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. Source Localization Framework
Observation model:
Noise
Sensor measurements
Array manifold matrix
Unknown “source density”
Cost functional (notional):
Data fidelity
Regularizing sparsity constraint
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-
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. l0 uniqueness conditions
Prefer the sparsest solution:
Let (i.e. we have L point sources)
When is ?
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 ?

9. l1 equivalence conditions
Consider the l1 problem:
Can we ever hope to get ?
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!

10. lp (p≤1) equivalence conditions
Consider the lp problem:
How about ?
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

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

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
Far-field
200 time samples
Uniform linear array with 8 sensors
DOAs: 65, 70
SNR = 10 dB

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. Multiple harmonics – high SNR
Beamforming
Underlying spectrum
Proposed

24. Multiband example – low SNR
Beamforming
MUSIC
Capon’s method (MVDR)
Proposed
Each signal covers some narrow-band -– signals are NOT pure harmonics.

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)