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

Area of Science:

• Acoustics
• Signal Processing
• Computational Mathematics

Background:

• Spherical harmonic transforms are crucial for analyzing data on spheres.
• Oversampling data can improve transform robustness but requires careful analysis.
• Noise sensitivity in transforms impacts accuracy in applications like sound field reconstruction.

Purpose of the Study:

• To investigate how spherical oversampling affects the robustness of spherical harmonic transforms.
• To quantify the relationship between oversampling and noise sensitivity reduction.
• To establish a rule for selecting appropriate sampling parameters in acoustic scenarios.

Main Methods:

• Monte Carlo simulations were used to evaluate error propagation and noise sensitivity.
• Random spherical functions served as ground truth data.
• Numerical experiments were conducted under mathematical and acoustic conditions.

Main Results:

• A quantitative 'sensitivity decrease rule' was derived: sensitivity decreases by -10 log10η (dB) with increasing sampling inefficiency η = Q/(L+1)².
• This rule was shown to be applicable across various spherical sampling schemes (Fibonacci spirals, equiangular sampling, spherical t-designs).
• The rule was validated in acoustic case studies, including sound field interpolation and reconstruction.

Conclusions:

• Spherical oversampling demonstrably increases the robustness of spherical harmonic transforms.
• The derived sensitivity decrease rule provides a practical method for parameter selection.
• Applying this rule can enhance the accuracy and reliability of acoustic signal processing techniques.