It is noteworthy that the coefficients without elevational dependency are identical to an azimuthal Fourier transform. The first Spherical Harmonic functions up to order 3 are illustrated in the figure below with the radius indicating the functions absolute value per angle and the color indicating the sign. An arbitrary pressure distribution on the sphere can be described as a linear combination of these basis functions. These basis functions have an angular dependency. Spherical Harmonics are used as a set of basis functions in order to transform the pressure distribution on the surface of the sphere into a more abstract representation. The simple spherical geometry allows for a implicit solution of the integral. In mathematical terms this is denoted as the Kirchhoff-Helmholtz Integral. Since the sound wave propagation underlies the principles of the wave equation, the knowledge of the pressure on the surface is sufficient to uniquely determine the pressure in the entire solenoidal interior of the sphere. The idea behind Ambisonics is to describe the sound pressure distribution on the surface of an imaginary sphere around a reference point in the sound field. The basic idea of Ambisonics is attributable to Michael Gerzon but was not substantially impelled until the extension of first order B-Format to Higher Order Ambisonics (HOA) in the mid 90s due to researchers such as Jérôme Daniel amongst others. The sound recorded using a spherical microphone array can easily be transformed into a domain which makes the sound field representation independent of the actual microphone setup. The great advantage of spherical setups is the rotational invariance.
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