Channel Assignment on Infinite Sets under Frequency-Distance Constraints

Large-scale radio systems for multiple users, such as those used in mobile phone networks, are often based on a cellular structure (Hale 1980; Macario 1997; MacDonald 1979). The service area is divided into cells and each cell is serviced by one transmitter, which communicates with users within the cell using a particular radio channel or set of channels. In any particular application the available channels are uniformly spaced in the spectrum, justifying integer labellings of these channels. It is often assumed, at least as a starting position, that the transmitters all have the same power, are omnidirectional, and are laid out like the vertices of a triangular lattice: in this case, the natural cells form a hexagonal pattern. Suppose that a radio receiver is tuned to a signal on channel c0, broadcast by the transmitter for its cell (typically, the closest transmitter). Reception will be degraded if there is excessive interference from other transmitters in the vicinity. First there is ‘co-channel’ interference due to re-use of channel c0 at nearby sites; but there are also contributions from sites using channels near c0, since in practice neither transmitters nor receivers operate exclusively within the frequencies of their assigned channels. One possible method to decide if an assignment of channels to transmitters is acceptable is as follows. First we agree on a suitable radio propagation model, and a collection of ‘test’ receiver positions. We then combine the effects of co-channel interference and other interferences at the test receiver positions, and determine whether the corresponding signal to interference ratios