A Sparse Signal Reconstruction Perspective for Source Localization with Sensor Arrays

1. A Sparse Signal Reconstruction Perspective for Source Localization with Sensor Arrays Müjdat Çetin Stochastic Systems Group, M.I.T. SensorWeb MURI Review Meeting September 22, 2003

2. Problem setup • Source localization based on passive sensor measurements • Context: Acoustic sensors, narrowband/wideband signals, sources in far-field/near-field, any array configuration • Issues: • Resolution • Robustness to noise • Limited observation time • Multipath, correlated sources • Model uncertainties • Our approach: View the problem as one of imaging a “source density” over the field of regard • Ill-posed inverse problem (overcomplete basis representation) • Favor sparse fields with concentrated densities

3. What we presented last year • Source localization framework using lp-norm-based sparsity constraints (far and near-field) • Special Quasi-Newton method for numerical solution • Preliminary experimental performance analysis • Joint source localization and self-calibration for moderate sensor location uncertainties

4. Data fidelity Regularizing sparsity constraint Source Localization Framework • Cost functional (notional): • Observation model: Noise Sensor measurements Array manifold matrix Unknown “source density” • Role of the regularizing constraint : • Preservation of strong features (source densities) • Preference of sparse source density field • Can resolve closely-spaced radiating sources

5. - - - - - - - An example • Uniform linear array with 8 sensors • Uncorrelated sources • DOAs: 50, 60 • SNR = 5 dB

6. Progress since then • Theoretical analysis of lp regularization • SVD-based approach for combining and summarizing multiple data samples • Optimization based on Second Order Cone (SOC) Programming • Adaptive grid refinement • Detailed performance analysis • Automatic parameter choice • Improved self-calibration procedure • Interactions • ARL: Brian Sadler, Ananthram Swami • Ohio State: Randy Moses

7. Theoretical analysis – setup • Basic problem: find an estimate of , where • Underdetermined -- non-uniqueness of solutions • Regularize by preferring sparse estimates • When does lp regularization yield the right solution? • Theoretical result: We can obtain the right solution by lp regularization if the actual spatial spectrum is sparse enough • Significance: • Conditions for performance guarantees and limits • Conditions for tractable solution of a combinatorial problem • Insights into the choice of regularizing constraints

8. Number of non-zero elements in s • Definition: A is called rank-K unambiguous if any set of K columns is linearly independent, but this is not true for K+1. • Assume A is rank-K unambiguous (for some K). • Thm. 1: • Small number of sources  exact solution by l0 optimization • K ≈ number of sensors • This is a hard combinatorial optimization problem. • What can we say about more tractable formulations like l1 ? l0 uniqueness conditions • Prefer the sparsest solution: • Let (i.e. we have L point sources) • When is ?

9. Definition: Maximum absolute dot product of columns • Thm. 2: • Small number of sources  exact solution by l1 optimization • More restrictive than the l0 condition • Can solve a combinatorial optimization problem by linear programming! l1 equivalence conditions • Consider the l1 problem: • Can we ever hope to get ?

10. Definition: • Thm. 3: • Small number of sources  exact solution by lp optimization • Less restrictive conditions for smaller p! • Smaller p  more sources can be resolved • As p0 we recover the l0 condition, namely lp(p≤1) equivalence conditions • Consider the lp problem: • How about ?

11. Thm. 4: (assuming rank(Y)=L) • Improves upon the single-sample l0 condition • Implication for array processing:guarantee for exact solution if # sources < # sensors ! Multi-sample l0 condition • Multiple snapshots: • Consider the l0 problem: • When is ?

12. Dealing with multiple snapshots • How to process multiple time samples efficiently and synergistically? • Similar problem of multiple frequency snapshots • View data as cloud of T points in a Q-dimensional subspace • Take the SVD of the data matrix • Summarize data using Q largest singular vectors • Best performance when Q is the number of sources (no catastrophic consequences in the case of other choices)

13. SVD-based formulation • Represent data by the largest Q singular vectors • This leads to: • Finally, we obtain the cost functional: • Natural and effective way of summarizing information contained in multiple data samples

14. Optimization by SOCP (for p=1) • Express the optimization problem as a second order cone program: • Solve by an efficient interior point algorithm Linear cost in auxiliary variables Quadratic, linear, and SOC constraints

15. Adaptive Grid Refinement • Goal: alleviate the effects of the grid, with reasonable computation • Find initial location estimates on a coarse grid • Make the grid finer around previous estimates and obtain source locations on the new grid • Iterate to required precision

16. Narrowband, uncorrelated sources – high SNR • DOAs: 65, 70 • SNR = 10 dB • Far-field • 200 time samples • Uniform linear array with 8 sensors

17. Narrowband, uncorrelated sources – low SNR • Far-field • 200 time samples • Uniform linear array with 8 sensors • DOAs: 65, 70 • SNR = 0 dB

18. Narrowband, correlated sources • Far-field • 200 time samples • Uniform linear array with 8 sensors • DOAs: 63, 73 • SNR = 20 dB

19. Robustness to limitations in data quantity Single time-sample processing • Uncorrelated sources • Uniform linear array with 8 sensors • DOAs: 43, 73 • SNR = 20 dB

20. Resolving many sources • 7 uncorrelated sources • Uniform linear array with 8 sensors

21. Estimator Variance and the CRB • Correlated sources • Uniform linear array with 8 sensors • DOAs: 43, 73 • Each point on curve average of 50 trials

22. Multiband/wideband case • Formulate the source localization problem in the frequency domain • Two options: • Independent processing at each frequency • Joint, coherent processing of all data • Current wideband methods are mostly incoherent • Our framework allows seamless coherent processing • Uses all data in synergy • Allows the incorporation of prior information about the temporal spectrum

23. Beamforming Proposed Multiple harmonics – high SNR Underlying spectrum

24. Beamforming MUSIC Capon’s method (MVDR) Proposed Multiband example – low SNR

25. Summary • Regularization-based framework for source localization with passive sensor arrays • Superior source localization performance • Superresolution • Reduced artifacts • Robustness to resource limitations • SNR • Observation time • Available aperture • Ability to handle correlated signals e.g. due to multipath effects • Adaptation to signal structures of interest to the Army (including multiband harmonic sources)

26. Where to from here? • Spatially-distributed sources • Extension through the use of a different overcomplete basis • Dynamic environment, mobile sources • Promising approach due to its robustness to limitations in observation time • Incorporating prior information on more complicated wideband spectra • Cyclostationary signals • Experiments with measured data (possibly through ARL)