Compressive Beamforming Based on Multiconstraint Bayesian Framework

Abstract

Compressive sensing (CS) is a promising technique recognized for its merits in recovering sparse signals with enhanced resolution entailing specific constraints. In recasting the CS model within a Bayesian framework, its formulation can be interpreted and solved under various prior assumptions that may correspond to, e.g., the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> or reweighted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> constraint. Bayesian CS (or Bayesian sparse learning, BSL) achieves improved resolution and robustness compared with the deterministic CS. Therefore, BSL has been invoked to solve the single and multisnapshot beamforming models for direction-of-arrival (DOA) estimation. However, the recovery performance deteriorates for complicated signals because of nonsparsity. In this article, a multiconstraint BSL approach is proposed to solve the multisnapshot beamforming model (termed M-MCRBSL), which reconstructs the amplitude and DOA of the source simultaneously. With a proper assembly of constraints, the source can be represented sparsely and the transformation coefficients can be recovered accurately in multiple sparse domains. As demonstrated in the simulations, M-MCRBSL outperforms other state-of-the-art multisnapshot beamforming methods gauged by both the normalized mean square error (NMSE) and the structural similarity (SSIM) index. In particular, a deficiency sensitivity experiment is devised to elaborate the feasibility of the invoked invertibility approximation. As attested with an underwater acoustics experiment, M-MCRBSL with joint identity and Haar wavelet constraints achieves improved performance in terms of enhanced resolution and suppressed interference sources.