## 2 System Model
### Relay Transmission  
In Figure 2 is represented the proposed architecture of the system. The relay is considered to be ideal since it can communicate with all active users simultaneously and yet treat them separately as there is only one user active at each time instant, in the attempt to minimize possible interferences between active users. This assumption has been made in order to analyze the performance achieved by the base station when the information reaches it through different paths--a direct path and an indirect path, via relay. The noise present on the channel is modeled as additive white Gaussian noise.  
The spread signal is sent through the two paths mentioned above to reach the base station [53; 54], thus the signal received by relay from the source  
\[\mathbf{y}_{SR,k}=\sqrt{p}J_{k}\mathbf{s}_{k}+\mathbf{n}_{SR,k},k=1,2,\ldots,K \tag{17}\]  
Figure 2: The architecture of the proposed system.  
and the signal received at the destination from the source is  
\[\mathbf{y}_{SD,k}=\sum_{k=1}^{K}\sqrt{P}\mathbf{H}_{k}\mathbf{s}_{k}\ +\ \mathbf{n}_{SD,k} \tag{18}\]  
where \(P\) denotes the power of the signal that is being transmitted; \(\mathbf{J}_{k}\) represents the channel matrix of the \(k\)th mobile terminal (MT), from the MT to relay; \(\mathbf{s}_{k}\) represents the spread MUSA signal sent by the \(k\)th MT; \(\mathbf{n}_{SR,k}\) represents the Gaussian noise vector of the \(k\)th MT, from the MT to relay with zero mean and variance \(\varepsilon\left[\mathbf{n}_{SR,k}\mathbf{n}_{SR,k}^{H}\right]=\sigma_{SR,k}^{2}\); \(\mathbf{H}_{k}\) represents the channel matrix of the \(k\)th MT, from MT to BS; \(\mathbf{n}_{SD,k}\) represents the Gaussian noise vector of the \(k\)th MT, from the MT to BS with zero mean and variance \(\varepsilon\left[\mathbf{n}_{SD,k}\mathbf{n}_{SD,k}^{H}\right]=\sigma_{SD,k}^{2}\).  
If the DF protocol is taken into consideration, the signal that reaches the BS coming from the relay is described as  
\[\mathbf{y}_{RD,k}=\sum_{k=1}^{K}\sqrt{\widetilde{P}_{r}}\mathbf{F}_{k}\mathbf{y}_{SR,k}\ +\ \mathbf{n}_{RD,k} \tag{19}\]  
and is chosen that  
\[P\ +\ P_{r}=1 \tag{20}\]  
where \(P_{r}\) denotes the power of the signal that is being transmitted from the relay to BS and \(\widetilde{P}_{r}=P_{r}\) if the relay correctly decoded the signal from MT; \(\mathbf{F}_{k}\) represents the channel matrix of the \(k\)th MT, from relay to BS; \(\mathbf{n}_{RD,k}\) represents the Gaussian noise vector of the \(k\)th MT, from relay to BS with zero mean and variance \(\varepsilon\left[\mathbf{n}_{RD,k}\mathbf{n}_{RD,k}^{H}\right]=\sigma_{RD,k}^{2}\).  
At the base station, there will be two synchronous copies of the same transmitted signal, taking into account the two paths assumed in system configuration. To recover the original data, a maximum ratio combiner is being used whose output is  
\[\mathbf{y}=\alpha_{1}\mathbf{y}_{SD}\ +\ \alpha_{2}\mathbf{y}_{RD} \tag{21}\]  
The channel coefficients, \(\alpha_{1}\) and \(\alpha_{2}\), are described as  
\[\alpha_{1}=\frac{\sqrt{P}\mathbf{H}_{k}^{*}}{\sigma_{SD,k}} \tag{22}\]  
and  
\[\alpha_{2}=\frac{\sqrt{P_{r}}\mathbf{F}_{k}^{*}}{\sigma_{RD,k}^{2}} \tag{23}\]  
with \(k=1,2,\ldots,K\).