Optimum transmission through the Gaussian multiple access channel

In this paper we study the optimality of particular points in the capacity region of Gaussian multiple access channels (GMACs) with various power constraints. The points of interest maximize general rate objectives that arise in practical communication scenarios. Achieving these points constitutes the task of jointly optimizing time-sharing parameters, input covariance matrices and the order of decoding used by the successive interference cancellation receiver. To approach this problem Carathéodory´s theorem is invoked to represent time-sharing and decoding orders jointly as a finite-dimensional matrix variable. This variable enables us to use variational inequalities to extend results pertaining to problems with linear objectives to more general, potentially nonconvex, problems. In particular, it is shown that for arbitrary objectives, if the power constraints are convex, it suffices for each user to use only one covariance matrix in all its allocated time slots. On the other hand, for arbitrary power constraints, if the objective is linear no time-sharing is necessary. These results significantly reduce the design complexity and render optimal signalling over GMAC more amenable to implementation in various practical scenarios.