On the EM Degrees of Freedom in Scattering Environments

By considering the 2D problem of electrical line sources radiating in the presence of perfectly conducting cylinders and decomposing the real power flow on a circumscribing observation circle separating the transmit nodes from the receive nodes, simple formulas are derived for the electromagnetic degrees of freedom in scattering environments for a network of nodes communicating with each other. The locations, magnitudes, and phases of the line sources are assumed to be independent and identically distributed (i.i.d.) random variables. Similarly, the locations of the scatterers in the region outside the observation circle are assumed to be i.i.d. random variables. The exact scattering problem is cast in the form of an integral equation, where the Fourier coefficients of the scatter current density are the unknowns. Based on the Born approximation that is valid for mild scatter densities and asymptotic analysis, a closed form expression is derived for the number of degrees of freedom in scattering environments. The benefit of observing near-fields in the determination of degrees of freedom is included in the numerical examples considered. If the power per source and/or the number of sources within the circumscribing circle are made to increase algebraically with the size of the circle, it is shown that scattering environments can offer much higher degrees of freedom than what are available in free-space.