# User-Centric Cell-Free Massive MIMO with Low-Resolution ADCs for Massive Access
## 2 System Model
### Uplink Training Phase  
During the coherence interval \(\tau_{c}\), UEs transmit pilot sequences to the APs for channel estimation. Additionally, because it is assumed to be in TDD mode, only channel state information (CSI) is required through the uplink training phase. The orthogonal pilot signals of duration \(\tau_{p}\) are denoted as \(\phi_{1},\ldots,\phi_{\tau_{p}}\), which all pilot signals satisfy \(\|\phi_{t}\|^{2}=1\). Since we assume a massive access scenario with a large number of UEs, we consider \(K>\tau_{p}\), which means the orthogonal pilot length \(\tau_{p}\) is less than the number of UEs. Therefore, some UEs should share the same pilot sequence. The set of pilot indices is denoted by \(i_{k}\in 1,\ldots,\tau_{p}\), and the set of UEs that share the same pilot with UE \(k\) and include UE \(k\) is denoted as \(D_{k}\).  
The received signal \(\mathbf{y}_{i_{k},l}^{p}\in\mathbb{C}^{N}\) at AP \(l\) from the pilot signal \(\phi_{i_{k}}\) with index \(i_{k}\) is given as follows:  
\[\mathbf{y}_{i_{k},l}^{p}=\sum_{k\in D_{k}}\sqrt{\tau_{p}\rho_{k}}\mathbf{g}_{ kl}+\mathbf{n}_{i_{k},l}, \tag{9}\]  
where \(\rho_{k}\) denotes the pilot transmission power of UE \(k\), and \(\mathbf{n}_{i_{k},l}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N})\in\mathbb{C}^ {N}\) denotes the thermal noise. Also, the received pilot signal at the AP is quantized before transmission to the CPU, (9) is expressed as follows:  
\[\widetilde{\mathbf{y}}_{i_{k},l}^{p} =Q_{\alpha}\left(\mathbf{y}_{i_{k},l}^{p}\right), \tag{10}\] \[\approx\alpha\left(\sum_{k\in D_{k}}\sqrt{\tau_{p}\rho_{k}} \mathbf{g}_{kl}+\mathbf{n}_{i_{k},l}\right)+\widehat{\mathbf{q}}_{\alpha}.\]  
If the MMSE estimation of the channel employed in [2] is used, the estimated channel is as follows:  
\[\widehat{\mathbf{g}}_{kl}=\alpha\sqrt{\tau_{p}\rho_{k}}\mathbf{R}_{kl}\Psi_{ kl}^{-1}\widetilde{\mathbf{y}}_{i_{k},l}^{p}. \tag{11}\]  
where the correlation matrix of (10) is as follows:  
\[\mathbf{\Psi}_{i_{k},l}=\mathbb{E}\bigg{\{}\widetilde{\mathbf{y}}_{i_{k},l}^{ p}\Big{(}\widetilde{\mathbf{y}}_{i_{k},l}^{p}\Big{)}^{H}\bigg{\}}=\sum_{k\in D _{k}}\alpha^{2}\rho_{k}\tau_{p}\mathbf{R}_{kl}+\sigma^{2}\mathbf{I}_{N}+ \mathbf{R}_{\widehat{\mathbf{q}}_{\alpha}\widehat{\mathbf{q}}_{\alpha}}\in \mathbb{C}^{N\times N}. \tag{12}\]  
Therefore, the channel estimate is distributed as follows:  
\[\widehat{\mathbf{g}}_{kl}\sim\mathcal{CN}\left(0,\alpha^{2}\rho_{k}\tau_{p} \mathbf{R}_{kl}\Psi_{kl}^{-1}\mathbf{R}_{kl}\right)\in\mathbb{C}^{N}. \tag{13}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(b\) & \(1\) & \(2\) & \(3\) & \(4\) & \(5\) \\ \hline \(\rho_{\alpha}\) & 0.3634 & 0.1175 & 0.03454 & 0.009497 & 0.002499 \\ \(\alpha\) & 0.6366 & 0.8825 & 0.96546 & 0.990503 & 0.997501 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The distortion parameter of the additive quantization noise model (derived from [31]).  
And estimation error, \(\widetilde{\mathbf{g}}_{kl}=\mathbf{g}_{kl}-\widetilde{\mathbf{g}}_{kl}\), is independent with \(\widetilde{\mathbf{g}}_{kl}\) and is distributed as follows:  
\[\widetilde{\mathbf{g}}_{kl}\sim\mathcal{CN}\Big{(}0,\mathbf{R}_{kl}-\alpha^{2} \rho_{k}\tau_{p}\mathbf{R}_{kl}\mathbf{r}_{kl}^{-1}\mathbf{R}_{kl}\Big{)}\in \mathbb{C}^{N}. \tag{14}\]