Spherical Harmonic Expansion of Fisher-Bingham Distribution and 3-D Spatial Fading Correlation for Multiple-Antenna Systems

This paper considers the 3-D spatial fading correlation (SFC) resulting from an angle-of-arrival (AoA) distribution that can be modeled by a mixture of Fisher-Bingham distributions (FB-distributions) on the sphere. By deriving a closed-form expression for the spherical harmonic transform for the component FB-distributions, with arbitrary parameter values, we obtain a closed-form expression of the 3-D SFC for the mixture case. The 3-D SFC expression is general and can be used in arbitrary multiantenna array geometries and is demonstrated for the cases of a 2-D uniform circular array (UCA) and a 3-D regular dodecahedron array (RDA). In computational aspects, we use recursions to compute the spherical harmonic coefficients and give pragmatic guidelines on the truncation size in the series representations to yield machine precision accuracy results. The results are further corroborated through numerical experiments to demonstrate that the closed-form expressions yield the same results as significantly more computationally expensive numerical integration methods.