## 1 Introduction

American engineers such as Metzger, Zoellner, Beall, and Hale recognized the significance of abstract mathematical graph-coloring problems in the 1970s and 1980s, pioneering an extended study of the *Frequency Assignment Problem*. Metzger's papers are difficult to obtain but *Zoellner and Beall* [1977], *Hale* [1980], and *Hale* [1981] give a good overview of this early work. Latter studies have been led by academia in the main, and an excellent body of work exists in the academic literature, see *Aardal et al.* [2007], *Leese and Hurley* [2002], and *Hurley et al.* [1997], for example, with some advanced algorithms now available and demonstrated on benchmark data sets. In general, professional practice and the wider engineering and scientific communities have not yet embraced these ideas fully and much of the expertise in this subdiscipline resides within academia, typically in the Computer Science or Mathematics fields.

A recent study by the authors [*Flood and Allen*, 2013] has investigated the nature of optimal frequency assignments and, in particular, the conflict between local and global spectral efficiency. This shows how the frequency assignment problem can be extended to include equipment selection and other practical planning considerations familiar to frequency assignment engineers.

The research first of all focused on the development of Integer Programming (IP) formulations with a minimum span objective function; that is, a set of frequency assignment and equipment selection problems were analyzed in order to find the smallest span of frequencies required to resolve a set of frequency assignment requests while satisfying the frequency separation constraints used to mitigate harmful interference. The problems were exposed to these formulations using a standard IP solver which is exhaustive and delivers exact, optimal solutions.

The study showed, contrary to intuition, that by using lower-order modulation equipment and so doubling the bandwidth requirement on selected links relative to a higher-order modulation equipment selection, we can, in some cases, actually reduce the overall span of frequencies required for a network frequency assignment. In other cases, it is possible to double the bandwidth requirement on a subset of links while maintaining the span obtained when higher-order modulation equipment is ubiquitous, thus improving the interference environment.

In this paper we investigate the development of *heuristics*. Using the exact solutions as a benchmark, these programs aim to optimize equipment selection and frequency assignment in a simulated off-line environment. The results obtained from using a range of equipment selection criteria are reported on here.

### 1.1 Inequalities in the Radio Interference Environment

Microwave fixed link operators require a high quality of service, and links are normally deployed in spectrum subject to detailed procedures such as the noise-limited frequency assignment methodology, for example, [*Flood and Bacon*, 2006]. The links are planned in order that a standard data rate is supported and, often, the planner has a choice of radio system types, each utilizing a particular modulation scheme. In Europe, for example, when seeking to resolve a standard data rate, the planner often has a choice between exactly two radio systems: one using a relatively lower order and one a relatively higher-order modulation scheme [*ETSI*, 2010].

There are well-established trade-offs between modulation, bandwidth, equivalent isotropic radiated power (EIRP), and protection ratios [*Leuenberger*, 1986], [*Farrar and Hinkle*, 1988]. While the higher-order modulation equipment requires less bandwidth to resolve a specific data rate on an isolated link, the higher *S*/*N* associated with these schemes means that the radios have higher receiver sensitivity levels (RSLs) and radiate at higher powers assuming that the higher RSL at the distant end of the link is the start point for calculation of EIRP. They require larger protection ratios in the radio interference environment when compared to equipment utilizing a relatively lower order of modulation.

These inequalities between radio systems can be given a more precise expression when we consider the radio system parameters used in the frequency assignment process. The assigner is primarily concerned with characterizing the radio as an interferer and as a victim in the radio interference environment. In both cases, we can actually use a receiver parameter to model the inequalities between radio system types.

The EIRP at a link end, expressed in dBW, is used to model interference at its source and a sophisticated frequency assignment method may include an EIRP assignment. Here RSL (dBW) is the *receiver sensitivity level* of the radio system at the distant end of the link and the start point for our calculation of EIRP at a link end. Working from RSL, we take account of all gains and losses on the wanted path arriving at an EIRP that, typically, is the minimum value required to support the link's availability requirement.

*I* (dBW) is the threshold for single-entry interference, and this parameter is used to characterize the victim receiver in pairwise interference scenarios. We have used a value for *I* associated with a noise limited frequency assignment criterion where the fully faded wanted signal is exposed to a model of the median interferer [*Ofcom*, 2012].

Using the syntax Mbit/s in MHz to describe the data rate and bandwidth of a radio system, Table 1 sets out values for RSL and *I*, illustrating the inequalities between the radio systems used in our study. Clearly, these inequalities are significant with ranges for RSL and *I* of 16.5 dB and 13.8 dB, respectively. Further, we can see that for a particular data rate, the higher-order modulation option uses half the bandwidth of the lower-order solution and, in general, has a higher RSL and a lower *I*.

Mbit/s in MHz | RSL (dBW) | I (dBW) |
---|---|---|

8 in 3.5 | −105.5 | −138.4 |

8 in 7 | −106.5 | −132.9 |

2 × 8 in 7 | −99.5 | −132.4 |

2 × 8 in 14 | −103.5 | −129.9 |

34 in 14 | −96.5 | −129.4 |

34 in 28 | −100.5 | −126.9 |

51 in 14 | −95.5 | −129.0 |

51 in 28 | −97.5 | −130.4 |

155 in 28 | −90 | −128.9 |

155 in 56 | −92.5 | −125.4 |