## 2 System Description  
We consider a downlink TDMA-RIS-NOMA point-to-point communication system that contains one AP with a single antenna; '\(M\)' devices, each equipped with a single antenna; and '\(n\)' RIS made up of '\(N\)' meta-atoms, each as shown in Figure 2.  
There are '\(n\)' numbers of RIS present in the proposed system model, as shown in Figure 1. We assume that the distances among the RISs are large enough to mitigate the possibility of interference from one RIS to another RIS. Here, the TDMA system is followed with '\(m\)' time slots. During one time slot, only one device is active in a TDMA group to access the corresponding RIS. Hence, a point-to-point communication model is formed between AP-RIS-user devices. One RIS is dedicated to one TDMA group. The entire communication period \(T_{C}\) is divided into multiple equal duration time slots of '\(t\)' milliseconds, as in a TDMA system and shown in Figure 3. During any time  
Figure 1: Motivation behind proposed work. slot '\(n\)' devices are active from '\(n\)' different TDMA groups. These active '\(n\)' devices make a downlink NOMA cluster. There are \(m\) devices in a TDMA group. Hence, there are '\(m\)' NOMA clusters. Each slot is assigned to a different cluster of user devices. The cluster formation is conducted as per the QoS (data-rates) requirements. When a new time slot starts, only one user device is communicating through one RIS. The number of user devices active in any time slot is equal to the number of RIS deployed (\(n\)) in the system. During the present time slot, these \(n\) active devices receive a common broadcasted composite NOMA signal through different wireless channels (i.e., AP-to-RIS-to-user device). This received NOMA composite signal is decoded at the receiver (user device). The realization of time synchronization and frequency distribution among different resource blocks are conducted as described in [16] for NB-IoT systems. Figure 3: Time slot allocation for different clusters in TDMA-RIS-NOMA downlink system. Figure 2: Overview of the TDMA-RIS assisted downlink NOMA system. Let \(D_{ll}\) denote the \(i\)th device of the \(l\)th cluster, which is scheduled in \(l\)th time slot. Further, \(N\) is the number of meta-atoms allocated to the device '\(D_{ll}\)' such that these \(N\) meta-atoms are tuned to reflect the incoming signal from AP towards the user device '\(D_{ll}\)'. Let \(h_{irl}\) denote the channel coefficient between \(D_{ll}\) and \(r\)th meta-atoms of \(i\)th RIS allocated to \(D_{ll}\). The \(g_{irl}\) indicates the channel gain between \(r\)th meta-atom of \(i\)th RIS allocated to \(D_{il}\) and AP. Here, \(i\in\{1,2,\ldots,n\}\)\(r\in\{1,2,3,\ldots,N\}\). Assume \(\mathbf{g_{ii}}\) and \(\mathbf{h_{ii}}\) are the corresponding channel vectors for AP to RIS and RIS to the device \(D_{il}\), respectively. The \(\mathbf{g_{ii}}\) and \(\mathbf{h_{ii}}\) have \(N\) total RIS elements, which generate the maximum channel gain (CG). If the symbol \(\mathbf{\Theta_{l}}\) denotes the reflectivity-matrix of RIS for all \(D_{ll}\)s, then equivalent channel gain \(h_{eq_{ii}}\) is given by  
\[h_{eq_{ii}}=\mathbf{h_{ii}^{T}}\mathbf{\Theta_{l}g_{ii}}=\alpha\sum_{r=1}^{N}( h_{irl})(g_{irl})e^{i\theta_{rl}}, \tag{1}\]  
where \(\mathbf{\Theta_{ii}}=\alpha\times diag(e^{i\theta_{ll}},e^{i\theta_{2}}, \cdots,e^{i\theta_{N}})\) is \(N\times N\) diagonal matrix for \(i\)th device \(D_{ll}\). \(\mathbf{h_{ii}}\) and \(\mathbf{g_{ii}}\) are the \(N\times 1\) matrix containing the channel gain coefficients. The \(\mathbf{h_{ii}^{T}}\) denotes the transpose of matrix \(\mathbf{h_{ii}}\). The parameter \(\alpha\in[0,1]\) is responsible for altering the amplitude of the reflected signal from the meta-atom. The optimal value of phase shifts is calculated using the point-to-point downlink model \(\theta_{rl}=-\angle(h_{irl}\times g_{irl})\forall r\in[1,N]\), which causes the in-phase addition of all the paths [17; 18]. We assume that the positions of RISs and AP are fixed so the distance between RIS and AP is also fixed.