# Performance Analysis of Ambient Backscatter NOMA Systems
## Appendix A Proof of Theorem 1  
Based on (8), the OP of the \(d\)-th user is written as  
\[P_{d} =1-\Pr\Bigl{\{}|h_{d}|^{2}>\psi_{d}^{*}\Bigl{(}\rho|\beta|^{2}|f|^ {2}|g_{d}|^{2}\tau_{d}+\alpha\rho|h_{I}|^{2}+1\Bigr{)}\Bigr{\}}\ \ \, \tag{10}\] \[=\Pr\Bigl{\{}|h_{d}|^{2}<\psi_{d}^{*}\Bigl{(}\rho|\beta|^{2}|f|^{ 2}|g_{d}|^{2}\tau_{d}+\alpha\rho|h_{I}|^{2}+1\Bigr{)}\Bigr{\}}\ \ \,\]  
where \(\tau_{d}=(1+\kappa_{sd}^{2})\), \(\omega=1\), \(\psi_{d}^{*}=\max\{\psi_{1},\psi_{2}\cdots\psi_{d}\}\), \(\psi_{d}=\frac{r_{ind}}{\rho(a_{d}-r_{ind}\sum_{i=q}^{q}+a_{i}+\kappa_{sd}^{2})}\), \(\psi_{D}=\frac{r_{ind}}{\rho(a_{D}+\kappa_{SD}^{2})}\), with \(a_{d}>r_{ind}(\sum_{i=q+1}^{D}a_{i}+\kappa_{sd}^{2})\), \(f_{|h_{I}|^{2}}(z)=\frac{1}{\lambda_{hl}}e^{-\frac{z}{\lambda_{hl}}}\). The channel cascade PDF is \(f_{|f^{2}|g_{d}|^{2}}(y)\), which can be derived by [44]. \(K_{0}(x)\) is the zero-order Bessel function modified for the second kind:  
\[f_{|f^{2}|g_{d}|^{2}}(y)=\frac{2}{\lambda_{f}\lambda_{gd}}K_{0}\Biggl{(}2\sqrt {\frac{y}{\lambda_{f}\lambda_{gd}}}\Biggr{)}. \tag{11}\]  
The effective channel gain ordering among the BS and the users, which obeys the Rayleigh distribution, is \({|h_{1}|}^{2}\leq\cdots\leq{|h_{d}|}^{2}\leq\cdots\leq{|h_{D}|}^{2}\). We analyze the key functions of the CDF \(F_{|h_{d}|^{2}}(x)\).  
We define X = \({|h_{d}|}^{2}\), and that X follows an exponential distribution function. The CDF \(F_{|h_{d}|^{2}}(x)\) of channel gain as a function of \(f_{|\tilde{h}_{d}|^{2}}(x)\) is as follows:  
\[F_{|h_{d}|^{2}}(x)=\phi_{d}\sum_{l=0}^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l} \biggl{(}F_{|\tilde{h}_{d}|^{2}}(x)\biggr{)}^{d+l}, \tag{12}\]  
where \(F_{|\tilde{h}_{d}|^{2}}(x)\) denotes the CDF of the unordered channel gain. \(f_{|\tilde{h}_{d}|^{2}}(x)\) is the PDF of the channel between BS and users, which can be calculated from (10):  
\[f_{|\tilde{h}_{d}|^{2}}(x)=\frac{1}{\lambda_{hd}}e^{-\frac{x}{\lambda_{hd}}}, \tag{13}\]  
where \(f_{|\tilde{h}_{d}|^{2}}(x)\) denotes the PDF of the unordered gain in the channel. Based on the above analysis, the channel gain CDF \(F_{|\tilde{h}_{d}|^{2}}(x)\) can be written as  
\[F_{|\tilde{h}_{d}|^{2}}(x)=1-e^{-\frac{x}{\lambda_{hd}}}. \tag{14}\]  
Upon substituting (14) into (12), then the effective channel gain \(F_{|h_{d}|^{2}}(x)\) can be expressed as  
\[F_{|h_{d}|^{2}}(x)=\phi_{d}\sum_{l=0}^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l} \Bigl{(}1-e^{-\frac{x}{\lambda_{hd}}}\Bigr{)}^{d+l}. \tag{15}\]Combining (A1), (A2), and (A6), the proof for the \(d\)-th user's \(P_{d}\) with ipSIC can be further given by  
\[P_{d}= \int_{0}^{\infty}\int_{0}^{\infty}\phi_{d}\sum_{l=0}^{D-d}\binom{D- d}{l}\frac{(-1)^{l}}{d+l}\left(1-e^{-\frac{\phi_{d}^{2}\left(\varphi_{d}\beta^{2} \gamma_{d}+\alpha p_{l}+1\right)}{\lambda_{hd}}}\right)^{d+l}\] \[\times\frac{2}{\lambda_{f}\lambda_{gd}}K_{0}\left(2\sqrt{\frac{y} {\lambda_{f}\lambda_{gd}}}\right)\frac{1}{\lambda_{hI}}e^{-\frac{z}{\lambda_ {hI}}}dydz.\] (A7)  
Based on (A7), we can obtain  
\[P_{d}= \int_{0}^{\infty}\int_{0}^{\infty}\phi_{d}\sum_{l=0}^{D-d}\binom{D -d}{l}\frac{(-1)^{l}}{d+l}\left(\sum_{r=1}^{d+l}C_{d+l}^{\tau}(-1)^{r}e^{- \frac{\phi_{d}^{2}\tau^{2}}{\lambda_{hI}}}e^{-\frac{\phi_{d}^{2}\rho^{2}\beta ^{2}\left(1+2\right)\tau}{\lambda_{hd}}}\right.\] \[\left.\times e^{-\frac{\phi_{d}^{2}\phi_{d}}{\lambda_{hd}}}+1 \right]\frac{2}{\lambda_{f}\lambda_{gd}}K_{0}\left(2\sqrt{\frac{y}{\lambda_{f }\lambda_{gd}}}\right)\frac{1}{\lambda_{hI}}e^{-\frac{z}{\lambda_{hI}}}dydz,\] (A8)  
where the integral can be resolved by [44].  
\[\int_{0}^{\infty}e^{-\kappa x}K_{0}(2\theta\sqrt{x})dx=-\frac{e^{\frac{\theta^ {2}}{\kappa}}}{2\kappa}Ei(-\frac{\theta^{2}}{\kappa}).\] (A9)  
Thus, (9) can be obtained by substituting (A9) into (A8). The proof is finished.