Two-dimensional multi-frequency synchronous grid-free compressive beamforming

Two-dimensional grid-free compressive beamforming with rectangular planar microphone arrays has attracted considerable attention due to its wide source identification region (the hemispherical space in front of arrays), high accuracy (overcoming basis mismatch), and robust performance. However, the uniform distribution of microphones tends to result in spatial aliasing, i.e., the existing single-frequency methods fail to achieve accurate localization for the sources with wavelengths shorter than twice the minimum spacing between microphones. This paper proposes a two-dimensional multi-frequency synchronous grid-free compressive beamforming method, aimed at mitigating spatial aliasing and extending its applicability beyond the sampling limit imposed by the array geometry. First, a two-dimensional multi-frequency synchronous atomic norm minimization problem is formulated and then converted into an equivalent dual problem. Subsequently, a spatial spectral function is constructed from the solution of the dual problem to extract source positions and strengths. Simulation and experiment results show that when the spatial Nyquist criterion is satisfied, both the proposed multi-frequency synchronous method and existing single-frequency methods can accurately identify sources, with the proposed method reducing strength quantification error by over 80% under strong interference. When the spatial Nyquist criterion is violated, the proposed method can effectively suppress spatial aliasing inherent in the existing single-frequency methods and accurately identify common engineering noise sources with high-frequency components, such as electric motor noise, gear whine, and bearing noise.