Abstract
Microphone and speaker array designs have increasingly diverged from simple topologies due to diversity of physical host geometries and use cases. Effective beamformer design must now account for variation in the array’s acoustic radiation pattern, spatial distribution of target and noise sources, and intended beampattern directivity. Relevant tasks such as representing complex pressure fields, specifying spatial priors, and composing beampatterns can be efficiently synthesized using spherical harmonic (SH) basis functions. This paper extends the expansion of common stationary covariance functions onto the SHs and proposes models for encoding magnitude functions on a sphere. Conventional beamformer designs are reformulated in terms of magnitude density functions and beampatterns along SH bases. Applications to speaker far-field response fitting, cross-talk cancelation design, and microphone beampattern fitting are presented.
Highlights
Acoustic array designs have undergone many iterations of improvement as physical simulation data becomes more accurate and abundant
Cross-validation experiments show that the mixture of encodings (MoE)-radial basis function (RBF) models have superior bias-variance trade-off compared to direct spherical harmonic (SH) fitting via the truncated singular value decomposition (TSVD) method
Spatial density functions and beampattern targets are designed using MoE and product of encodings (PoE) methods. This enabled conventional beamforming to be reformulated in terms of probabilistic directivity index (PDI) and probablistic steering vectors sampled from the SH encoded far-field response and spatial density
Summary
Acoustic array designs have undergone many iterations of improvement as physical simulation data becomes more accurate and abundant. Magnitude function synthesis on a sphere is less wellknown in beamforming but appears in interpolation and Bayesian methods such as Kriging [32] and Gaussian processes [28] We show how these techniques that specify covariance or kernel functions can relate acoustic directivity with spherical probablistic density functions to better design, fit, and evaluate beampatterns. This improves upon several prototyping stages which we list as the following contributions:. The Legendre polynomial Pl(cos ) of the angular distance 0 ≤ ≤ π is expanded into products of SHs evaluated at free coordinates (θ, φ) w.r.t
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