Bounds on the Spatial Richness of Multipath

In this paper wireless multipath elds are modelled using classes of orthogonal func- tional expansions based on the solutions to the Helmholtz (wave) equation. These expansions permit a multipath eld|generated by any num- ber of neareld, fareld, specular and diuse mul- tipath reections|to be modelled to any preci- sion in a region of interest. Two expansions are provided, one suitable for multipath elds which show no variation with height, and the other suit- able for general 3D elds. We establish that the dimensionality of the functional expansion, and thereby the multipath eld, scales with the size of the boundary of the region in space. When the region of interest is small in wavelengths, typical of antenna arrays apertures in practice, then the multipath eld can be modelled by only a small number, given by the dimensionality, of arbitrar- ily located sources. Multipath eld synthesis can be done using combinations of arbitrarily located distinct fareld, neareld, point and spatially dis- tributed sources. These results establish rigorous bounds on the spatial richness of multipath.