On Dimensionality for Sparse Multipath

We give a 2WT style result for the degrees of freedom of multipath signals that pass through spatially limited (sparse) scattering environments. The dimensionality scales with the circumference of the scattering region, and the total communications path length. We provide a direct comparison to the time-frequency case, where space replaces time. This is a rigorous wavefield examination of previous heuristic geometric arguments.