Contribution
Symmetries in Complex-Valued Spherical Harmonic Processing of Real-Valued Signals
* Presenting author
Abstract:
Spherical harmonics (SHs) are widely used in audio and acoustics to describe and render directional properties of sound fields. Acoustic problems are typically based on real-valued functions of time, most commonly describing sound pressure in space. In this case, an SH expansion can be performed using complex or real-valued SH definitions. Although choosing a real-valued definition reduces the computational load and the required storage, complex-valued definitions are often in use as many operations such as translations and rotations, and properties like recurrence relations can be formulated mathematically more straightforwardly in the complex domain. This work explores the inherent symmetries in complex-valued SH expansions of real signals in the time and frequency domain and highlights redundancies similar to Hermitian symmetry in Fourier transforms. We examine the symmetry properties of SHs and circular harmonics (CHs) and provide relations for the conversion between real-valued and complex-valued SH and CH coefficients as an alternative. The properties can be leveraged to limit computations to the non-redundant coefficients, significantly reducing computational complexity and storage requirements in algorithms using complex-valued SHs.