# 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  
                      
Most common form is Minimum Variance Distortionless
Response (MVDR) beamformer (aka Capon beamformer)
[Capon, 1969]

K
1
Given cross-spectral matrix Rx        Rx 
K
x x   i
H
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( )
x

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
(ABF)

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

Improvements:
   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
Questions?

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