  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 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.
