The Information Carried by Scattered Waves: Near-Field and Nonasymptotic Regimes

Abstract

The number of spatial degrees of freedom of the field radiated in a two-dimensional setting by a time-harmonic, arbitrary square-integrable current density and in the presence of a random distribution of scattering elements is determined. It is shown that the active power associated to the $k$ th singular value of the near field in the presence of scatterers external to the cut presents a heavy tail decay as a function of its index, rather than the usual exponential attenuation occurring beyond a critical index term observed in free space. This near-field information gain due to scattering was recently anticipated by Janaswamy using a stochastic source model, it is extended here to arbitrary sources, and it is shown to disappear in the limit of large radiating systems. It is also shown that the same information gain and asymptotic cut-off occurs for the singular values of the field radiating in free space. Collectively, these results show that while the presence of scatterers external to the cut can increase the number of channels that can be exploited for communication by the active power in the near field, they do not change the number of channels associated to the field, nor the asymptotic behavior of the number of degrees of freedom.