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DA-MUSIC: Data-Driven DoA Estimation via Deep Augmented MUSIC Algorithm

Julian P. Merkofer, Guy Revach, Nir Shlezinger, Tirza Routtenberg, Ruud J. G. van Sloun

arXiv:2109.10581v5eess.SPcs.LG

TL;DR

MUSIC’s model assumptions limit its reliability for coherent and broadband sources, motivating a data-driven augmentation that preserves its subspace structure. DA-MUSIC combines neural networks with MUSIC and outperforms benchmark methods across reported settings, including seismic data.

  • Problem

    MUSIC relies on an accurate signal model and assumptions such as non-coherent, narrowband sources, while DoA estimation must also address coherent and broadband signals.

    Source: §I · p. 2

  • Method

    DA-MUSIC augments MUSIC’s structure with neural networks that estimate a pseudo covariance matrix and identify spectrum peaks.

    Source: §I · p. 2, C. Discussion · pp. 7–8

  • Results

    Across reported settings, DA-MUSIC outperforms model-based and data-driven benchmarks and shows strong resolution, including on real-world seismic signals.

    Source: 1) Known Number of Sources · p. 9, §V · p. 13, C. Non-Synthetic Data: Azimuth… · p. 13

  • Takeaways & Limitations

    DA-MUSIC provides a hybrid DoA estimator reported to operate with unknown source counts and broadband signals while adapting to various scenarios.

    Source: §V · p. 13

    The augmentation depends on the array geometry because steering vectors are used in its computation.

    Source: C. Discussion · p. 8

Abstract

from arXiv · show

Direction of arrival (DoA) estimation of multiple signals is pivotal in sensor array signal processing. A popular multi-signal DoA estimation method is the multiple signal classification (MUSIC) algorithm, which enables high-performance super-resolution DoA recovery while being highly applicable in practice. MUSIC is a model-based algorithm, relying on an accurate mathematical description of the relationship between the signals and the measurements and assumptions on the signals themselves (non-coherent, narrowband sources). As such, it is sensitive to model imperfections. In this work we propose to overcome these limitations of MUSIC by augmenting the algorithm with specifically designed neural architectures. Our proposed deep augmented MUSIC (DA-MUSIC) algorithm is thus a hybrid model-based/data-driven DoA estimator, which leverages data to improve performance and robustness while preserving the interpretable flow of the classic method. DA-MUSIC is shown to learn to overcome limitations of the purely model-based method, such as its inability to successfully localize coherent sources as well as estimate the number of coherent signal sources present. We further demonstrate the superior resolution of the DA-MUSIC algorithm in synthetic narrowband and broadband scenarios as well as with real-world data of DoA estimation from seismic signals.

I. INTRODUCTION

DoA estimation matters across sensor-array applications, but model-based MUSIC depends on narrowband, non-coherent assumptions and can fail on coherent or difficult-to-resolve signals. DA-MUSIC augments MUSIC with neural networks to improve robustness while retaining its structure.

I. INTRODUCTION

  • MUSIC’s narrowband signal model limits reliable localization of coherent signals and resolution of closely spaced signals when samples or SNR are insufficient.The introduction also surveys coherent broadband methods that focus covariances across frequency bins and incoherent methods that combine per-frequency DoA estimates.Source: §I · p. 1
  • Prior neural DoA estimators include model-agnostic networks and hybrid methods, but cited approaches retain model assumptions, depend on covariance labels, or trade interpretability for highly parameterized models.The introduction also describes broadband single-source CNN and regression approaches and support-vector regression on narrowband components.Source: §I · pp. 1–2
  • DA-MUSIC is proposed to address the difficulty of resolving multiple signals that may be broadband and coherent, combining MUSIC’s structure with data-driven neural networks.Its recurrent network estimates a pseudo covariance matrix from measurements, and another network acts as a peak finder.Source: §I · p. 2
  • DA-MUSIC estimates an unknown, varying number of sources, including coherent sources, and adds a learned classifier to categorize signal and noise eigenvectors.The introduction reports 98% accuracy versus MUSIC’s 89% in the same non-coherent scenario.Source: §I · p. 2

A. Signal Model

  • The preliminaries distinguish narrowband and broadband signal models; the narrowband model relates array measurements to incident signals through steering vectors and records observations across snapshots.The steering vectors specify the array geometry.Source: §II · p. 2, A. Signal Model · p. 2, 1) Narrowband · pp. 2–3

2) Broadband:

The broadband model accounts for propagation delays across array elements, then expresses the frequency-domain measurements in a form analogous to the narrowband model.

2) Broadband:

  • Broadband measurements incorporate the propagation delay between each array element and a reference element.The delay is defined for source d at element m relative to element m = 1.Source: 2) Broadband · p. 3
  • In the frequency domain, the delayed measurement relationship can be written in vector-matrix form analogous to the narrowband model.For a uniform linear array, steering vectors depend on frequency and angle, with propagation velocity denoted by c.Source: 2) Broadband · p. 3

B. Problem Formulation

The problem formulation estimates source directions from array measurements using available training data and possibly mismatched array geometry. The reviewed model-based and data-driven methods have trade-offs in broadband handling, source-count flexibility, and dependence on model or covariance estimates.

B. Problem Formulation

  • DoA estimation recovers source angles from array measurements under far-field, uniform-propagation assumptions, with known but possibly mismatched array geometry.The data-driven formulation assumes access to training pairs of observations and DoA angles, or synthetic training data when ground-truth angles are unavailable.Source: B. Problem Formulation · p. 3

1) MB Narrowband Estimators:

  • Conventional beamforming maximizes steered response, while subspace methods such as MUSIC identify signal and noise subspaces but rely on model assumptions and are generally narrowband.Spatial smoothing can decorrelate coherent sources, but its effect is not obvious when their number is unknown.Source: 1) MB Narrowband Estimators · p. 3

2) MB Broadband Estimators:

  • Incoherent broadband methods combine separate frequency-bin estimates, whereas coherent methods focus frequency-bin covariances into a single narrowband covariance.Incoherent processing grows more computationally complex with each frequency bin; coherent methods can be biased over larger angular sectors and may require additional assumptions.Source: 2) MB Broadband Estimators · p. 4

3) DD Estimators:

  • A multi-label CNN can handle a potentially unknown and varying source count, but its number of classes grows exponentially as sources increase.It uses real, imaginary, and phase channels of the sample covariance to predict a grid of directions.Source: 3) DD Estimators · p. 4
  • CNNs trained to estimate a discretized MUSIC spectrum inherit MUSIC’s drawbacks and remain dependent on underlying model assumptions.The cited approaches use sample covariance inputs, including a spatially smoothed single-snapshot covariance in one method.Source: 3) DD Estimators · p. 4
  • A covariance-reconstruction network predicts a full covariance matrix from subarray covariance inputs for subsequent MUSIC recovery, but cannot react to MUSIC’s downstream performance.The method remains dependent on its subarray estimate.Source: 3) DD Estimators · p. 4
  • A hybrid framework uses FTMR matrices and EVD-derived eigenvalues and eigenvectors for source counting and a pseudo-MUSIC spectrum, but localization depends on selecting noise eigenvectors.That dependency makes the method less robust to an incorrect source-count estimate.Source: 3) DD Estimators · p. 4

D. MUSIC Algorithm

MUSIC estimates DoAs from the noise subspace obtained by eigendecomposing the empirical covariance matrix. Its performance depends on the signal model and covariance estimate, and it cannot reliably handle coherent sources.

D. MUSIC Algorithm

  • MUSIC takes the empirical covariance matrix of received measurements, applies EVD, and categorizes eigenvectors into signal and noise subspaces before forming a spatial spectrum.The spectrum has peaks at DoA angles.Source: D. MUSIC Algorithm · p. 4

1) Formal Derivation:

  • The eigenvectors associated with the smallest eigenvalues form the noise subspace, orthogonal to the signal subspace; MUSIC selects dominant spectrum peaks as estimated DoAs.The method estimates the number of signals from the multiplicity of the minimum covariance eigenvalue.Source: 1) Formal Derivation · p. 5
  • MUSIC is inherently narrowband and cannot reliably estimate coherent-source DoAs or their count when correlated signals become indistinguishable from noise in the covariance matrix.The algorithm is also sensitive to the accuracy of the empirical covariance estimate.Source: 1) Formal Derivation · p. 5

E. Coherent Broadband

DA-MUSIC preserves MUSIC’s flow while replacing sensitive steps with learned components. Coherent broadband processing focuses frequency-bin covariances for narrowband DoA recovery, though CSS focusing has initialization and subspace-alignment limitations.

E. Coherent Broadband

  • CSS focuses frequency-bin covariances into one covariance for narrowband DoA recovery by aligning signal subspaces across bins.The focusing matrices are chosen for a reference frequency and focusing angles.Source: E. Coherent Broadband · p. 5
  • CSS requires initial focusing values and may be sensitive to them; subspace alignment is not guaranteed to yield a viable covariance without disrupting the noise subspace.Source: E. Coherent Broadband · p. 5

A. Architecture

  • DA-MUSIC retains MUSIC’s structure while using neural networks to address model-sensitive subspace estimation and spectrum-to-DoA conversion.The architecture learns measurement correlations with an RNN and uses a neural peak-finder for DoA recovery.Source: §III · pp. 5–6, A. Architecture · p. 6

1) Known Number of Sources:

  • For a known source count, a GRU maps the input signal to a pseudo-covariance matrix, whose eigenvectors support subspace selection and spatial-spectrum estimation.Source: 1) Known Number of Sources · p. 6
  • An MLP converts uniformly sampled spatial-spectrum values into DoA angles, enabling resolution beyond the spectrum’s evaluation grid and end-to-end gradient-based training.Source: 1) Known Number of Sources · p. 6

2) Varying Number of Sources:

For varying source counts, DA-MUSIC adds a learned source-count classifier and a flexible noise-subspace selector, then emits a maximum-size ordered set of DoA estimates. Its supervised training uses permutation-invariant periodic angle error.

2) Varying Number of Sources:

  • For an unknown source count, a classifier maps eigenvalues to one of M −1 possible source-count classes instead of relying on their conventional magnitude inspection.Source: 2) Varying Number of Sources · p. 6

Computing noise subspace:

  • A learned selector assigns each eigenvector a probability of belonging to the noise subspace, allowing flexible selection when eigenvectors are difficult to distinguish.It weights the eigenvectors to form the selected noise subspace.Source: Computing noise subspace · pp. 6–7
  • The DoA output layer has M −1 slots, ordered by likelihood; the estimated source count determines how many leading outputs are retained.Source: Computing noise subspace · p. 7

B. Training Procedure

2) Unknown Number of Sources:

DA-MUSIC retains MUSIC’s subspace structure while replacing mismatch-sensitive components with neural modules; its training separates variable-source angle estimation from source-count classification.

2) Unknown Number of Sources:

  • For variable source counts, the RMSPE loss compares only the first D_u estimated angles with the true angles, while a separate classifier is trained with categorical cross-entropy.The remaining estimated angles are ignored when computing this loss.Source: 2) Unknown Number of Sources · p. 7
  • The classifier’s gradient is blocked from reaching the EVD, and the DoA estimator remains independent of the classifier.This permits using other classifiers or source-count estimation schemes.Source: 2) Unknown Number of Sources · p. 7

C. Discussion

  • Replacing empirical covariance estimation with an RNN lets DA-MUSIC learn a pseudo covariance from measurements and reduces the coherent-source problems of MUSIC.The learned covariance also supports subspace-based DoA recovery.Source: C. Discussion · p. 8
  • End-to-end learning lets DA-MUSIC form a focused pseudo covariance for broadband signals and learn to overcome array-geometry mismatch without renewed training.The spectrum still uses steering vectors, so the augmentation remains dependent on array geometry.Source: C. Discussion · p. 8
  • The RNN processes time-domain measurements, the internal EVD supports subspace division and source counting, and an MLP peak finder enables end-to-end training and learned angular resolution.The recurrent design can handle different signal durations; real-time response to changing DoAs is left for future work.Source: C. Discussion · p. 8

IV. NUMERICAL EVALUATIONS

The narrowband synthetic evaluations compare DA-MUSIC with conventional and learned estimators across source types and operating conditions. The reported results cover benchmark accuracy, resolution, geometry mismatch, SNR, and snapshot count.

A. Synthetic Narrowband

1) Known Number of Sources:

  • As the separation of two non-coherent sources approaches Δθ ≈ 0.1 radians, MUSIC collapses while DA-MUSIC maintains low RMSPE.Fig. 7 compares localization error as the angular separation varies.Source: Figure 7 · p. 9, 1) Known Number of Sources · p. 9
  • With corrupted steering-vector geometry, DA-MUSIC is reported to overcome the mismatch from data.The mismatch is introduced by zero-mean Gaussian noise on each steering-vector element.Source: Figure 8 · p. 9, 1) Known Number of Sources · p. 9
  • DA-MUSIC notably outperforms all considered benchmarks for both coherent and non-coherent sources, including non-coherent sources for which MUSIC is designed.The comparison reports RMSPE with a constant, known source count and T = 200 snapshots.Source: Table II · p. 9, 1) Known Number of Sources · p. 9
  • For five non-coherent signals at T = 200 snapshots, DA-MUSIC’s error is low and steady at positive SNR settings, decreasing slowly as SNR rises.The evaluated SNR range is [−20, 20] dB.Source: Figure 9 · p. 9, 1) Known Number of Sources · p. 10
  • With fewer snapshots, estimator performance degrades; data-driven estimators trained only at T = 200 nevertheless operate on shorter sequences during inference.The figure varies the number of snapshots for five signals.Source: Figure 10 · p. 9, 1) Known Number of Sources · p. 10

unknown and varying number of sources

The supplied passage identifies an RMSPE evaluation for a varying, unknown number of non-coherent signals.

unknown and varying number of sources

  • Fig. 13 reports RMSPE for a varying, unknown number of non-coherent signals with D ∈ {2, ..., 5}.The caption identifies the evaluation scope but does not state comparative outcomes.Source: Figure 13 · p. 10

2) Unknown and Varying Number of Sources:

The evaluations consider localization with an unknown, varying source count and synthetic broadband scenarios. The source-count results include a snapshot-dependent classifier trade-off, while the broadband figures report RMSPE for three scenarios.

2) Unknown and Varying Number of Sources:

  • For unknown, varying source counts, MUSIC’s coherent-source count-estimation limitation also harms localization performance.The reported RMSPE averages over D ∈ {2, ..., 5} for coherent and non-coherent signals.Source: Figure 14 · p. 10, 2) Unknown and Varying Number of Sources · p. 11
  • Source-count accuracy is evaluated across snapshot counts for non-coherent and coherent signals; DA-MUSIC’s classifier depends on snapshot count, while retroactive classifier training gives the most accurate predictions but requires separate training for each T.The DD estimators are trained at T = 200 in the described evaluation.Source: 2) Unknown and Varying Number of Sources · pp. 10–11

B. Synthetic Broadband

  • The synthetic broadband evaluation reports RMSPE for known D = 2 in each of three scenarios.The setup uses the broadband system model and scales the DA-MUSIC GRU to provide ten times more parameters for the more complex scenarios.Source: Figure 15 · p. 11, Figure 16 · p. 11, Figure 17 · p. 11

1) Broadband Scenario 1:

The three broadband scenarios use different carrier and OFDM constructions. DA-MUSIC performs best across them except in Scenario 3 at very large snapshot counts, and handles frequency ranges beyond its training range with fixed complexity.

1) Broadband Scenario 1:

  • Scenario 1 forms broadband signals by modulating narrowband signals on different carrier frequencies.The source values and carrier frequencies are randomly drawn from the stated distributions.Source: 1) Broadband Scenario 1 · p. 11

2) Broadband Scenario 2:

  • Scenario 2 uses OFDM signals on a shared carrier frequency, with bandwidth Δf_d = f_max − f_min.The OFDM subcarrier values are randomly drawn for each of the K subcarriers.Source: 2) Broadband Scenario 2 · pp. 11–12

3) Broadband Scenario 3:

  • Scenario 3 combines OFDM with different carrier frequencies; DA-MUSIC handles larger bandwidths and carrier frequencies than training with fixed complexity.The training signals have carrier frequencies and bandwidths within 0–1000 Hz, while broadband MUSIC requires more IFBs as the frequency range grows.Source: 3) Broadband Scenario 3 · p. 12
  • DA-MUSIC gives the most accurate estimates across the three broadband scenarios, except in Scenario 3 at T > 10^3, where broadband MUSIC is slightly better.Classic narrowband MUSIC fails on these broadband signals; DA-MUSIC is trained only at T = 200 yet performs similarly with more snapshots.Source: 3) Broadband Scenario 3 · p. 12

C. Non-Synthetic Data: Azimuth Estimation in Seismic Arrays

DA-MUSIC is evaluated on GERES seismic-array data and compared with model-based estimators. It outperforms those estimators in the reported comparison, and the conclusion describes its operation across source counts, bandwidths, and array-geometry mismatches.

C. Non-Synthetic Data: Azimuth Estimation in Seismic Arrays

  • The GERES evaluation uses data from a 25-seismometer array in the Bavarian Forest, with 2,816 events split into 2,534 training and 282 test events.The data were collected from October to December 2021, using 100-second sliding windows shifted by 25 seconds.Source: C. Non-Synthetic Data: Azimuth… · p. 12
  • DA-MUSIC outperforms the model-based estimators on seismic data, with errors comparable to the 0.4243 rad average from expert analysis.The comparison uses simplified, approximated model parameters; the model-based algorithms require prior elevation and frequency information to operate reliably.Source: C. Non-Synthetic Data: Azimuth… · p. 13, Table V · p. 13

V. CONCLUSIONS

  • The conclusion describes DA-MUSIC as operable with an unknown number of sources and broadband signals, and robust to severe array-geometry mismatches.It also reports resolution capability relative to model-based and data-driven benchmarks in various settings.Source: §V · p. 13
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