uncorrelated with ${s_d}_k$ and ${s_u}_k$, respectively. Therefore, both the desired signal and the so-called *effective noise* are uncorrelated. Now, recalling the fact that uncorrelated Gaussian noise represents the worst case, from a capacity point of view, and that the complex-valued fast fading random variables characterizing the propagation channels between different pairs of - connections are independent, the and achievable rates (measured in bits per second per Hertz) for $k$ can be obtained as stated in the following theorems:

An achievable rate of k using the analog precoders ${\boldsymbol}{W}_m^{RF}$, for all $m\in\{1,\ldots,M\}$, and the baseband precoder ${\boldsymbol}{W}_d^{BB} = \hat{{\boldsymbol}{G}}^*\left(\hat{{\boldsymbol}{G}}^T \hat{{\boldsymbol}{G}}^*\right)^{-1}$ is ${R_d}_k=\log_2\left(1+{\operatorname{SINR}_d}_k\right)$, with $${\operatorname{SINR}_d}_k=\frac{\upsilon_k}{\sum_{k'=1}^K \upsilon_{k'} \varpi_{kk'} + \sigma_{\eta_{dk}}^2},
   \label{eq:SINRdk}$$ where $$\sigma_{\eta_{dk}}^2=\sum_{m=1}^M {\sigma_q^2}_{dm} \operatorname{tr}\left({\boldsymbol}{R}_{mk}^{RF}\right) + \sigma_d^2,$$ and $$\varpi_{kk'}=\left[\operatorname{diag}\left(\mathbb{E}\left\{{{\boldsymbol}{W}_d^{BB}}^H \tilde{{\boldsymbol}{g}}_k^* \tilde{{\boldsymbol}{g}}_k^T{\boldsymbol}{W}_d^{BB}\right\}\right)\right]_{k'}.$$ \[theo:theorem\_DL\]

See Appendix \[app:Appendix\_1\].

An achievable rate for the $k$th in the Cell-Free Massive system with limited capacity fronthaul links and using detection, for any $M$, $N$ and $K$, is given by ${R_u}_k=\log_2\left(1+{\operatorname{SINR}_u}_k\right)$, with $${\operatorname{SINR}_u}_k=\frac{P_u \omega_k}{P_u \sum_{k'=1}^K \omega_{k'} \delta_{kk'} + \sigma_{\eta_{uk}}^2},
   \label{eq:SINRuk}$$ where $$\delta_{kk'}=\left[\operatorname{diag}\left(\mathbb{E}\left\{\tilde{{\boldsymbol}{G}}^H{{\boldsymbol}{w}_{uk}^{BB}}^H {\boldsymbol}{w}_{uk}^{BB} \tilde{{\boldsymbol}{G}}\right\}\right)\right]_{k'}$$ with ${\boldsymbol}{w}_{uk}^{BB}$ denoting the $k$th row of ${\boldsymbol}{W}_u^{BB}$, or, equivalently, $$\delta_{kk'}=\left[\operatorname{diag}\left(\mathbb{E}\left\{{\boldsymbol}{W}_u^{BB} \tilde{{\boldsymbol}{g}}_{k'} \tilde{{\boldsymbol}{g}}_{k'}^H{{\boldsymbol}{W}_u^{BB}}^H\right\}\right)\right]_k,$$ and $$\sigma_{\eta_{uk}}^2=\sum_{m=1}^M\left({\sigma_q^2}_{um} +\sigma_u^2(N)\right){\nu_u}_{mk},$$ with $${\nu_u}_{mk}=\left[\operatorname{diag}\left(\mathbb{E}\left\{{\boldsymbol}{W}_{u\,m}^{BB} {{\boldsymbol}{W}_{u\,m}^{BB}}^H\right\}\right)\right]_k.$$ \[theo:theorem\_UL\]

See Appendix \[app:Appendix\_2\].

Fronthaul capacity consumption {#sec:fronthaul_capacity}
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The quantization process performed at the $m$th can 