# Performance Analysis of a New Non-Orthogonal Multiple Access Design for Mitigating Information Loss
## 4 Performance Analysis
### Exact BER of Conventional NOMA
\begin{split}\text{P}_{1}(e|(X^{1})^{\prime})=&\frac{1}{2} \bigg{[}\frac{1}{4}\text{P}\Big{(}(X^{1}_{1,A_{1}})^{\prime}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\geq\sqrt{\frac{ \epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{1,A_{1}})^{\prime}\Big{)}+\text{P}\Big{(} n_{\text{I}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{1,A_{1}})^{ \prime}\Big{)}\bigg{\}}\\ &+\frac{1}{4}\text{P}\Big{(}(X^{1}_{1,B_{1}})^{\prime}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\leq-\sqrt{\frac{ \epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{1,B_{1}})^{\prime}\Big{)}+\text{P}\Big{(} n_{\text{I}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{1,B_{1}})^{ \prime}\Big{)}\bigg{\}}\\ &+\frac{1}{4}\text{P}\Big{(}(X^{1}_{2,A_{1}})^{\prime}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\geq\sqrt{\frac{ \epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{2,A_{1}})^{\prime}\Big{)}+\text{P}\Big{(} n_{\text{I}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{2,A_{1}})^{ \prime}\Big{)}\bigg{\}}\\ &+\frac{1}{4}\text{P}\Big{(}(X^{1}_{2,B_{1}})^{\prime}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\leq-\sqrt{\frac{ \epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{2,B_{1}})^{\prime}\Big{)}+\text{P}\Big{(} n_{\text{I}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\big{|}(X^{1}_{2,B_{1}})^{ \prime}\Big{)}\bigg{\}}\bigg{]},\end{split}  \tag{22}\]  
where \(\frac{1}{4}\text{P}((X^{1}_{j,A_{1}})^{\prime})\) and \(\frac{1}{4}\text{P}((X^{1}_{j,B_{1}})^{\prime})\) are the probabilities that user 1 correctly detects the \(j\)-th bit of \(s_{2}\) as given in (13). If the fundamental theorem of conditional probability, \(\text{P}(A|\text{B})=\text{P}(A\cap\text{B})/\text{P}(\text{B})\), is applied, (22) can be redefined as follows:  
\[ \begin{split}\text{P}_{1}(e|(X^{1})^{\prime})=&\frac {1}{8}\bigg{[}\text{P}\bigg{(}\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\leq n_{\text{ Q}}\leq\sqrt{\frac{\epsilon_{2}}{2}}h_{1}+\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \bigg{)}+\text{P}\bigg{(}n_{\text{Q}}\leq-\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \bigg{)}\\ &+\text{P}\bigg{(}n_{\text{I}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{ 1}\bigg{)}\\ &\times\bigg{\{}\text{P}\bigg{(}n_{\text{Q}}\leq\sqrt{\frac{ \epsilon_{2}}{2}}h_{1}+\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\bigg{)}+\text{P}\bigg{(} n_{\text{Q}}\leq\sqrt{\frac{\epsilon_{2}}{2}}h_{1}-\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \bigg{)}\bigg{\}}\\ &+\text{P}\bigg{(}\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\leq n_{\text{ I}}\leq\sqrt{\frac{\epsilon_{2}}{2}}h_{1}+\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \bigg{)}+\text{P}\bigg{(}n_{\text{I}}\leq-\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \bigg{)}\\ &+\text{P}\bigg{(}n_{\text{Q}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{ 1}\bigg{)}\\ &\times\bigg{\{}\text{P}\bigg{(}n_{\text{I}}\leq\sqrt{\frac{ \epsilon_{2}}{2}}h_{1}+\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\bigg{)}+\text{P} \bigg{(}n_{\text{I}}\leq\sqrt{\frac{\epsilon_{2}}{2}}h_{1}-\sqrt{\frac{ \epsilon_{1}}{2}}h_{1}\bigg{)}\bigg{\}}\bigg{]}.\end{split}  \tag{23}\]  
For the purpose of simplifying (23), we define the SNR for the C or C\({}^{\prime}\) signal point given in Figure 5a as follows:  
\[\gamma_{\text{C}}=\frac{\epsilon_{1}{|h_{1}|}^{2}}{\sigma^{2}}. \tag{24}\]  
Then, the error probability of \(s_{1}\), given that user 1 correctly detects \(s_{2}\), is expressed as follows:  
\[\text{P}_{1}(e|(X^{1})^{\prime})= \frac{1}{4}\big{[}Q(\sqrt{\gamma_{\text{C}}})\times\big{\{}4-Q \big{(}\sqrt{\gamma_{\text{A}_{1}}}\big{)}-Q\big{(}\sqrt{\gamma_{\text{B}_{1}}} \big{)}\big{\}}-Q\big{(}\sqrt{\gamma_{\text{A}_{1}}}\big{)}\big{]}. \tag{25}\]  
Figure 5b,c show the constellations of \(s_{1}\) after user 1 incorrectly detects \(s_{2}\) and performs SIC. These represent the cases where the first bit of \(s_{2}\) is in error and where the second bit is in error, respectively. The expressions at points D and E, contrary to the case of \(X^{1}\), represent the in-phase and quadrature components of the signal, respectively, where terms containing \(\epsilon_{1}\) or \(\epsilon_{2}\) are not removed, but instead added to the signals in Figure 4: \((\pm\sqrt{\epsilon_{1}/2}-2\sqrt{\epsilon_{1}/2})h_{1}+n_{1}\).