Beamforming with double-sided, acoustically hard planar arrays (2019)

Several types of microphone arrays have been proposed for the purpose of capturing spatial audio signals. When the aim is to capture a full 3D sound field, the most obvious geometry is that of a sphere, and this is the array shape which has received the most attention from authors in this field. The wave equation and its solutions can be separated into the spherical coordinates, and for an internal problem like an open microphone array, the solutions are linear combinations of the well-known ambisonic basis functions. These lend themselves well to the analysis and optimization of spherical arrays. One important observation is that the radial part of these solutions is described by the spherical Bessel functions and that these oscillate around zero. This leads to the problem that a single-radius open array (i.e. one which does not perturb the sound field) will not be able to function over a wide frequency band.An open multi-radius array is able to overcome the problem by simultaneously sampling the Bessel functions at different points, chosen such that not all of them are zero at the same time. Another solution is to sample the sound field with a mixture of zeroth- and first-order microphones. Since the radial pressure gradient is maximal where the pressure crosses zero, these different microphone types can complement each other to produce a continuous coverage over a wide frequency range. However, the most popular solution to the problem is to mount zeroth-order microphones on a hard spherical shell. The scattering off the shell is maximal at those frequencies where the pressure of the incident field is zero at the radius of the shell. The resulting frequency response of the total sound field on the shell is a smooth function without zero crossings.Another geometry which has been studied for the same purpose is the planar array. This effectively samples the sound field in the x-y plane. The shape of the ambisonic basis functions is such that about half of them are zero in this plane. As with the spherical arrays, there are three possible solutions to this problem; the microphones can be replaced with a mixture of zeroth- and first-order microphones, the array can be comprised of several parallel layers of microphones, or a scattering surface can be introduced in the x-y plane, with zeroth-order microphones placed on both sides. A paper concerning this last solution has only recently been published. So far, these arrays have only been studied with regards to their suitability for capturing ambisonic signals.Among the studies that have been published concerning spherical arrays, several authors have proposed beamforming methods that perform better than methods that use ambisonic signals as an intermediate representation of the sound field. In particular, it has been shown that effective beamforming can be achieved beyond the frequencies where the ambisonic signals become unusable due to spatial aliasing.The current paper proposes beamforming methods for acoustically hard planar arrays and compares these with methods using intermediate ambisonic signals. The methods are tested numerically and experimentally on a 17 cm diameter disc-shaped array with 84 microphones in a 7-by-6-by-2 configuration (6 rings of 7 microphones on each side). The implications of these techniques on the optimal microphone layout of the array are addressed.