Intrinsic limits of dimensionality and richness in random multipath fields

We study the dimensions or degrees of freedom of farfield multipath that is observed in a limited, source-free region of space. The multipath fields are studied as solutions to the wave equation in an infinite-dimensional vector space. We prove two universal upper bounds on the truncation error of fixed and random multipath fields. A direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension. For circular and spherical spatial regions, we show that this finite dimension is proportional to the radius and area of the region, respectively. We use the Karhunen-Loeve (KL) expansion of random multipath fields to quantify the notion of multipath richness. The multipath richness is defined as the number of significant eigenvalues in the KL expansion that achieve 99% of the total multipath energy. We establish a lower bound on the largest eigenvalue. This lower bound quantifies, to some extent, the well-known reduction of multipath richness with reducing the angular power spread of multipath angular power spectrum.