# NOMA Clustering for Improved Multicast IoT Schemes
## 3 System Model and Problem Formulation
### Formation of Coalitions
\(y_{0}=y_{0}+1\),
3. for\(\text{UE}_{j}\in\mathcal{C}\):do \(\text{UE}_{j}\) create its senders preference list from \(\{\text{UE}_{0},\text{UE}_{1},\dots\text{UE}_{j-1}\}\), noted as \(\{\text{S}_{1},\text{S}_{2},\dots\text{S}_{j}\}\), with \(\text{S}_{1}\) the most preferred sender. \(f\) a bijective function from \([0,j]\) to \([1,j-1]\) defined as: \(f(x)=y\), with \(y\) is the index of \(\text{S}_{k}\) in the coalition. for\(k\in[1,j]\):do if\(y_{f(k)}<x_{f(k)}\)then \(\text{UE}_{j}\) connects to \(\text{S}_{k}\) \(y_{f(k)}=y_{f(k)}+1\); BREAK;  end if  end for return Transmission tree. ```

**Algorithm 2** Transmission Tree Construction. At the end of the transmission tree construction, multicast messages are routed through CH to all \(\text{UE}_{i}\in\mathcal{C}\). In the following, we will evaluate the performance of the proposed approach in terms of spectral efficiency and energy efficiency. ## 4 Spectrum and Energy Efficiency Analysis

### Sum Rate

In this subsection, we calculate the system's sum rate as a base framework for our analysis. We denote by \(R\) and \(\overline{R}\), respectively, the instantaneous and the average rates. We start by defining the rate of the users connected to the cellular system (i.e., to BS, including users in coalition and lone users), then we evaluate the rate of the D2D users. Accordingly, the cellular users sum rate can be written as

\[R^{\text{Cell}}=\tau\cdot\left[R^{\text{CH}}+R^{\mathcal{C}}\right], \tag{1}\]where \(R^{\text{CH}}\) is the rate of all coalition heads, and \(R^{\mathcal{L}}\) is the rate of lone users not connected to any coalition, served by the BS in an OMA mode. The factor \(\tau\) is used to account only for the ratio of time used to route data; excluding the time used to complete the coalition formation stage. We define \(T\) as the coherence time [22] over which the fading channel remains invariant. \(T=uT_{s}\), \(T_{s}\) is the symbol duration, and \(u\) is the total number of symbols per block. Hence, we can write \(\tau=\frac{u-u^{\prime}}{u}\), with \(u^{\prime}\) the number of symbols used during the coalition formation stage. We assume that perfect successive interference cancellation (SIC) is performed at the user level, the sum rate of coalition heads is written as:

\[R^{\text{CH}}=B_{n}\sum_{i=1}^{M/2}(\log_{2}(1+\gamma_{i})+\log_{2}(1+\gamma_{ M-i+1})), \tag{2}\]

with \(\gamma_{i}=\frac{a_{i}P_{n}H_{i}}{B_{n}N_{0}}\) and \(\gamma_{M-i+1}=\frac{a_{M-i+1}P_{n}H_{M-i+1}}{a_{i}P_{n}H_{M-i+1}+B_{n}N_{0}}\) being the signal-to-interference-plus-noise ratios (SINRs) of the \(i\)-th NOMA pair \([\text{CH}_{i},\text{CH}_{M-i+1}]\), \(B_{n}\) is the bandwidth assigned to a NOMA pair or a lone user, given by \(B_{n}=\frac{B}{M/2+l}\), with \(B\) the total available bandwidth at BS, and \(l\) the number of lone users. The total transmission power of BS is as well equally partitioned among the \(M/2\) NOMA pairs and the \(l\) lone users, i.e., \(P_{n}=\frac{P}{M/2+l}\), \(a_{i}\) and \(a_{M-i+1}\) are the optimal power coefficients for the \(i\)-th NOMA pair \([\text{CH}_{i},\text{CH}_{\text{M}-i+1}]\) (with \(a_{M-i+1}=1-a_{i}\)). On the other hand, for the lone users, the sum rate can be expressed as

\[R^{\mathcal{L}}=B_{n}\sum_{k=1}^{l}\log_{2}\!\left(1+\frac{P_{n}H_{k}}{B_{n}N _{0}}\right). \tag{3}\]

By combining (2) and (3), we obtain the expression of the cellular users sum rate in (1). Similarly, we write the rate of a D2D user \(\text{UE}_{k}\) within a given coalition as

\[R_{k}^{\text{D2D}}=\tau^{\prime}\log_{2}\!\left(1+\frac{P_{k}G_{k}}{N_{0}} \right), \tag{4}\]

with \(G_{k}\) the channel gain of the D2D link between \(\text{UE}_{k}\) and its source transmitter, and \(P_{k}\) the D2D transmit power. The factor \(\tau^{\prime}\) is used to account for the ration of channel use needed to form the coalition and the transmission tree of that given coalition (denoted by \(\tau_{\text{tr}}\)), i.e., \(\tau^{\prime}=\tau+\tau_{\text{tr}}\).