Minimizing Great-Circle Distance Ratios of Undesired and Desired Signal Paths on a Spherical Earth

Preventing interference among mobile radio (MR) nets often requires spectrum managers to observe channel-assignment rules based on minimum ratios of the great-circle distances traversed by desired and undesired (potentially interfering) signals. Finding the true minimum values of these ratios, rather than the simply computed worst-case estimates, is necessary to utilize the spectrum efficiently. This paper provides an analysis of the case where the MRs operate in or above circular service areas on the surface of a spherical Earth. The analysis provides an accurate and efficient way, which is less complicated than performing the calculations on the sphere, to compute true minimum distance ratios. The method uses stereographic projection to transform the original minimization problem into a simpler problem of minimizing a ratio of Euclidean distances, which is expressed as a function of a single real variable, over the boundaries of discs (i.e., circles) in the complex plane. In a large, complicated problem, this methodology has been shown to reduce the computational time by two thirds.