Gaunt coefficients for complex and real spherical harmonics with applications to spherical array processing and Ambisonics

TL;DR

This paper defines and derives Gaunt coefficients for complex and real spherical harmonics, providing a matrix-based approach and MATLAB code for applications in spherical array processing and Ambisonics.

Contribution

It introduces a unified definition of Gaunt coefficients for both complex and real spherical harmonics, including a new derivation for real SHs and practical computational tools.

Findings

Provides a matrix formalism for spherical harmonic multiplication

Derives Gaunt coefficients for real spherical harmonics

Includes MATLAB code for computing coefficients up to specified SH orders

Abstract

Acoustical signal processing of directional representations of sound fields, including source, receiver, and scatterer transfer functions, are often expressed and modeled in the spherical harmonic domain (SHD). Certain such modeling operations, or applications of those models, involve multiplications of those directional quantities, which can also be expressed conveniently in the SHD through coupling coefficients known as Gaunt coefficients. Since the definition and notation of Gaunt coefficients varies across acoustical publications, this work defines them based on established conventions of complex and real spherical harmonics (SHs) along with a convenient matrix form for spherical multiplication of directionally band-limited spherical functions. Additionally, the report provides a derivation of the Gaunt coefficients for real SHs, which has been missing from the literature and can be used directly in spatial audio frameworks such as Ambisonics. Matlab code is provided that can compute all coefficients up to user specified SH orders. Finally, a number of relevant acoustical processing examples from the literature are presented, following the matrix formalism of coefficients introduced in the report.