Optimising the signal-to-noise ratio

A long-term aim of current research into the frequency assignment problem is to identify features that would enable us to classify instances of the problem in terms of certain easily-computed parameters. Although there are good mathematical reasons why this may be impossible in full generality, experimental evidence is emerging which suggests that instances which occur in practice may be amenable to such classiication 1]. One parameter that contributes to the speciication of an instance of the problem is the signal-to-noise ratio (SNR). Recently some experimental work 2] has been done on the question of nding the minimum SNR for randomly-generated arrays of transmitters. The results suggest that, for a given connguration, the minima occur on the boundaries of what is known as the Voronoi partition. When the number of transmitters is small, and they are 'regularly-spaced', it is intuitively clear that a result of this kind should hold. But it is possible to conceive conngurations in which one powerful transmitter causes complex variations in the SNR within the neighbourhood of a cluster of less powerful transmitters. The question is further complicated by the practical diiculty of specifying an exact propagation law for radio waves under operational conditions. In this paper we shall state general conditions under which it can be proved that the minima of the signal-to-noise ratio must lie on the boundaries of the Voronoi partition. This result is signiicant for two reasons. 1. The complexity of calculating the minimum in any speciic instance is reduced by an order of magnitude.