in Section \[sec:Conclusion\].

*Notation*: Vectors and matrices are denoted by lower-case and upper-case boldface symbols. The $q$-dimensional identity matrix is represented by $\boldsymbol{I}_q$. The operator $\det({\boldsymbol}{X})$ represents the determinant of matrix ${\boldsymbol}{X}$, $\operatorname{tr}({\boldsymbol}{X})$ denotes its trace, $\|{\boldsymbol}{X}\|_F$ is its Frobenius norm, whereas ${\boldsymbol}{X}^{-1}$, ${\boldsymbol}{X}^T$, ${\boldsymbol}{X}^*$ and ${\boldsymbol}{X}^H$ denote its inverse, transpose, conjugate and conjugate transpose (also known as Hermitian), respectively. With a slight abuse of notation, the operator $\operatorname{diag}(\boldsymbol{x})$ is used to denote a diagonal matrix with the entries of vector $\boldsymbol{x}$ on its main diagonal, and the operator $\operatorname{diag}(\boldsymbol{X})$ is used to denote a vector containing the entries in the main diagonal of matrix ${\boldsymbol}{X}$. The expectation operator is denoted by $\mathbb{E}\{\cdot\}$. Finally, $\mathcal{CN}({\boldsymbol}{m},{\boldsymbol}{R})$ denotes a circularly symmetric complex Gaussian vector distributions with mean ${\boldsymbol}{m}$ and covariance ${\boldsymbol}{R}$, $\mathcal{N}(0,\sigma^2)$ denotes a real valued zero-mean Gaussian random variable with standard deviation $\sigma$, and $\mathcal{U}[a,b]$ represents a random variable uniformly distributed in the range $[a,b]$.

System model {#sec:System_model}
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Let us consider a cell-free massive system where a coordinates the communication between $M$ and $K$ single-antenna randomly distributed in a large area. Each of the communicates with the via error-free fronthaul links with and capacities ${C_F}_d$ and ${C_F}_u$, respectively. Baseband processing 