Spherical Harmonics Symmetries

(Franz Zotter)

The spherical harmonics exhibit some basic types of symmetries, which can be exploited in order to reduce the set of Ambisonics channels in incomplete sphere layouts.

Mainly, this option is based on the ambiguity arising if only parts of the spheres are used as range for the harmonics. The spherical harmonics are only orthonormal on a complete sphere.

Most obviously, and most suitably, the following symmetries can be successfully exploited when dealing with rotational symmetry or hemispheres:

• rotational symmetry around the z-axis
• mirror (even) symmetry with respect to the y-axis
• mirror (even) symmetry with respect to the z-axis