Fast Fourier-based deconvolution for three-dimensional acoustic source identification with solid spherical arrays

Being capable of demystifying the acoustic source identification result fast, Fourier-based deconvolution has been studied and applied widely for the delay and sum (DAS) beamforming with two-dimensional (2D) planar arrays. It is, however so far, still blank in the context of spherical harmonics beamforming (SHB) with three-dimensional (3D) solid spherical arrays. This paper is motivated to settle this problem. Firstly, for the purpose of determining the effective identification region, the premise of deconvolution, a shift-invariant point spread function (PSF), is analyzed with simulations. To make the premise be satisfied approximately, the opening angle in elevation dimension of the surface of interest should be small, while no restriction is imposed to the azimuth dimension. Then, two kinds of deconvolution theories are built for SHB using the zero and the periodic boundary conditions respectively. Both simulations and experiments demonstrate that the periodic boundary condition is superior to the zero one, and fits the 3D acoustic source identification with solid spherical arrays better. Finally, four periodic boundary condition based deconvolution methods are formulated, and their performance is disclosed both with simulations and experimentally. All the four methods offer enhanced spatial resolution and reduced sidelobe contaminations over SHB. The recovered source strength approximates to the exact one multiplied with a coefficient that is the square of the focus distance divided by the distance from the source to the array center, while the recovered pressure contribution is scarcely affected by the focus distance, always approximating to the exact one.