# Performance Analysis of a New Non-Orthogonal Multiple Access Design for Mitigating Information Loss
## 4 Performance Analysis
### Exact BER of Conventional NOMA
The simplified expression of (15) is as follows:  
\[\mathrm{P}_{2}(e)=\frac{1}{2}\big{[}\mathrm{Q}\big{(}\sqrt{\tau_{\mathrm{A}_{ 2}}}\big{)}+\mathrm{Q}\big{(}\sqrt{\tau_{\mathrm{B}_{2}}}\big{)}\big{]}, \tag{16}\]  
where \(\gamma_{i}\) is the SNR for the \(i\) signal points given in Figure 4. For \(i=\mathrm{A}_{k},\mathrm{B}_{k}\), \(\gamma_{i}\) can be expressed as follows:  
\[ \begin{split}\gamma_{\mathrm{A}_{k}}&=\frac{(\sqrt {\epsilon_{2}}+\sqrt{\epsilon_{1}})^{2}{|h_{k}|}^{2}}{\sigma^{2}},\\ \gamma_{\mathrm{B}_{k}}&=\frac{(\sqrt{\epsilon_{2}} -\sqrt{\epsilon_{1}})^{2}{|h_{k}|}^{2}}{\sigma^{2}}.\end{split}  \tag{17}\]  
Then, the average BER at user 2 can be obtained as follows:  
\[\overline{\mathrm{P}_{2}(e)}= \frac{1}{2}\bigg{[}\int_{0}^{\infty}\mathrm{Q}\big{(}\sqrt{ \tau_{\mathrm{A}_{2}}}\big{)}f_{\gamma_{\mathrm{A}_{2}}}\big{(}\gamma_{ \mathrm{A}_{2}}\big{)}\mathrm{d}\gamma_{\mathrm{A}_{2}}+\int_{0}^{\infty} \mathrm{Q}\big{(}\sqrt{\tau_{\mathrm{B}_{2}}}\big{)}f_{\gamma_{\mathrm{B}_{2}}} \big{(}\gamma_{\mathrm{B}_{2}}\big{)}\mathrm{d}\gamma_{\mathrm{B}_{2}}\bigg{]}, \tag{18}\]  
where \(f_{\gamma_{i}}\) is the probability density function (PDF) of SNR. In this paper, assuming a Rayleigh fading channel, the PDF of the SNR is defined as follows:  
\[f_{\gamma_{i}}(\gamma)=\frac{1}{\overline{\gamma_{i}}}e^{-\gamma/\overline{ \gamma_{i}}},\quad\gamma\geq 0, \tag{19}\]  
where \(\overline{\gamma_{i}}\) denotes the average of \(\gamma_{i}\). Using ([10], Equations (13.3)-(7)) and (19), (18) is given as follows:  
\[\overline{\mathrm{P}_{2}(e)}=\frac{1}{4}\bigg{(}2-\sqrt{\frac{\overline{\tau_ {\mathrm{A}_{2}}}}{2+\overline{\gamma_{\mathrm{A}_{2}}}}}-\sqrt{\frac{ \overline{\tau_{\mathrm{B}_{2}}}}{2+\overline{\gamma_{\mathrm{B}_{2}}}}}\bigg{)}. \tag{20}\]  
Next, user 1 decodes \(s_{1}\) after performing SIC to remove \(s_{2}\). The error probability of user 1 depends on whether \(s_{2}\) is detected correctly or incorrectly. Therefore, the error probability of user 1 is defined as follows:  
\[\mathrm{P}_{1}(e)=\mathrm{P}_{1}(e|(X^{1})^{\prime})+\mathrm{P}_{1}(e|X^{1}), \tag{21}\]where \((\cdot)^{\prime}\) denotes the complement of \((\cdot)\). Figure 5 shows the constellation of \(s_{1}\) after user \(1\) correctly detects \(s_{2}\) and performs SIC. The expressions for each signal are derived from the signal expressions in Figure 4 with the terms containing \(e_{2}\) removed. Therefore, the signal at each point is expressed as \((\pm\sqrt{e_{1}/2}\pm j\sqrt{e_{1}/2})h_{1}+n_{1}\). For the correct detection of the bits in \(s_{1}\), the points C or C\({}^{\prime}\) on the in-phase and quadrature axes should not fall outside their respective decision regions. User \(1\) incorrectly detects the first bit of \(s_{1}\) if either \(\sqrt{e_{1}/2}h_{1}+n_{\text{Q}}\leq 0\) or \(-\sqrt{e_{1}/2}h_{1}+n_{\text{Q}}\geq 0\), and incorrectly detects the second bit of \(s_{1}\) if either \(\sqrt{e_{1}/2}h_{1}+n_{1}\leq 0\) or \(-\sqrt{e_{1}/2}h_{1}+n_{1}\geq 0\). Hence, the bit error probability of user \(1\), under the condition that no error occurs when detecting \(s_{2}\) symbols, is given as follows:  
Figure 5: Constellations of \(s_{1}\) signals after performing SIC for conventional NOMA: (**a**) when user \(1\) correctly detects \(s_{2}\), (**b**) when user \(1\) incorrectly detects the first bit of \(s_{2}\), and (**c**) when user \(1\) incorrectly detects the second bit of \(s_{2}\). \[