## 2 Materials and Methods
### Optimization Algorithm for Planning  
Within frequency planning, the main goal is that the set of channels allocated to each cell minimizes interference. For a given geographic position, the carrier to interference ratio (\(C/I\)) measures the ratio between the cell's signal power level and the sum of power levels of cells operating in the same frequency. A system is only capable of operating above a minimum value, \(C/I_{min}\). An interference region, depicted in Figure 6, is characterized by \(C/I\) levels below \(C/I_{min}\), an area where communication is not possible in the evaluated frequency. The proposed algorithm evaluates, for each unplanned cell \(c\), the interference experienced when operating on each of the available BCCHs, the best server cell, \(c_{f,b}\), shall correspond to the cell with strongest received signal level, \(P_{rx}\). Remaining cells operating in \(f\) with signal present in pixel \(\mathbf{p}\) are the set of non-best server cells, \(\pm c_{f,nb}\). Thus, a given covered pixel \(\mathbf{p}\) of \(c\) is labeled as interfered if \(C/I\) is below \(C/I_{min}\):  
\[\frac{P_{rx}(\mathbf{p},c_{f,b})}{\sum_{c_{f,n}\in\mathbf{c_{f,nb}}}P_{rx}(\mathbf{p},c_{ f,nb})}<(C/I)_{min} \tag{1}\]  
where \(P_{rx}(\mathbf{p},c_{f,b})\) (mW) is the received signal power level in pixel \(\mathbf{p}\) of best server cell \(c_{f,b}\), \(P_{rx}(\mathbf{p},c_{f,nb})\) (mW) is the received signal power level in pixel \(\mathbf{p}\) of non-best server cell \(c_{f,nb}\), belonging to \(\mathbf{c}_{f,nb}\) set; and \((C/I)_{min}\) is the minimum \(C/I\) level that guarantees interference free communication. In GSM this value is typically taken as 9 dB. For the unplanned cells set, \(\mathbf{c}\), and the set of sorted frequencies, \(\mathbf{f}\), \(\mathbf{D}\) matrix is built. It has dimension \(dim(\mathbf{c})\times dim(\mathbf{f})\), having for each cell and frequency the number of pixels with interference; i.e., where condition (1) is observed. The objective is to minimize interference, given mathematically by:  
\[min\ \sum_{c\in\mathbf{c}}\sum_{f\in\mathbf{f}}\mathbf{D}_{cf}\times\mathbf{X}_{cf}^{T} \tag{2}\]  
where \(\mathbf{X}\) is a Boolean matrix with similar dimensions as \(\mathbf{D}\). Figure 6: Illustration of the received signal level decay with distance for two neighboring cells: the corresponding interference region is identified. When various cells of different sites shall be planned, the amount of possible interference created by unplanned cells that share the coverage area is also minimized, in case these cells get the same BCCH:  
\[min\ \forall f\in\mathbf{f}\ and\ \forall c_{1},c_{2}\in\mathbf{c}:\left(\mathbf{X}_{c_{1f}}+ \mathbf{X}_{c_{2}f}\right)\times\mathbf{U}_{c_{1}c_{2}} \tag{3}\]  
where \(\mathbf{U}_{c_{1}c_{2}}\) is \"1\" if cells \(c_{1}\) and \(c_{2}\) interfere, and \"0\" otherwise. Two cells operating in the same BCCH interfere with each other when there are pixels covered by \(c_{1}\) that are interfered by \(c_{2}\). Nevertheless, this restriction shall be relaxed. In fact, the solution to this optimization problem may not be the best one or might even return a solution as impossible. Ideally, \(\mathbf{U}_{c_{1}c_{2}}\), would become an integer, representing the number of pixels in \(c_{1}\) interfered by \(c_{2}\). This minimization was originally a condition, but after some research on integer programming heuristics, we concluded that this way the optimization would be much faster (from hours to under a second, when optimizing 50 cells). When planning frequencies, certain rules must be followed. These are expressed by a set of restrictions to the optimization problem.