Using spherical harmonics for modeling antenna patterns

A Generalized Spatial Correlation Model for 3D MIMO Channels Based on the Fourier Coefficients of Power Spectrums

“…2, the spherical coordinates of the wave vector are , and the spherical coordinates of i.e., the vector along are . Combining the addition theorem in (21) with (22), such that and using the resulting expression in (17) would expand to yield (23). Proposition 1: For a uniform linear array of antenna ports with arbitrary antenna patterns and for arbitrary angular distributions such that the and , the correlation between any pair of Tx antennas ports can be expanded in a systematic way to yield (24), where and .…”

“…where (θ 1 , φ 1 ) and (θ 2 , φ 2 ) are the spherical coordinates of the vectors v and x respectively, then the Legendre polynomial of argument cos(γ) is given by (21), where P m n are the associated Legendre polynomials.…”

“…2, the spherical coordinates (θ 1 , φ 1 ) of the wave vector kv are (θ, φ), and the spherical coordinates (θ 2 , φ 2 ) of x i.e., the vector along d t (s − s ) are ( π 2 , π 2 ). Combining the addition theorem in (21) with (22), such that cos γ=sin φ sin θ and using the resulting expression in (17) would expand ρ t (s − s ) to yield (23).…”

“…However, with the advent of smart antennas arXiv:1504.05455v1 [cs.IT] 21 Apr 2015 [19], it is important to take into account the characteristics and patterns of the directional antennas that are being widely deployed now. These antenna patterns have often been described in literature using spherical harmonics [20], [21], but this has not been done in the context of spatial correlation. A recent approach to incorporate these patterns in the correlation function was proposed in [22], but the method could not admit a closed-form solution for this case and succumbed to numerical integration methods.…”