Ultraspherical/Gegenbauer polynomials to unify 2D/3D Ambisonic directivity designs

TL;DR

The paper introduces axisymmetric Ambisonic directivity design using ultraspherical Gegenbauer polynomials to provide a dimension-agnostic framework for 2D/3D panning. It derives the axisymmetric Laplacian eigenproblem, establishes orthogonality and transforms, and defines continuous directivity patterns across dimensions with explicit metrics (P, E, Q, $\bm r_V$, $\bm r_E$, FBR). It then develops and analyzes multiple design families (basic/maximum-DI, max-$r_E$, supercardioid, in-phase, Butterworth-like, cap functions) and provides closed-form or computable expressions for their coefficients $a_n$ and performance, along with discretization guidance via $t$-designs on circles and spheres. The work connects 2D Ambisonic circular designs (via Chebyshev polynomials) and 3D spherical designs (via Legendre polynomials) through addition theorems and unified orthogonal bases, enabling uniform derivations of directivity patterns and optimal discretizations. Together, these results offer analytic tools for axisymmetric beam pattern synthesis and discretization across arbitrary dimensions, with practical recipes for achieving desired directivity and front-to-back characteristics in spherical microphone arrays and Ambisonic arrays.

Abstract

This report on axisymmetric ultraspherical/Gegenbauer polynomials and their use in Ambisonic directivity design in 2D and 3D presents an alternative mathematical formalism to what can be read in, e.g., my and Matthias Frank's book on Ambisonics or Jérôme Daniel's thesis, Gary Elko's differential array book chapters, or Boaz Rafaely's spherical microphone array book. Ultraspherical/Gegenbauer polynomials are highly valuable when designing axisymmetric beams and understanding spherical t designs, and this report will shed some light on what circular, spherical, and ultraspherical axisymmetric polynomials are. While mathematically interesting by themselves already, they can be useful in spherical beamforming as described in the literature on spherical and differential microphone arrays. In this report, these ultraspherical/Gegenbauer polynomials will be used to uniformly derive for arbitrary dimensions D the various directivity designs or Ambisonic order weightings known from literature: max-DI/basic, max-rE , supercardioid, cardioid/inphase. Is there a way to relate higher-order cardioids and supercardioids? How could one define directivity patterns with an on-axis flatness constraint?

Figures (9)

• Figure 1: Ultraspherical polynomials $\mathcal{P}_n(x)=\frac{n!}{(2\alpha)^{\overline{n}}}C_n^{(\alpha)}(x)|_{1}=1$ for $\mathrm{D}=2,3,4$, $\alpha=\frac{D-2}{2}$, i.e. Chebyshev polynomials $T_n(x)$ for $\mathrm{D}=2$, and Legendre polynomials $P_n(x)$ for $\mathrm{D}=3$. They are orthogonal wrt. the weight $w(x)=\sqrt{1-x^2}^{\mathrm{D}-3}$.
• Figure 2: Basic and max-$r_\mathrm{E}$, $\mathrm{D}=2,3$ (solid,dashed), $\mathrm{N}=1,2,5$ (green,red,blue).
• Figure 3: Supercardioid and in-phase, $\mathrm{D}=2,3$ (solid,dashed), $\mathrm{N}=1,2,5$ (blue,red).
• Figure 4: Examples for maximally flat designs with $\mathrm{L}+1$ flatness degrees on axis, and a zero with $\mathrm{M}+1$ flatness degrees for $\mathrm{N}=\mathrm{L}+\mathrm{M}+1$ and for $\mathrm{D}=2$ and $\mathrm{D}=3$.
• Figure 5: Cap functions for $\mathrm{D}=2,3$ (solid,dashed), and $\mathrm{N}=7,30$ (blue,red).