# Performance Analysis of Ambient Backscatter NOMA Systems
## 3 OP Analysis
### Diversity Analysis  
For deeper insight, we obtained the diversity order of the OP in the AmBC-NOMA systems, which described the rate at which the OP decreased with increasing SNR. Hence, the diversity orders \(D_{n}\), \(n\in\{d,BD\}\) can be defined as  
\[D_{n}=-\lim_{\rho\to\infty}\frac{\log_{2}(P_{n}^{\infty}(\rho))}{\log_{2}\rho}, \tag{16}\]  
where \(P_{n}^{\infty}(\rho)\) denotes the OP for the \(n\)-th user at high SNRs. The asymptotic outage behaviors for users and BD were analyzed.  
**Corollary 4**.: _The diversity order of the \(d\)-th user is calculated by (9); when \(\rho\to\infty\), the asymptotic OP expression of the \(d\)-th user in the case of ipSIC for AmBC-NOMA systems is shown as_  
\[P_{d,\infty}^{pSIC}=\phi_{d}\sum_{l=0}^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l }\Bigg{[}\sum_{r=1}^{d+l}C_{d+1}^{r}\times(-1)^{r+1}Q_{d}e^{Q_{d}}\mathrm{Ei} (-Q_{d})\frac{1}{\lambda_{hI}\xi_{1}}+1\Bigg{]}. \tag{17}\]  
**Corollary 5**.: _When \(\varpi=0\) and \(\rho\to\infty\), the asymptotic OP expression of the \(d\)-th user in the case of pSIC for AmBC-NOMA systems is given by_  
\[P_{d,\infty}^{pSIC}=\phi_{d}\sum_{l=0}^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l }\times\Bigg{[}\sum_{r=1}^{d+l}C_{d+1}^{r}(-1)^{r+1}Q_{d}e^{Q_{d}}\mathrm{Ei} (-Q_{d})+1\Bigg{]}. \tag{18}\]  
**Remark 1**.: _By substituting (17) and (18) into (16), the diversity orders for the \(d\)-th user with the ipSIC/pSIC in the AmBC-NOMA systems are equal to zero._  
**Corollary 6**.: _Similar to the procedure in (5), the asymptotic OP expression of the orthogonal user for the AmBC-OMA systems can be expressed as_  
\[P_{d,\infty}^{OMA}=1+Q_{d}^{OMA}e^{Q_{d}^{OMA}}\mathrm{Ei}\Big{(}-Q_{d}^{OMA }\Big{)}. \tag{19}\]  
**Corollary 7**.: _At high SNRs, the asymptotic OP expression of the BD in the case of ipSIC for AmBC-NOMA systems can be given by_  
\[P_{BD,\infty}^{ipSIC}= 1+\phi_{D}\sum_{l=0}^{D-d}\sum_{r=1}^{D}\binom{D-d}{l}\frac{(-1) ^{l}}{d+l}C_{D}^{r}\times(-1)^{r+1}[(e^{\tau_{1}}\Delta_{b1}\epsilon_{1}- \Delta_{b2})\,-\frac{\pi\varphi_{y}}{N\lambda_{f}\lambda_{gd}}\] \[\times\,\sum_{n=1}^{N}\Big{(}\epsilon_{1}e^{\xi_{1}}-\epsilon_{2} e^{\xi_{2}}\Big{)}K_{0}(\Lambda_{b})\sqrt{1-x_{n}{}^{2}}\Bigg{]}. \tag{20}\]  
**Corollary 8**.: _For \(\varpi=0\), the asymptotic OP expression of the BD in the case of pSIC for the AmBC-NOMA systems can be derived as_  
\[P_{BD,\varpi}^{pSIC}= 1+\phi_{D}\sum_{l=0}^{D-d}\sum_{r=1}^{D}\binom{D-d}{l}\frac{(-1)^ {l}}{d+l}C_{D}^{r}\times(-1)^{r+1}[(e^{\tau_{1}}\Delta_{b1}-\Delta_{b2})\,- \frac{\pi\varphi_{y}}{N\lambda_{f}\lambda_{gd}}\] \[\times\,\sum_{n=1}^{N}\left(1-e^{\xi_{2}}\right)K_{0}(\Lambda_{b })\sqrt{1-{x_{n}}^{2}}\Bigg{]}. \tag{21}\]  
**Remark 2**.: _Substituting (20) and (21) into (16), the diversity order of the BD for AmBC-NOMA systems is zero, which is due to the residential interference from ipSIC/pSIC._  
**Remark 3**.: _According to Corollaries 4 and 5, and Corollaries 7 and 8, we can determine that RHIs and ipSIC increase the OP of AmBC-NOMA systems, which has a detrimental effect on the performance of the system. The OP of the \(d\)-th user and the BD gradually resembles a constant, as the transmitting SNR rises. The system has a diversity order of zero, which leads to the appearance of error floors. Therefore, the diversity orders of users and the BD in AmBC-NOMA systems can be formulated as_  
\[D_{d}^{ipSIC}=D_{d}^{pSIC}=D_{BD}^{ipSIC}=D_{BD}^{pSIC}=0. \tag{22}\]