# Performance Analysis of a New Non-Orthogonal Multiple Access Design for Mitigating Information Loss
## 4 Performance Analysis
### Exact BER of Conventional NOMA
In Figure 5b, user 1 incorrectly detects the first bit of \(s_{1}\) if either  
\((-\sqrt{\epsilon_{1}/2}-2\sqrt{\epsilon_{2}/2})h_{1}+n_{\text{Q}}\geq 0\) or \((\sqrt{\epsilon_{1}/2}-2\sqrt{\epsilon_{2}/2})+n_{\text{Q}}\leq 0\), and incorrectly detects the second bit of \(s_{1}\) if either \(\sqrt{\epsilon_{1}/2}h_{1}+n_{\text{I}}\leq 0\) or \(-\sqrt{\epsilon_{1}/2}h_{1}+n_{\text{I}}\geq 0\). Similarly, in Figure 5c, user \(1\) incorrectly detects the first bit of \(s_{1}\) if either \(\sqrt{\epsilon_{1}/2}h_{1}+n_{\text{Q}}\leq 0\) or \(-\sqrt{\epsilon_{1}/2}h_{1}+n_{\text{Q}}\geq 0\), and incorrectly detects the second bit of \(s_{1}\) if either \((-\sqrt{\epsilon_{1}/2}-2\sqrt{\epsilon_{2}/2})h_{1}+n_{\text{I}}\geq 0\) or \((\sqrt{\epsilon_{1}/2}-2\sqrt{\epsilon_{2}/2})+n_{\text{I}}\leq 0\). Hence, the bit error probability of user \(1\), under the condition that error occurs when detecting \(s_{2}\) symbols, is given as follows:  
\[ \begin{split}\text{P}_{1}(e|X^{1})&=\frac{1}{2} \bigg{[}\frac{1}{4}\text{P}\Big{(}X^{1}_{1,A_{1}}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\geq 2\sqrt{\frac{ \epsilon_{2}}{2}}h_{1}+\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\Big{|}X^{1}_{1,A_{1} }\Big{)}+\text{P}\Big{(}n_{\text{I}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \Big{|}X^{1}_{1,A_{1}}\Big{)}\bigg{\}}\\ &+\frac{1}{4}\text{P}\Big{(}X^{1}_{1,\text{B}_{1}}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\leq 2\sqrt{\frac{ \epsilon_{2}}{2}}h_{1}-\sqrt{\frac{\epsilon_{1}}{2}}h_{1}\Big{|}X^{1}_{1,\text{ B}_{1}}\Big{)}+\text{P}\Big{(}n_{\text{I}}\geq\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \Big{|}X^{1}_{1,\text{B}_{1}}\Big{)}\bigg{\}}\\ &+\frac{1}{4}\text{P}\Big{(}X^{1}_{2,A_{1}}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\geq\sqrt{\frac{ \epsilon_{1}}{2}}h_{1}\Big{|}X^{1}_{2,A_{1}}\Big{)}+\text{P}\Big{(}n_{\text{I}} \geq 2\sqrt{\frac{\epsilon_{2}}{2}}h_{1}+\sqrt{\frac{\epsilon_{1}}{2}}h_{1} \Big{|}X^{1}_{2,A_{1}}\Big{)}\bigg{\}}\\ &+\frac{1}{4}\text{P}\Big{(}X^{1}_{2,\text{B}_{1}}\Big{)}\\ &\times\bigg{\{}\text{P}\Big{(}n_{\text{Q}}\geq\sqrt{\frac{ \epsilon_{1}}{2}}h_{1}\Big{|}X^{1}_{2,\text{B}_{1}}\Big{)}+\text{P}\Big{(}n_{ \text{I}}\leq 2\sqrt{\frac{\epsilon_{2}}{2}}h_{1}-\sqrt{\frac{\epsilon_{1}}{2}}h_{1 }\Big{|}X^{1}_{2,\text{B}_{1}}\Big{)}\bigg{\}}\bigg{]},\end{split}  \tag{26}\]  
where \(\frac{1}{4}\text{P}(X^{1}_{i,A_{1}})\) and \(\frac{1}{4}\text{P}(X^{1}_{i,B_{1}})\) are the probabilities that user \(1\) incorrectly detects the \(j\)-th bit of \(s_{2}\) as given in (13). Similar to (23), (26) can be redefined as follows:  
\[\begin{split}\text{P}_{1}(e|X^{1})=&\frac{1  
By substituting (25) and (29) into (21), the error probability of user 1 is obtained as follows:  
\[ \begin{split}\mathrm{P}_{1}(e)=& Q(\sqrt{\gamma_{\mathrm{C }}})+\frac{1}{4}\big{[}-Q\big{(}\sqrt{\gamma_{\mathrm{A}_{1}}}\big{)}+Q\big{(} \sqrt{\gamma_{\mathrm{B}_{1}}}\big{)}+Q(\sqrt{\gamma_{\mathrm{D}}})-Q(\sqrt{ \gamma_{\mathrm{E}}})\big{]}.\end{split}  \tag{30}\]  
Then, the average BER at user 1 can be obtained as follows:  
\[ \begin{split}\overline{\mathrm{P}_{1}(e)}=&\int_{0}^ {\infty}Q(\sqrt{\gamma_{\mathrm{C}}})f_{\gamma_{\mathrm{C}}}(\gamma_{\mathrm{C }})\mathrm{d}\gamma_{\mathrm{C}}\\ &+\frac{1}{4}\bigg{[}-\int_{0}^{\infty}Q\big{(}\sqrt{\gamma_{ \mathrm{A}_{1}}}\big{)}f_{\gamma_{\mathrm{A}_{1}}}\big{(}\gamma_{\mathrm{A}_{ 1}}\big{)}\mathrm{d}\gamma_{\mathrm{A}_{1}}+\int_{0}^{\infty}Q\big{(}\sqrt{ \gamma_{\mathrm{B}_{1}}}\big{)}f_{\gamma_{\mathrm{B}_{1}}}\big{(}\gamma_{ \mathrm{B}_{1}}\big{)}\mathrm{d}\gamma_{\mathrm{B}_{1}}\\ &+\int_{0}^{\infty}Q(\sqrt{\gamma_{\mathrm{D}}})f_{\gamma_{ \mathrm{D}}}(\gamma_{\mathrm{D}})\mathrm{d}\gamma_{\mathrm{D}}-\int_{0}^{ \infty}Q(\sqrt{\gamma_{\mathrm{E}}})f_{\gamma_{\mathrm{E}}}(\gamma_{\mathrm{E }})\mathrm{d}\gamma_{\mathrm{E}}\bigg{]}.\end{split}  \tag{31}\]  
Using ([10], Equations (13.3)-(7)) and (19), (31) is given as follows:  
\[