A method for converting between cylindrical and spherical harmonic representations of sound fields

Spherical microphone and circular microphone arrays are useful for sampling sound fields that may be resynthesized with loudspeaker arrays. Spherical microphone arrays are desirable because of their ability to capture three-dimensional sound fields, however it is often more practical to construct loudspeaker arrays in the form of a closed circle located in the horizontal plane. This leads to a spatial undersampling as such a circular sampling can only yield a perfect representation of a height-invariant sound field. This paper investigates the consequences of such spatial undersampling by converting between cylindrical and spherical harmonic decompositions of solutions to the wave equation. We show analytically and via numerical simulations that 1) the result of the spatial undersampling is a purely horizontally propagating sound field, and 2) the ratio of travelling and standing components in the undersampled sound field varies depending on the incidence colatitude. The conversion is also used in a beamforming scenario and shows that the beamformer response becomes increasingly omnidirectional as the source moves away from the horizontal plane.