Spatial Continuum Extensions of Asymmetric Gaussian Channels (Multiple Access and Broadcast)

This paper proposes a new model called \emph{spatial continuum asymmetric channels} to study the channel capacity region of asymmetric scenarios in which either one source transmits to a spatial density of receivers or a density of transmitters transmit to a unique receiver.This approach is built upon the classical broadcast channel (BC) and multiple access channel (MAC). For the sake of consistency, the study is limited to Gaussian channels with power constraints and is restricted to the asymptotic regime (zero-error capacity).The reference scenario comprises one base station (BS) in Tx or Rx mode, a spatial random distribution of nodes (resp. in Rx or Tx mode) characterized by a probability spatial density $u(x)$ and a request for a quantity of information with no delay constraint. This system is modeled as an $\infty-$user asymmetric channel (BC or MAC). To derive the properties of this model, a spatial discretization is performed and the equivalence with either a BC or MAC is established. A discretization sequence is then defined to refine infinitely the approximation. Achievability and capacity results are obtained in the limit of this sequence. The uniform capacity is then defined as the maximal symmetric achievable rate at which the distributed users can transmit/receive with no delay constraint.The capacity region is also established as the set of information distributions that are achievable. The tightness of these limits and their practical interest are briefly illustrated and discussed.