of detection, the uses the detection matrix $${\boldsymbol}{W}_u^{BB}=\left(\hat{{\boldsymbol}{G}}^H \hat{{\boldsymbol}{G}}\right)^{-1}\hat{{\boldsymbol}{G}}^H = {{\boldsymbol}{W}_d^{BB}}^T$$ or, equivalently $${\boldsymbol}{W}_{u m}^{BB}=\left(\hat{{\boldsymbol}{G}}^H \hat{{\boldsymbol}{G}}\right)^{-1}\hat{{\boldsymbol}{G}}_m^H = {{\boldsymbol}{W}_{d m}^{BB}}^T,\ \forall m,$$ to jointly process the vector ${\boldsymbol}{z}_u=\left[{{\boldsymbol}{z}_u}_1^T\,\ldots\,{{\boldsymbol}{z}_u}_M^T\right]^T$ and obtain the vector of detected samples $$\begin{split}
   {\boldsymbol}{y}_u=&{\boldsymbol}{W}_u^{BB}{\boldsymbol}{z}_u=\sqrt{P_u}{\boldsymbol}{W}_u^{BB} {\boldsymbol}{G} {\boldsymbol}{\Omega}^{1/2} {\boldsymbol}{s}_u + {\boldsymbol}{\eta}_u \\
           =&\sqrt{P_u} {\boldsymbol}{\Omega}^{1/2} {\boldsymbol}{s}_u + \sqrt{P_u}{\boldsymbol}{W}_u^{BB} \tilde{{\boldsymbol}{G}} {\boldsymbol}{\Omega}^{1/2} {\boldsymbol}{s}_u + {\boldsymbol}{\eta}_u,
\end{split}$$ where ${\boldsymbol}{\eta}_u={\boldsymbol}{W}_u^{BB}\left({\boldsymbol}{q}_u + {\boldsymbol}{n}_u\right)$. Again, the first term denotes the useful received signal, the second term contains the interference terms due to the use of imperfect , and the third term includes both the quantification and thermal noise samples. The detected sample corresponding to the symbol transmitted by the $k$th can then be obtained as $${y_u}_k =\sqrt{P_u} \omega_k^{1/2} {s_u}_k + \sqrt{P_u}\left[{\boldsymbol}{W}_u^{BB} \tilde{{\boldsymbol}{G}} {\boldsymbol}{\Omega}^{1/2} {\boldsymbol}{s}_u\right]_k + {\eta_u}_k,
\label{eq:yuk}$$ where $[{\boldsymbol}{x}]_k$ denotes the $k$th entry of vector ${\boldsymbol}{x}$.

Achievable rates {#sec:Achievable_rates}
================

Analysis techniques similar to those applied, for instance, in [@Hassibi03; @Yang13; @Interdonato16; @Marzetta16; @Ngo17; @Nayebi17], are used in this section to derive and achievable rates. In particular, the sum of the second and third terms on the of , for the case, and , for the case, are treated as *effective noise*. The additive terms constituting the *effective noise* are, in both and cases, mutually uncorrelated, and 