# Performance Analysis of Rate Splitting in Massive MIMO Systems with Low Resolution ADCs/DACs
## Appendix A Proof of Theorem 1  
Firstly, we derive the asymptotic SINR associated with the common message. Based on (7) and (16), the desired signal power of the common message is derived as [33]  
\[|\mathbf{h}_{k}^{H}\mathbf{w}_{c}|^{2} =|\{\varrho\mathbf{h}_{k}^{H}\mathbf{\hat{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{ \mathbf{   }}}}}}}}}}}}}|^{2},\] \[\overset{(a)}{=}|\{\varrho(\mathbf{\hat{\mathbf{\mathbf{h}}}}_{k }^{H}+\mathbf{e}_{k}^{H})\mathbf{\hat{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{    \mathbf{    }}}}}}}}}}}}}}|^{2},\] \[=\varrho^{2}M^{2}\lambda_{k}^{2}. \tag{10}\]  
where we use \(\mathbf{w}_{c}=\varrho\mathbf{\hat{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{    \mathbf{     }}}}}}}}}}}}}}}\) and \(\varrho\) is defined as \(\varrho=1/\sqrt{M\sum_{k=1}^{K}\lambda_{k}}\)[33]. Step \((a)\) follows from (7) where \(\mathbf{h}_{k}=\mathbf{\hat{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{      \mathbf{      }}}}}}}}}}}}}}}\). We obtain the result in (A) based on the fact that the estimated and error channels are uncorrelated. \begin{table \begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Quantization Bits**} & \multicolumn{2}{c}{**SE [bps/Hz]**} & \multicolumn{2}{c}{**EE [Mbit/]**} \\ \cline{2-5}  & **RS** & **Non-RS** & **RS** & **Non-RS** \\ \hline
1 & 5.55 & 5.55 & 11.01 & 11.01 \\ \hline
2 & 8.22 & 7.52 & 14.95 & 13.68 \\ \hline
3 & 10.13 & 8.14 & 17.27 & 13.90 \\ \hline
4 & 11.48 & 8.33 & 18.01 & 13.06 \\ \hline
5 & 12.21 & 8.38 & 17.22 & 11.82 \\ \hline
6 & 12.48 & 8.40 & 15.38 & 10.35 \\ \hline
7 & 12.56 & 8.40 & 13.13 & 8.78 \\ \hline
8 & 12.58 & 8.40 & 10.79 & 7.21 \\ \hline
9 & 12.59 & 8.41 & 8.58 & 5.72 \\ \hline
10 & 12.59 & 8.41 & 6.60 & 4.41 \\ \hline \hline \end{tabular} 
\end{table}
Table 1: Comparison of RS and non-RS schemes for various quantization bits. To derive the term associated with interference from the private messages, we first rewrite the interference term as  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]  
We derive the first part of (10), \(|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}\), as  
\[|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2} =\kappa^{2}|\mathbf{h}_{k}^{H}\mathbf{\Sigma}\mathbf{h}_{k}|^{2},\] \[\overset{(a)}{=}\kappa^{2}\left|\frac{\mathbf{h}_{k}^{H}\mathbf{ \Sigma}_{k}\mathbf{\hat{h}}_{k}}{1+\mathbf{h}_{k}^{H}\mathbf{\Sigma}_{k} \mathbf{\hat{h}}_{k}}\right|^{2},\] \[\overset{(b)}{=}\kappa^{2}\frac{\mathbf{\hat{h}}_{k}^{H}\mathbf{ \Sigma}_{k}\mathbf{\hat{h}}_{k}}{1+\mathbf{\hat{h}}_{k}^{H}\mathbf{\Sigma}_{k} \mathbf{\hat{h}}_{k}}\right|^{2},\] \[=\kappa^{2}\frac{\mathcal{A}_{k}^{2}}{(1+\mathcal{A}_{k})^{2}}, \tag{11}\]  
where \(\mathbf{\Sigma}\triangleq(\mathbf{\hat{H}}\mathbf{\hat{H}}^{H}+\delta\mathbf{ I}_{M})^{-1}\), \(\mathbf{\Sigma}_{k}\triangleq(\mathbf{\hat{H}}\mathbf{\hat{H}}^{H}-\mathbf{ \hat{h}}_{k}\mathbf{\hat{h}}_{k}^{H}+\delta\mathbf{I}_{M})^{-1}\), and \(\mathcal{A}_{k}=\mathbf{\hat{h}}_{k}^{H}\mathbf{\Sigma}_{k}\mathbf{\hat{h}}_{k}\), respectively. We used the definition of the RZF precoder in (20). Therefore, \(\mathbf{w}_{k}\) is expressed as \(\mathbf{w}_{k}=\mathbf{\Sigma}\mathbf{\hat{h}}_{k}\). Step \((a)\) is derived by using the matrix inversion lemma [33]. Step \((b)\) follows directly from (7) and the fact that the estimated and error channels are uncorrelated.