Robust High-Dimensional MVDR Beamforming under Heavy-Tailed Noise via Spiked Covariance Modeling

1. INTRODUCTION

• The SMI beamformer can deteriorate with limited snapshots or outlier-contaminated, heavy-tailed data because it substitutes the sample covariance for the unknown inverse covariance.MVDR minimizes output power subject to a distortionless response constraint toward the look direction.Source: §1 · p. 1
• Robust scatter estimation reduces the influence of impulsive observations, but high-dimensional spectral distortions in eigenvalues and eigenvectors are amplified by MVDR matrix inversion.Diagonal loading regularizes robust scatter estimates but does not directly exploit the low-rank signal-plus-interference structure, and can remain suboptimal.Source: §1 · p. 1
• The proposed method combines Maronna’s robust scatter estimator with spiked covariance techniques to address both high-dimensional spectral distortion and non-Gaussian snapshot contamination.It extends the random-matrix-optimized MVDR framework in to impulsive and heavy-tailed noise.Source: §1 · p. 1
• Simulations under elliptically distributed noise show stable performance and outperformance of competing beamformers over a broad range of signal-to-noise ratios.The preceding SCM-based spiked-covariance approach remains sensitive to impulsive and heavy-tailed observations.Source: §1 · p. 1

2.1. Signal model and optimal MVDR beamforming

• The classical MVDR beamformer minimizes array output power while preserving the signal of interest, but its solution requires an estimate of the unknown inverse covariance matrix.The array model has N sensors and m < N narrow-band sources; its noise is modeled as a positive texture scaling a unitarily invariant vector.Source: §2.1 · p. 1

2.2. Sample-based implementation of MVDR beamforming with robust scatter estimator

• The plug-in MVDR beamformer uses an estimated inverse covariance, and imperfect estimation increases normalized total output power relative to the oracle optimum.The SMI version can incur substantial output-power loss with heavy-tailed noise or when dimension and sample size are of the same order.Source: §2.2 · p. 2
• Maronna’s robust scatter estimator reduces sensitivity to outliers and heavy-tailed samples, but does not remove finite-sample spectral distortions when sensor and snapshot counts are comparable.For a proper choice of α, the estimator corresponds to the maximum-likelihood scatter estimator under multivariate Student-distributed observations.Source: §2.2 · p. 2
• The proposed optimized robust precision estimator exploits the low-rank spiked structure and is designed to minimize MVDR output power while addressing heavy-tailed noise and high-dimensional finite-sample effects.It builds on Maronna’s robust scatter estimate rather than relying on generic covariance estimation alone.Source: §2.2 · p. 2

3.1. Robust MVDR beamforming based on spiked covariance models

• The estimator hard-clips noise-subspace eigenvalues and optimizes shrinkage coefficients for the leading spike-associated eigenvalues to minimize normalized MVDR output power.The population spike strengths and eigenvectors define the signal subspace, while the remaining eigenvalues are set to 1/σ^2.Source: §3.1 · p. 2

3.2. Robust Deterministic Equivalent ¯ρ(w) and the optimal w∗

• Under the stated large-system assumptions, the analysis derives a deterministic equivalent of output power and characterizes the oracle weights that minimize it.The asymptotic regime has N/n→c>0 and a fixed number of spikes.Source: §3.2 · pp. 2–3

3.3. Estimated optimal weights ˆw∗and proposed algorithm

• The practical method consistently estimates the oracle weights from leading robust-scatter eigenvalues and eigenvectors, yielding the proposed robust spiked MVDR beamformer.The resulting construction is summarized in Algorithm 1.Source: §3.3 · p. 3

4. NUMERICAL SIMULATIONS

• Simulations use a half-wavelength-spaced uniform linear array and normalized Student-type texture with β = 5; smaller β produces more impulsive snapshots.The baseline experiments average 200 independent Monte Carlo trials.Source: §4 · pp. 3–4
• With n = 2N, empirical E[ρ(ŵ*)] converges toward the deterministic equivalent as N and n increase, remains close to oracle ρ_min, and is below SMI across the tested N range.Fig. 1 reports the ρ-performance comparison at SNR = 5 dB and INR = 30 dB.Source: Figure 1 · p. 4, §4.1 · p. 4

4.2. Beamformer Performance and Comparison with Baselines

• The proposed beamformer consistently achieves the highest output SINR among sample-based methods and remains closest to oracle MVDR; it is comparable to robust diagonal loading at low SNR and robust eigen-subspace at high SNR.The comparison is reported in Fig. 2(b).Source: Figure 2 · p. 4, §4.2 · p. 4
• At SNR = 5 dB, proposed hR,MVDRopt places clear nulls toward interferers while maintaining a stable look-direction response; robust baselines suppress interference less, while hSMI exhibits higher sidelobes and noise floor.Source: §4.2 · p. 4

• The method constructs an MVDR-oriented precision matrix estimator by combining Maronna’s robust scatter estimator with spiked covariance modeling.Source: §5 · p. 4
• Random matrix theory yields a deterministic equivalent of MVDR output power, closed-form asymptotically optimal shrinkage weights, and a fully sample-based implementation.Source: §5 · p. 4
• Under elliptically distributed noise, simulations show stronger interference suppression and higher output SINR than competing methods.Source: §5 · p. 4