# User-Centric Cell-Free Massive MIMO with Low-Resolution ADCs for Massive Access
## 2 System Model
### Channel Model and Quantization Model  
Because the antennas have non-uniform radiation patterns and the physical propagation environment affects certain spatial directions, the channels are generally spatially correlated [2; 28]. Therefore, the channel in this paper utilizes the correlated Rayleigh block fading model used in [2]. The channel between the AP \(l\) and the UE \(k\) is represented as follows:  
\[\mathbf{g}_{kl}=\sqrt{\beta_{kl}}\mathbf{h}_{kl}, \tag{1}\]  
where \(\beta_{kl}\) represents the large-scale fading factor, which includes path loss and shadowing effects, and \(\mathbf{h}_{kl}\in\mathbb{C}^{N}\) is an independent and identically distributed (i.i.d.) small-scale fading variable with the \(\mathcal{CN}(0,1)\) distribution. Under the assumption of block fading channels, it is assumed that the CSI remains constant during coherence intervals \(\tau_{c}\). Among these intervals, a portion of duration \(\tau_{p}\) is allocated for uplink training, while the remainder \(\tau_{c}-\tau_{p}=\tau_{u}\) is allocated for uplink data transmission. Unlike conventional cell-free massive MIMO research that uses high-resolution or perfect-resolution ADCs, this study assumes user-centric massive MIMO that considers fronthaul capacity constraints and employs low-resolution ADCs. In the signal transmission process, APs and CPUs that use low-resolution ADCs introduce the issue of nonlinear compression of signals [26]. We refer to \(Q_{\alpha}(\mathbf{y})\) as the mathematical representation of the generic scalar quantizer result for signal \(\mathbf{y}\in\mathbb{C}^{N}\). The classical approach dealing with such nonlinear characteristics uses Bussgang decomposition as follows [29]:  
\[\widetilde{\mathbf{y}}=Q_{\alpha}(\mathbf{y})=\mathbf{F}_{\alpha}\mathbf{y}+ \mathbf{q}_{\alpha}, \tag{2}\]  
where \(\mathbf{F}_{\alpha}\in\mathbb{C}^{N\times N}\) can be obtained from the linear minimum mean square error (MMSE) estimation of \(\widetilde{\mathbf{y}}\) as follows:  
\[\mathbf{F}_{\alpha}=\mathbb{E}\big{(}\widetilde{\mathbf{y}}\mathbf{y}\big{)} \Big{(}\mathbb{E}\Big{\{}\mathbf{y}\mathbf{y}^{H}\Big{\}}\Big{)}^{-1}=\mathbf{ R}_{\widetilde{\mathbf{y}}\mathbf{y}}\mathbf{R}_{\mathbf{y}\mathbf{y}}^{-1}, \tag{3}\]  
and the quantization noise \(\mathbf{q}_{\alpha}\in\mathbb{C}^{N}\) has the correlation matrix as follows:  
\[ \begin{split}\mathbf{R}_{\mathbf{q}_{\alpha}\mathbf{q}_{\alpha}}& =\mathbb{E}\big{\{}(\widetilde{\mathbf{y}}-\mathbf{F}_{\alpha} \mathbf{y})\big{\}}\\ &=\mathbf{R}_{\widetilde{\mathbf{y}}\widetilde{\mathbf{y}}}- \mathbf{R}_{\widetilde{\mathbf{y}}\widetilde{\mathbf{y}}}\mathbf{F}_{\alpha}^ {H}-\mathbf{FR}_{\mathbf{y}\widetilde{\mathbf{y}}}+\mathbf{FR}_{\mathbf{y} \mathbf{y}}\mathbf{F}_{\alpha}^{H}\\ &=\mathbf{R}_{\widetilde{\mathbf{y}}\widetilde{\mathbf{y}}}- \mathbf{R}_{\widetilde{\mathbf{y}}\widetilde{\mathbf{y}}}\mathbf{R}_{ \widetilde{\mathbf{y}}\widetilde{\mathbf{y}}}^{-1}\mathbf{R}_{\widetilde{ \mathbf{y}}\widetilde{\mathbf{y}}}.\end{split}  \tag{4}\]  
Generic scalar quantizers approximate the covariance matrix through approximation methods. In [30], the covariance matrix of such quantizers was approximated using the distortion factor \(\rho_{\alpha}=(1-\alpha)=1/\text{SQNR}_{\alpha}\) based on the signal-to-quantization noise ratio (SQNR).