## 2 Related Work
### Spectral Efficiency of the System  
Under the completely centralized processing setup, APs send all pilot signals from user equipment to the CPU to perform channel estimation, combining, and the data detection process. The APs are only responsible for interfacing the UE and CPU, as shown in Figure 3 below. The overall channel of UE k is represented as \(\mathrm{h}_{k}=\left[\mathrm{h}_{1k}^{T},\ldots,\mathrm{h}_{Mk}^{T}\right]^{T}\in \mathbb{C}^{MLN}\), where \(\mathrm{h}_{Mk}\in\mathbb{C}^{LN}\) is the channel matrix between AP M and UE K, with L and N denoting the number of antennas at the AP and UE, respectively. The mean of the collective channel estimate for UE K is denoted as \(\overline{\mathrm{h}}_{k}=\left[\overline{\mathrm{h}}_{1k}^{T}e^{\varphi_{1k} },\ldots,\overline{\mathrm{h}}_{Mk}^{T}e^{\varphi_{Mk}}\right]^{T}\), where \(\varphi_{Mk}\) represents the phase shift in the LoS component of the wireless channel. The full covariance matrix of the collective channel estimate for UE K is denoted as \(\mathrm{R}_{k}=\mathrm{diag}(\mathrm{R}_{1k},\ldots,\mathrm{R}_{Mk})\). For fully centralized implementation, it is assumed that \(\varphi_{Mk}\) is available at the CPU, so channel estimates can be obtained by PA-MMSE estimators at the CPU. The collective channel estimate for UE k can be constructed as \(\mathrm{f}_{k}=\left[\mathrm{f}_{1k}^{T},\ldots,\mathrm{f}_{Mk}^{T}\right]^{T}\), where the conditional mean and covariance matrix of the channel estimate given \(\varphi_{Mk}\) is given by  
\[\mathbb{E}\left\{\hat{\mathrm{h}}_{k}\mid e^{\varphi_{k}}\right\}=\overline{ \mathrm{h}}_{k} \tag{10}\]  
\[\mathrm{Cov}\left\{\hat{\mathrm{f}}_{k}\mid e^{\varphi_{k}}\right\}=\tau_{p} \mathrm{R}_{k}\overline{\mathrm{F}}_{k}\overline{\mathrm{F}}_{k}^{-1} \overline{\mathrm{F}}_{k}^{H}\mathrm{R}_{k} \tag{11}\]  
Here, \(\tau_{p}\) represents the pilot length. \(\overline{\mathrm{F}}_{k}=\mathrm{diag}(\underbrace{\mathrm{\overline{F}}_{k },\ldots,\mathrm{\overline{F}}_{k}}_{M})\) is a diagonal matrix where \(\overline{\mathrm{F}}_{k}\) is the precoding matrix for UE K, and \(\Psi_{k}^{-1}=\mathrm{diag}(\Psi_{1k}^{-1},\ldots,\Psi_{Mk}^{-1})\) is a diagonal matrix where VMK is the variance of the estimation error of \(\mathrm{H}_{Mk}\). The received signal available at the CPU can be expressed as  
\[\underbrace{ \begin{bmatrix}y_{1}\\ \vdots\\ y_{M}\end{bmatrix} }_{=y}=\sum_{k=1}^{K} \begin{bmatrix}H_{1k}\\ \vdots\\ H_{Mk}\end{bmatrix} F_{k}x_{k}+\underbrace{ \begin{bmatrix}n_{1}\\ \vdots\\ n_{M}\end{bmatrix} }_{=n} \tag{12}\]  
The received signal at the AP can be expressed as shown in Equation (12) and can be rewritten as  
\[\mathrm{y}=\sum_{k=1}^{K}\mathrm{H}_{k}\mathrm{F}_{k}\mathrm{x}_{k}+\mathrm{n} \tag{13}\]  
To decode the data symbol vector \(\mathrm{x}_{k}\) for UE k, an arbitrary receive combining matrix \(\mathrm{V}_{k}\in\mathbb{C}^{LM\times N}\) can be designed by the CPU based on the collective channel estimates. The receive combining matrix Vk is a complex matrix of size \(\mathrm{LM}\times\mathrm{N}\), where \(\mathrm{LM}\) is the number  
Figure 3: Full centralized processing schemes [20]. of antennas at the AP and N is the number of antennas at the UE k. The decoded signal \(\mathrm{x}_{k}\) obtained as  
\[\hat{\mathrm{x}}_{k}=\mathrm{V}_{k}^{H}\mathrm{H}_{k}\mathrm{F}_{k}\mathrm{x}_{k }+\sum_{l\
eq k}^{K}\mathrm{V}_{k}^{H}\mathrm{H}_{l}\mathrm{F}_{l}\mathrm{x}_{l }+\mathrm{V}_{k}^{H}\mathbf{n} \tag{14}\]  
Based on \(\hat{\mathrm{x}}_{k}\), it is possible to derive the achievable SE for UE k in a CF M-MIMO system using standard capacity lower bounds [27; 28]. The feasible SE for UE k using the PA-MMSE estimator is given by Equation (7) for the fully centralized processing with a given combining vector Vk defined in Equation (9) for MMSE combining.