# Simulated Annealing for Resource Allocation in Downlink NOMA Systems in 5G Networks
## 2 NOMA System Model
According to [PERSON]'s capacity formula, the throughput of subchannel \(n\) is given below:  
\[R_{n}=B_{\mathrm{sc}}\ \sum_{i=1}^{U_{n}}\log_{2}\Biggl{(}1+\frac{P_{m,n} \left|G_{m,n}\right|^{2}}{1+\sum_{i=1,j\
eq m}^{U_{n}}P_{i,n}\ G_{m,n}}\Biggr{)} =B_{\mathrm{sc}}\ \sum_{i=1}^{U_{n}}\log_{2}(1+SINR_{m,n}) \tag{4}\]  
The user \(U_{m,n}\) receives interference from the other user, where the interferences \(SINR_{m,n}\), also denoted as \(\gamma_{m,n}\), are expressed as follows:  
\[\gamma_{m,n}=\sum_{i=1,i,j\
eq m}^{U_{n}}P_{i,n}\ G_{m,n} \tag{5}\]  
The NOMA's basic principle is to perform SIC at one receiver to reduce the interference between the multiplexed users on the same subchannel. The SIC technique enables the user with stronger CRNNs to decode the user's signal with poorer CRNNs and cancel it from the superimposed message. However, the user with the poorer CRNNs decodes the intended signal and treats the other signal as noise. Moreover, the increasing order of CRNNs achieves the optimal decoding order for appropriate SIC performance. Hence, any user can decode the other user's signal successfully and correctly based on the CRNNs order. In power domain multiplexing, the NOMA system allocates more power to the user with lower CRNN to maximize the throughput and achieve balanced Throughput between the multiplexed users. For instance, the two users \(U_{1,n}\) and \(U_{2,n}\) are multiplexed on the same subchannel \(n\) with \(|G_{1,n}|^{2}\geq|G_{2,n}|^{2}\), and the user fairness is ensured by the power allocation scheme of \(P_{1,n}\leq P_{2,n}\) based on the NOMA power allocation protocol [4; 24; 23]. In Figure 2, SIC application is performed for each subchannel, where each user can extract the intended signal. Consider that the users allocated on the same subchannel \(n\) are \(U_{U_{n},n}\). The user multiplexing scheme is adopted based on CRNNs in order to assign users on the same subchannel as \(|G_{1,n}|\geq|G_{2,n}|\geq\ldots\geq|G_{m,n}|\geq|G_{m+1,n}|\geq\ldots\geq|G _{U_{n},\ n}|\). The user \(UE_{m}\) decodes and cancels the interference power from the user with a weaker channel condition as \(UE_{m+1}\)-\(UE_{m+2},\ldots,UE_{U_{n}}\), according to the optimal SIC decoding order where the interference is cancelled successfully. However, during the decoding process user \(UE_{m}\) experiences a weaker channel condition and will not be able to remove the interference from the user \(UE_{m-1}\), who experiences better channel conditions, so it treats the interference power as noise. The standard NOMA design is assumed to allocate more power ratio to the user with a weak channel gain and less power ratio to the user with a strong channel gain. In the first step, SIC will decode the data signal of the user with a weak channel gain as more power is assigned. In the second step, the second data signal is decoded for the user with a strong channel gain, where the lower power ratio is assigned. The optimal SIC application depends on the power ratio for the multiplexed users, where the power level difference is the main factor in reducing the SIC complexity. Therefore, the user \(U_{m,n}\) performs SIC, while SINR is expressed as follows:  
\[SINR_{m,n}{}^{SIC}=\frac{P_{m,n}\ G_{m,n}}{1+\sum_{i=1,i\
eq m}^{m-1}P_{i,n}G_ {m,n}} \tag{6}\]