The Minimum Principle of SINR: A Useful Discretization Tool for Wireless Communication

Theoretical study of optimization problems in wireless communication often deals with tasks that concern a single point. For example, the power control problem requires computing a power assignment guaranteeing that each transmitting station s_i is successfully received at a single receiver point r_i. This paper aims at addressing communication applications that require handling two-dimensional tasks (e.g., guaranteeing successful transmission in entire regions rather than at specific points).The natural approach to two-dimensional optimization tasks is to discretize the optimization domain, e.g., by sampling points within the domain. The straightforward implementation of the discretization approach, however, might incur high time and memory requirements, and moreover, it cannot guarantee exact solutions.The alternative proposed and explored in this paper is based on establishing the minimum principle¹ for the signal to interference and noise ratio (SINR) function with free space path loss (i.e., when the signal decays in proportion to the square of the distance between the transmitter and receiver). Essentially, the minimum principle allows us to reduce the dimension of the optimization domain without losing anything in the accuracy or quality of the solution. More specifically, when the two-dimensional optimization domain is bounded and free from any interfering station, the minimum principle implies that it is sufficient to optimize the SINR function over the boundary of the domain, as the "hardest"points to be satisfied reside on the boundary and not in the interior.We then utilize the minimum principle as the basis for an improved discretization technique for solving two-dimensional problems in the SINR model. This approach is shown to be useful for handling optimization problems over two dimensions (e.g., power control, energy minimization); in providing tight bounds on the number of null cells in the reception map; and in approximating geometric and topological properties of the wireless reception map (e.g., maximum inscribed sphere).The minimum principle, as well as the interplay between continuous and discrete analysis presented in this paper, are expected to pave the way to future study of algorithmic SINR in higher dimensions.