Body of Mathematics for Analysis of Three-Dimensional Multipath Wireless Channels: Spherical Harmonics

Abstract

The commonly accepted visualization of the electromagnetic field in the receiving area consists in thinking of it as a sum of plane waves (rays) produced by various sources. However, such conceptualization is convenient for small quantities of rays only. In case of a multipath channel with a limited-size receiving area, there is good reason to represent the field as an integration of spherical harmonic fields. The following reasoning substantiates the advantage of such conceptualization. First, spherical harmonics allow a vector-to-scalar transition in field description (i.e. a change-over from the strength vectors E and H to electric and magnetic potentials). Second, spherical harmonics make an orthogonal system of functions on the sphere. The coefficients of the electromagnetic field expansion into spherical harmonics allow us to represent a continuous field in terms of a discrete set of complex numbers.