Optimum Passive Beamforming in Relation to Active-Passive Data Fusion

1. 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

2. 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/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) = a(θ) (aka array pattern) • Beamformer output: yn = wH(θ)xn • Direction of arrival estimation: precision limited by length of array

4. 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.

5. 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, Φ, 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]:

6. Bayesian Updates • Posterior PDF: • Differential entropy: • Entropy improvement: • Expected entropy improvement: • Expected error in DOA estimate:

7. 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 wHRxw subject to wHa(θ)=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:

8. Sensitivity to mismatch • With limited computational resources how can we solve this problem? Mismatch of 2 degrees [Li, et al, 2003]

9. 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

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

11. 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)

12. 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.

13. Cued Beams with RCB Prior probability Maximum Response Axes Wide beams in areas of low probability Narrow beams in areas of high prior probability

14. Results – Entropy Improvement

15. Results – Expected DOA Error

16. 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 • γ term in likelihood function is SNR dependent • The value of γ 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 γ RCB could outperform ABF

17. Beamformers used in a Bayesian Tracker (time permitting)

18. Questions?