# User-Centric Cell-Free Massive MIMO with Low-Resolution ADCs for Massive Access
## 3 Spectral Efficiency  
In this section, we derive the closed-form SE expression for user-centric cell-free massive MIMO with low-resolution ADCs. Since the CPU does not have knowledge of channel estimation, we utilize the use-and-then-forget (UatF) bound to derive the achievable SE [2]. The process of deriving achievable SE is similar to [2], but the received signal is given by (24). In particular, the first term of (24) is treated as the desired signal, while the second, third, and fourth terms are treated as effective noise. In fact, the signals received from other UEs, as well as quantization and thermal noise, are uncorrelated with the desired signal and also uncorrelated among the effective noise terms [2; 7; 26]. **Lemma 1**. : _The achievable spectral efficiency for the kth user in a user-centric massive MIMO with low-resolution ADCs is as follows:_  
\[\text{SE}_{k}=\bigg{(}1-\frac{\tau_{p}}{\tau_{c}}\bigg{)}log_{2}(1+\text{ SINR}_{k}), \tag{25}\]  
_where \(\text{SINR}_{k}\) is given as follows:_  
\[\text{SINR}_{k}=\frac{\alpha^{2}\rho_{k}\big{|}\mathbb{E}\big{\{}\mathbf{v}_{k }^{H}\mathbf{W}_{k}^{H}\mathbf{D}_{k}^{H}\mathbf{g}_{k}\big{\}}\big{|}^{2}}{ \alpha^{2}\sum_{j=1}^{K}\rho_{j}\Gamma_{ik}^{(2)}-\alpha^{2}\rho_{k}\big{|} \Gamma_{k}^{(1)}\big{|}^{2}+\alpha^{2}\sigma^{2}\Gamma_{k}^{(3)}+\Gamma_{k}^{ (4)}}, \tag{26}\]  
_where_  
\[\Gamma_{k}^{(1)}=\mathbb{E}\Big{\{}\mathbf{v}_{k}^{H}\mathbf{W}_{k}^{H} \mathbf{D}_{k}^{H}\mathbf{g}_{k}\Big{\}} \tag{27}\]\[\Gamma_{ik}^{(2)}=\mathbb{E}\bigg{\{}\Big{|}\mathbf{v}_{k}^{H}\mathbf{W}_{k}^{H} \mathbf{D}_{k}\mathbf{g}_{j}\Big{|}^{2}\bigg{\}} \tag{28}\]  
\[\Gamma_{k}^{(3)}=\mathbb{E}\bigg{\{}\Big{\|}\mathbf{D}_{k}\mathbf{W}_{k}^{H} \mathbf{v}_{k}\Big{\|}^{2}\bigg{\}} \tag{29}\]  
\[\Gamma_{k}^{(4)}=\mathbb{E}\bigg{\{}\Big{|}\mathbf{v}_{k}^{H}\mathbf{W}_{k}^{H} \mathbf{D}_{k}\mathbf{\bar{n}}_{q}\Big{|}^{2}\bigg{\}}. \tag{30}\]  
_The \(\mathrm{SINR}_{k}\) can be simplified as follows:_  
\[\mathrm{SINR}_{k}=\frac{\alpha^{2}\rho_{k}\big{|}\mathbf{w}_{k}^{H}\mathbf{f} _{k}\big{|}^{2}}{\mathbf{w}_{k}^{H}\Big{(}\alpha^{2}\sum_{j=1}^{k}\rho_{j} \mathbf{\Lambda}_{ki}^{(1)}-\alpha^{2}\rho_{k}\mathbf{f}_{k}\mathbf{f}_{k}^{H }+\alpha^{2}\sigma^{2}\mathbf{\Lambda}_{k}^{(2)}+\mathbf{\Lambda}_{k}^{(3)} \Big{)}\mathbf{w}_{k}}, \tag{31}\]  
_where_  
\[\mathbf{w}_{k}=[w_{k1},\dots,w_{kl}]^{\top} \tag{32}\] \[\mathbf{f}_{k}=\Big{[}\mathbb{E}\Big{\{}\mathbf{v}_{kl}^{H} \mathbf{D}_{kl}\mathbf{g}_{kl}\Big{\}},\dots,\mathbb{E}\Big{\{}\mathbf{v}_{kl }^{H}\mathbf{D}_{kl}\mathbf{g}_{kl}\Big{\}}\Big{]}^{T}\] \[\mathbf{\Lambda}_{ki}^{(1)}=\Big{[}\mathbb{E}\Big{\{}\mathbf{v}_{ kl}^{H}\mathbf{D}_{kl}\mathbf{g}_{j}\mathbf{g}_{jm}^{H}\mathbf{D}_{km}\mathbf{v}_{km} \Big{\}}:l,m=1,\dots,L\Big{]}\] \[\text{SINR}_{k}=\frac{\alpha^{2}\rho_{k}\big{|}\mathbb{E}\big{\{}\mathbf{v}_{k }^{H}\mathbf{W}_{k}^{H}\mathbf{D}_{k}^{H}\mathbf{g}_{k}\big{\}}\big{|}^{2}}{ \alpha^{2}\sum_{j=1}^{K}\rho_{j}\Gamma_{ik}^{(2)}-\alpha^{2}\rho_{k}\big{|} \Gamma_{k}^{(1)}\big{|}^{2}+\alpha^{2}\sigma^{2}\Gamma_{k}^{(3)}+\Gamma_{k}^{ (4)}}, \tag{26}\]0 \[\text{SINR}_{k}=\frac{\alpha^{2}\rho_{k}\big{|}\mathbb{E}\big{\{}\mathbf{v}_{k }^{H}\mathbf{W}_{k}^{H}\mathbf{D}_{k}^{H}\mathbf{g}_{k}\big{\}}\big{|}^{2}}{ \alpha^{2}\sum_{j=1}^{K}\rho_{j}\Gamma_{ik}^{(2)}-\alpha^{2}\rho_{k}\big{|} \Gamma_{k}^{(1)}\big{|}^{2}+\alpha^{2}\sigma^{2}\Gamma_{k}^{(3)}+\Gamma_{k}^{ (4)}}, \tag{26}\]1  
Proof of Lemma 1.: It follows the similar approach as in ([2], Th. 4.4), but for the received signal in (24). The structure of Equation (31) represents the generalized Rayleigh quotient with respect to \(\mathbf{w}_{k}\). Therefore, the maximum value of \(\mathrm{SINR}_{k}\) is achieved as follows ([2], Lem.