This paper studies the behavior of the sum-rate of outdoor multi-user multiple input multiple output systems, where a uniform linear array is utilized at the base station (BS), as the number of BS antennas and the number of users increase. Two schemes are studied for the downlink, the zero-forcing and the maximum ratio transmission (MRT). To begin with, the channel matrix is described deterministically using direct scattering theory. Its matrix elements are the Fourier coefficients of functions that have a physical significance, being connected to the geometry of the environment. For the case when the energy is not arriving at the BS from every spatial direction, we prove for the ZF scheme that the achievable sum-rate behavior is sublinear in the number of antennas at the BS. For the MRT scheme, we prove that there is a link between the sum-rate and the distribution of the users in the city. To numerically evaluate the sum-rate for a given city, schematically described by its streets and its buildings, we introduce a model for the Fourier coefficients of the aforementioned functions, which mixes geometrical and statistical ingredients. We randomly distribute high buildings that will act as strong scatterers and look at the influence of their number on the sum-rate. With the ZF scheme, we perform the calculations for the case where the number of BS antennas,    $M_{T}$   , is equal to the number of users,    $M_{R}$   , and for the case where    $M_{T}$    is larger than    $M_{R}$   . We observe that the sum-rate first increases linearly with the number of BS antennas, and becomes sublinear after    $M_{T}$    reaches a certain value. The threshold depends on the number of high buildings. With the MRT scheme, the calculations confirm the theoretical predictions, and the sum-rate is greater if the users are spread over all of the city.
['Francois Bentosela', 'Horia D. Cornean', 'Arman Farhang', 'Nicola Marchetti']
On the Sublinear Behavior of Massive Multi-User MIMO Sum-Rates for Deterministic Channel Models