# Performance Analysis of Rate Splitting in Massive MIMO Systems with Low Resolution ADCs/DACs
## 2 System Model
### Downlink Data Transmission Using Rate-Splitting Scheme  
During the DL data transmission, the BS applies the RS scheme for data transmission to the \(K\) users. The transmitter structure is depicted in Figure 3a. Denote the uncoded messages intended for the \(k\)th user as \(W_{k}\), \(\forall\ k=\{1\cdots K\}\). Firstly, the BS splits each user's  
Figure 2: Receiver structure of BS for UL channel estimation. message, \(W_{k}\), into two parts: a common part, \(W_{c,k}\), and a private part, \(W_{p,k}\) using the message splitter. Next, the common parts from all users are encoded into a single common stream, \(s_{c}\) using a common codebook. This enables all users to decode the common stream and extract their messages. Also, each private part, \(W_{p,k}\) is individually encoded into a private stream, \(s_{k}\). Then, properly designed linear precoders are applied to the encoded common and private streams, respectively. The linearly precoded signals then pass through the low-resolution DACs and RF chain for transmission. The transmit signal of the RS scheme before DAC quantization is expressed as  
\[\mathbf{x}=\sqrt{P_{c}}\mathbf{w}_{c}s_{c}+\sum_{k=1}^{K}\sqrt{P_{k}}\mathbf{ w}_{k}s_{k}, \tag{9}\]  
where \(\mathbf{x}\), \(s_{c}\sim\mathcal{CN}(0,1)\), and \(s_{k}\sim\mathcal{CN}(0,1)\) denote the superimposed signal, the common stream, and the private message of the \(k\)th user, respectively [31]. Furthermore, \(\mathbf{w}_{c}\) and \(\mathbf{w}_{k}\) represent the precoders for the common message and private message of the \(k\)th user, respectively, and the powers allocated to the common message and the private message of the \(k\)th user are denoted respectively by \(P_{c}\) and \(P_{k}\). For the private message, we assume equal power allocation to all \(K\) users. Hence, \(P_{c}\) and \(P_{k}\) can be further expressed as \(P_{c}=P(1-t)\) and \(P_{k}=Pt/K\triangleq P_{d}\ \forall k\) where \(P\) represents the total DL power budget and \(t\in(0,1]\) is the power allocation factor. Consequently, the received signal at the \(k\)th user is given by  
\[y_{k}=\mathbf{h}_{k}^{H}\mathbf{x}+n_{k}, \tag{10}\]  
where \(n_{k}\) is the AWGN at the \(k\)th node distributed as \(n_{k}\sim\mathcal{CN}(0,1)\). Figure 3: Illustration of RS scheme at BS transmitter and user (**a**) Illustration of RS scheme at BS transmitter [5] (**b**) Receiver structure of RS scheme at \(k\)th user. Since the BS is equipped with low-resolution DACs, the transmit signal, \(\mathbf{x}\), is distorted by quantization noise. Using the AQNM, the transmit signal after DAC quantization, \(\hat{\mathbf{x}}\), takes the form  
\[\hat{\mathbf{x}}=\mathcal{Q}(\mathbf{x})=\alpha_{D}\mathbf{x}+ \mathbf{n}_{D}, \tag{11}\]  
where \(\mathcal{Q}(\cdot)\) denotes the quantization operation and \(\alpha_{D}\) is a linear gain which depends on the number of quantization bits, \(b_{D}\). Please see Table 1 for values of \(\alpha_{D}\)[19]. The additive Gaussian quantization noise denoted by \(\mathbf{n}_{D}\) is uncorrelated with \(\mathbf{x}\). The covariance matrix of \(\mathbf{n}_{D}\) is expressed as  
\[\mathbf{R}_{\mathbf{n}_{D}} =\alpha_{D}(1-\alpha_{D})\text{diag}\Big{(}\mathbb{E}\{\mathbf{x }\mathbf{x}^{H}\}\Big{)},\] \[=\alpha_{D}(1-\alpha_{D})\text{diag}\Bigg{(}P_{c}\mathbf{w}_{c} \mathbf{w}_{c}^{H}+P_{d}\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H} \Bigg{)}. \tag{12}\]  
To satisfy the power constraint of the signal before quantization \(\mathbf{x}\), a normalization factor of \(1/\sqrt{\alpha_{D}}\) is applied to (11) [19]. At the \(k\)th user, the transmitted signal is decoded as shown in Figure 3b. Firstly, the user decodes the common stream by treating all private streams as noise. Then, the user can recover its common message. Next, the user removes the decoded common message from the composite signal by SIC. After this, the user decodes its own private message. Using this process, each user obtains its desired message by combining the decoded common and private messages.