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Deep Networks for Direction-of-Arrival Estimation in Low SNR

Georgios K. Papageorgiou, Mathini Sellathurai, Yonina C. Eldar

arXiv:2011.08848v1eess.SPcs.LGcs.SDeess.AS

TL;DR

The paper addresses robust DoA estimation under extreme noise, where low-SNR covariance estimates make conventional methods unreliable. It introduces a CNN using multi-channel covariance data and multi-label classification, reporting robust low-SNR estimation, off-grid accuracy within grid resolution, and joint inference of unknown source counts.

  • Problem

    Low-SNR DoA estimation is difficult because sample covariance estimates can substantially deviate from the true manifold matrix, while existing approaches may require parameter tuning.

  • Method

    A 2D CNN is trained on multi-channel complex covariance data as a multi-label classifier, using low-SNR training and varying-source training for joint DoA and source-count prediction.

  • Results

    The CNN demonstrates robust low-SNR DoA estimation across snapshots, angular separations, and SNR mismatches, with predicted angles within grid resolution and successful unknown-source inference.

  • Takeaways & Limitations

    The approach provides a parameter-tuning-free DoA estimator that can operate with sample covariance data and infer source number jointly with directions.

Abstract

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In this work, we consider direction-of-arrival (DoA) estimation in the presence of extreme noise using Deep Learning (DL). In particular, we introduce a Convolutional Neural Network (CNN) that is trained from mutli-channel data of the true array manifold matrix and is able to predict angular directions using the sample covariance estimate. We model the problem as a multi-label classification task and train a CNN in the low-SNR regime to predict DoAs across all SNRs. The proposed architecture demonstrates enhanced robustness in the presence of noise, and resilience to a small number of snapshots. Moreover, it is able to resolve angles within the grid resolution. Experimental results demonstrate significant performance gains in the low-SNR regime compared to state-of-the-art methods and without the requirement of any parameter tuning. We relax the assumption that the number of sources is known a priori and present a training method, where the CNN learns to infer the number of sources jointly with the DoAs. Simulation results demonstrate that the proposed CNN can accurately estimate off-grid angles in low SNR, while at the same time the number of sources is successfully inferred for a sufficient number of snapshots. Our robust solution can be applied in several fields, ranging from wireless array sensors to acoustic microphones or sonars.

I. INTRODUCTION

The paper targets robust DoA estimation in low SNR using a CNN trained on multi-channel covariance data. It addresses parameter tuning and unknown source-count assumptions while evaluating robustness across challenging conditions.

  • Prior methods: Existing MUSIC and sparse-recovery methods estimate directions through subspace or optimization procedures, often requiring grid searches or parameter tuning.Sparse methods may require tuning based on snapshots, SNR, or both, while MUSIC uses a specified angular grid.
  • Motivation: Low-SNR DoA estimation is challenging because sample covariance estimates can deviate substantially from the true manifold matrix.The paper identifies robustness in extreme noise as the central gap motivating its approach.
  • Proposed approach: The proposed CNN uses multi-channel data from complex covariance information, 2D convolutions, and multi-label classification to predict multiple angular directions.Training uses the true covariance matrix, while prediction uses the sample covariance estimate.
  • Unknown source count: The assumption of a known source count is relaxed by training with varying numbers of sources so the CNN predicts source number jointly with DoAs.The varying-source strategy requires only a known maximum source count during training.
  • Evaluation: Simulations compare the CNN with state-of-the-art methods and the CRLB, reporting low-SNR gains, snapshot resilience, SNR-mismatch robustness, and source-count inference.The evaluation includes off-grid angles and varying angular separations.

III. DOA ESTIMATION WITH MULTIPLE MEASUREMENT VECTORS (MMV)

This section summarizes multiple-measurement-vector DoA estimators and discusses their respective advantages and disadvantages.

  • MMV DoA estimators: The paper reviews MMV-based DoA estimators to compare their strengths and weaknesses.The section is presented as a brief summary of the estimator families used for comparison.

A. Multiple Signal Classification (MUSIC)

MUSIC separates signal and noise subspaces to locate DoAs through peaks on a discretized angular grid. Its practical accuracy degrades when low-SNR covariance estimates obscure that separation.

  • MUSIC: MUSIC separates signal and noise subspaces through covariance-matrix eigendecomposition.The signal subspace corresponds to the leading eigenvectors, while the remaining eigenvectors span the noise subspace.
  • MUSIC: The MUSIC pseudo-spectrum is evaluated over discretized grid angles, with peaks identifying candidate source directions.The denominator becomes zero or small when a grid steering vector matches a true source direction.
  • Limitations: In practice, replacing the true covariance matrix with its sample estimate can cause subspace swaps and inaccurate DoA estimates at low SNR.Low SNR makes signal and noise eigenvalues difficult to distinguish.
  • Limitations: MUSIC also has finite angular resolution from grid discretization and computational cost from repeated grid searches.These limitations arise even apart from low-SNR subspace-estimation errors.
  • R-MUSIC: R-MUSIC replaces MUSIC’s spectral search with polynomial rooting and often performs better than MUSIC in the low-SNR regime.The root-based method derives DoAs from selected polynomial roots rather than directly scanning the angular grid.

C. Compressed Sensing: mixed ℓ2,1-norm minimization

Compressed-sensing DoA estimation represents angular sources on a grid and recovers sparse source coefficients through mixed ℓ2,1-norm optimization, with dimensionality reduction used for many snapshots.

  • Compressed sensing replaces the continuous DoA domain with grid points and constructs a dictionary from the corresponding array-manifold vectors.The dictionary has size N × |G|, where each column is an array response at one grid angle.
  • Grid mismatch introduces a quantization error that is absorbed into the effective noise matrix before convex optimization.The effective noise is defined as Ē = E + H, where H is the quantization error.
  • The source matrix is recovered using a mixed ℓ2,1-norm objective, which promotes sparsity across angular rows while accounting for multiple snapshots.The mixed norm is described as preserving sparsity and offering greater robustness than greedy ℓ0-based approaches.
  • When the snapshot count is high, singular value decomposition reduces the data to an N × R representation before solving the compressed-sensing problem.Here R = rank(Y), and the reduced problem is solved in the lower-dimensional space.
  • The ℓ2,1-SVD method reconstructs the source matrix and estimates DoAs from the power of its rows using standard convex-optimization packages.The reduced-space solution is transformed back before row powers are computed.

D. Multi-layer Prerceptron (MLP)

Prior work used a multitask autoencoder and parallel multilayer DNNs for two-source DoA estimation, with training performed separately at each SNR.

  • The prior MLP framework estimates two sources while accounting for array imperfections.It combines a multitask autoencoder acting as spatial filters with parallel multilayer DNNs for spatial-spectrum estimation.
  • The network is trained independently at each SNR and is not always successful in resolving DoAs.The reported resolution performance depends on the operating conditions described in the surrounding discussion.
  • The proposed work instead formulates DoA estimation as multilabel classification, using convolutional feature extraction followed by fully connected DoA inference on a selected grid.This section introduces the task formulation and points to later descriptions of data management and architecture.

A. Data Management and Labeling

The method represents covariance estimates as three-channel inputs and trains a CNN to map them to binary angle labels on a predefined grid using multilabel classification.

  • A. Data Management and Labeling: DoA prediction is modeled as multilabel classification over a discrete angular grid whose resolution and limits determine the output size.For φmax = Gρ, the grid contains 2G + 1 points spanning the selected angular interval.
  • A. Data Management and Labeling: The CNN input is an N × N × 3 real-valued tensor containing covariance real parts, imaginary parts, and phase entries.The first two channels encode Re{Ry} and Im{Ry}; the third encodes ∠{Ry}.
  • A. Data Management and Labeling: For a 1° grid spanning −60° to 60°, each training label is a 121-dimensional binary vector with ones at the source-angle locations.The label independently marks each selected grid angle.
  • A. Data Management and Labeling: Training examples pair each covariance-derived input with its binary label vector to form the supervised dataset.The dataset is written as pairs (X(i), z(i)).
  • B. The Proposed CNN’s Architecture: The CNN uses four 2D convolutional stages with 256 filters, followed by batch normalization, ReLU activations, flattening, fully connected layers, and dropout.Pooling is omitted because information loss resulted in poor performance; dropout randomly sets weights to zero with probability 20%.
  • B. The Proposed CNN’s Architecture: A final sigmoid layer outputs an independent probability for each of the K labels rather than using softmax.Independent labels can simultaneously receive values equal or close to 1.
  • B. The Proposed CNN’s Architecture: The network is trained by back-propagation on a multilabel objective using binary cross-entropy loss.The trainable parameters are optimized over the supervised training dataset.

V. TRAINING APPROACH

The CNN is trained using true covariance-matrix data on a 1° grid from −60° to 60°, enabling prediction from sample covariance estimates formed from sufficient snapshots.

  • V. TRAINING APPROACH: Training uses a 1° angular grid from −60° to 60°, producing 121 output grid points after binary label transformation.The grid is defined by ρ = 1° and φmax = 60°.
  • V. TRAINING APPROACH: The training data come from the true covariance matrix rather than individual noisy sample estimates.This reduces the number of training examples required and supports prediction from any sufficient collection of snapshots used to estimate the sample covariance.

A. Fixed Number of Sources

The CNN is trained on multi-channel covariance-derived data across low SNRs, with fixed-source and mixed-source training strategies. Low-SNR training is intended to support prediction at higher SNRs, while mixed-source labels allow source-count variation.

  • A. Fixed Number of Sources: For fixed-source training, examples from all SNRs are used jointly, with K = 2 sources in the described setup.
  • A. Fixed Number of Sources: Low-SNR training from −20 to 0 dB is sufficient for prediction at higher SNRs in the reported observations.The training range uses 5 dB increments and produces 36,300 training examples.
  • B. Mixed Number of Sources: For varying source counts, the numerical study sets Kmax = 3 and assigns each example a 121 × 1 binary label vector with one to three active entries.The training set contains 295,361 examples for G = 60 angle-grid points.

VI. SIMULATION RESULTS

The simulations evaluate the CNN against MUSIC, R-MUSIC, ℓ2,1-SVD, and MLP across low-SNR off-grid DoA settings. The experiments vary SNR, snapshots, and angular configurations under a common ULA and grid resolution.

  • VI. SIMULATION RESULTS: The evaluation compares the CNN with MUSIC, R-MUSIC, ℓ2,1-SVD, and MLP, including settings where the source count is known or mixed.
  • VI. SIMULATION RESULTS: All experiments use a 16-element half-wavelength-spaced ULA, with 1° grid resolution for the CNN, MUSIC, and ℓ2,1-SVD.R-MUSIC is gridless and is expected to perform optimally at high SNR.
  • VI. SIMULATION RESULTS: The CNN is evaluated using empirical RMSE, with scenarios including off-grid source directions and sample covariance estimates from finite snapshots.
  • 1) Direction-of-Arrival Estimation and Errors:: At −10 dB SNR with 2,000 snapshots, the CNN attains the smallest errors and performance comparable to gridless R-MUSIC on off-grid angles.
  • 1) Direction-of-Arrival Estimation and Errors:: At 0 dB SNR with 200 snapshots, the CNN gains over subspace-based methods and performs similarly to ℓ2,1-SVD without threshold tuning.
  • 1) Direction-of-Arrival Estimation and Errors:: Across SNR values for two off-grid sources and 1,000 snapshots, the CNN outperforms subspace-based methods in the low-SNR regime.

2) RMSE versus the SNR:

The reported RMSE experiments examine SNR, snapshot count, and source separation under difficult off-grid, low-SNR conditions. The CNN remains robust with few snapshots and closely spaced sources, while grid-based methods encounter resolution floors.

  • 2) RMSE versus the SNR:: At low SNR, the CNN has RMSE close to robust ℓ2,1-SVD while requiring no parameter tuning; at high SNR, only gridless R-MUSIC attains the CRLB.The CNN’s no-tuning advantage is relevant when SNR is unknown or varies slightly.
  • 3) RMSE versus the number of snapshots T: For up to T = 500 snapshots at −10 dB SNR, the CNN demonstrates robust behavior compared with MUSIC and R-MUSIC.
  • 3) RMSE versus the number of snapshots T: For large snapshot counts, grid-based estimators floor because finite grid resolution limits their performance.
  • 2) RMSE versus the SNR:: The ℓ2,1-SVD comparisons use threshold values that vary with snapshot count, whereas the CNN does not depend on such parameter fine-tuning.
  • 3) RMSE versus the number of snapshots T: For 1° ≤ ∆θ < 4° at −10 dB SNR and T = 500, the CNN and ℓ2,1-SVD resolve closely spaced angles while subspace methods fail to provide accurate estimates.Their RMSE remains below 0.65° across the tested angular separations.

4) RMSE versus the angle separation

The experiments evaluate DoA robustness under SNR mismatches and unknown source counts, using RMSE and Hausdorff distance where predicted sets may differ in size.

  • Robustness to SNR mismatches: The CNN shows enhanced robustness under SNR mismatches, while ℓ2,1-SVD exhibits larger error dispersion and requires SNR-specific threshold tuning.The threshold values η = 60 and η = 400 were optimized for 0 dB and −10 dB, respectively.
  • Robustness to SNR mismatches: 48% improvement: the CNN achieves 0.74° RMSE versus 1.42° for ℓ2,1-SVD at an actual SNR of −11.549 dB.This experiment uses two sources separated by 4° and T = 1,000 snapshots.
  • Number of Sources Unknown: Without known K, the CNN is evaluated with Hausdorff distance because predicted DoA sets may have different cardinalities from the true sets.Mean and maximum Hausdorff distance are used over the testing set.
  • Number of Sources Unknown: At −10 dB, varying-angle tests produce three, two, and four classification errors for K = 1, 2, and 3, respectively.The tests use T = 3,000 snapshots and confidence levels 0.88, 0.84, and 0.71.
  • Number of Sources Unknown: At 0 dB, K = 1 and K = 2 have no source-count errors, while K = 3 has two false negatives.The remaining K = 1 and K = 2 errors arise from grid mismatch, described as quantization errors.

VII. CONCLUSIONS AND FUTURE WORK

The paper presents a 2D-filter CNN for low-SNR DoA prediction and evaluates it across varied operating conditions. It also supports inference of an unknown source count jointly with DoAs, with angles estimated within grid resolution.

  • Conclusions and Future Work: The proposed CNN models DoA estimation as on-grid multi-label classification using 2D filters and multi-channel input data.Two training strategies address fixed and varying numbers of sources.
  • Conclusions and Future Work: Future work includes architectures with finer grids for identifying multiple targets.The proposed approach currently uses an on-grid formulation.
  • Conclusions and Future Work: The evaluation covers fixed and varying directions, SNR, snapshot count, angular separation, and SNR mismatches against state-of-the-art methods.The Cramér-Rao lower bound is also provided as a benchmark.
  • Conclusions and Future Work: The results indicate enhanced robustness, resilience across snapshot counts, and the ability to resolve closely spaced angles in low SNR.These outcomes are reported for off-grid angles under multiple experimental setups.
  • Conclusions and Future Work: The varying-source training approach jointly identifies the unknown number of sources and their DoAs with high probability.The predicted angles are sufficiently accurate within the grid’s resolution.
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