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Multiple and single snapshot compressive beamforming

Peter Gerstoft, Angeliki Xenaki, Christoph F. Mecklenbräuker

arXiv:1503.02339v2math.STcs.IT

TL;DR

The paper addresses sparse DOA estimation for single and multiple array snapshots, including coherent arrivals and underdetermined angular grids. It formulates CS reconstruction as ℓ1-regularized least squares with a MAP interpretation, then evaluates resolution against conventional methods. CS gives higher resolution than CBF for single snapshots and than MVDR/MUSIC for multiple snapshots, while resolving coherent multipath arrivals in real data.

  • Problem

    DOA estimation requires localizing multiple sources from noisy array measurements in an underdetermined problem, with limited prior performance evaluation for multiple snapshots.

  • Method

    The paper uses LASSO-based ℓ1-regularized least squares to obtain MAP sparse source estimates for single and multiple snapshots.

  • Results

    CS provides higher resolution than CBF for single snapshots and than MVDR/MUSIC for multiple snapshots, and resolves coherent multipath arrivals in real data.

  • Takeaways & Limitations

    CS supplies high-resolution acoustic imaging for both single- and multiple-snapshot data, including coherent wave arrivals.

Abstract

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For a sound field observed on a sensor array, compressive sensing (CS) reconstructs the direction-of-arrival (DOA) of multiple sources using a sparsity constraint. The DOA estimation is posed as an underdetermined problem by expressing the acoustic pressure at each sensor as a phase-lagged superposition of source amplitudes at all hypothetical DOAs. Regularizing with an $\ell_1$-norm constraint renders the problem solvable with convex optimization, and promoting sparsity gives high-resolution DOA maps. Here, the sparse source distribution is derived using maximum a posteriori (MAP) estimates for both single and multiple snapshots. CS does not require inversion of the data covariance matrix and thus works well even for a single snapshot where it gives higher resolution than conventional beamforming. For multiple snapshots, CS outperforms conventional high-resolution methods, even with coherent arrivals and at low signal-to-noise ratio. The superior resolution of CS is demonstrated with vertical array data from the SWellEx96 experiment for coherent multi-paths.

I. INTRODUCTION

The paper frames DOA estimation as a sparse underdetermined reconstruction problem and evaluates compressive sensing for single and multiple snapshots. CS is presented as resolving coherent arrivals and achieving higher resolution than conventional and high-resolution beamformers.

  • I. INTRODUCTION: DOA estimation localizes multiple sources from noisy array measurements by solving an underdetermined linear problem with a sparsity constraint.The sensing model represents sensor measurements using hypothetical source amplitudes across discretized directions.
  • I. INTRODUCTION: CS uses convex optimization to promote sparse solutions and produce high-resolution acoustic imaging.The approach is contrasted with traditional and subspace-based estimators, particularly for single-snapshot data.
  • I. INTRODUCTION: Multiple-snapshot CS promotes spatial sparsity by combining snapshots with an ℓ2-norm and spatial samples with an ℓ1-norm.The formulation is designed to handle partially coherent arrivals and arbitrary numbers of snapshots.
  • I. INTRODUCTION: CS achieves higher resolution than MUSIC and MVDR, including scenarios that favor these classical high-resolution methods.The paper emphasizes this comparison as a central performance result.
  • I. INTRODUCTION: Coherent arrivals arise from multipath propagation, which high-resolution beamformers cannot resolve in the described ocean-acoustic setting.The paper motivates CS partly through coherent multipath arrivals produced by refraction, diffraction, scattering, ducting, and reflection.
  • I. INTRODUCTION: The study evaluates single- and multiple-snapshot performance using simulated and real data, addressing a performance-evaluation gap for multiple snapshots.The real-data validation uses a complex multipath shallow-water environment and compares CS with CBF and MVDR.
  • I. INTRODUCTION: The formulation assumes plane-wave propagation, narrowband processing, known sound speed, and a one-dimensional uniform linear array.The source direction is represented by DOA over the angular interval −90° to 90°.

A. Sparse reconstruction with compressive sensing

The compressive-sensing formulation recovers sparse DOA supports by replacing nonconvex ℓ0 minimization with convex ℓ1-regularized least squares. Its regularization parameter controls the trade-off between data fit and sparsity, while conventional ℓ2 regularization produces nonsparse solutions.

  • A. Sparse reconstruction with compressive sensing: DOA recovery seeks the nonzero entries of a sparse source vector from an underdetermined sensing model with M ≪ N.The fine angular grid creates more candidate directions than sensors, making prior sparsity information necessary.
  • A. Sparse reconstruction with compressive sensing: Traditional ℓ2-regularized least squares minimizes source energy rather than sparsity, yielding a non-sparse solution.The regularization parameter controls the relative importance of data fit and the ℓ2 norm.
  • A. Sparse reconstruction with compressive sensing: Conventional beamforming is associated with the ℓ2 solution and is robust to noise but has low resolution and sidelobe artifacts.This motivates comparing sparse CS reconstructions with conventional beamforming.
  • A. Sparse reconstruction with compressive sensing: The ℓ0 formulation directly seeks few nonzero components but is nonconvex and computationally intractable for moderate dimensions.Compressive sensing uses ℓ1 minimization as a tractable sparse-reconstruction alternative under suitable sparsity and sensing conditions.
  • A. Sparse reconstruction with compressive sensing: The LASSO path tracks how the active set changes with µ, with new active indices appearing at path transition values.After active DOAs are identified, the corresponding source amplitudes are estimated using the active steering vectors.
  • A. Sparse reconstruction with compressive sensing: LASSO minimizes a least-squares data-fit term regularized by the ℓ1 norm, shrinking source coefficients toward zero as µ increases.For each sparsity level, an appropriate µ balances fit quality against the desired sparsity.

B. MAP estimate via LASSO

The paper interprets LASSO-based sparse DOA reconstruction probabilistically as a MAP estimate. A Gaussian likelihood models measurement noise, while a Laplacian-like prior on source coefficients promotes sparsity.

  • B. MAP estimate via LASSO: The unconstrained LASSO formulation enables a Bayesian interpretation by imposing a sparsity-promoting prior on the source vector.The statistical interpretation treats both unknowns and observations as stochastic processes.
  • B. MAP estimate via LASSO: Bayes’ theorem combines the posterior, data likelihood, prior distribution, and marginal data distribution to define the MAP estimate.The marginal distribution of the data is omitted from MAP optimization because it is independent of the model x.
  • B. MAP estimate via LASSO: An iid complex Gaussian noise model gives a complex Gaussian likelihood centered at Ax with covariance σ^2I.This likelihood connects the measurement-fit term to the assumed observation model.
  • B. MAP estimate via LASSO: A Laplacian-like iid prior on the complex source coefficients encourages sparsity through heavy tails and a sharp peak at zero.The prior supplies the probabilistic basis for the ℓ1 penalty.
  • B. MAP estimate via LASSO: The LASSO estimate is therefore interpreted as the MAP estimate under the stated Gaussian likelihood and Laplacian-like prior.The formulation imposes no restriction on source phases, which are assumed uniformly distributed on [0, 2π).

III. MULTIPLE-SNAPSHOT DOA ESTIMATION

The multiple-snapshot formulation aggregates stationary sensor measurements and imposes row sparsity across source amplitudes. Its direct use of measured pressure avoids covariance-matrix limitations affecting methods such as MVDR and MUSIC.

  • Multiple-snapshot model: Stationary multiple snapshots are collected into measurement and noise matrices, while source amplitudes form a matrix with columns indexed by snapshots.The source matrix exhibits row sparsity because stationary sources retain the same DOAs across snapshots.
  • Sparse recovery: The formulation sums source energy across snapshots and applies a Laplacian-like prior to promote sparse source locations.Source amplitudes may vary across snapshots even though their spatial support remains fixed.
  • Comparison with covariance methods: MVDR requires at least M snapshots because its inverse sample covariance matrix must be full rank.This requirement follows from the covariance-matrix inversion in the MVDR weights.
  • Comparison with covariance methods: MVDR and MUSIC degrade with few snapshots or coherent sources, whereas CS directly uses the measured pressure Y.Their performance depends on eigenvalues and signal-subspace structure in the sample covariance matrix.

IV. REGULARIZATION PARAMETER SELECTION

The LASSO regularization parameter controls the trade-off between sparsity and data fit, so selecting it determines reconstruction quality. The LASSO path provides a way to identify an adequate value.

  • Regularization trade-off: The regularization parameter µ controls the balance between solution sparsity and data fit, determining reconstruction accuracy.It is also called the LASSO shrinkage parameter.
  • Regularization trade-off: Large µ produces a very sparse solution with poor data fit, whereas decreasing µ improves fit while reducing sparsity.At µ = 0, the formulation becomes the unconstrained least-squares solution.
  • Parameter selection: The LASSO path can be used to select µ from the evolution of the solution as the regularization parameter changes.This approach relies on properties of the path rather than source or noise statistics.

A. The LASSO path

The LASSO path tracks the solution continuously as µ decreases and uses sparsity changes to guide regularization selection. Convexity and subgradient conditions characterize when components become active.

  • The LASSO path: As µ evolves from infinity to zero, the LASSO solution follows a continuous, piecewise-smooth path.The path describes how the solution changes across regularization values.
  • The LASSO path: Singularity points in the LASSO path coincide with changes in solution sparsity and indicate an adequate choice for µ.These points provide the basis for selecting the regularization parameter.
  • Path computation: The full solution path is obtained by repeatedly solving the convex LASSO problem for different µ values.The implementation uses CVX interior-point solvers to obtain global solutions.
  • Optimality conditions: For nonzero coefficients, the ℓ1 subgradient is a unit vector in the coefficient direction; for zero coefficients, its magnitude is at most one.Thus, the subgradient has uniformly bounded infinity norm.
  • Optimality conditions: A component becomes active when its beamformed residual reaches the boundary set by µ.The active and inactive conditions are expressed through |r_i| = µ and |r_i| ≤ µ, respectively.
  • Multiple snapshots: For multiple snapshots, the beamformed residuals and the µ values marking sparsity changes are determined analogously.The residual construction uses the estimated multiple-snapshot source matrix.

B. Algorithm for the LASSO path

The proposed algorithm estimates the LASSO solution path and unbiased source amplitudes for a desired sparsity level. It uses residual peaks and coherence-based approximations to select regularization values.

  • Algorithm: Table I gives a fast iterative algorithm for solving the LASSO problem at a desired sparsity level K and estimating unbiased complex source amplitudes.The algorithm targets the LASSO formulation and amplitude estimator described in the paper.
  • Regularization selection: The dual method identifies regularization values for different sparsity levels from the dual solution r.Starting from the zero solution at large µ, components activate as residual terms reach the boundary.
  • Residual evaluation: Residuals for candidate steering vectors are evaluated from the current active set to continue the solution path.The procedure uses progressively stronger approximations when selecting µ.
  • Approximation condition: The approximations used for µ selection are valid when the steering vectors in the final active set are sufficiently incoherent.One approximation corresponds to the conventional beamformer for a single snapshot, but its peaks are not the CS solution.
  • Algorithm parameters: The procedure selects residual peaks using peak(r, k) and sets F = 0.9.The peak operator identifies the kth residual peak used by the algorithm.
  • Related extensions: The generalized LASSO replaces the sparsity penalty with ∥Dx∥1 to impose structural or geometric constraints.Examples include block sparsity for continuous sources and adaptive weighting for moving-source DOA tracking.

C. Regularization parameter selection via the LASSO path

The LASSO path links the regularization parameter µ to sparsity, residual fit, and active DOA components. Decreasing µ improves data fit but can activate noisy or basis-coherent components, so the path helps select a suitable sparsity level.

  • Regularization trade-off: The LASSO objective trades the ℓ1-norm of the estimate against the ℓ2-norm data-fitting error as µ varies.For 1.54 > µ > 0.02, decreasing µ shifts emphasis from sparse solutions toward reducing the model residual.
  • LASSO path: For large µ, the solution is zero; as µ decreases, components activate at singular points where residual magnitudes reach µ.The first component appears at µ = 1.76, while further components activate at lower thresholds.
  • LASSO path: µ ≤ 1.14 yields two active components, µ ≤ 0.38 yields three, and µ ≤ 0.18 activates many noisy components.At very low µ, the data-fitting term dominates and sparsity deteriorates.
  • Parameter selection: A predefined K-sparse solution can be selected by decreasing µ to the Kth singular point, while any value between adjacent singular points gives the same sparsity.Once active DOAs are identified, unbiased amplitudes are estimated separately.
  • Residual interpretation: For µ = 0.1, five potential locations reach the residual boundary although only two sources remain after unbiased amplitude estimation.Residuals decrease as µ is reduced, but boundary hits can exceed the true source count.
  • Grid effects: A denser grid can recover the correct bin as µ decreases, but basis coherence may initially offset estimates and can cause components to activate or disappear.In the 1° grid example, the initial estimate is 1° from the strongest DOA before the correct bin is recovered.

V. DOA ESTIMATION ERROR EVALUATION

DOA error depends strongly on source separation, source strength, and estimator assumptions. CS is comparable to conventional methods for well-separated sources but improves resolution when sources are closely spaced, including in single-snapshot settings.

  • Evaluation procedure: The synthetic evaluation pairs estimated and true DOAs to minimize realization-wise RMSE before computing ensemble error.The data use a fixed SNR, while random source phases enable covariance-based methods to resolve incoherent sources.
  • Estimator comparison: CBF has low resolution and sidelobes, while MVDR and MUSIC reduce these problems for multiple snapshots but require additional conditions.MVDR additionally assumes incoherent arrivals and at least as many snapshots as sensors, L ≥ M.
  • Estimator comparison: Exhaustive search provides a maximum-likelihood benchmark but requires evaluating N!/K!(N − K)! steering-vector combinations.For N = 361, this means 77,000 combinations for K = 2 and 7,700,000 for K = 3, making larger-K cases impractical.
  • Single-snapshot evaluation: For well-separated sources of similar magnitude, CBF, CS, and exhaustive search have similar DOA error.Across 1000 noise realizations and several SNRs, CS and CBF show comparable RMSE in the two-source scenario.
  • Single-snapshot evaluation: CBF cannot resolve the closely spaced sources at −3° and 2°, whereas CS performs much better because it retains high resolution from one snapshot.CBF errors also include sidelobe-induced estimates near −65°.

B. Multiple Snapshot

With multiple snapshots, CS operates directly on observations and resolves closely spaced sources at lower SNR than the covariance-based comparisons in these simulations. Its advantage comes with substantially higher computational cost and remains bounded by the tested assumptions and scenarios.

  • Multiple-snapshot formulation: CS, CBF, and CS operate directly on multiple observations, whereas MVDR and MUSIC use a sample covariance matrix averaged over L snapshots.The simulations keep source magnitudes invariant while sampling source phases independently across snapshots.
  • Scope and assumptions: The reported simulations assume incoherent arrivals for MVDR, while CS does not require that assumption; broader conditions need further simulation.The authors specifically note that colored noise, unknown source counts, and random source locations require additional study.
  • Three-source scenario: With L = 50 snapshots at 0 dB SNR, CS localizes three sources well while CBF and MVDR do not.CS performs well down to 2.5 dB SNR, whereas MVDR performs well only above 10 dB in this scenario.
  • Closer sources: When the weak sources move closer to [−2, 1]°, MVDR fails below 20 dB and MUSIC fails at a level 10 dB higher than in the preceding scenario.CS fails only below 5 dB and localizes the weak source with increased spread.
  • Problem scale: The multiple-snapshot inversion contains 1000 equations for 18,050 complex-valued variables, making the sparsity constraint essential.The dimensions arise from 20 sensors, 50 snapshots, and 361 azimuths.
  • Computational cost: CS requires several orders of magnitude more CPU time than the beamforming methods.The exhaustive-search approach is also computationally impractical as the number of sources grows.

VI. EXPERIMENTAL RESULTS

Experimental SWellEx-96 data show that CS resolves coherent multipath arrivals with higher resolution and fewer artifacts than conventional beamformers in both single- and multiple-snapshot processing.

  • Experimental setup: The SWellEx-96 experiment used a 64-element vertical array to evaluate CS against conventional beamformers in a complex shallow-water multipath environment.The array had 1.875 m intersensor spacing and covered depths from 94.125 to 212.25 m.
  • Experimental setup: The deep-towed source transmitted nine tones from 112 to 388 Hz during recordings spanning the closest point of approach.The processed recording contained 87 overlapping snapshots of 2.7 seconds each.
  • Frequency characterization: CBF detected arrivals at transmitted and additional frequencies associated with the shallow source, weaker deep-source tones, and the tow-ship.The multiple-snapshot spatial spectrum covered 50–400 Hz.
  • Single-snapshot processing: For single snapshots, CS improved resolution and significantly reduced sidelobe- and noise-related artifacts relative to CBF.CBF showed six significant peaks but suffered from low resolution and artifacts.
  • Multiple-snapshot processing: For multiple snapshots, MVDR failed to detect coherent multipath arrivals, whereas CS and CBF identified consistent peaks and CS provided improved resolution.The multiple-snapshot solution emphasizes stationary paths while allowing varying source power.
  • Overall findings: Across the evaluation, CS achieved higher resolution than CBF for single snapshots and than MVDR/MUSIC for multiple snapshots, with relative gains increasing as DOAs approached each other.The real-data example also demonstrated resolution of multiple coherent wave arrivals from multipath propagation.
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