# User-Centric Cell-Free Massive MIMO with Low-Resolution ADCs for Massive Access
## 2 System Model
### Uplink Data Transmission Phase  
During the data transmission phase, the APs receive uplink data from all UEs. The signal received by AP \(l\) from all UEs can be expressed as follows:  
\[\mathbf{y}_{l}=\sum_{k=1}^{K}\mathbf{g}_{kl}s_{k}+\mathbf{n}_{l}\in\mathbb{C }^{N}, \tag{15}\]  
where \(s_{k}\sim\mathcal{CN}(0,p_{k})\) represents the received signal transmitted from UE \(k\), which the power is \(p_{k}\), and \(\mathbf{n}_{l}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N})\in\mathbb{C}^{N}\) denotes the received noise. In this study, it is assumed that local computational tasks are performed at the AP level to avoid CPU overload due to the assumption of a large-scale network. Furthermore, under the assumption of a user-centric cell-free massive MIMO system, each AP supports subsets of UEs. Therefore, in this study, the association between APs and UEs is expressed as follows:  
\[\mathcal{M}_{k}=\{l:\mathbf{S}_{kl}=1,l\in\{1,\dots,L\}\}, \tag{16}\]  
and  
\[\mathcal{A}_{l}=\{k:\mathbf{S}_{kl}=1,k\in\{1,\dots,K\}\}, \tag{17}\]  
where \(\mathcal{M}_{k}\) represents the subset of APs supporting UE \(k\), and \(\mathcal{A}_{l}\) represents the subset of UEs supported by AP \(l\). And the matrix \(\mathbf{S}\in\mathbb{R}^{L\times K}\) denotes the connectivity between APs and UEs in the user-centric cell-free massive MIMO system. Specifically, when UE \(k\) is linked to AP \(l\), the corresponding entry \(S_{kl}\) is equal to 1; otherwise, it remains 0. Furthermore, we define the block-diagonal matrix \(\mathbf{D}_{k}=\text{diag}(\mathbf{D}_{k}^{1},\dots,\mathbf{D}_{k}^{L})\), where the diagonal matrix \(\mathbf{D}_{k}^{l}\in\mathbb{C}^{N\times N}\) represents the antenna configuration between AP \(l\) and UE \(k\). The diagonal elements of \(\mathbf{D}_{k}^{l}\) indicate whether the antennas of AP \(l\) are serving UE \(k\). Specifically, if AP \(l\) is serving UE \(k\), the diagonal elements are set to 1 (forming an identity matrix \(\mathbf{I}_{N}\)), indicating that all \(N\) antennas are used to serve this UE. Otherwise, the diagonal elements are set to 0 (forming a zero matrix \(\mathbf{0}_{N}\)). Furthermore, to reduce the load on the fronthaul link, APs use low-resolution ADCs to send compressed data to the CPU. Therefore, Equation (15) can be quantized and expressed as follows:  
\[\widetilde{\mathbf{y}}_{l}=\alpha\left(\sum_{k=1}^{K}\mathbf{g}_{kl}s_{k}+ \mathbf{n}_{l}\right)+\mathbf{n}_{q}\in\mathbb{C}^{N}, \tag{18}\]  
where \(\mathbf{n}_{q}\in\mathbb{C}^{N}\) represents Gaussian quantization noise, and the covariance matrix is expressed as follows:  
\[\mathbf{R}_{\mathbf{n}_{q}}=\alpha(1-\alpha)\text{diag}\left(\sum_{k=1}^{K} \rho_{k}\mathbf{R}_{kl}+\sigma^{2}\mathbf{I}_{N}\right)\in\mathbb{C}^{N\times N}. \tag{19}\]  
Therefore, each AP can obtain a local estimation for the signal \(s_{k}\), as described in [21], as follows:  
\[ \begin{split}\widetilde{s}_{kl}&=\mathbf{v}_{kl}^{ H}\mathbf{D}_{k}^{l}\widetilde{\mathbf{y}}_{l}\\ &=\alpha\mathbf{v}_{kl}^{H}\mathbf{D}_{k}^{l}\mathbf{g}_{kl}s_{k} +\alpha\mathbf{v}_{kl}^{H}\mathbf{D}_{k}^{l}\sum_{j=1,j\
eq k}^{K}\mathbf{g}_{ jl}s_{j}+\alpha\mathbf{v}_{kl}^{H}\mathbf{D}_{k}^{l}\mathbf{n}_{l}+\mathbf{v}_{kl}^{H} \mathbf{D}_{k}^{l}\mathbf{n}_{q},\end{split}  \tag{20}\]  
where \(\mathbf{v}_{kl}\in\mathbb{C}^{N}\) represents the combining vector chosen at AP \(l\) for UE \(k\), and the above equation applies to any combining vector [21]. In [32], the maximum ratio (MR) combining with vector \(\mathbf{v}_{kl}^{MR}=\mathbf{g}_{kl}^{H}\) was considered. Also, the local partial MMSE (LP-MMSE) with perfect resolution ADCs at the APs was investigated in [21]. \[\mathbf{v}_{kl}=\rho_{k}\Bigg{(}\sum_{j\in\mathcal{A}_{l}}\rho_{k}\Big{(} \widehat{\mathbf{g}}_{j}\widehat{\mathbf{g}}_{jl}^{H}+\mathbf{C}_{jl}\Big{)}+ \sigma^{2}\mathbf{I}_{N}\Bigg{)}^{-1}\widehat{\mathbf{g}}_{kl} \tag{21}\]  
Furthermore, the local MMSE (L-MMSE) combining with low-resolution ADCs at the APs was considered in [14].