Intrinsic finite dimensionality of random multipath fields

We study the dimensions or degrees of freedom of random multipath fields in wireless communications. Random multipath fields are presented as solutions to the wave equation in an infinite-dimensional vector space. We prove a universal bound for the dimension of random multipath field in the mean square error sense. The derived maximum dimension is directly proportional to the radius of the two-dimensional spatial region where the field is coupled to. Using the Karhunen-Loeve expansion of multipath fields, we prove that, among all random multipath fields, Isotropic random multipath achieves the maximum dimension bound. These results mathematically quantify the imprecise notion of rich scattering that is often used in multiple-antenna communication theory and show that even the richest scatterer (isotropic) has a finite intrinsic dimension.