# A Light Signaling Approach to Node Grouping for Massive MIMO IoT Networks
## 7 Numerical Evaluation  
We conducted a numerical evaluation to test the performance of our optimization and approximation algorithm, along with two reference algorithms, one of which uses the full CSI, on randomly generated instances of our problem scenario, consisting of a set of single-antenna nodes and an accompanying channel matrix. We compared the performance of the algorithms in terms of the achieved minimum, maximum, and average SINR. We also compared the performance of our approximation with the optimal solution in terms of the proportion of the optimal objective function value obtained by the approximation. All the source code for our implementations of the optimization problem, approximation algorithm, channel model, and experiments is available online [33]. The parameters for our numerical evaluation are given in Table 2. We generated instances consisting of between 15 and 36 nodes, along with a massive MIMO base station with 100 antennas, and 12 pilots available in each coherence block. These base station parameters are based on the Lund University massive MIMO (LuMaMi) testbed [34; 35]. The numbers of nodes and pilots were also chosen such that there were always fewer pilots than nodes, necessitating node grouping. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Number of nodes & \(|K|\) & \(15\dots 36\), step 3 \\ Number of instances for each number of nodes & - & 20 \\ Number of pilots & \(P\) & 12 \\ Number of groups & \(|\mathcal{G}|\) & 3 \\ Number of base station antennas & \(M\) & 100 \\ Type of antenna array & - & Uniform rectangular array \\ Precoding & - & Maximum ratio combining \\ Carrier frequency & - & 2.47 GHz \\ Number of channel clusters & \(N_{\text{cl}}\) & 4 \\ Number of channel components per cluster & \(N_{\text{ray}}\) & 10 \\ Distribution of cluster central angles & - & Uniform 45–135 degrees \\ Distribution of component angles within each cluster & - & Exponential, \(\mu=7.5\) degrees \\ (vertical and horizontal) & - & Exponential, \(\mu=0\), \(\sigma=1\) \\ Component small scale fading distribution & - & Complex normal, \(\mu=0\), \(\sigma=1\) \\ Signal-to-noise ratio (for all nodes) & - & 20 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for the numerical evaluation. We partitioned the nodes into three groups, and so as the number of nodes increased, the number of nodes per group increased accordingly. The number of nodes per group is the most important parameter for determining the achievable SINR, since with more nodes per group the density of nodes is greater and the angular differences between them are lower. Meanwhile, the overall number of nodes is the most important parameter in terms of the running time of the partitioning algorithms. For each number of nodes, we generated and tested 20 problem instances. The instances were generated using a channel model based on the Saleh-Valenzuela model [36] and its adaption to MIMO channels [37]. The channel model and problem instance generation will be further detailed in Section 7.1. As an output, the model produces the dominant direction and angular spectrum spread for each node, along with the complete channel matrix, giving the channel from each node to each base station antenna. Using the dominant directions and angular spectrum spreads, we then ran our optimization MMAD, given in Section 5.2, on each problem instance. For this purpose, we used the AMPL modeling language [38], along with the CPLEX optimization solver [39], and the optimization was run on a 12-core server with 2.2 GHZ Intel Xeon ES-2420 CPUs and 24 GiB RAM. The optimal solution was then compared with three other node partitioning algorithms. An overview of the partitioning algorithms is given in Table 3. The first comparison algorithm was our approximation algorithm, given in Algorithm 1.