Estimating a robustness increase in spherical harmonic transforms resulting from oversampling on a sphere

This study explores the increase in robustness, or decrease in sensitivity, of spherical harmonic transforms resulting from spherical oversampling. To analyze the error propagation through spherical harmonic transforms under spherical oversampling, the transformation sensitivity to noise contamination, such as thermal noise, was evaluated through Monte Carlo simulations that employ random spherical functions as the ground truth. Numerical experiments were first conducted under a mathematical scenario, revealing a quantitative connection between the degree of oversampling and the sensitivity decrease in spherical harmonic transforms. This connection is termed the sensitivity decrease rule, which states that the sensitivity decreases by -10 log10η (dB) with increasing sampling inefficiency η=Q/(L+1)2, where Q is the number of sampling points and L is the maximum order of the spherical functions to be sampled. This rule applies to various spherical sampling schemes, including Fibonacci spirals, equiangular sampling, and spherical t-designs. Numerical case studies then examined the applicability of the rule to acoustic problems, such as sound field interpolation on a sphere and sound field reconstruction using a spherical microphone array, suggesting that the rule facilitates an appropriate selection of lower truncation orders to enhance the robustness of the interpolation and reconstruction.