received by the $K$ can be rewritten as $$\begin{split}
   {\boldsymbol}{y}_d=&\sum_{m=1}^M {\boldsymbol}{H}_m^T {\boldsymbol}{W}_m^{RF} {\boldsymbol}{W}_{d\,m}^{BB} {\boldsymbol}{\Upsilon}^{1/2} {\boldsymbol}{s}_d \\
            &+ \sum_{m=1}^M {\boldsymbol}{H}_m^T {\boldsymbol}{W}_m^{RF} {{\boldsymbol}{q}_d}_m + {\boldsymbol}{n}_d \\
           =&\ {\boldsymbol}{G}^T {\boldsymbol}{W}_d^{BB}{\boldsymbol}{\Upsilon}^{1/2} {\boldsymbol}{s}_d + {\boldsymbol}{\eta}_d,
\end{split}
\label{eq:yd}$$ where ${\boldsymbol}{G}=[{\boldsymbol}{G}_1^T\,\ldots\,{\boldsymbol}{G}_M^T]^T$, with ${\boldsymbol}{G}_m={{\boldsymbol}{W}_m^{RF}}^T {\boldsymbol}{H}_m$, representing the equivalent channel matrix between the $K$ and the $M$ , including the RF precoding/decoding matrices, and $${\boldsymbol}{\eta}_d  ={\boldsymbol}{G}^T {\boldsymbol}{q}_d + {\boldsymbol}{n}_d,$$ with ${\boldsymbol}{q}_d=[{{\boldsymbol}{q}_d}_1^T \ldots {{\boldsymbol}{q}_d}_M^T]^T$, includes the thermal noise as well as the quantization noise samples received from all the in the network. Now, using the classical baseband precoder to harness the spatial multiplexing, we have that $${\boldsymbol}{W}_d^{BB}=\hat{{\boldsymbol}{G}}^*\left(\hat{{\boldsymbol}{G}}^T \hat{{\boldsymbol}{G}}^*\right)^{-1}$$ or, equivalently, $${\boldsymbol}{W}_{d\,m}^{BB} = \hat{{\boldsymbol}{G}}_m^*\left(\hat{{\boldsymbol}{G}}^T \hat{{\boldsymbol}{G}}^*\right)^{-1}\ \forall m,$$ where we have assumed that ${\boldsymbol}{G}=\hat{{\boldsymbol}{G}}+\tilde{{\boldsymbol}{G}}$ and ${\boldsymbol}{G}_m=\hat{{\boldsymbol}{G}}_m+\tilde{{\boldsymbol}{G}}_m$. Consequently, the signal received by the $k$th can be expressed as $$\begin{split}
   {y_d}_k = &{\boldsymbol}{g}_k^T \hat{{\boldsymbol}{G}}^*\left(\hat{{\boldsymbol}{G}}^T \hat{{\boldsymbol}{G}}^*\right)^{-1} {\boldsymbol}{\Upsilon}^{1/2}{\boldsymbol}{s}_d + {\eta_d}_k \\
           = &\left(\hat{{\boldsymbol}{g}}_k^T + \tilde{{\boldsymbol}{g}}_k^T\right) \hat{{\boldsymbol}{G}}^*\left(\hat{{\boldsymbol}{G}}^T \hat{{\boldsymbol}{G}}^*\right)^{-1} {\boldsymbol}{\Upsilon}^{1/2}{\boldsymbol}{s}_d + {\eta_d}_k \\
           = &\sqrt{\upsilon_k} {s_d}_k + \tilde{{\boldsymbol}{g}}_k^T \hat{{\boldsymbol}{G}}^*\left(\hat{{\boldsymbol}{G}}^T \hat{{\boldsymbol}{G}}^*\right)^{-1} {\boldsymbol}{\Upsilon}^{1/2}{\boldsymbol}{s}_d + {\eta_d}_k
\end{split}
\label{eq:ydk}$$ where ${\eta_d}_k = {\boldsymbol}{g}_k^T {\boldsymbol}{q}_d + {n_d}_k$. The first term denotes the useful received signal, the second term contains the interference terms due to the use of imperfect (pilot contamination), and the third term encompass both the quantification and thermal noise samples.

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