## 2 System Model
The received signal at relay S can be expressed as  
\[r_{\text{R}}=h_{\text{SR}}s_{\text{S}}+h_{\text{LI}}s_{\text{R}}+w_{\text{R}} \tag{4}\]  
where \(w_{\text{R}}\sim\mathcal{CN}(0,\sigma_{\text{R}}^{2})\) denotes the complex circularly symmetric additive white Gaussian noise (AWGN), and \(h_{\text{LI}}\sim\mathcal{CN}(0,\beta_{\text{R}})\) is the residual LI coefficient after some interference cancellation methods. In the second phase, the relay forwards the signal \(s_{\text{R}}\) with the AF protocol to \(N\) users \(\text{U}_{1},\cdots,\text{U}_{N}\). By the AF relaying, the relay amplifies the received signal by a factor \(A_{\text{R}}\) with a processing delay \(\tau\). The transmit signal \(s_{\text{R}}\) of the relay can be expressed as  
\[s_{\text{R}}[n]=\sqrt{P_{\text{R}}}A_{\text{R}}r_{\text{R}}[n-\tau] \tag{5}\]  
where \(P_{\text{R}}\) is the transmit power of the relay. The instantaneous transmit power of R should be \(P_{\text{R}}\), which is shown as \[ \begin{split} P_{\text{R}}&=\mathbb{E}|s_{\text{R}}|^{2}\\ &=P_{\text{R}}A_{\text{R}}^{2}\mathbb{E}|r_{\text{R}}|^{2}\\ &=P_{\text{R}}A_{\text{R}}^{2}\Big{(}\mathbb{E}|h_{\text{SR}}|^{2 }P_{\text{S}}+\beta_{R}P_{\text{R}}+\sigma_{\text{R}}^{2}\Big{)}\end{split}  \tag{6}\]  
Hence, the amplification factor can be denoted as  
\[A_{\text{R}}=\frac{1}{\sqrt{P_{\text{S}}|h_{\text{SR}}|^{2}+P_{\text{S}}\sigma_ {\text{SR}}^{2}+P_{\text{R}}\beta_{\text{R}}+\sigma_{\text{R}}^{2}}} \tag{7}\]  
Then, the received signal at user U\({}_{n}\) can be expressed as  
\[r_{n}=h_{n}s_{\text{R}}+w_{n} \tag{8}\]  
where \(w_{n}\sim\mathcal{CN}(0,\sigma_{\text{U}}^{2})\) denotes the AWGN at user \(n\). The signal received by the user is first fed to the ADC for processing with the quantization, as shown in Figure 2. As described in [17], the output of a low-resolution ADC corresponding to input \(r_{n}(m)\) can be expressed as  
\[\bar{r}_{n}(m)=\alpha r_{n}(m)+\bar{n}_{r} \tag{9}\]  
where the values of \(\alpha\) is determined by the quantization bits of the ADC, as illustrated in Table 1.  
Let \(b\) be the number of quantization bits. When \(b>5\), the parameter \(\alpha\) is given by  
\[\alpha=1-\frac{\pi\sqrt{3}}{2}2^{-2b} \tag{10}\]  
and when the quantization bit \(b\leq 5\), the value of \(\alpha\) is given as follows:  
\(\bar{n}_{r}\) denotes the additive quantization noise, which has a complex normal distribution with zero mean and variance,  
\[\mathbb{E}|\bar{n}_{r}|^{2}=\alpha(1-\alpha)\mathbb{E}|r_{n}(m)|^{2} \tag{11}\]  
The power of the quantized signal obtained by the user \(n\) can be expressed as  
\[ \begin{split}\mathbb{E}|\bar{r}_{n}|^{2}&=\mathbb{ E}|\alpha r_{n}|^{2}+\mathbb{E}|\bar{w}_{n}|^{2}\\ &=\alpha^{2}\mathbb{E}|r_{n}|^{2}+\alpha(1-\alpha)\mathbb{E}|r_{ n}|^{2}\\ &=\alpha\mathbb{E}|r_{n}|^{2}\end{split}  \tag{12}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Quantization Bits** & \(\pi\) \\ \hline
1 & 0.6366 \\
2 & 0.8825 \\
3 & 0.96546 \\
4 & 0.990503 \\
5 & 0.997501 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the \(\alpha\) parameter corresponding to different quantization bit numbers. Figure 2: System model of low-resolution ADCs. The quantized signal obtained by the user \(n\) is given by  
\[ \begin{split}\hat{r}_{n}&=\alpha r_{n}+\bar{n}_{r}\\ &=\alpha(h_{n}s_{\text{R}}+w_{n})+\bar{n}_{r}\\ &=\alpha(h_{n}\sqrt{P_{\text{R}}}A_{\text{R}}r_{\text{R}}+w_{n}) +\bar{n}_{r}\\ &=\alpha[h_{n}\sqrt{P_{\text{R}}}A_{\text{R}}(l_{\text{SR}}s_{ \text{S}}+h_{\text{LI}}s_{\text{R}}+w_{\text{R}})+w_{n}]+\bar{n}_{r}\\ &=\alpha\Big{[}(\hat{h}_{n}+\epsilon_{n})\sqrt{P_{\text{R}}}A_{ \text{R}}\big{(}(\hat{h}_{\text{SR}}+\epsilon_{\text{SR}})s_{\text{S}}+h_{\text {LI}}s_{\text{R}}+w_{\text{R}}\big{)}+w_{n}\Big{]}+\bar{n}_{r}\\ &=\alpha\Bigg{[}(\hat{h}_{n}+\epsilon_{n})\sqrt{P_{\text{R}}}A_{ \text{R}}\bigg{(}(\hat{h}_{\text{SR}}+\epsilon_{\text{SR}})\Big{(}\sqrt{P_{ \text{S}}}\sum_{n=1}^{N}\sqrt{a_{n}}x_{n}\Big{)}+h_{\text{LI}}s_{\text{R}}+w_{ \text{R}}\bigg{)}+w_{n}\Bigg{]}+\bar{n}_{r}\\ \end{split}