# Performance Analysis of Rate Splitting in Massive MIMO Systems with Low Resolution ADCs/DACs
## 2 System Model
### Uplink Channel Estimation  
Through UL training, the BS estimates the UL channel vectors of the users. The BS can acquire the DL CSI by exploiting channel reciprocity enabled by the TDD operation. For a coherence time interval of length \(\tau\), we use \(\tau_{p}\) sequence for the channel estimation. The remaining \((\tau_{d}=\tau-\tau_{p})\) sequence is used for DL transmission In Internet of Things (IoT) scenarios, discontinuous transmissions may exist. In this case, the coherence time interval can be written as \(\tau=\tau_{p}+\tau_{d}+\tau_{n}\) where \(\tau_{n}\) denotes the time interval when the channel is not used. In this work, we consider the case where \(\tau_{n}=0\) so that the rest of the coherence time interval is used for data transmission [29]. We assume that \(\tau_{p}\geq K\) orthogonal pilot sequences are transmitted to avoid pilot contamination.  
Figure 2 shows the receiver structure of the BS in the UL channel estimation phase. Before passing through the low-resolution ADCs in Figure 2, the received pilot signal at the BS is expressed as  
\[\mathbf{Y}_{p}=\sqrt{\tau_{p}P_{p}}\mathbf{H}\mathbf{\Phi}_{p}+\mathbf{N}_{p}, \tag{3}\]  
where \(P_{p}\) and \(\mathbf{N}_{p}\) denote the pilot power and the additive white Gaussian noise (AWGN) at the BS with i.i.d. \(\mathcal{CN}(0,1)\) elements. The mutually orthogonal pilot sequence \(\mathbf{\Phi}_{p}\in\mathbb{C}^{K\times\tau_{p}}\) is transmitted from the \(K\) users such that \(\mathbf{\Phi}_{p}\mathbf{\Phi}_{p}^{H}=\mathbf{I}_{K}\).  
Figure 1: A massive MIMO system with \(M\)-antenna BS and \(K\) users.  
Since the BS is equipped with low-resolution ADCs, this affects the quality of the channel estimates. We adopt the well-known AQNM to characterize the effect of low-resolution quantizers. In MIMO systems, where the separate ADCs operate at each antenna, a lot of quantized signals are summed to generate decision variables for data symbols. Therefore, due to central limit theorem, the QN at the decision variable can be well approximated Gaussian even with very low-resolution ADCs. This also holds for the QN due to low-resolution DACs. From [30], the Gaussian approximation provides a lower capacity bound of the quantized system. The AQNM was shown to be analytically tractable and accurate in the low- and medium-SNR regions [16; 17]. After passing through the low-resolution ADC quantizers, the resulting signal at the BS becomes  
\[\hat{\mathbf{Y}}_{q}=\mathcal{Q}(\hat{\mathbf{Y}}_{p})=\alpha_{A}\sqrt{\tau_{ p}}P_{p}\mathbf{H}\mathbf{\Phi}_{p}+\alpha_{A}\mathbf{N}_{p}+\mathbf{N}_{q}, \tag{4}\]  
where \(\mathcal{Q}(\cdot)\) and \(\alpha_{A}\) denote the quantization operation and the linear gain of the ADC quantizer, respectively. The exact values of \(\alpha_{A}\), which depend on the quantization bits, \(b_{A}\), which be found in [19]. The matrix \(\mathbf{N}_{q}\) denotes the quantization noise which is uncorrelated with \(\mathbf{Y}_{p}\). The covariance matrix of \(\mathbf{N}_{q}\) is expressed as  
\[\mathbf{R}_{\mathbf{N}_{q}}=\mathbb{E}\{\mathbf{N}_{q}\mathbf{N}_{q}^{H}\}= \alpha_{A}(1-\alpha_{A})\text{diag}\Big{(}\mathbf{Y}_{q}\mathbf{Y}_{q}^{H} \Big{)}. \tag{5}\]  
Using the minimum mean square error (MMSE) estimator [2], the channel matrix \(\mathbf{H}\) is decomposed as [2]  
\[\mathbf{H}=\hat{\mathbf{H}}+\mathbf{E}, \tag{6}\]  
where \(\hat{\mathbf{H}}\) and \(\mathbf{E}\) represent the estimated channel matrix and the estimation error matrix of \(\mathbf{H}\), respectively. The channel from the BS to the \(k\)th user, \(\mathbf{h}_{k}\), is further written as  
\[\mathbf{h}_{k}=\hat{\mathbf{h}}_{k}+\mathbf{e}_{k}, \tag{7}\]  
where the elements of \(\hat{\mathbf{h}}_{k}\) and \(\mathbf{e}_{k}\) are independent Gaussian random variables with zero mean and variances denoted by \(\lambda_{k}\) and \(\epsilon_{k}\), respectively, given as [19]  
\[\lambda_{k}=\frac{\alpha_{A}\tau_{p}P_{p}\beta_{k}^{2}}{1+\tau_{p}P_{p}\beta_{ k}},\quad\text{and}\quad\epsilon_{k}=\beta_{k}-\lambda_{k}. \tag{8}\]