Symmetries in Complex-Valued Spherical Harmonic Processing of Real-Valued Signals

Paper in proceeding, 2025

Spherical harmonics (SHs) are widely used in audio and acoustics to represent sound fields and their spatial information. While SH expansions of real-valued functions can equivalently employ real- or complex-valued definitions of SH basis functions, real-valued definitions reduce computational load and storage needs. Researchers, however, often prefer a complex definition for its concise handling of operations like rotations and translations, which comes at the expense of partially redundant computations in practical implementations. To mitigate this downside, this work examines symmetries in expansions of real signals in terms of complex SHs in both the time and frequency domain and identifies redundancies akin to Hermitian symmetry in Fourier transforms. The resulting collection of the symmetry properties of SHs and circular harmonics (CHs) can be leveraged to limit computations to the non-redundant coefficients, significantly reducing the computational complexity and storage requirements in algorithms using complex-valued SHs.