# Performance Analysis of a New Non-Orthogonal Multiple Access Design for Mitigating Information Loss
## 4 Performance Analysis
### Exact BER of Conventional NOMA  
In the conventional NOMA system, both user 1 and user 2 decode the received signal to detect \(s_{2}\). Additionally, user 1 detects and removes \(s_{2}\) to decode its own signal, \(s_{1}\). Therefore, the error probability for \(s_{2}\) at user \(k\) is first calculated. In QPSK modulation, since each symbol consists of 2 bits, the probability that user \(k\) fails to decode \(s_{2}\) can be expressed as the average of the error probabilities for the first and second bits of \(s_{2}\), as follows:  
\[\mathrm{P}(X^{k})=\frac{1}{2}\Big{[}\mathrm{P}(X^{k}_{1})+\mathrm{P}(X^{k}_{2} )\Big{]}, \tag{12}\]  
where \(X_{j}\) denotes the event that user \(k\) incorrectly detects \(s_{2}\) and \(X^{k}_{j}\) denotes the event that user \(k\) incorrectly detects the \(j\)-th bit of \(s_{2}\) (\(j=1,2\)) [8]. Figure 4 shows the constellation diagram of the signals received by each user in a conventional NOMA system. Each point represents the combination of signals \(s_{1}\) and \(s_{2}\) under SC. It is denoted as \(\{s_{1},s_{2}\}\), and the received signals at each point are expressed by (5) as \((\pm\sqrt{\alpha_{1}P_{t}/2}\pm j\sqrt{\alpha_{1}P_{t}/2})h_{k}+(\pm\sqrt{ \alpha_{2}P_{t}/2}\pm j\sqrt{\alpha_{2}P_{t}/2})h_{k}+n_{k}\). In Figure 4, an error occurs if any point on the constellation diagram deviates from its appropriate decision region. It is evident that the first bit of the QPSK symbol is determined by the quadrature part, and the second bit is determined by the in-phase part. For instance, if the actual first bit of \(s_{2}\) is 1, in order to detect the first bit correctly, points A\({}_{k}\) and B\({}_{k}\) should be located below the boundary of the decision region. The second bit can also be approached in the same way as before. That is, since user \(k\) mistakenly decides bit 1 as 0 when point A\({}_{k}\) or B\({}_{k}\) is greater than 0, \(\mathrm{P}(X^{k}_{1})\) and \(\mathrm{P}(X^{k}_{2})\) can be expressed as follows:  
Figure 4: Constellation of signals received by user \(k\) for conventional NOMA. \[ \begin{split}\mathrm{P}(X_{1}^{k})&=\frac{1}{2}\Big{[} \mathrm{P}\Big{(}X_{1,\mathrm{A}_{k}}^{k}\Big{)}+\mathrm{P}\Big{(}X_{1, \mathrm{B}_{k}}^{k}\Big{)}\Big{]}\\ \mathrm{P}(X_{2}^{k})&=\frac{1}{2}\Big{[}\mathrm{P} \Big{(}X_{2,\mathrm{A}_{k}}^{k}\Big{)}+\mathrm{P}\Big{(}X_{2,\mathrm{B}_{k}}^{ k}\Big{)}\Big{]},\end{split}  \tag{13}\]  
where \(X_{j,i}^{k}\) denotes the event that user \(k\) incorrectly detects the \(j\)-th bit of \(s_{2}\) due to the position of point \(i\) (\(i=\mathrm{A}_{k},\mathrm{B}_{k}\)). By substituting (5) and the QPSK symbols, (13) can be expressed as follows:  
\[ \begin{split}\mathrm{P}(X_{1}^{k})=&\frac{1}{2} \bigg{[}\mathrm{P}\bigg{(}n_{\mathrm{Q}}\geq\sqrt{\frac{\epsilon_{2}}{2}}h_{k} +\sqrt{\frac{\epsilon_{1}}{2}}h_{k}\bigg{)}+\mathrm{P}\bigg{(}n_{\mathrm{Q}} \geq\sqrt{\frac{\epsilon_{2}}{2}}h_{k}-\sqrt{\frac{\epsilon_{1}}{2}}h_{k} \bigg{)}\bigg{]},\\ \mathrm{P}(X_{2}^{k})=&\frac{1}{2}\bigg{[}\mathrm{P} \bigg{(}n_{\mathrm{I}}\geq\sqrt{\frac{\epsilon_{2}}{2}}h_{k}+\sqrt{\frac{ \epsilon_{1}}{2}}h_{k}\bigg{)}+\mathrm{P}\bigg{(}n_{\mathrm{I}}\geq\sqrt{ \frac{\epsilon_{2}}{2}}h_{k}-\sqrt{\frac{\epsilon_{1}}{2}}h_{k}\bigg{)}\bigg{]},\end{split}  \tag{14}\]  
where \(n_{\mathrm{I}}\) and \(n_{\mathrm{Q}}\) denote the in-phase and quadrature components of the noise, respectively, and \(\epsilon_{k}\) is the symbol energy, \(\alpha_{k}\mathrm{P}_{\mathrm{t}}\)[9]. By substituting (14) into (12), the error probability of user 2 is defined as follows:  
\[ \begin{split}\mathrm{P}_{2}(e)=&\mathrm{P}(X^{2}) \\ =&\frac{1}{2}\bigg{[}\mathrm{Q}\Big{(}\frac{(\sqrt{ \epsilon_{2}/2}+\sqrt{\epsilon_{1}/2})h_{2}}{\sqrt{\sigma^{2}/2}}\Big{)}+ \mathrm{Q}\Big{(}\frac{(\sqrt{\epsilon_{2}/2}-\sqrt{\epsilon_{1}/2})h_{2}}{ \sqrt{\sigma^{2}/2}}\Big{)}\bigg{]},\end{split}  \tag{15}\]  
where \(\mathrm{Q}(\cdot)\) denotes the Gaussian \(Q\) function.