Diagonally Loaded LMS Beamforming for Underwater Acoustic Arrays: Steady-State Limits and Robustness in Anisotropic, Impulsive, and Non-Stationary Noise

Abstract

Adaptive array processing is a key enabler of interference-resilient underwater acoustic (UWA) links, yet the noise seen by UWA receivers is spatially anisotropic, impulsive, and non-stationary, conditions under which low-complexity stochastic-gradient beamformers are poorly characterized. This paper analyzes the Griffiths-type least-mean-squares beamformer with diagonal loading and a power-normalized step size (LMS-DL) on a ten-element half-wavelength line array, and benchmarks it against sample-matrix-inversion (SMI-DL), normalized LMS (NLMS-DL), and exponentially weighted recursive-least-squares (EW-RLS-DL) beamformers. A closed-form approximation of the steady-state output signal-to-interference-plus-noise ratio (SINR) of LMS-DL is derived. It shows that the misadjustment is driven by the total output power, so the SINR saturates at 10log₁₀(2/γ) dB for step-size parameter γ; Monte Carlo simulations show the prediction is conservative and accurate to within 0.75 dB at SNR ≤ 10 dB. For a 10 dB signal and interferers at 20 and 40 dB, LMS-DL settles 3.3 dB below the optimum after about 10⁴ snapshots, whereas SMI-DL comes within 1 dB in 681 snapshots. In Cron–Sherman surface-generated noise the optimum gains 7.4 dB over white noise; SMI-DL recovers 6.4 dB of this gain but LMS-DL only 2.5 dB. Symmetric α-stable noise with α = 1.2 costs LMS-DL 5.6 dB, a loss that per-snapshot normalization (NLMS-DL) reduces to 0.3 dB. LMS-DL tracks a moving 40 dB interferer (worst-case loss 4.1 dB) but not a moving 20 dB interferer (13.3 dB). A link budget with Thorp absorption and Wenz noise shows that converged LMS-DL extends the QPSK range at a bit-error rate (BER) of 10⁻³ from 3.4 km to 15.2 km (optimum 16.0 km), but leaves an error floor of 2–5×10⁻⁵ at short range. The analysis yields design rules for the step size and for SNR-dependent loading in UWA array receivers.