Boundary Matching Filters for Spherical Microphone and Loudspeaker Arrays

Conversion of microphone array signals into loudspeaker array signals is an essential process in high-definition spatial audio. This paper presents the theory of boundary matching filters (BMFs) for spherical array signal conversion. BMFs adapt the physical boundary conditions used during recording to the ones required for reproduction by relying on a theoretical framework provided by the Kirchhoff-Helmholtz integral equation (KHIE). Computationally, array signal conversion is performed in a transform domain where sound fields are represented in terms of spherical harmonic functions. Related research on transform-domain signal conversion filters is interpreted in the context of the KHIE. The case of a rigid recording boundary and an open reproduction boundary is addressed. The proposed rigid-to-open BMFs provide a suitable basis for designing gain-limited filters to deal with the problem of excessive gains at certain frequency bands, observed when using high-resolution arrays. Spatial, spectral, and temporal effects in sound field reconstruction when finite numbers of transducers are used in anechoic conditions are investigated analytically and exemplified numerically. Results show that the proposed gain-limited rigid-to-open BMFs outperform the existing gain-limited filters based on Tikhonov regularization because they reduce the spatial discretization effects and yield impulse responses that are more localized around their main peaks.