# Performance Analysis of Ambient Backscatter NOMA Systems
## Appendix B Proof of Theorem 2
\[I_{b}=\Pr\Bigl{\{}\psi_{D}^{*}\Bigl{(}\rho|\beta|^{2}|f|^{2}| \beta_{d}|^{2}(1+\kappa_{SD}^{2})+\alpha\rho|h_{I}|^{2}+1\Bigr{)}<|h_{D}|^{2}< \frac{|f|^{2}|\gamma_{D}|^{2}}{\psi_{BD}\kappa_{SD}^{2}}-\frac{\alpha\varphi| h_{I}|^{2}+1}{\rho\kappa_{SD}^{2}}\Bigr{\}}\] \[= \underbrace{\int_{0}^{\infty}\int_{0}^{\infty}\phi_{D}\sum_{l=0} ^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l}\left(1-e^{-\frac{\left(\frac{1}{\psi _{BD}\rho\kappa_{SD}^{2}}\gamma_{D}-\frac{\alpha\varphi_{l}+1}{\rho\kappa_{SD} ^{2}}\right)}{\lambda_{hd}}}\right)^{d+l}\frac{2}{\lambda_{f}\lambda_{gd}}K_{0 }\left(2\sqrt{\frac{y}{\lambda_{f}\lambda_{gd}}}\right)\frac{1}{\lambda_{hI}}e ^{-\frac{z}{\lambda_{hI}}}dydz}_{l_{1}}\] \[- \underbrace{\int_{0}^{\infty}\int_{0}^{\infty}\phi_{D}\sum_{l=0} ^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l}\left(1-e^{-\frac{\phi_{D}^{2}\left( \rho\beta^{2}\gamma_{d}\left(1+\kappa_{SD}^{2}\right)+\alpha\varphi_{l}+1 \right)}{\lambda_{hd}}}\right)^{d+l}\frac{2}{\lambda_{f}\lambda_{gd}}K_{0} \left(2\sqrt{\frac{y}{\lambda_{f}\lambda_{gd}}}\right)\frac{1}{\lambda_{hI}}e ^{-\frac{z}{\lambda_{hI}}}dydz}_{l_{2}}\] (A11) \[+ \underbrace{\int_{0}^{\infty}\int_{0}^{\phi_{d}}\phi_{D}\sum_{l=0 }^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l}\left(1-e^{-\frac{\phi_{D}^{2}\left( \rho\beta^{2}\gamma_{d}\left(1+\kappa_{SD}^{2}\right)+\alpha\varphi_{l}+1 \right)}{\lambda_{hd}}}\right)^{d+l}\frac{2}{\lambda_{f}\lambda_{gd}}K_{0} \left(2\sqrt{\frac{y}{\lambda_{f}\lambda_{gd}}}\right)\frac{1}{\lambda_{hI}}e ^{-\frac{z}{\lambda_{hI}}}dydz}_{l_{b}_{b_{3}}}\] \[- \underbrace{\int_{0}^{\infty}\int_{0}^{\phi_{d}}\phi_{D}\sum_{l=0 }^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l}\left(1-e^{-\frac{\left(\frac{1}{\psi _{BD}\rho\kappa_{SD}^{2}}\gamma_{D}-\frac{\alpha\varphi_{l}+1}{\rho\kappa_{SD} ^{2}}\right)}{\lambda_{hd}}}\right)^{d+l}\frac{2}{\lambda_{f}\lambda_{gd}}K_{0 }\left(2\sqrt{\frac{y}{\lambda_{f}\lambda_{gd}}}\right)\frac{1}{\lambda_{hI}}e ^{-\frac{z}{\lambda_{hI}}}dydz}_{l_{b_{4}}}\]By using some mathematical manipulations, \(I_{b1}\) can be further given by  
\[I_{b1}= \int_{0}^{\infty}\int_{0}^{\infty}\phi_{d}\sum_{l=0}^{D-d}\binom{D-d }{l}\frac{(-1)^{l}}{d+l}\left[\sum_{r=1}^{4+l}C_{d+l}^{r}(-1)^{r}\times e^{- \frac{\Psi_{2}^{\star}\rho^{\star}}{\lambda_{M}}}e^{-\frac{\Psi_{2}^{\star}\rho ^{\star}\beta^{2}\left(1+\kappa_{D}^{2}\right)r}{\lambda_{M}}}e^{-\frac{\Psi_{ 2}^{\star}\rho\psi_{2}}{\lambda_{M}}}+1\right]\] \[\times\frac{2}{\lambda_{f}\lambda_{gd}}K_{0}\Bigg{(}2\sqrt{\frac{ y}{\lambda_{f}\lambda_{gd}}}\frac{1}{\lambda_{h_{1}}}e^{-\frac{\kappa_{g}}{ \lambda_{h1}}}dydz. \tag{101}\]  
On the basis of (101), \(I_{b1}\) is given by  
\[I_{b1}= \phi_{D}\sum_{l=0}^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l}[\sum_ {r=1}^{d+l}C_{d+l}^{r}(-1)^{r+1}e^{-\frac{\Psi_{2}^{\star}\rho^{\star}}{ \lambda_{M}}}\times Q_{b1}e^{Q_{b1}}\text{Ei}(-Q_{b1})\int_{0}^{\infty}e^{- \frac{\Psi_{2}^{\star}\rho\psi_{2}}{\lambda_{M}}}\frac{1}{\lambda_{h_{1}}}e^{- \frac{\kappa_{g}}{\lambda_{h1}}}dz+1]\] \[= \phi_{D}\sum_{l=0}^{D-d}\binom{D-d}{l}\frac{(-1)^{l}}{d+l}[\sum_ {r=1}^{d+l}C_{d+l}^{r}(-1)^{r+1}\times e^{-\frac{\Psi_{2}^{\star}\rho^{\star}}{ \lambda_{M}}}Q_{b1}e^{Q_{b1}}\text{Ei}(-Q_{b1})\frac{1}{\lambda_{h1}\xi_{b1}}+1]. \tag{102}\]  
Based on the above calculation, \(I_{b2}\) can be respectively written as  
\[I_{b}=\Pr\Bigl{\{}\psi_{D}^{*}\Bigl{(}\rho|\beta|^{2}|f|^{2}| \beta_{d}|^{2}(1+\kappa_{SD}^{2})+\alpha\rho|h_{I}|^{2}+1\Bigr{)}<|h_{D}|^{2}< \frac{|f|^{2}|\gamma_{D}|^{2}}{\psi_{BD}\kappa_{SD}^{2}}-\frac{\alpha\varphi| h_{I}|^{2}+1}{\rho\kappa_{SD}^{2}}\Bigr{\}}\]0  
\[I_{b}=\Pr\Bigl{\{}\psi_{D}^{*}\Bigl{(}\rho|\beta|^{2}|f|^{2}| \beta_{d}|^{2}(1+\kappa_{SD}^{2})+\alpha\rho|h_{I}|^{2}+1\Bigr{)}<|h_{D}|^{2}< \frac{|f|^{2}|\gamma_{D}|^{2}}{\psi_{BD}\kappa_{SD}^{2}}-\frac{\alpha\varphi| h_{I}|^{2}+1}{\rho\kappa_{SD}^{2}}\Bigr{\}}\]1 \[I_{b}=\Pr\Bigl{\{}\psi_{D}^{*}\Bigl{(}\rho|\beta|^{2}|f|^{2}| \beta_{d}|^{2}(1+\kappa_{SD}^{2})+\alpha\rho|h_{I}|^{2}+1\Bigr{)}<|h_{D}|^{2}< \frac{|f|^{2}|\gamma_{D}|^{2}}{\psi_{BD}\kappa_{SD}^{2}}-\frac{\alpha\varphi| h_{I}|^{2}+1}{\rho\kappa_{SD}^{2}}\Bigr{\}}\]2  
where \(I_{b3}\) can be calculated using the Gauss-Chebyshev integral function.