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)