Spatial Continuum Model: Toward the Fundamental Limits of Dense Wireless Networks

This paper proposes a new model called 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 in Tx or Rx mode, a spatial random distribution of nodes (resp. in Rx or Tx mode) characterized by a probability spatial density of users u(x) where each of them requests a quantity of information with no delay constraint, thus leading to a requested rate spatial density ρ(x). This system is modeled as an ∞-user asymmetric channel (BC or MAC). To derive the fundamental limits of this model, a spatial discretization is first proposed to obtain an equivalent BC or MAC. Then, a specific sequence of discretized spaces is defined to refine infinitely the approximation. Achievability and capacity results are obtained in the limit of this sequence while the access capacity region D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ω</sub> (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> ) is defined as the set of requested rates spatial densities ρ(x) that are achievable with a transmission power P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> . The uniform capacity defined as the maximal symmetric achievable rate is also computed.