# Performance Analysis of Ambient Backscatter NOMA Systems
## 2 System Model
### Signal Model  
The data signal \(x(t)\) at the BS was represented as \(x(t)=\sum_{i=1}^{D}\sqrt{a_{i}P_{s}}x_{i}(t)\), where \(x_{i}(t)\) was the unit power signal of the \(i\)-th user, and satisfied \(\mathbb{E}\{x_{i}^{2}(t)\}=1\). To ensure user fairness, the corresponding power allocation factor \(a_{i}\) met the relationship \(a_{1}\geqslant\cdots\geqslant a_{d}\geqslant\cdots\geqslant a_{D}\), with \(\sum_{i=1}^{D}a_{i}=1\). \(P_{s}\) denoted the normalized transmission power [38].  
Next, the BS sent data signals to the downlink users. In addition, the data signals were used to excite circuits in the uplink BD, which generated and sent a backscatter signal \(c(t)\) to the NOMA users, with \(\mathbb{E}\{c_{i}^{2}(t)\}=1\). Hence, all users were capable of receiving signals from both the backscatter links and the direct links, where the received signal at the \(i\)-th user was formulated as  
\[y_{i}=h_{i}(x(t)+\eta_{si})+\beta fg_{i}(x(t)c(t)+\eta_{si})+n_{i}(t). \tag{1}\]  
We expressed \(h_{i}\sim\mathcal{CN}(0,\lambda_{h,i})\), \(f\sim\mathcal{CN}\Big{(}0,\lambda_{f}\Big{)}\), and \(g_{i}\sim\mathcal{CN}\big{(}0,\lambda_{g,i}\big{)}\) as the channel coefficients between the BS and the \(i\)-th user, between the BS and the BD, and between the BD and the \(i\)-th user, respectively. \(\beta\) was the reflected coefficient at the BD, which was used to send different signals, and to normalize the signal \(c(t)\). The additive white Gaussian noise of the radio link at the \(i\)-th user was denoted by \(n_{i}\sim\mathcal{CN}\big{(}0,\sigma^{2}\big{)}\). The impairment factor of the hardware facilities among the BS, the users, and the BD was indicated by \(\eta_{si}\sim\mathcal{CN}\big{(}0,\kappa_{si}^{2}P_{s}\big{)}\), \(\kappa_{si}^{2}\)[37]. Due to the large fading of the backscatter links, the farthest user was decoded first, and then the other signals were decoded with the assistance of the SIC; therefore, for the \(d\)-th user, the received signal-plus-interference-to-noise ratio (SINR) for detecting the \(q\)-th user's information is written as  
\[\gamma_{d\to q}=\frac{\rho|{h_{d}}|^{2}a_{d}}{\rho|{h_{d}}|^{2}\vartheta_{d}+ \rho|\beta|^{2}|f|^{2}|g_{d}|^{2}\tau_{d}+\bar{w}\rho|{h_{I}}|^{2}+1}, \tag{2}\]  
where \(\tau_{d}=(1+\kappa_{sd}^{2})\), \(\vartheta_{d}=\sum_{i=q+1}^{D}a_{i}+\kappa_{sd}^{2}\), \(\rho=\frac{P_{s}}{N_{0}}\) represented the transmitting SNR at the BS. The residual interference level for SIC was \(\bar{w}\in[0,1]\). More specifically, \(\bar{w}=0\) and \(\bar{w}\
eq 0\) indicated the switchover between pSIC and ipSIC, respectively. On the premise of not losing generality, since Rayleigh channel fading was anticipated for the channel in the case of ipSIC, the corresponding channel fading coefficient was expressed as \(h_{I}\sim\mathcal{CN}(0,\lambda_{I})\). The  
Figure 1: The system model of AmBC-assisted downlink NOMA.  
SIC technique was adopted at the \(D\)-th user, to remove the signals from the previous \((D-1)\) user before analyzing its signal. The received SINR of the \(D\)-th user was provided by  
\[\gamma_{D}=\frac{\rho|h_{D}|^{2}a_{D}}{\rho|h_{D}|^{2}\kappa_{SD}^{2}+\rho|\beta |^{2}|f|^{2}|g_{D}|^{2}\tau_{D}+\bar{w}\rho|h_{I}|^{2}+1}. \tag{3}\]  
Leveraging the SIC technology, the backscatter signal could be decoded by the \(D\)-th user. The SINR of the BD can be written as  
\[\gamma_{BD}=\frac{\rho|\beta|^{2}|f|^{2}|g_{D}|^{2}}{\rho|h_{D}|^{2}\kappa_{SD }^{2}+\bar{w}\rho|h_{I}|^{2}+1}. \tag{4}\]