Capacity Region of Two-users Weak Gaussian Interference Channel

Computing capacity of Gaussian Interference Channel (GIC) is complex since knowledge of input distributions is needed to find the mutual information terms in closed forms, which should be optimized over input distributions and associated resource allocation. The optimum solution may require dividing the available resources among several GIC (each called a "constituent region", hereafter) and apply time-sharing among them. The current article focuses on a single constituent region (meaning the constraints on resources are all satisfied with equality) for a 2-users weak GIC. It is shown that, by relying on a different, intuitively straightforward, interpretation of the underlying optimization problem, one can determine the encoding/decoding strategies in the process of computing the optimum solution. This is based on gradually moving along the boundary of the capacity region in infinitesimal steps, where the solution for the end point in each step is constructed and optimized relying on the solution at the step's starting point. This approach enables proving Gaussian distribution is optimum over the entire boundary, and also allows finding simple closed form solutions describing different parts of the capacity region. The solution for each constituent 2-users GIC coincides with the optimum solution to the Han Kobayashi (HK) system of constraints with i.i.d. (scalar) Gaussian inputs. Although the article is focused on 2-users weak GIC, the proof for optimality of Gaussian distribution is independent of the values of cross gains, and thereby is universally applicable to strong, mixed and Z interference channels, as well as to GIC with more than two users. In addition, the procedure for the construction of boundary is applicable for arbitrary cross gain values, by re-deriving various conditions that have been established assuming cross gains being less than one.