# Performance Analysis of Rate Splitting in Massive MIMO Systems with Low Resolution ADCs/DACs
## 3 Performance Analysis
### Asymptotic Analysis  
In this section, we derive the asymptotic SINRs for the common and private messages of the \(k\)th user, respectively. Due to the TDD protocol, the precoders \(\mathbf{w}_{c}\) and \(\mathbf{w}_{k}\) are designed using the estimated channel obtained in the UL training phase. Similar to [12; 13; 14], we consider the MF beamformer and RZF precoders for the common and private messages, respectively. In the asymptotic regime, when the number of BS antennas tends to infinity, the common message precoder is designed as a linear combination of the estimated channel vectors between the BS and the users [32]. The BS can assign different weights to the channel vectors to maximize the precoder performance. In this work, we consider the MF beamformer with equal weights to all users for analytic tractability [12]. The common message precoder is given by  
\[\mathbf{w}_{c}=\varrho\hat{\mathbf{f}}_{c}, \tag{19}\]  
where \(\hat{\mathbf{f}}_{c}=\sum\limits_{k=1}^{K}\hat{\mathbf{f}}_{k}\) and \(\varrho=1/\sqrt{\mathbb{E}\{||\hat{\mathbf{f}}_{c}||^{2}\}}\) denotes the normalization constant. For the private messages, we use the RZF precoder to strike a balance between the desired signal energy and the amount of interference leakage to other users. If we define a precoder matrix \(\mathbf{W}\) such that the \(k\)th column of \(\mathbf{W}\) is equal to \(\mathbf{w}_{k}\), then the RZF precoder is expressed as  
\[\mathbf{W}=\kappa(\mathbf{\hat{H}}\mathbf{\hat{H}}^{H}+M\delta\mathbf{I}_{M}) ^{-1}\mathbf{\hat{H}}, \tag{20}\]  
where \(\delta\) is the regularization parameter that minimizes the interference to other users. A typical value of \(\delta\) is \(\delta=K/(MP)\)[12]. This parameter will be optimized in later section. The normalization constant, \(\kappa\), is given by  
\[\kappa=\sqrt{\frac{K}{\text{Tr}\{\mathbf{\hat{H}}^{H}(\mathbf{\hat{H}}\mathbf{ \hat{H}}^{H}+M\delta\mathbf{I}_{M})^{-2}\mathbf{\hat{H}}\}}}. \tag{21}\]  
Based on the precoders, it is difficult to derive the SINRs in finite system dimensions. Therefore, we analyze the SINRs in the large system limit where \(K\) and \(M\rightarrow\infty\) but the user loading ratio \(\theta=K/M\) remains fixed [33]. Closed-form solutions for the asymptotic SINRs are derived in the following theorem. **Theorem 1**. : _By considering the effect of low-resolution DACs and ADCs at the BS and imperfect CSI, the asymptotic SINRs of the common message and private message of the kth user are respectively derived as_  
\[\mathsf{SINR}_{k,c}=\frac{\alpha_{D}P_{c}\varrho^{2}M^{2}\lambda_{ k}^{2}(1+f(\theta,\xi))^{2}}{K\alpha_{D}P_{d}\lambda_{k}f(\theta,\xi)\Big{[}1+ \frac{\xi}{\delta}\big{(}1+f(\theta,\xi)\big{)}^{2}\Big{]}+K\alpha_{D}P_{d} \lambda_{k}\Big{[}1+\frac{\xi_{k}}{\lambda_{k}}\big{(}1+f(\theta,\xi)\big{)}^{2 }\Big{]}+\Omega}, \tag{22}\]  
_and_  
\[\mathsf{SINR}_{k,p}=\frac{K\alpha_{D}P_{d}\lambda_{k}f(\theta, \xi)\Big{[}1+\frac{\xi}{\delta}\big{(}1+f(\theta,\xi)\big{)}^{2}\Big{]}}{K \alpha_{D}P_{d}\lambda_{k}\Big{[}1+\frac{\xi_{k}}{\lambda_{k}}\big{(}1+f( \theta,\xi)\big{)}^{2}\Big{]}+\Omega}, \tag{23}\]  
_where_  
\[f(\theta,\xi)=\frac{1}{2}\Bigg{[}\sqrt{\frac{(1-\theta)^{2}}{ \xi^{2}}+\frac{2(1+\theta)}{\xi}+1}+\frac{(1-\theta)}{\xi}-1\Bigg{]}, \tag{24}\]  
\(\Omega=[(1-\alpha_{D})P\beta_{k}+1](1+f(\theta,\xi))^{2}\)_,_ \(\varrho=1/\sqrt{M\sum_{k=1}^{K}\lambda_{k}}\)_, and_ \(\xi=\delta/M\)_, respectively._  
Proof. : Please refer to Appendix A. **Remark 1**. : _From (22) and (23), we observe the impact of the various parameters on the SINRs of the common and private messages, respectively. As expected, the SINRs of the common and private messages are an increasing function of the number of BS antennas, linear gain of the ADC/DAC resolution, and transmit power of the common message. The SINR of the common message is a decreasing function of the number of users. Therefore, the performance of the RS scheme degrades rapidly for higher number of users and high-QN.