This article shows that the capacity region of a 2-users weak Gaussian interference channel is achieved using Gaussian code-books. The approach relies on traversing the boundary in incremental steps. Starting from a corner point with Gaussian code-books, and relying on calculus of variation, it is shown that the end point in each step is achieved using Gaussian code-books. Optimality of Gaussian code-books is first established by limiting the random coding to independent and identically distributed scalar (single-letter) samples. Then, it is shown that the optimum solution for vector inputs coincides with the single-letter case. It is also shown that the maximum number of phases needed to realize the gain due to power allocation over time is two. It is also established that the solution to the Han-Kobayashi achievable rate region, with single letter Gaussian random code-books, achieves the optimum boundary.

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