Two-Dimensional Off-Grid Beamforming Acoustic Source Identification for Planar Microphone Arrays via Unfolding Delay-and-Sum

The classical sparsity-based source identification method encounters the basis mismatch problem due to discretizing the focus region and assuming that acoustic sources are on-grid. This limitation results in degraded performance when sources do not align precisely with grid points. The off-grid method based on the first-order Taylor expansion offers a solution to this problem. However, when using coarse search grids, the Taylor expansion of the transfer function vectors at the grid points does not approximate the actual transfer function vectors well, leading to the performance deterioration of the off-grid model. To address basis mismatch, this article introduces a 2-D off-grid acoustic source identification method based on unfolding delay-and-sum (OG-UDAS). This approach develops a novel off-grid delay-and-sum (DAS) model by leveraging the orthogonality between the transfer function vector obtained from the first-order Taylor expansion and the noise matrix (NM). OG-UDAS is a quadratic programming problem that is unfolded into a polynomial optimization function with respect to the acoustic source off-grid deviation. Subsequently, by minimizing the objective function, the closed-form solutions for source off-grid deviation are obtained using the difference method (DM). The strength estimate of the source is obtained using the least-squares method (LSM). Using the estimated off-grid deviation and strength estimation can be alternately learned iteratively to the actual source. Both simulations and experiments demonstrate that OG-UDAS effectively alleviates the basis mismatch problem, even with a limited number of microphones. Compared to existing off-grid methods, this approach achieves accurate acoustic source identification even with coarse search grids.