1. 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

2. 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
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3. Passive Beamforming A form of spatial filtering
Narrowband delay-and-sum beamformer
Planar wavefront, linear array
Suppose 2N+1 elements
Sampled array output: xn = a(θ)sn + vn
Steering vector: w(θ)
Beamformer output: yn = wH(θ)xn
Direction of arrival estimation: precision limited by length of array

4. 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 w*Rxw subject to w*a(θ)=1:
Direction of arrival estimation: much more precise, but very sensitive to mismatch

5. 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

6. 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

7. 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

8. Questions?