Discrete Inputs in Gaussian Interference Networks: Performance Analysis and Approximate Optimality

Most of the communication and information theoretic literature assumes idealistic channel models, such as transmitters and receivers have almost inexhaustible memory and computing power, are fully synchronized, have global knowledge of all the communication protocols, number of users and state of the channel, etc. Clearly, in real world applications these assumptions are too idealistic. It is therefore of interest to understand fundamental limits of more practical models where such assumptions are relaxed, in which case classical techniques are no longer sufficient. This work focuses on Gaussian interference networks. Traditionally, Gaussian codes are considered for such settings because they are optimal in point-to-point setting and in a number of multiuser scenarios. However, non-Gaussian codes are known to outperform Gaussian ones in some asynchronous multi-user channels. Regrettably, such conclusions have been either drawn from numerical evaluations, which are difficult to generalize analytically, or existence proofs that show that discrete inputs are optimal but give little information about the actual optimal input distribution. Therefore, this work develops a theoretical framework and tools for establishing the performance of non-Gaussian/discrete input codes in Gaussian multi-user settings. The dissertation consists of two parts. In the first part, the necessary machinery required for evaluating performance of discrete inputs is developed. The framework is rooted in novel connections between information theory, additive combinatorics and number theory. Many of the tools developed bridge the gap between these three seemingly unrelated branches of mathematics and can be of interest on their own. In the second part, the developed tools are used to evaluate the performance of discrete inputs in several important channel models and scenarios. For example, quite surprisingly, it is shown that the simple transmission strategy that treats interference as noise is approximately optimal for the two-user Gaussian interference channel, even in the presence of asynchronism and with the lack of codebook knowledge at the receivers. Another application considered in this work is communications under a disturbance constraint measured by the minimum square error of estimating the interfering signal. Also, in this case, discrete inputs are shown to be approximately optimal.