Shadow Area and Degrees-of-Freedom for Free-Space Communication

The number of degrees-of-freedom (NDoF) in a communication system is limited by the number of antenna ports, element shapes, positions, and the propagation environment. As the number of antenna elements increases within a given region, the NDoF eventually saturates due to correlation of the radiated fields. The maximal NDoF can be determined numerically for communication between two regions using singular value decomposition of a channel model representing wave propagation between densely sampled sources at the transmitter and fields at the receiver. This paper provides a straightforward analytical estimate of the NDoF for arbitrarily shaped transmitter and receiver regions. The analysis show that the NDoF for electrically large regions is approximated by the mutual shadow area of the regions, measured in wavelengths. Several setups illustrate the results, which are then compared with numerical evaluations of the singular values of the propagation channel. These new analytical expressions also simplify to previously established results based on Weyl's law and the paraxial approximation.