A Sparse Signal Reconstruction Perspective for Source Localization with Sensor Arrays
Müjdat ÇetinStochastic Systems Group, M.I.T.
SensorWeb MURI Review Meeting September 22, 2003
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
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
Source Localization Framework
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
Observation model:
Sensor measurementsArray manifold matrix Unknown “source density”
Noise
An example
Uniform linear array with 8 sensors Uncorrelated sources DOAs: 50, 60 SNR = 5 dB
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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
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
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 ?
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!
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 p0 we recover the l0 condition, namely
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 !
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)
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
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
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
Narrowband, uncorrelated sources – high SNR
DOAs: 65, 70 SNR = 10 dB
Far-field 200 time samples Uniform linear array with 8
sensors
Narrowband, uncorrelated sources – low SNR
DOAs: 65, 70 SNR = 0 dB
Far-field 200 time samples Uniform linear array with 8
sensors
Narrowband, correlated sources
DOAs: 63, 73 SNR = 20 dB
Far-field 200 time samples Uniform linear array with 8
sensors
Robustness to limitations in data quantity
Uncorrelated sources Uniform linear array with 8 sensors
DOAs: 43, 73 SNR = 20 dB
Single time-sample processing
Resolving many sources
7 uncorrelated sources Uniform linear array with 8 sensors
Estimator Variance and the CRB
Correlated sources Uniform linear array with 8 sensors DOAs: 43, 73 Each point on curve average of 50 trials
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
Multiple harmonics – high SNRBeamforming
Proposed
Underlying spectrum
Multiband example – low SNR Beamforming
Capon’s method (MVDR)
MUSIC
Proposed
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)
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)