[Paper Review] Sparse Bayesian Learning for DOA Estimation in Heteroscedastic Noise

Peter Gerstoft, Santosh Nannuru+2 more|arXiv (Cornell University)|Nov 8, 2017

Direction-of-Arrival Estimation Techniques41 references3 citations

TL;DR

This paper proposes a sparse Bayesian learning (SBL) framework for direction-of-arrival (DOA) estimation in heterogeneous noise environments where noise variance varies across sensors and snapshots. By jointly estimating source DOAs, powers, and heteroscedastic noise variances using a stochastic maximum likelihood approach, the method outperforms conventional high-resolution and phase-only methods, especially for closely spaced sources, with significant RMSE reduction in simulations at low SNR and multi-snapshot scenarios.

ABSTRACT

The paper considers direction of arrival (DOA) estimation from long-term observations in a noisy environment. In such an environment the noise source might evolve, causing the stationary models to fail. Therefore a heteroscedastic Gaussian noise model is introduced where the variance can vary across observations and sensors. The source amplitudes are assumed independent zero-mean complex Gaussian distributed with unknown variances (i.e. the source powers), inspiring stochastic maximum likelihood DOA estimation. The DOAs of plane waves are estimated from multi-snapshot sensor array data using sparse Bayesian learning (SBL) where the noise is estimated across both sensors and snapshots. This SBL approach is more flexible and performs better than high-resolution methods since they cannot estimate the heteroscedastic noise process. An alternative to SBL is simple data normalization, whereby only the phase across the array is utilized. Simulations demonstrate that taking the heteroscedastic noise into account improves DOA estimation.

Motivation & Objective

• Address the limitation of stationary noise models in long-term DOA estimation, where noise variance evolves over time and across sensors.
• Develop a robust DOA estimation framework that accounts for spatiotemporal noise heteroscedasticity, which degrades performance in conventional methods.
• Improve DOA estimation accuracy for weak, closely spaced sources by jointly estimating source powers, DOAs, and noise variances using a parametric Bayesian model.
• Demonstrate that SBL with heteroscedastic noise estimation outperforms data normalization (phase-only processing) and standard SBL in challenging noise environments.

Proposed method

• Formulate a multi-snapshot multiple measurement vector (MMV) model where sensor array data is modeled as a linear combination of steering vectors corrupted by heteroscedastic complex Gaussian noise.
• Assume source amplitudes follow a zero-mean complex Gaussian prior with unknown variances (source powers), and noise variances are independently distributed across sensors and snapshots.
• Apply sparse Bayesian learning (SBL) with maximum a posteriori (MAP) estimation to jointly infer DOAs, source powers, and noise variance parameters.
• Incorporate a stochastic maximum likelihood (SML) approach to estimate noise variances, enabling robust estimation even with limited snapshots.
• Use a fast SBL algorithm augmented with per-snapshot and per-sensor noise variance estimation, avoiding reliance on eigenvalue decomposition or EM-based methods.
• Compare performance against phase-only processing (CBF-Phase), standard SBL, and EM-based noise estimation, using RMSE and histogram analysis of DOA peak locations.

Experimental results

Research questions

• RQ1 Can SBL with heteroscedastic noise modeling improve DOA estimation accuracy compared to stationary noise models in long-term observations?
• RQ2 How does joint estimation of source powers, DOAs, and per-snapshot/per-sensor noise variances affect performance for closely spaced sources?
• RQ3 To what extent does phase-only processing (magnitude normalization) remain effective when sources are closely spaced and noise is nonstationary?
• RQ4 How do different noise estimation strategies (SBL vs. EM) impact convergence and DOA estimation accuracy in low-SNR and multi-snapshot scenarios?

Key findings

• SBL with heteroscedastic noise estimation (SBL3) achieves significantly lower RMSE than CBF-Phase and standard SBL, especially at low SNR (e.g., -15 dB) and for three closely spaced sources.
• The histogram of top three DOA peaks for SBL3 is tightly concentrated around true DOAs, indicating superior localization accuracy compared to other methods.
• SBL3 estimates noise standard deviations with low bias (mean deviation of 0.007) and good variance, even when estimating from a single observation per sensor-snapshot pair.
• The EM-based noise estimation in SBL2 leads to a mean noise variance ratio (σ²_Est / σ²_T) of ~0.5 at SNR -10 dB, causing earlier performance degradation compared to SBL2 with direct SML estimation.
• For three sources at [-3°, 2°, 50°], SBL3 reduces RMSE by up to 5 dB compared to SBL2 with EM noise estimation, demonstrating robustness in nonstationary noise.
• With 50 snapshots and SNR 0 dB, SBL3 maintains accurate noise variance estimation (mean deviation near zero), confirming stability in multi-snapshot settings.