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Robust Multi-Source DoA Estimation under Weather-Induced Distortion: Structured Covariance Methods and Performance Bounds

Chenyang Yan, Geert Leus, Mats Bengtsson

arXiv:2609.07484v1eess.SP

TL;DR

Rain-induced phase and amplitude distortions disrupt covariance assumptions used in classical DoA estimation. The paper develops structured-covariance calibration and CRLB analyses for single and multiple sources; simulations report improved accuracy and source resolvability under rain compared with conventional baselines.

  • Problem

    Rain-induced phase and amplitude distortions disrupt the covariance structure assumed by classical subspace DoA methods.

    Source: §I · p. 1

  • Method

    The paper models distortion covariance as an unknown real-valued Toeplitz matrix and develops single- and multi-source calibration methods and CRLBs.

    Source: §I · p. 1

  • Results

    Under rain, structured-covariance methods improve DoA accuracy and recover source resolvability compared with conventional baselines, with alternating LASSO most robust as angular separation decreases.

    Source: §VIII · p. 12

  • Takeaways & Limitations

    The framework addresses rain-distorted DoA estimation relevant to automotive sensing, intelligent transportation, and safety monitoring.

    Source: §I · p. 1

    Future work should address remaining multi-source non-identifiability, broader propagation and array settings, model mismatch, calibration errors, limited snapshots, and validation with measured adverse-weather radar data.

    Source: §VIII · p. 12

Abstract

from arXiv · show

Direction-of-arrival (DoA) estimation in adverse weather is degraded by propagation-induced phase and amplitude distortions that violate the covariance structure assumed by classical subspace methods. Motivated by a physics-based model of rain propagation, we develop a structured covariance formulation for rain-distorted arrays in both single- and multi-source scenarios. A key ingredient is to relax the physically parameterized distortion model and instead model the distortion covariance as an unknown real-valued Hermitian Toeplitz matrix. For the single-source case, we derive a closed-form covariance-matching calibration algorithm and provide structured-covariance and physics-informed Cramer-Rao lower bounds (CRLBs). For the multi-source case, we prove that the original per-source distortion model is non-identifiable and introduce a collinear approximation across angles. Under this approximation, we propose three multi-source calibration methods: alternating LASSO, joint LASSO, and a joint nuclear-norm formulation. Simulations under heavy rain demonstrate improved DoA accuracy and source resolvability compared with conventional baselines, highlighting the benefit of structured covariance modeling and calibration for safety-critical sensing.

I. INTRODUCTION

Rain-induced phase and amplitude perturbations challenge covariance assumptions in conventional DoA estimation, motivating a structured covariance approach. The paper develops single- and multi-source calibration methods, bounds, and a rain-distortion measurement model.

I. INTRODUCTION

  • Rain can produce stochastic phase and amplitude distortions that perturb array response and explain why classical subspace covariance assumptions fail.The paper focuses on rain, while noting that the covariance-based formulation may apply to broader random-media scenarios.Source: §I · p. 1
  • The paper models distortion covariance as an unknown real-valued Toeplitz matrix, with single-source calibration, multi-source methods, and CRLB analyses.The multi-source contribution includes three calibration methods; the paper also identifies non-identifiability in the original formulation.Source: §I · pp. 1–2

II. DISTORTION MODEL UNDER ADVERSE WEATHER

  • The wavefront model treats normalized fluctuations at separated points as zero-mean, jointly circularly symmetric complex Gaussian variables, with empirical correlation parameters tied to rain rate and frequency.The cited empirical model is reported valid for R ≤ 500 m and 0.1 ≤ d ≤ 8.Source: §II · p. 2
  • As range, separation, or rain rate increases, the phase-difference PDF becomes less concentrated around 0° and the magnitude-ratio PDF shifts away from unity.The phase-difference PDFs remain symmetric about zero.Source: §II · p. 2

III. MEASUREMENT MODEL

  • The measurement model assumes far-field propagation and a ULA, with rain fluctuations applied at the first encountered array element's reference plane and negligible beyond it.The empirical correlation model limits total array aperture to 8λ0, corresponding to at most 17 antennas at half-wavelength spacing.Source: §III · p. 2

A. Snapshot-Domain Model

  • Each sensor snapshot combines the distortion-free steering response with time-varying rain-distortion amplitude and phase, path loss, and spatially white Gaussian noise.The distortion amplitude and phase are assumed independent and identically distributed across snapshots for each sensor.Source: A. Snapshot-Domain Model · p. 3
  • In the multi-source model, distortion statistics depend on source angle because incidence changes the effective sensor-pair separation on the reference wavefront.Source: A. Snapshot-Domain Model · p. 3

B. Covariance-Domain Model

The covariance-domain model describes array observations through source and rain-distortion covariance terms under assumptions on sources and noise. Rain distortion varies with arrival direction and rain conditions.

B. Covariance-Domain Model

IV. DOA ESTIMATION USING COVARIANCE MATCHING

The section estimates directions from the sample covariance while treating weather distortion covariances as nuisance parameters. It uses covariance matching, with LS/WLS formulations and distinct reparameterizations for single- and multi-source cases.

IV. DOA ESTIMATION USING COVARIANCE MATCHING

  • GLS/COMET matches the sample covariance to a parametric covariance model using second-order statistics.Source: §IV · p. 4
  • DoAs are estimated from the sample covariance while weather-distortion covariances are treated as nuisance parameters.Source: §IV · p. 4
  • Path-loss and source-power quantities cannot be separated from covariance data alone, so the single- and multi-source cases use different reparameterizations.Source: §IV · p. 4
  • WLS accounts for nonuniform covariance-entry variance and is asymptotically equivalent to Gaussian maximum-likelihood weighting, but costs more than LS.Source: §IV · p. 4

V. CALIBRATION ALGORITHMS

The calibration framework extends the single-source procedure to WLS and develops an approach for multi-source estimation. For single-source calibration, it estimates a combined Hermitian Toeplitz covariance and decouples its phase and magnitude components.

V. CALIBRATION ALGORITHMS

  • The framework generalizes the single-source procedure to WLS and extends calibration and estimation to the multi-source scenario.Source: §V · p. 5

A. Single-Source Calibration

  • Single-source calibration estimates the combined Hermitian Toeplitz matrix, then decouples it into steering and distortion covariance estimates.Source: A. Single-Source Calibration · p. 5
  • The covariance-coefficient fit has a closed-form solution, and setting the weighting matrix to identity gives the unweighted LS estimator.Source: A. Single-Source Calibration · p. 5

B. Multiple-Source Calibration

The original multi-source model is generally non-identifiable when each source has an independent angle-dependent distortion covariance. A near-collinearity observation motivates a shared distortion template and three calibration approaches.

B. Multiple-Source Calibration

  • For N ≥ 2, independent angle-dependent distortion covariances make the multi-source model generally non-identifiable from the covariance alone.The covariance has 2M − 1 real degrees of freedom, while the distortion matrices introduce NM nuisance parameters in addition to source powers and DoAs.Source: B. Multiple-Source Calibration · pp. 5–6

1) Collinear Approximation for Multi-Source Calibration:

  • Cosine similarity exceeds 0.998 across all angle pairs from −90° to 90° for 50 mm/hr rain, 200 m propagation, and M = 8 antennas.This near-collinearity motivates approximating angle-specific distortion matrices as scalar multiples of the 0° reference matrix.Source: 1) Collinear Approximation for… · p. 6, Figure 4 · p. 6
  • The collinear approximation absorbs angle-dependent proportionality, path loss, and source power into a sparse angular-grid power vector.Nonzero grid entries indicate active DoAs, and their magnitudes represent effective reweighted powers.Source: 1) Collinear Approximation for… · p. 6
  • The multi-source methods are alternating LASSO, joint LASSO, and joint nuclear-norm calibration, trading fidelity to the original criterion against computational simplicity.Alternating LASSO retains structurally consistent subproblems but can be computationally demanding; joint formulations reduce outer iterations but are more sensitive to modeling mismatch.Source: 1) Collinear Approximation for… · pp. 6–7

2) Alternating-Optimization Method with LASSO:

The alternating method represents multi-source DoAs through a sparse angular-grid power vector and alternates between power estimation and distortion calibration. It also debiases active powers before updating the distortion parameters.

2) Alternating-Optimization Method with LASSO:

  • The sparse grid vector’s nonzero entries identify active DoAs, while their magnitudes encode effective reweighted powers.The distortion covariance is parameterized using real-valued Toeplitz basis matrices, with its diagonal entry fixed to remove scale ambiguity.Source: 2) Alternating-Optimization Method… · p. 7
  • With these parameterizations, the cost in (54) is bi-convex in γ and c, motivating alternating optimization that updates γ and c in turn while holding the other fixed.The formulation also imposes γ ≥ 0 and c ≥ 0; its LASSO regularization uses μ > 0 to control sparsity of γ.Source: 2) Alternating-Optimization Method… · p. 7
  • After LASSO support selection, the method refits active coefficients without the ℓ1 penalty, then updates distortion parameters by constrained least squares until convergence.The debiasing refit uses nonnegative least squares on the active support.Source: 2) Alternating-Optimization Method… · p. 7

3) Joint Optimization Method with LASSO:

The joint LASSO estimates sparse grid powers and inverse distortion parameters together, while the nuclear-norm alternative relaxes the signal covariance and promotes low rank. Optional refits reduce shrinkage after either joint estimation approach.

3) Joint Optimization Method with LASSO:

  • Joint LASSO constrains inverse-distortion coefficients to increase monotonically with lag, stabilizing estimation and suppressing oscillatory solutions.The constraint is motivated by expected decay of distortion correlation with sensor separation.Source: 3) Joint Optimization Method with LASSO · pp. 7–8
  • The joint LASSO estimates the sparse power vector and inverse-distortion parameters in a convex program, then optionally refits both on the selected support.The refit keeps the DoA support fixed while reducing bias in power and distortion estimates.Source: 3) Joint Optimization Method with LASSO · p. 8

4) Joint Optimization Method with Nuclear Norm:

  • The nuclear-norm formulation relaxes the undistorted signal covariance as a Hermitian Toeplitz covariance and promotes low rank to reflect a small source count.After estimation, an optional subspace-constrained refit removes the nuclear-norm penalty, and MUSIC extracts DoAs from the refitted covariance.Source: 4) Joint Optimization Method with… · pp. 8–9

VI. PERFORMANCE BOUNDS AND IDENTIFIABILITY

The section defines structured-covariance and physics-informed CRLBs for single-source estimation. The bounds are restricted to that setting because the original multi-source model is non-identifiable.

VI. PERFORMANCE BOUNDS AND IDENTIFIABILITY

  • The structured-covariance CRLB applies the relaxed model used by the single-source calibration method, whereas the physics-informed CRLB uses the physical distortion covariance and empirical correlation law.The physics-informed bound also uses the angle-dependent separation model.Source: §VI · p. 9
  • The bounds are restricted to the single-source case because independent angle-dependent distortion covariances make the original multi-source model non-identifiable.The section uses the Slepian–Bangs formula and eliminates nuisance parameters through a Schur complement.Source: §VI · p. 9

A. Structured-Covariance CRLB

The section derives two single-source bounds: one for the relaxed structured-covariance model and one using the physics-informed distortion covariance. The empirical parameters are locally identifiable under the considered operating conditions, but translating them into physical quantities requires a calibrated mapping.

A. Structured-Covariance CRLB

  • The SC–CRLB treats the real symmetric Toeplitz nuisance vector c separately from the direction parameter θ in the relaxed single-source covariance model.The section identifies this as the bound associated with the structured-covariance model.Source: A. Structured-Covariance CRLB · p. 9

B. Physics-Informed CRLB

  • The PI–CRLB constrains the distortion covariance with the physical model and reparameterizes it using κ and a3.The physics-informed covariance uses the empirical correlation law and angle-dependent separation.Source: B. Physics-Informed CRLB · p. 10
  • The empirical parameters are locally identifiable under the considered single-source conditions, but converting κ and a3 into rain rate and propagation range requires a calibrated mapping.The paper gives tabulated coefficients in [14, Table II] as an example of such a mapping.Source: B. Physics-Informed CRLB · p. 10

VII. NUMERICAL RESULTS AND PERFORMANCE ANALYSIS

Simulations show that calibration improves single-source accuracy and that structured-covariance methods recover multi-source directions under rain distortion. Alternating LASSO is most robust as angular separation decreases, while the study identifies remaining scope and validation needs.

A. Single-Source Case

B. Multiple-Source Case

  • Under heavy rain, calibrated MUSIC resolves two sources at their correct angles, whereas MUSIC on the distorted covariance does not.The evaluated two-source case uses 1,000 snapshots and a 10 dB source-power difference.Source: B. Multiple-Source Case · p. 11, Figure 10 · p. 12
  • Alternating LASSO reconstructs the 10 dB source-power difference more accurately than joint LASSO in the two-source experiment.Alternating LASSO estimates powers 1.03 and 0.10; joint LASSO estimates 0.91 and 0.11.Source: B. Multiple-Source Case · p. 11
  • With T=1000 snapshots, for two symmetrically placed sources, at 8° separation alternating LASSO maintains RMSE close to 1°, versus around 2.5° for nuclear-norm and about 9° for joint LASSO.Source: Figure 11 · p. 12, B. Multiple-Source Case · p. 12

VIII. CONCLUSION

  • The study identifies remaining non-identifiability in general multi-source formulations and calls for broader array-setting, robustness, and measured-data validation.The stated extensions include beyond narrowband far-field ULAs and testing model mismatch, calibration errors, and limited snapshots.Source: §VIII · p. 12

APPENDIX A COVARIANCE SENSITIVITIES FOR LEMMA 2

The appendices provide covariance sensitivities for forming the Fisher information matrices and deriving both the structured-covariance and physics-informed bounds.

APPENDIX A COVARIANCE SENSITIVITIES FOR LEMMA 2

  • The general CRLB calculation uses the Slepian–Bangs formula and a Schur complement, with the appendices listing the required covariance sensitivities.The structured-covariance calculation takes the Schur complement over c.Source: Appendix A · p. 13
  • For the structured-covariance model, the appendix supplies the angle and covariance-parameter sensitivities used to form the bound.The resulting vectorized sensitivities are substituted into the Fisher information calculation before eliminating c.Source: Appendix A · p. 13

APPENDIX B COVARIANCE SENSITIVITIES FOR LEMMA 3

  • For the physics-informed model, parameter sensitivities are substituted into the Fisher information calculation before the Schur complement over nuisance parameters.The appendix refers to the angle sensitivity given in (83).Source: Appendix B · p. 13
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