On Deterministic Models for Gaussian Networks

In this thesis we study wireless networks modeled by the additive white Gaussian noise (AWGN) model. The AWGN capacity region of most network topologies is unknown, which means that the optimal transmission scheme is unknown as well. This motivates the search for capacity approximations and for approximately optimal schemes. Deterministic channel models have been proposed as means to approximate the AWGN model within a constant additive gap. We consider two particular models, the linear finite-field model (LFFM) and the discrete superposition model (DSM). In the first part of the thesis we utilize the LFFM to design transmission schemes for layered relay networks in the AWGN model. We show that if a transmission scheme in the LFFM satisfies a certain set of coordination constraints, it can be translated to the AWGN model. A form of hierarchical modulation is used to build multiple transmission layers. By analyzing the performance in the AWGN model, we show that the AWGN rate is at most a constant gap below the LFFM rate. In the second part, we use the DSM to approximate the capacity and secrecy capacity of AWGN networks. First, we prove that the DSM capacity of some topologies is within a constant gap to the corresponding AWGN capacity. The topologies are given by the partially cognitive interference channel (PCIFC), a class of multiple-unicast networks, and a class of relay networks with secrecy constraints, respectively. Then, we approximate the capacity in the DSM. We bound the capacity of the point-to-point channel, the capacity regions of the multipleaccess channel and the broadcast channel, as well as the secrecy capacity of parallel relay networks (PRN) with an orthogonal eavesdropper and conventional relays. Furthermore, we find inner bounds on the capacity region of the PCIFC. This approach yields achievable rate regions for the PCIFC in the AWGN model and the AWGN secrecy capacity of the PRN within a constant gap.