# Performance Analysis of Rate Splitting in Massive MIMO Systems with Low Resolution ADCs/DACs
## Appendix A Proof of Theorem 1
Next, we derive the second part of (10), \(\sum\limits_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}\), as  
\[\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2} =\kappa^{2}\sum_{j\
eq k}^{K}\mathbb{E}\{|\mathbf{h}_{k}^{H}\mathbf{ \Sigma}\mathbf{\hat{h}}_{k}|^{2}\},\] \[=\kappa^{2}|\mathbf{h}_{k}^{H}\mathbf{\Sigma}\mathbf{\hat{H}}_{k} |^{2},\] \[\overset{(a)}{=}\kappa^{2}\frac{\mathbf{h}_{k}^{H}\mathbf{ \Sigma}_{k}\mathbf{\hat{H}}_{k}\mathbf{\hat{H}}_{k}^{H}\mathbf{\Sigma}_{k} \mathbf{h}_{k}}{(1+\mathbf{\hat{h}}_{k}^{H}\mathbf{\Sigma}_{k}\mathbf{\hat{h} }_{k})^{2}},\] \[=\kappa^{2}\frac{(\mathcal{B}_{k}+\mathcal{E}_{k})}{(1+\mathcal{ X}_{k})^{2}}, \tag{12}\]  
where \(\mathbf{\hat{H}}_{k}=\mathbf{\hat{H}}\mathbf{\hat{H}}^{H}-\mathbf{\hat{h}}_{k} \mathbf{\hat{h}}_{k}^{H}\), \(\mathcal{B}_{k}=\mathbf{\hat{h}}_{k}^{H}\mathbf{\Sigma}_{k}\mathbf{\hat{H}}_{ k}\mathbf{\hat{H}}_{k}^{H}\mathbf{\Sigma}_{k}\mathbf{\hat{h}}_{k}\), and \(\mathcal{E}_{k}=\mathbf{e}_{k}^{H}\mathbf{\Sigma}_{k}\mathbf{\hat{H}}_{k} \mathbf{\hat{H}}_{k}^{H}\mathbf{\Sigma}_{k}\mathbf{e}_{k}\), respectively. Step \((a)\) is derived by using the matrix inversion lemma [33]. Based on the results in [34; 36], in the large system limit where \(K\) and \(M\rightarrow\infty\) but the user loading ratio \(\theta=K/M\) is constant, \(\mathcal{A}_{k}\) converges to a deterministic value given by \(f(\theta,\xi)\) in (24). \(f(\theta,\xi)\) is obtained as the solution of the equation  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]0  
Following the similar approach [34; 36], we can show that  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]1  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]2  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]3where  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]4  
Using (12), the QN term due to the effect of the low-resolution DACs is written as  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]5 \[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]6  
For \(\Gamma_{1}\) and \(\Gamma_{2}\), we derive the analytic results in the asymptotic regime. In the asymptotic regime, \(\Gamma_{1}\) can be approximated as  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]7 \[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]8  
For \(\Gamma_{2}\), we have  
\[\sum_{k=1}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}=|\mathbf{h}_{k}^{H}\mathbf{w }_{k}|^{2}+\sum_{j\
eq k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}. \tag{10}\]9 \[|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2} =\kappa^{2}|\mathbf{h}_{k}^{H}\mathbf{\Sigma}\mathbf{h}_{k}|^{2},\]0  
Step \((a)\) follows from using the asymptotic property in [Lemma 5] [24]. By substituting (13)-(13) into (16), we can obtain the SINR of the common message in (22). Finally, by combining (14)-(13), we have the SINR of the private message given by (23). This completes the proof.