# NOMA Clustering for Improved Multicast IoT Schemes
## 3 System Model and Problem Formulation  
### System and Channel Models  
We consider a downlink system, where a NOMA-capable base station BS sends multicast data to \(N\) D2D-enabled end nodes labelled as \(\Gamma=\)[UE\({}_{1}\),UE\({}_{2}\),...,UE\({}_{N}\)]. For clear understanding, we list the notations of this paper in Table 1.  
In the orthogonal mode, the BS uses \(N\) orthogonal channels to send the same message to the \(N\) users during the multicast phase Sun and Xu (2020). Such a configuration is resource consuming and inefficient, especially in the context of ultra-dense 6G scenarios, and, thanks to the signal superposition in the power domain NOMA for example, the system can be optimized. However, as the number of end users increases, defining NOMA-clusters and computing the corresponding power allocation coefficients brings significant computing and system complexity. Our model is based on the concept of \"_coalitions_\", we illustrate it in Figure 1 and define it as follows:  
* First, according to the procedure that we detail in Section 3.2, the base station BS selects a sub-group of users, referred to as \"_heads_\", which will be served in a NOMA mode;
* Second, clusters of users (i.e., _coalitions_) are formed around these heads according to the procedure detailed in Section 3.3, then, a transmission tree is constructed as described in Section 3.4 to establish D2D paths through the coalition head and connecting all the users within the coalition so they can be reached and served. We also note that, at the end of the coalition formation stage, users that are left outside of all coalitions (referred to as _\"lone\"_ users) are served by the BS directly, in an OMA mode to avoid additional computation time at the BS level at this stage. Hence, bandwidth resources at the BS are shared among cellular users only (i.e., coalition heads and lone users), with an objective of maximizing the system rate and SE. The rest of the users, associated to coalitions, get the service via D2D links in an out-band mode, and are, thus, not impacting the BS resources, nor influencing the cellular users rate through interferences. We assume that the global channel state information (CSI) is available at the BS, and that all channels are assumed to be independent but not necessarily identically distributed. We consider that the users \(\text{UE}_{i}\), \(i=1,\ldots,N\), are ordered according to their noise-normalized global channel gains, i.e., \(H_{i}/N_{0}\), where \(H_{i}=|h_{i}|^{2}d_{i}^{-\alpha_{i}}\), and the instantaneous channel coefficients \(|h_{i}|\) are modeled following a zero-mean, unit-variance, Rayleigh distribution, \(d_{i}\) is the distance between \(\text{UE}_{i}\) and \(\text{BS}\), and \(\alpha_{i}\) is the path loss exponent. \(N_{0}\) is the power spectral density of the zero-mean additive white Gaussian noise (AWGN), assumed the same over all channels. \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Definition** \\ \hline \(\Gamma\) & The set of D2D-enabled nodes \\ \(N\) & Total number of nodes \\ \(M\) & Number of Coaliton heads (CHs) \\ \(\mathcal{C}_{i}\) & Set of UEs forming a coalition around \(\text{CH}_{i}\) \\ C & Set of all coalition heads \\ \(H_{i}\) & Channel gain between \(\text{CH}_{i}\) and the base station \\ \(G_{ij}\) & Channel gain between nodes \(i\) and \(j\) \\ \(x_{i}\) & Maximum number of users that \(\text{UE}_{i}\) can serve \\ \(y_{i}\) & Number of connected D2D users to \(\text{UE}_{i}\) \\ \(\mathcal{L}\) & Set of Lone users \\ \(l\) & Number of Lone users \\ \(R\) & Instantaneous rate \\ \(\overline{R}\) & Average rate \\ \(T\) & Coherence time \\ \(T_{s}\) & Symbol time \\ ee & Energy efficiency \\ se & Spectrum efficiency \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of Notations. Figure 1: Illustration of the system model. On the other hand, we denote by \(G_{ij}\) the channel gain between \(\text{UE}_{i}\) and \(\text{UE}_{j}\). It is modeled similarly to \(H_{i}\) as \(G_{ij}=|g_{ij}|^{2}d_{ij}^{-\alpha_{ij}}\).