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

TL;DR

This paper derives a closed-form spherical harmonic expansion for Fisher-Bingham distributions on the sphere to model 3D spatial fading correlation in multi-antenna systems, enabling efficient analysis across various array geometries.

Contribution

It introduces a novel closed-form spherical harmonic expansion for Fisher-Bingham distributions, facilitating accurate and efficient modeling of 3D spatial fading correlation in multi-antenna systems.

Findings

Closed-form expression for 3D spatial fading correlation derived.

Method applicable to arbitrary array geometries demonstrated.

Numerical validation confirms accuracy and efficiency.

Abstract

This paper considers the 3D spatial fading correlation (SFC) resulting from an angle-of-arrival (AoA) distribution that can be modelled by a mixture of Fisher-Bingham distributions on the sphere. By deriving a closed-form expression for the spherical harmonic transform for the component Fisher-Bingham distributions, with arbitrary parameter values, we obtain a closed-form expression of the 3D-SFC for the mixture case. The 3D-SFC expression is general and can be used in arbitrary multi-antenna array geometries and is demonstrated for the cases of a 2D uniform circular array in the horizontal plane and a 3D regular dodecahedral array. 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.