# Novel Resource Allocation Techniques for Downlink Non-Orthogonal Multiple Access Systems
## 3 System Model and Performance Metrics
### System Model  
The system model examines a downlink cellular wireless NOMA architecture with a BS located at the cell center, as shown in Figure 1. The signal of BS is assigned consistently to \(M\) users within the cell radius \(R\) with \(S\) subchannels. The symbols \(w\) = {1, 2,..., W} and \(s\) = {1, 2,..., S} denote the groups of users and subchannels, respectively. UE\({}_{ws}\) represents the \(w\)th user in the \(s\)th subchannel within the NOMA system. \(B_{w}\) is the system bandwidth, and \(B_{sch}\) = \(B_{w}\)/S is the assigned bandwidth for each subchannel. \(P_{\text{max}}\) and the \(P_{\text{s}}\) denote the maximum transmit power for BS and the assigned power for each subchannel, respectively. With the assumption that the BS has an ideal knowledge of CSI, \(M_{s}\) represents the number of the paired users on \(s\)th subchannel and \(S_{i}\) represents their superposed signals. The superposition of the user's modulated symbols takes place in each of the \(S\) subchannels. Hence, the BS transmission symbols are expressed in Equations (1) and (3).  
Figure 1: The proposed downlink NOMA system model for single-cell transmission.  
\[x_{s}=\sum_{q=1}^{M}\sqrt{P_{q,s}s_{q}} \tag{1}\]  
\(S_{q}\) represents the modulated symbol of \(q\)th user, in which E [\(|S_{q}|^{2}=1\)]. The transmit power allocated to the \(q\)th user on the \(s\)th subchannel is denoted by \(P_{q,s}\). The received signal for UE\({}_{\text{in},s}\) is given as in [28].  
\[y_{w,s}=h_{w,s}x_{s}+z_{w,s} \tag{2}\]  
\[=\sqrt{P_{w,s}}h_{w,s}s_{w}+\sum_{i=1,i\
eq w}^{M}\sqrt{P_{i,s}}h_{w,s}s_{i}+n _{w,s} \tag{3}\]  
where \(h_{w,s}=g_{w,s}(d_{w})^{-\beta}\) represents the channel gain from the BS to the \(w\)th user, \(g_{w,s}\) denotes the Rayleigh fading coefficient, \(d_{w}\) is the distance between \(w\)th user and the BS, \(\beta\) is the path-loss exponent and \(n_{w,s}\sim\)C\(\mathcal{N}\big{(}0,\sigma_{s}^{2}\big{)}\) represents the additive white Gaussian noise (AWGN), with zero mean and variance \(\sigma_{s}^{2}\). The power spectral density is symbolized by \(N_{0}\). The power constraints for each subchannel and the BS are limited to \(\sum_{i=1}^{M_{s}}P_{w,s}=P_{s}\) and \(\sum_{s=1}^{S}P_{s}=P_{\text{max}}\), respectively, where \(P_{w,s}\) is the power of UE\({}_{\text{in},s}\). Let \(H_{w,s}=\big{|}h_{w,s}\big{|}^{2}/N_{0}\) be the channel gain-to-noise ratio (CNR) of UE\({}_{\text{in},s}\). The arrangement of the users in a subchannel is based on their CNR as shown in Equation (4).  
\[H_{1,s}\geq H_{2,s}\geq H_{3,s}\geq\cdots\geq H_{M,s} \tag{4}\]  
Basically, high transmission power is allocated to users with low channel gain, and low transmission power is assigned to users with high channel gain [29]. For two users pairing with \(H_{1,s}\geq H_{2,s}\), the BS allocates the power as \(P_{1,s}<P_{2,s}\). The SIC process is deployed for the signal separation in a multiplexed user's subchannels considering the available user channel information. Hence, the strong user (UE\({}_{1,s}\)) decodes first and removes the weak user (UE\({}_{2,s}\)) signal before decoding its signal. On the contrary, the weak user (UE\({}_{2,s}\)) directly decodes its signal without adopting SIC. With the consideration of SIC at the receiver, the received signal to interference plus noise ratio (SINR) for UE\({}_{\text{in},s}\) is represented in Equation (5).  
\[SINR_{w,s}=\frac{P_{w,s}H_{w,s}}{1+\sum_{l=1}^{w-1}P_{l,s}H_{w,s}} \tag{5}\]