Planewave decomposition represents a sound field as the superposition of cylindrical or spherical
waves. The decomposition enables a threedimensional spatial analysis of an enclosed volume only
using the pressure data on the surface of the microphone. The data extracted from the decomposition
can be applied to beamforming and spatial audio reproduction techniques. This paper will focus on the
necessary tools to calculate the decomposition using a spherical array, spherical Fourier transform, and
spherical convolution.

The spherical Fourier transform of a function $f(\theta,\phi)$ is defined as:

$\mathcal{S}\{f(\theta, \phi)\}=\int_0^{2\pi}\int_0^{pi}f(\theta,\phi){Y_n^m}^{*}(\theta,\phi)sin\theta d\theta d\phi = f_{nm} \\ \\ \mathcal{S}^{-1}\{f_{nm}\}=\sum_{n=0}^\infty\sum_{m=-n}^nY_n^m(\theta,\phi)=f(\theta, \phi)$

where $\mathcal{S}$ and $\mathcal{S}^{-1}$ are forward and inverse transforms, respectively. Spherical harmonics, ${Y_n^m}(\theta,\phi)$, are an orthonormal basis that describes the angular dependence of a given function. The orthonormality states that the inner product of two spherical harmonics is equal to one only if the order n and degree m are the same while all being normalized. This completeness property gives an insight into the linear independence of spherical harmonics and its ability to approximate any square-integrable function as a sum of the eigenfunctions.

Spherical convolution between functions f and k is defined

$f(\gamma)*k(\gamma) = \int_0^{2\pi}\int_0^{2\pi}\int_0^{\pi} f(R_z(\theta)R_y(\phi)R_z(\Psi))k(R_z^{-1}(\theta)R_y(\phi)R_z(\Psi)) R_z(\theta)R_y^{-1}(\phi)R_z^{-1}(\Psi)sin\theta \, d\theta\, d\phi\, d\Psi$

where * denotes convolution and $R_z(\theta)R_y(\phi)R_z(\Psi)$ represents rotation about the y and z axes. Typical convolution involves the product of a reversed and shifted function with another function; however, spherical convolution rotates the function in all directions instead of shifting. The multiplication property also holds in the spherical domain, where convolution in the time domain transforms to multiplication in the spherical domain, but the operation is no longer commutative. The Fourier coefficients m=0 degree are the only values of the rotated function that participate in the convolution, making the order of the functions significant during the convolution.