## 4 Outage Probability Analysis  
We define the outage event \(A_{i,j}=\{\mathrm{SINR}_{i,j}\leq\gamma_{\mathrm{thj}}\}\), where \(\gamma_{\mathrm{thj}}\) is the target threshold for the \(j\)th user. Then, event \(A_{i,j}^{c}\) means that the \(i\)th user cannot decode the \(j\)th user's signal. An outage occurs for the \(i\)th user when any of \(A_{i,j}^{c}\) occurs. The probability of an outage for the \(i\)th user can be expressed as  
\[\mathcal{P}_{\text{out}}^{i}=\text{Pr}\bigg{(}\bigcup_{j=1}^{i}A_{i,j}^{c} \bigg{)}=1-\text{Pr}(A_{i,1}\cdots A_{i,i}). \tag{27}\]  
**Theorem 2**. : _A closed-form expression of the probability of an outage for user \(i\) (\(\mathcal{P}_{\text{out}}^{i}\)) can be obtained as_  
\[ \begin{split}\mathcal{P}_{\text{out}}^{i}=1-\frac{N!\beta_{ \text{RU}}^{-1}}{(i-1)!(N-i)!}\sum_{s=0}^{i-1}& 2\binom{i-1}{s}(-1)^{s}\mathrm{e}^{-\big{(} \frac{\Gamma_{2}\beta_{\text{RU}}^{*}}{f_{\text{SR}}\mathbb{I}^{3}}+\chi_{i,s} \delta_{i}^{*}\big{)}}\\ &\times\sqrt{\frac{\phi_{i}}{\chi_{i,s}}}\,\mathrm{K}_{1}\bigg{(} 2\sqrt{\phi_{i}\chi_{i,s}}\bigg{)}\end{split}  \tag{28}\]  
_where \(\chi_{i,s}\), \(\phi_{i}\), and \(\delta_{i}^{*}\) are defined in (32), and \(K_{1}(\cdot)\) denotes the first-order modified Bessel function of the second kind._  
Proof. : By replacing (15) in \(A_{i,j}\), it can be rewritten as  
\[A_{i,j}=\bigg{\{}|\hat{h}_{i}|^{2}\geq\delta_{j},|\hat{h}_{\text{SR}}|^{2}\geq \frac{(\Gamma_{2}|\hat{h}_{i}|^{2}+\Gamma_{4})\delta_{j}}{\Gamma_{3}(|\hat{h} _{i}|^{2}-\delta_{j})}\bigg{\}}, \tag{29}\]  
where \(\delta_{j}\coloneqq\frac{\gamma_{\mathrm{thj}}\Gamma_{3}}{\Gamma_{1}(\alpha _{ij}-\beta_{j}\gamma_{\mathrm{thj}})}\). By noting that \(\delta_{j}/(|\hat{h}_{i}|^{2}-\delta_{j})\) is monotone in \(\delta_{j}\), we can obtain that  
\[A_{i,1}\cdots A_{i,i}=\bigg{\{}|\hat{h}_{i}|^{2}\geq\delta_{i}^{*},|\hat{h}_{ \text{SR}}|^{2}\geq\frac{(\Gamma_{2}|\hat{h}_{i}|^{2}+\Gamma_{4})\delta_{i}^{ *}}{\Gamma_{3}(|\hat{h}_{i}|^{2}-\delta_{i}^{*})}\bigg{\}}, \tag{30}\]  
where \(\delta_{i}^{*}=\max(\delta_{1},\cdots,\delta_{i})\). From Figures 3 and 4, it can be seen that when the transmission power of the second stage is fixed, the outage probability gradually decreases, and the ergodic rate gradually increases as the transmission power of the first stage is increased. For different users, the obtained results have shown the existence of a floor at moderate and high SNR levels, which is due to channel estimation errors and residual LI. It is also noted that both the outage probability and ergodic rate of user 3 outperforms user 2 and user 1. User 1 has the worst performance. This happens because NOMA assigns the largest power factor to the user with the worst channel conditions, so the first user has the highest outage probability and the lowest ergodic rate. Figures 5 and 6 show the effect of transmit power variation on the outage probability and ergodic rate of the system in the second transmission stage. The performance improves as the transmit power increases under moderate or high SNRs. For both the outage probability and ergodic rate, there are SNR boundaries corresponding to maximum points of curves. The performance decreases for all users when the SNR increases over boundaries. This is because the SINR of the relay is dominated by a large residual LBL. Figures 7 and 8 show the experimental results on the effect of LI gain level on the system performance. It is obvious to see that as the LI gain increases, the performance of the system progressively deteriorates, which is shown in the figures as an increase in the outage probability and a decrease in the ergodic rate. This is due to the higher interference level, which makes the signal relatively smaller. Therefore, a practical transceiver design would adopt a combination of the RF domain, analog domain, and digital domain's self-interference cancellations to mitigate LI. If it is hard to ensure that LI is suppressed within an acceptable range, it is better to use HD mode.