the so-called pilot contamination, a widely studied phenomenon in the context of collocated massive systems [@Elijah16].

Channel estimation {#channel_estimation}
------------------

Channel estimation is known to play a central role in the performance of massive schemes [@Lu14] and also in the specific context of cell-free architectures [@Ngo17]. The estimation filter for the channel between the $k$th active and the $m$th can be calculated as $$\begin{split}
   {\boldsymbol}{D}_{mk}&=\arg\min_{{\boldsymbol}{D}}\mathbb{E}\left\{\left\|{\boldsymbol}{g}_{mk}-{\boldsymbol}{D} {{\boldsymbol}{Y}_p}_m {\boldsymbol}{\varphi}_k^*\right\|^2\right\} \\
                          &= \sqrt{\tau_p P_p} {\boldsymbol}{R}_{mk}^{RF} {\boldsymbol}{Q}_{mk}^{-1},
\end{split}$$ where $${\boldsymbol}{R}_{mk}^{RF} = \mathbb{E}\left\{{\boldsymbol}{g}_{mk}  {\boldsymbol}{g}_{mk}^H\right\}={{\boldsymbol}{W}_m^{RF}}^T  {\boldsymbol}{R}_{mk} {{\boldsymbol}{W}_m^{RF}}^*,$$ and $${\boldsymbol}{Q}_{mk}=\tau_p P_p\sum_{k'=1}^K {\boldsymbol}{R}_{mk'}^{RF} \left|{\boldsymbol}{\varphi}_{k'}^T {\boldsymbol}{\varphi}_k^*\right|^2 + \sigma_u^2(N) {\boldsymbol}{I}_{L_A}.$$ Hence, the corresponding estimated channel vector can be expressed as $$\hat{{\boldsymbol}{g}}_{mk}={\boldsymbol}{D}_{mk} {{\boldsymbol}{Y}_p}_m {\boldsymbol}{\varphi}_k^* = \sqrt{\tau_p P_p} {\boldsymbol}{R}_{mk}^{RF} {\boldsymbol}{Q}_{mk}^{-1} {{\boldsymbol}{Y}_p}_m {\boldsymbol}{\varphi}_k^*.$$ The channel vector estimates can be shown to be distributed as ${\hat{{\boldsymbol}{g}}}_{mk} \sim \mathcal{CN}\left({\boldsymbol}{0},{\hat{{\boldsymbol}{R}}}_{mk}^{RF}\right)$, where $$\hat{{\boldsymbol}{R}}_{mk}^{RF}\triangleq \tau_p P_p {\boldsymbol}{R}_{mk}^{RF}{\boldsymbol}{Q}_{mk}^{-1}{{\boldsymbol}{R}_{mk}^{RF}}^H.$$ Furthermore, the channel vector ${\boldsymbol}{g}_{mk}$ can be decomposed as ${\boldsymbol}{g}_{mk}=\hat{{\boldsymbol}{g}}_{mk}+\tilde{{\boldsymbol}{g}}_{mk}$, where $\tilde{{\boldsymbol}{g}}_{mk}$ is the channel estimation error, which is statistically independent of both ${\boldsymbol}{g}_{mk}$ and $\hat{{\boldsymbol}{g}}_{mk}$.

Downlink payload data transmission
----------------------------------

Let us define ${\boldsymbol}{s}_d=\left[{s_d}_1 \ldots {s_d}_K\right]^T$ as the $K \times 1$ vector of symbols jointly (cooperatively) transmitted from the to the , such that $E\left\{{\boldsymbol}{s}_d{\boldsymbol}{s}_d^H\right\}={\boldsymbol}{I}_K$. Let us also define ${\boldsymbol}{x}_m=\mathcal{P}_m\left({\boldsymbol}{s}_d\right)$ as the $N \times 1$ vector of signals transmitted from 