Maximum sum rates via analysis of 2-user interference channel achievable rates region

Treating the interference as noise, the paper studies the first derivative of the frontiers which trace the achievable rates region of the 2-user interference channel. The achievable rates region in this case was found to be the convex hull of the union of two regions, each is bounded by a log-defined line. Those log-defined lines are characterized by holding one of the transmitters at full power, while the other transmitter sweeps its full power range [1]. Maximizing the sum rates for the 2-user interference channel translates to the study of the first intersection point with lines of slope -1 approaching the rates region from positive infinity. The paper achieves the same result reported in [2], that the maximum sum rates solution is one of three points: one user transmitting with full power while the other user is silent, or both users transmitting at full power simultaneously. The result in [2] is achieved through analysis of the objective function, while the solution presented herein follows from analyzing the first derivative of the rates region frontiers.