# Performance Analysis of Ambient Backscatter NOMA Systems
## 3 OP Analysis
### OP Expression for the BD  
The backscatter signal \(c(t)\) from the BD will fail to be decoded unless the data signal \(x(t)\) is decoded correctly; therefore, the OP expression for the BD can be written as  
\[P_{out}^{BD}=1-\mathrm{Pr}\Big{\{}\gamma_{D}>\gamma_{thD},\gamma_{BD}>\gamma_ {th}^{BD}\Big{\}}, \tag{13}\]  
where \(\gamma_{th}^{BD}\) is the threshold rate of the BD.  
Theorem 3.2: _The approximate OP expression of BDs for AmBC-NOMA systems can be derived as_  
\[P_{BD}^{ipSIC}\approx 1+\phi_{D}\sum_{l=0}^{D-d}\sum_{r=1}^{D}\binom{D-d}{l}\frac{(-1) ^{l}}{d+l}C_{D}^{r}(-1)^{r+1}[(e^{\tau_{1}}\Delta_{b1}\epsilon_{1}-e^{\tau_{2} }\Delta_{b2}\epsilon_{2})\] \[-\frac{\pi\varphi_{y}}{N\lambda_{f}\lambda_{gd}}\sum_{n=1}^{N} \Big{(}\epsilon_{1}\epsilon^{2}-\epsilon_{2}\epsilon^{2}\Big{)}D_{0}(\Lambda_ {b})\sqrt{1-x_{n}{}^{2}}\Bigg{]}, \tag{14}\]  
_where \(\epsilon_{1}=\frac{1}{\lambda_{hl}\epsilon_{01}}\), \(\epsilon_{2}=\frac{1}{\lambda_{hl}\epsilon_{02}}\), \(\Lambda_{b}=\sqrt{\frac{2(x_{n}+1)\varphi_{y}}{\lambda_{f}\lambda_{gd}}}\), \(\tau_{1}=-\frac{\psi_{d}^{2}r}{\lambda_{hl}}\), \(\tau_{2}=\frac{1}{\lambda_{hl}\rho\rho\kappa_{sd}^{2}}\), \(\Delta_{b1}=Q_{b1}e^{Q_{b1}}\mathrm{Ei}(-Q_{b1})\), \(\Delta_{b2}=Q_{b2}e^{Q_{b2}}\mathrm{Ei}(-Q_{b2})\), \(x_{n}=\cos\left(\frac{2n-1}{2N}\pi\right)\), and \(N\) is a balance parameter of accuracy and complexity. \(Q_{b1}=\frac{\lambda_{hl}}{\lambda_{f}\lambda_{f}\lambda_{gd}^{2}\rho[\beta^{2 }\left(1+x_{l}^{2}\right)r}\), \(\zeta_{b1}=\frac{\psi_{d}^{2}\varphi\rho\lambda_{hl}r+\lambda_{hd}}{\lambda_{hl} \lambda_{hl}\lambda_{hl}}\), \(\xi_{1}=\frac{1}{\lambda_{hl}\rho\kappa_{sd}^{2}}-\frac{r(x_{n}+1)\varphi_{y}}{ 2\lambda_{hl}\rho\kappa_{sd}^{2}}\), \(Q_{b2}=\frac{\lambda_{hl}\varphi\lambda_{b2}\rho\kappa_{sd}^{2}}{\lambda_{f} \lambda_{gd}\kappa_{gd}^{2}}\), \(\xi_{2}=-\left(\frac{2\psi_{d}^{2}r+\psi_{d}^{2}\rho[\beta^{2}\left(1+x_{l}^{2} \right)r(x_{n}+1)\varphi_{y}}{2\lambda_{hl}\lambda_{hl}\rho\kappa_{sd}^{2}} \right)\), \(\zeta_{b2}=\frac{\lambda_{hl}\rho\kappa_{sd}^{2}-\lambda_{hl}}{\lambda_{hl} \lambda_{hl}\rho\kappa_{sd}^{2}}\), \(\varphi_{y}=\frac{\psi_{d}\rho\kappa_{sd}^{2}\psi_{y}^{2}+\psi_{b1}\rho\kappa_{sd} ^{2}}{1-\psi_{b2}\rho\kappa_{sd}^{2}\psi_{y}^{2}+\psi_{b2}\rho[\beta^{2}\left(1 +x_{n}^{2}\right)]}\), \(\psi_{b2}=\frac{\kappa_{b1}^{2}}{\rho\left(\left|\beta^{2}-\frac{\psi_{d}^{2}}{ \beta[\beta^{2}\left(1+x_{n}^{2}\right)]}\beta^{2}\kappa_{sd}^{2}\right)}\), \(r_{th}^{BD}=2^{R_{BD}}-1\)and \(R_{BD}\) represents the threshold rate of the BD. \(K_{0}(\cdot)\) is the second-kind modification of the zero-order Bessel function._  
Proof.: Refer to Appendix B.  
**Corollary 3**.: _When \(\varpi=0\), the approximate OP expression of the BD in the case of pSIC for the AmBC-NOMA systems is given by_  
\[P_{BD}^{pSIC}\approx 1+\phi_{D}\sum_{l=0}^{D-d}\sum_{r=1}^{D}\left(\begin{array}{c }D-d\\ l\end{array}\right)\frac{(-1)^{l}}{d+l}C_{D}^{r}(-1)^{r+1}[(e^{\tau_{1}}\Delta_ {b1}-e^{\tau_{2}}\Delta_{b2})\] \[-\frac{\pi\varphi_{y}}{N\lambda_{f}\lambda_{gd}}\sum_{n=1}^{N} \left(e^{\xi_{1}}-e^{\xi_{2}}\right)K_{0}(\Lambda_{b})\sqrt{1-x_{n}{}^{2}} \Bigg{]}. \tag{15}\]