Optimum Passive Beamforming in Relation to Active-Passive Data Fusion

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 Image from http://www.atlantic.drdc-rddc.gc.ca/factsheets/22_UDF_e.shtml

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 Rxand 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?