e                     
                                                                                       
   Suppose 2N+1 elements                                                               
                                                                          jkd sin 
   Sampled array output: xn = a(θ)sn + vn                         e                    
                                                                                        
   Steering vector: w(θ)                                w ( )           1            
    Beamformer output: yn = wH(θ)xn
                                                                   e jkd sin  
   Direction of arrival estimation: precision limited                                  
    by length of array                                                                 
                                                                   j ( N 1) kd sin  
                                                                  e                     
                                                                   e jNkd sin  
                                                                                        
       Adaptive Beamforming
Most common form is Minimum Variance Distortionless
Response (MVDR) beamformer (aka Capon beamformer)
[Capon, 1969]

Given cross-spectral matrix Rx        Rx 
                                               x x   i
and replica vector a(θ)                        i 1

Minimize w*Rxw subject to w*a(θ)=1:         R 1a( )
                                      w        x
                                         a( ) H R 1a( )

Direction of arrival estimation: much more precise,
but very sensitive to mismatch
       Cued Beams [Yudichak, et al, 2007]
Need to account for sensitivity of adaptive beamforming

Steer (adaptive) beams more densely in areas where the
prior probability density function (PDF) is large
   Cued beams are steered within a certain number of standard
    deviations from the mean of a Gaussian prior PDF

Use the beamformer output as a likelihood function

Use Bayes’ rule to generate a posterior PDF

   Need to fully cover bearing
   The use of the beamformer output as a likelihood function is ad hoc
Bayesian Beamformer [Bell, et al, 2000]
 Also assumes a priori PDF

 Beamformer is a linear combination of adaptive MVDR
 beamformers weighted by the posterior probability density
 function, p(θ|X)

 Computationally efficient, O(MVDR)

 The likelihood function they derive assumes Gaussian
 random processes and is therefore less ad hoc then using
 the beamformer output

 Difficult to extend their likelihood function to other classes
 of beamformers
  Robust Capon Beamformer
                    [Li, et al, 2003]

A natural extension of the Capon beamformer

Directly addresses steering vector uncertainty by
assuming an ellipsoidal uncertainty set:

minimize a*R-1a subject to (a-a0)*C-1 (a-a0) ≤ 1

Computationally efficient, O(MVDR)

When used with cued beams its use could guarantee
that bearing is fully covered

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