# A Light Signaling Approach to Node Grouping for Massive MIMO IoT Networks
## 7 Numerical Evaluation
### Channel Model  
To generate each problem instance, we used the channel model from [37], which is in turn based on the Saleh-Valenzuela model [36]. The model defines the channel matrix \(\mathbf{H}\) as  
\[\mathbf{H}=\sqrt{\frac{M}{N_{\mathrm{cl}}N_{\mathrm{ray}}}}\sum_{i=1}^{N_{ \mathrm{cl}}}\sum_{l=1}^{N_{\mathrm{ray}}}\alpha_{i\ell}\Lambda_{\mathrm{r}}( \phi^{\mathrm{r}}_{i\ell},\theta^{\mathrm{r}}_{i\ell})\Lambda_{\mathrm{t}} \big{(}\phi^{\mathrm{t}}_{i\ell},\theta^{\mathrm{t}}_{i\ell}\big{)}\mathbf{a}_ {\mathrm{r}}(\phi^{\mathrm{r}}_{i\ell},\theta^{\mathrm{r}}_{i\ell}). \tag{22}\]  
The channel matrix \(\mathbf{H}\) models the effect of all the nodes' signals arriving at the \(M\) receive antennas of the base station. At the nodes' side, single antennas are considered. The signals are modeled as multiple components, grouped into \(N_{\mathrm{cl}}\) different clusters. Clusters can, for example, be created by objects in the environment that the nodes' signals are reflected from (possibly multiple times) before arriving at the base station. Each cluster has a central angle of arrival in relation to the base station, around which its \(N_{\mathrm{ray}}\) constituent components propagate. The propagation effects (attenuation and phase rotations) are modeled in the complex gain \(\alpha_{i\ell}\) for each contributing ray. The receive array response is denoted by \(\mathbf{a}_{\mathrm{r}}(\phi^{\mathrm{r}}_{i\ell},\theta^{\mathrm{r}}_{i\ell})\) and is a function of the antenna array structure, the azimuth, and elevation angles \((\phi^{\mathrm{r}}_{i\ell},\theta^{\mathrm{r}}_{i\ell})\) of the arriving signal of the \(\ell\)th ray in the \(i\)th cluster. The expression of the array response for a uniform linear array (ULA) and a uniform rectangular array (URA) can be found in [41]. The antenna element gain of the receive and transmit antennas are denoted by the functions \(\Lambda_{\mathrm{r}}\big{(}\phi^{\mathrm{r}}_{i\ell},\theta^{\mathrm{r}}_{i \ell}\big{)}\) and \(\Lambda_{\mathrm{t}}\big{(}\phi^{\mathrm{t}}_{i\ell},\theta^{\mathrm{t}}_{i \ell}\big{)}\), respectively. For isotropic antennas, the antenna element gain is one (\(\Lambda=1\)), as the gain is independent of the azimuth and elevation angle. In our experiments, we used four clusters, each with ten components, and the cluster central angles were drawn from a uniform random distribution between 45 degrees and 135 degrees, where 90 degrees is perpendicular to the plane of the URA. The componentswere distributed around these central angles according to exponential distributions with a mean of 7.5 degrees, one in each of the vertical and horizontal planes, with a 0.5 probability of a given component lying in either direction from the central angle in each plane. Each component is subject to small-scale fading as \(\alpha_{i\ell}\) is following a complex normal distribution with a mean of 0 and a standard deviation of 1. Finally, the node channels are normalized such that the expected power of each node's signal is 1. In [37], clusters are generated homogeneously; however, our use case concerns channels that exhibit clear dominant directions [28]. To model this, for each node, we amplified the cluster with the highest power, by multiplying it by a scalar drawn uniformly randomly from the interval \([2.0,6.0]\). This could represent either the line-of-sight part of the node's channel, or a set of components reflected from a highly reflective surface, if there is no line-of-sight path between the base station and the node. The node channel was then re-normalized to ensure that the expected power was still 1.0. In this way, we generated channels with a more clear dominant component and lower angular spectrum spread than in [37]. The dominant direction for each node was taken to be the central angle of the dominant cluster. To compute the angular spectrum spread, we used the Rician K-factor, adjusted for the angular distribution of the generated clusters for each node.