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AI-Aided ESPRIT for Joint DoA Estimation and Uncertainty Extraction

Raz Zohar, Nir Shlezinger

arXiv:2609.17059v1eess.SP

TL;DR

DoA systems need reliable uncertainty alongside direction estimates, but classical analyses rely on restrictive assumptions and deep models usually provide only point predictions. The paper combines surrogate covariance recovery with ESPRIT-based uncertainty analysis and dedicated joint training, achieving accurate DoA estimates and reliable uncertainty across challenging scenarios.

  • Problem

    DoA estimation requires both source directions and confidence information, while classical methods rely on restrictive assumptions and deep models generally lack principled uncertainty characterization.

  • Method

    The framework combines learned surrogate covariance recovery with ESPRIT-based DoA estimation, full error-covariance characterization, and uncertainty-aware training.

  • Results

    The proposed approach achieves accurate DoA estimation together with reliable uncertainty characterization across coherent sources, limited snapshots, and array calibration errors.

  • Takeaways & Limitations

    Preserving the ESPRIT processing structure provides an interpretable route to uncertainty extraction while supporting operation in challenging regimes.

Abstract

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DoA estimation often requires not only accurate recovery of source directions, but also reliable characterization of the uncertainty in these estimates. While classical subspace methods such as ESPRIT provide principled uncertainty analyses, their performance and uncertainty quantification rely on restrictive assumptions. Recent deep learning approaches alleviate these limitations and enable robust. DoA estimation in challenging conditions, but generally provide point estimates and lack principled uncertainty characterization. In this work, we develop an AI-aided framework for joint DoA estimation and uncertainty quantification that combines the robustness of model-based deep learning with the analytical foundations of classical subspace methods. Building on AI surrogate covariance recovery, we extend existing ESPRIT uncertainty analyses to characterize the full covariance structure of the DoA estimation error and integrate this characterization into a subspace-oriented deep learning architecture. We further propose a dedicated learning strategy that jointly promotes accurate DoA recovery and faithful uncertainty estimation. The resulting methodology preserves the interpretable processing pipeline of classical subspace methods while enabling reliable operation in regimes where conventional approaches struggle. Our numerical studies demonstrate that the proposed framework consistently achieves accurate DoA estimation together with reliable uncertainty characterization across diverse challenging scenarios, including coherent sources, limited snapshots, and array calibration errors.

I. INTRODUCTION

DoA estimation requires both accurate source directions and reliable uncertainty characterization, but classical methods rely on restrictive assumptions while deep models generally provide only point estimates. The paper combines surrogate covariance recovery, ESPRIT-based analytical uncertainty, and uncertainty-aware training to address challenging operating conditions.

  • Motivation: Accurate DoA recovery alone is insufficient because downstream tracking, sensor fusion, channel estimation, and signal enhancement also require confidence information.
  • Limitations of Existing Methods: Classical subspace methods provide analytical uncertainty tools, but their performance and uncertainty characterizations depend on restrictive modeling and observation assumptions.
  • Related Work: Deep neural networks can operate where classical array-processing methods struggle, yet generally produce point estimates without principled predictive confidence.
  • Contributions: The paper targets joint DoA estimation and uncertainty quantification by extending ESPRIT analysis within a DNN-aided subspace pipeline.The framework combines surrogate covariance recovery with ESPRIT-based angle estimation and covariance reconstruction.
  • Contributions: The proposed uncertainty-aware framework preserves the interpretable subspace-processing chain while extending applicability to challenging regimes.
  • Contributions: Numerical studies cover coherent sources, limited snapshots, low SNRs, and array calibration errors, reporting accurate DoA recovery with reliable uncertainty characterization.

B. Problem Formulation

The problem is to map array measurements to DoA estimates and their full estimation-error covariance under coherent sources, miscalibration, and limited observations. The formulation uses subspace structure and ESPRIT rotational invariance, while recognizing that conventional methods become sensitive when their assumptions fail.

  • B. Problem Formulation: The estimator must map measurements X to estimated directions ˆθ and an error-covariance estimate ˆΣ.
  • B. Problem Formulation: The covariance estimate ˆΣ should represent both individual-direction reliability through diagonal entries and cross-direction error coupling through off-diagonal entries.
  • B. Problem Formulation: The target operating conditions include coherent sources, array miscalibration, and limited observations, which challenge conventional subspace methods.
  • B. Problem Formulation: A data-aided training design uses labeled measurements and ground-truth DoAs to improve robustness while retaining interpretable subspace-based uncertainty characterization.
  • 1) Subspace Methods:: Subspace methods decompose the covariance eigenspace into signal and noise subspaces, using their orthogonality to infer source directions.
  • 1) Subspace Methods:: Empirical covariance, coherent sources, limited snapshots, and array mismatch respectively expose subspace estimation to finite-sample and model-assumption failures.
  • 1) Subspace Methods:: ESPRIT exploits ULA rotational invariance by selecting two shifted subarrays and relating their signal subspaces through a nonsingular transformation.
  • 1) Subspace Methods:: The eigenvalues of the ESPRIT transformation encode the source directions, which are recovered from the transformation spectrum.

2) DNN-Based DoA Recovery:

DNN-aided DoA methods learn surrogate covariance representations while retaining ESPRIT’s interpretable subspace processing. This framework extends that pipeline to estimate both directions and their full error covariance under challenging conditions.

  • DNN-Based DoA Recovery: Surrogate covariance recovery preserves downstream ESPRIT processing while enabling DoA estimation in regimes where conventional subspace methods struggle.The learned covariance supports the subsequent subspace operation rather than replacing it with a fully black-box predictor.
  • DNN-Based DoA Recovery: The resulting architecture combines learned surrogate covariance recovery with ESPRIT-based angle estimation and perturbation-based covariance reconstruction.Selection matrices remain design parameters as long as they satisfy ESPRIT’s rotational invariance requirement.
  • Covariance Characterization: The proposed analysis uses ESPRIT’s subspace perturbations to characterize full DoA error covariance, including correlations between source estimates.The formulation builds on the shared sensitivity of recovered angles and subspace quantities to covariance perturbations.
  • Covariance Characterization: Existing ESPRIT uncertainty analyses assume calibrated arrays, non-coherent sources, and sufficiently many snapshots, limiting their direct applicability.They also commonly use non-overlapping subarrays and focus on per-direction mean-squared error rather than full error covariance.

2) Covariance Characterization:

The covariance characterization extends ESPRIT uncertainty analysis from marginal variances to the full covariance of DoA errors. The same estimated covariance can support both angle recovery and uncertainty extraction.

  • Covariance Characterization: Theorem 1 characterizes the full ESPRIT DoA error covariance, including inter-source correlations, under small covariance perturbations and stated regularity conditions.The conditions include an existing ESPRIT rotational matrix, order-one eigenvalues, small perturbations, and cos θ_i bounded away from zero.
  • Covariance Characterization: For non-overlapping ESPRIT subarrays with |λ_i| = 1, the full formulation reduces to the previously established variance-only result.This provides a consistency check against the earlier ESPRIT uncertainty analysis.
  • Uncertainty-Aware SubspaceNet: An estimated covariance matrix can be reused for ESPRIT DoA recovery and subsequent estimation of the full uncertainty covariance.Its diagonal entries represent marginal uncertainty, while off-diagonal entries represent coupling between source-estimation errors.

1) Inference:

During inference, the learned surrogate covariance replaces the empirical covariance in the ESPRIT pipeline. The resulting subspace quantities are reused to reconstruct uncertainty, including both marginal errors and inter-source correlations.

  • Inference: The inference procedure applies ESPRIT to the surrogate covariance and reuses its signal-subspace, rotational-matrix, and eigenvector quantities for uncertainty extraction.The pipeline computes the signal subspace, rotational matrix, and eigenvectors before reconstructing the predicted covariance.
  • Inference: Unknown covariance-perturbation moments are approximated by replacing the true covariance with the learned surrogate under a local zero-mean complex-Gaussian assumption.Wick’s factorization supplies the plug-in second-order moments used by the covariance characterization.
  • Inference: The predicted uncertainty covariance reports marginal DoA uncertainty on its diagonal and correlations between estimation errors in its off-diagonal entries.The overall computation is summarized as the uncertainty-aware SubspaceNet inference algorithm.

2) Source Matching and Losses:

Training matches unordered ESPRIT directions to ground truth and applies the same permutation to the predicted covariance. A two-stage objective then jointly optimizes angle accuracy and uncertainty calibration.

  • Source Matching and Losses: Source matching is required because ESPRIT returns unordered directions, and the identical assignment must permute the full covariance before training.Both diagonal and off-diagonal covariance entries depend on source ordering.
  • Source Matching and Losses: The uncertainty loss compares predicted covariance with empirical DoA error covariance after positive-semidefinite projection and diagonal regularization.This objective is designed to make the predicted covariance explain observed estimation errors.
  • Source Matching and Losses: Training first optimizes DoA accuracy, then fine-tunes the surrogate covariance using both DoA and uncertainty losses.The uncertainty weight μ can be fixed or gradually increased to avoid destabilizing the initial DoA-oriented solution.
  • Source Matching and Losses: The model learns a surrogate covariance whose induced ESPRIT estimate and perturbation-based covariance reconstruction are jointly reliable without an additional black-box uncertainty head.Uncertainty quantification remains tied to the preserved subspace-processing chain.

C. Discussion

The framework combines surrogate covariance recovery with ESPRIT to retain model-based processing while improving robustness and uncertainty characterization. Its added computational burden is confined to deterministic covariance post-processing, while broader array geometries and estimators remain future work.

  • C. Discussion: Surrogate covariance recovery lets the learned representation compensate for challenging conditions while subsequent ESPRIT processing preserves the interpretable model-based pipeline.The recovered covariance replaces the empirical covariance before ESPRIT and supports uncertainty estimation.
  • C. Discussion: The framework estimates full DoA-error covariance, whose diagonal entries describe marginal uncertainty and off-diagonal entries describe coupling between source errors.The same recovered covariance supports both DoA recovery and uncertainty quantification.
  • C. Discussion: The proposed method adds no DNN branches or ensembles, with extra cost limited to model-based post-processing scaling as O(M^2N^3 + M^3N^2), or O(M^2N^3) when M ≤ N.This is compared with SubspaceNet point estimation.
  • C. Discussion: The characterization is naturally suited to ESPRIT-compatible arrays with rotational invariance, while extensions to general planar, sparse, and near-field arrays remain future work.The authors also identify combining covariance recovery with conformal or Bayesian uncertainty tools as a possible extension.

IV. NUMERICAL STUDY

The numerical study evaluates USSN for DoA accuracy and uncertainty reliability across nominal and challenging operating conditions. It uses classical, learning-based, and data-driven uncertainty baselines with RMSPE, uncertainty loss, and ANEES diagnostics.

  • IV. NUMERICAL STUDY: USSN is evaluated for DoA accuracy and uncertainty reliability under nominal conditions and challenging cases C1–C3.The study includes coherent sources, limited snapshots, low SNRs, and array calibration errors.
  • IV. NUMERICAL STUDY: The comparisons include standard ESPRIT, conditional CRB, SubspaceNet Stage 1, USSN, TransMUSIC with uncertainty prediction, and a data-driven CNN.The conditional CRB serves as a reference under the corresponding idealized statistical model.
  • IV. NUMERICAL STUDY: RMSPE measures DoA accuracy, uncertainty loss evaluates predicted covariance against observed errors, and ANEES diagnoses covariance calibration, with lower RMSPE and uncertainty loss preferred.A well-calibrated covariance has ANEES approximately 1, reported as log(1)=0 in the figures.
  • IV. NUMERICAL STUDY: The experiments average metrics over K = 5000 test samples, and benchmark plots use CRB-based references for accuracy and uncertainty alongside the ideal ANEES value.The dashed reference curve marks these quantities across the three panels.

B. Results

USSN maintains accurate DoA estimation and more reliable uncertainty characterization across nominal, mismatched, multi-source, non-Gaussian, coherent, and combined challenging regimes. Its learned surrogate covariance is especially beneficial when classical ESPRIT assumptions fail.

  • B. Results: In the nominal calibrated non-coherent setting, USSN preserves accurate DoA estimation across SNR while uncertainty-aware training improves covariance calibration over point-estimation SubspaceNet.Classical ESPRIT remains a meaningful reference when the source covariance is full rank and the array model is correctly specified.
  • B. Results: Under array miscalibration, classical ESPRIT deteriorates as perturbation η increases, while USSN retains more informative covariance estimates.Classical uncertainty extraction becomes unreliable when perturbation-analysis assumptions are violated.
  • B. Results: As source count M increases, USSN maintains competitive accuracy and more reliable covariance estimates despite reduced angular separation and a growing M × M covariance output.The full covariance captures coupling between different DoA errors.

4) Geometry-Dependent Uncertainty:

The proposed framework produces sample-dependent uncertainty that reflects source geometry and remains useful across non-Gaussian, coherent, and miscalibrated-array scenarios. Compared with generic uncertainty methods, its full-covariance output supports a favorable accuracy–reliability tradeoff.

  • Geometry-Dependent Uncertainty: Source geometry changes estimation difficulty near endfire and for closely spaced sources, and USSN captures these changes in its predicted uncertainty.The resulting covariance estimate depends on source geometry and coupling between estimated directions, not only global parameters such as SNR.
  • Non-Gaussian Signals: The framework provides useful uncertainty estimates for 16-QAM signals despite covariance-moment approximations based on a local Gaussian assumption.This supports application beyond ideal Gaussian source models in practical array-processing and wireless-communication settings.
  • Coherent Sources: USSN provides accurate DoA estimates and meaningful uncertainty measures for coherent sources, where classical ESPRIT’s assumptions fail.The learned surrogate covariance restores a representation suitable for subspace processing, while uncertainty-aware training supports covariance reconstruction.
  • Coherent Sources and Miscalibration: USSN remains robust when source coherence and array miscalibration jointly violate full-rank covariance and nominal-array assumptions.Classical ESPRIT degradation is further amplified when both effects occur together.
  • Uncertainty-Method Comparison: USSN offers a favorable accuracy–reliability tradeoff against Monte-Carlo dropout and conformal prediction, while directly estimating full covariance.Gaussian negative log-likelihood and normalized ANEES align naturally with USSN, whereas conformal intervals require a surrogate diagonal-Gaussian interpretation.
  • Conclusion: Numerical results show accurate DoA recovery together with faithful uncertainty estimates across challenging operating conditions.The method leverages preserved subspace structure and adapted ESPRIT error analysis.

APPENDIX A PROOF OF THEOREM 1

The proof derives the full ESPRIT DoA-error covariance by propagating covariance perturbations through subspace, eigenvalue, and angle mappings. First-order expansions connect the estimated covariance to the resulting DoA perturbation and its covariance.

  • Proof Strategy: Theorem 1 propagates covariance perturbations through covariance-to-subspace, subspace-to-eigenvalue, and eigenvalue-to-DoA mappings.The analysis compares ESPRIT quantities computed from the true covariance with those computed from the estimated covariance.
  • Covariance Construction: The proof obtains the DoA-error covariance by defining eigenvalue perturbation covariance terms, expanding the total error outer product, and taking expectations.The resulting expression is completed by combining the covariance and pseudo-covariance of the principal first-order term with the higher-order approximation.
  • Covariance-to-Subspace Mapping: First-order subspace perturbation theory establishes a linear relation between covariance perturbation and recovered signal-subspace perturbation.This relation is then used to analyze the ESPRIT shift-invariance matrices and their perturbations.
  • Subspace-to-Eigenvalue Mapping: Each ESPRIT eigenvalue perturbation is expressed as a linear function of the signal-subspace perturbation, yielding a stacked covariance-to-eigenvalue operator.The eigenvalue perturbations are then related to the ideal ESPRIT eigenvalues before angle perturbations are formed.
  • Eigenvalue-to-DoA Mapping: A first-order Taylor expansion of the complex phase maps eigenvalue perturbations into DoA perturbations, including higher-order error terms.The source-wise perturbations are stacked and written compactly using the angle-dependent matrices defined in the analysis.

APPENDIX B PROOF OF COROLLARY 1

The corollary recovers the classical variance-only ESPRIT uncertainty formula from the proposed full covariance characterization. Retaining only the diagonal entry yields the non-overlapping-source special case.

  • Diagonal Specialization: The corollary follows by taking the i-th diagonal entry of the full DoA-error covariance expression.The eigenvalue perturbation covariance terms are substituted into the diagonal-entry formula.
  • Classical Special Case: The classical variance-only ESPRIT uncertainty expression is obtained by retaining only [Σ]i,i from the proposed covariance matrix.Thus, the established variance formula appears as a special case of the full covariance characterization.
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