A Sparse Signal Reconstruction Perspective for Source Localization with Sensor Arrays

Alan WillskyContributors: Dmitry Malioutov, Müjdat Çetin

SensorWeb MURI Review Meeting December 2, 2005

Source Localization Problem

Source localization based on passive sensor measurements

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

Contributions/Highlights Source localization framework using lp-norm-

based sparsity constraints Efficient techniques for synergistic use of all data

and efficient algorithms for numerical solution Theoretical results:

Justification based on properties of the data and human goals

Solution of combinatorial optimization problems by computationally feasible algorithms!

Extensive performance analysis on simulated data

Self-calibration for sensor location uncertainties Experimental results on ARL’s acoustic data

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

Data Processing and Optimization Algorithms

SVD-based approach for processing and using multiple time or frequency snapshots efficiently and synergistically

Two numerical optimization algorithms: One based on half-quadratic regularization One based on second-order cone

programming and interior point algorithms Fast multiresolution approach for iteratively

refining the search around likely source locations

Theoretical Results on lp regularization

Observations: Preferring the optimally sparse solution would involve l0-

norms But that requires solving combinatorial optimization

problems lp-norm-based techniques have been empirically

observed to yield solutions that look sparse Question: Can we ever get the optimally sparse

solution using lp–norms? Interestingly, the answer, as we have found out,

is YES! Provided that the actual spatial spectrum is sparse

enough This provides a rigorous characterization of the lp–

sparsity link As a result, we can solve a combinatorial optimization

problem by tractable algorithms!

Performance Analysis on Simulated Data

Narrow-band and wide-band signals Far-field and near-field sources Incoherent and coherent (due to multipath)

sources Linear, circular, cross, rectangular arrays Wide range of SNRs Wide range of the number of snapshots

We will show only a subset of this analysis

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

Multi-band example – low SNR

Capon’s method (MVDR)

MUSIC

Proposed

Underlying true spectrum

Extension to Self-calibration What if we don’t know the sensor locations

exactly? Extended our framework to include optimization

over sensor locations Setup for experiments:

Far-field case Narrowband signals Linear array with 15 sensors Two uncorrelated sources DOAs: 45, 75 SNR = 30 dB Sensor locations perturbed with a standard deviation of

1/3 of the nominal sensor spacing

Self-calibration Example

Moderate calibration errors can be compensated up to intrinsic ambiguities

Validation on real data provided by ARL

• Six acoustic sensor arrays in oval loop.

• Each has a circular array configuration with seven microphones.

• Tanks and trucks travel on oval loop or on nearby asphalt road.

ARL Field Experiment Setup

Sensor Configuration

Results on single-vehicle data

Results on multiple-vehicle data - I

Results on multiple-vehicle data - I

A temporal slice – (when the vehicles are closest)

Results on multiple-vehicle data – II (limited observations)

Results on multiple-vehicle data – III (limited bandwidth)

Results on multiple-vehicle data – IV (low SNR)

Summary Sparse signal reconstruction framework & algorithms for

source localization with passive sensor arrays Theoretical analysis justifying the formulation Extensive performance analysis

Superior source localization performance (Superresolution, Reduced artifacts)

Robustness to resource limitations(SNR, Observation time, Available aperture)

Self-calibration capability Fruitful interactions with ARL

Validation on ARL field data Adaptation to signal structures of interest to the Army

(including multiband harmonic sources) Collaboration with Dr. Brian Sadler & successful application

of this framework to estimation of sparse communication channels