# Pilot Sequence Allocation Schemes in Massive MIMO Systems Using Heuristic Approaches
## 4 Proposed Solutions
### Variable Neighbourhood Search
## 5 Simulations Results

We verify the applicability of the techniques presented in this manuscript to the pilot sequence assignment problem, as well as the impact of using them in terms of user SINR, system spectral efficiency and minimum and maximum user spectral efficiency. We developed a MATLAB script to create a dataset and made it public at github.com/ evertonalex/utfpr-pppsi-pilotontamination, accessed on 12 May 2022. We consider a seven hexagonal cell cluster geographically disposed such as presented in Figure 1. Each user in the cell is randomly positioned. However, to establish some physical limits, user positions can only be integers numbers pairs and must be contained within the limits of the hexagon, i.e., let \((x,y)\in\mathbb{Z}^{2}\) be some user position related to its base station which is fixed at the origin, \(x,y\sim\mathcal{U}\{1,R\}\), and the following conditions are also met:

1. \(|x|\leq R\)
2. \(|y|\leq\frac{\sqrt{3}}{2}\,R\)
3. \(\frac{\sqrt{3}}{2}\,|x|+|y|\geq\frac{\sqrt{3}}{2}\,R\)After their positions are randomly generated with respect to the origin, each user position is translated to their cell position. Moreover, this process replaces any users that overlap the same position with a new random one. Table 2 shows the parameters used in the aforementioned scenario. ### Heuristic Parameters Optimization

At this point, when the performance results are combined with the execution time results, such as observed in Figure 2 and 3, more iterations do not represent a more significant increase in performance. Hence, the best approach is to use as few iterations as possible in each of the algorithms due to the high cost and low increase in performance. All the simulations use the optimal parameters for each heuristic, and the number of iterations is set to 50. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Adopted Value(s)** \\ \hline Number of Cells \((L)\) & 7 \\ Number of Users per Cell \((K)\) & \(\{20,40,60\}\) \\ Cell Radius \((R)\) & \(\{100,250,500,1000\}\) (m) \\ Wavelength \((\lambda)\) & 8.56 (cm) \\ Reference Distance \((d_{0})\) & 10 (m) \\ Path Loss Exponent \((\gamma)\) & \(\{6,2\}\) \\ Shadowing Variance \((\sigma_{s}^{2})\) & \(\{6,10\}\) (dB) \\ Channel Bandwidth \((B)\) & 20 (MHz) \\ Transmission frequency & 3.5 GHz \\ Dataset instances & 1000 per set of parameters \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System parameters used in the computational simulations. Figure 1: Base stations, users and cells geographic disposition in our simulation scenario. User positions were randomly generated. Since each heuristic has its own set of parameters which may impact the overall performance, we investigated the algorithms performance in the following scenario: \(K=20\) users and \(R=1000\) m. The findings, for each heuristic, are presented in Table 3. The values which result in the best performance are marked in bold type. Figure 3: Time to complete the execution of a single instance in the dataset by each algorithm. We used a Google Cloud solution with 32 virtual CPUs, 32 GB of RAM, running MATLAB on Ubuntu 20. Figure 2: Spectral efficiency average by iterations. It is worth noting that although we tested different parameters for each heuristic, the results in terms of system spectral efficiency are where all are within a less than 1% margin from each other. For instance, the increment in the results for GA when \(G_{\max}\) goes from 100 to 200 is around 0.0019%. Additionally, the increment from using a 0.5 mutation rate to a 0.75 is around 0.0067%. In both cases, the difference means less than a bit per second per Hertz, which mean they are irrelevant system-wise. This pattern is also observed in the BPSO, SPV-PSO and SPVG-PSO (with or without VNS) results. The difference between the parameters is within the margin of error for the dataset size used in our work. Since the difference between the algorithms performance in terms of average SE is marginal, we further analyzed the minimal spectral efficiency case.