# Performance Analysis of a New Non-Orthogonal Multiple Access Design for Mitigating Information Loss
## 4 Performance Analysis
### Exact BER of Proposed Transmitter and Receiver  
In the proposed method, the transmitter sends two types of data: one is the information for user 2, and the other are the redundancy data used to transform the former data into the information for user 1. In this method, user 2 only requires the former data, while user 1 requires both the former and the latter data. Therefore, we proceed by separating the former and latter data to calculate the error probability. In this section, we assume \(N=2\). First, the constellation diagram of the former data is represented as shown in Figure 6.  
The received signals at each point are expressed as \((\pm\sqrt{\mathrm{P}_{\mathrm{t}}/2}\pm j\sqrt{\mathrm{P}_{\mathrm{t}}/2})h_{ k}+n_{k}\). User \(k\) incorrectly detects the first bit of \(s_{2}\) if \(-\sqrt{\mathrm{P}_{\mathrm{t}}/2}h_{k}+n_{\mathrm{Q}}\geq 0\), and incorrectly detects the  
Figure 6: Constellation of \(s_{2}\) signals received by user \(k\) for proposed receiver design. second bit of \(s_{2}\) if \(-\sqrt{P_{t}/2}h_{k}+n_{1}\geq 0\). Thus, the error probability for \(s_{2}\) at user \(k\) is expressed as follows:  
\[\bar{\mathrm{P}}(X^{k})=\frac{1}{2}\left[\mathrm{P}\left(n_{\mathrm{Q}}\geq \sqrt{\frac{P_{t}}{2}}h_{k}\right)+\mathrm{P}\left(n_{1}\geq\sqrt{\frac{P_{t}} {2}}h_{k}\right)\right]. \tag{33}\]  
To simplify (33), SNR at points \(\mathrm{F}_{k}\) can be expressed as follows:  
\[\gamma_{\mathrm{F}_{k}}=\frac{P_{t}|h_{k}|^{2}}{\sigma^{2}}. \tag{34}\]  
By substituting (34) into (33), the error probability of user 2 is defined as follows:  
\[\bar{\mathrm{P}}_{2}(e)=\bar{\mathrm{P}}(X^{2})=Q\big{(}\sqrt{\gamma_{\mathrm{ F}_{2}}}\big{)}. \tag{35}\]  
Then, the average BER at user 2 for the proposed receiver can be obtained as follows:  
\[\overline{\bar{\mathrm{P}}_{2}(e)}=\int_{0}^{\infty}Q\big{(}\sqrt{\gamma_{ \mathrm{F}_{2}}}\big{)}f_{\gamma_{\mathrm{F}_{2}}}\big{(}\gamma_{\mathrm{F}_{ 2}}\big{)}\mathrm{d}\gamma_{\mathrm{F}_{2}}. \tag{36}\]  
Using ([10], Equations (13.3)-(7)) and (19), (36) is given as follows:  
\[\overline{\bar{\mathrm{P}}_{2}(e)}= \frac{1}{2}\left(1-\sqrt{\frac{\gamma_{\mathrm{F}_{2}}}{2+\gamma _{\mathrm{F}_{2}}}}\right). \tag{37}\]  
Next, for user 1, the error probabilities of both \(s_{2}\) and \(s_{\mathrm{c}}\) should be considered. The error probability of \(s_{2}\) is the probability of incorrectly detecting \(b_{2}\), and the error probability of \(s_{\mathrm{c}}\) is the probability of incorrectly detecting \(b_{\mathrm{XOR}}\). Examining the error probability of user 1, the XOR bit operation results in 1 when the two bits are different, and 0 when they are the same. Due to this characteristic, if both \(b_{2}\) and \(b_{\mathrm{XOR}}\) are incorrectly detected, the final \(b_{1}\) obtained in Section 3.2 would be detected without error. Therefore, the error for user 1 occurs only when either \(s_{2}\) or \(s_{\mathrm{c}}\) has an error. The average error probability of user 1 in the proposed receiver is expressed as follows:  
\[\overline{\bar{\mathrm{P}}_{1}(e)}=\Big{(}1-\overline{\bar{\mathrm{P}}(X^{1}) }\Big{)}\overline{\bar{\mathrm{P}}(Y^{1})}+\Big{(}1-\overline{\bar{\mathrm{P} }(Y^{1})}\Big{)}\overline{\bar{\mathrm{P}}(X^{1})}, \tag{38}\]  
where \(Y^{1}\) denotes the event that user 1 incorrectly detects \(s_{\mathrm{c}}\). Due to (33) and (34), the error probability of \(s_{2}\) for user 1 is given as follows:  
\[\bar{\mathrm{P}}(X^{1})=Q\big{(}\sqrt{\gamma_{\mathrm{F}_{1}}}\big{)}. \tag{39}\]  
Then, the average error probability of \(s_{2}\) at user 1 is given as follows:  
\[ \begin{split}\overline{\bar{\mathrm{P}}(X^{1})}&= \int_{0}^{\infty}Q\big{(}\sqrt{\gamma_{\mathrm{F}_{1}}}\big{)}f_{\gamma_{ \mathrm{F}_{1}}}\big{(}\gamma_{\mathrm{F}_{1}}\big{)}\mathrm{d}\gamma_{ \mathrm{F}_{1}}\\ &=\frac{1}{2}\left(1-\sqrt{\frac{\gamma_{\mathrm{F}_{1}}}{2+ \gamma_{\mathrm{F}_{1}}}}\right).\end{split}