the $m$th , where $\mathcal{P}_m\left({\boldsymbol}{s}_d\right)$ is used to denote the mathematical operations (linear and/or non-linear) used to obtain ${\boldsymbol}{x}_m$ from ${\boldsymbol}{s}_d$. Note that this vector must comply with a power constraint $\mathbb{E}\left\{\left\|{\boldsymbol}{x}_m\right\|_F^2\right\}\leq \overline{P}_m$, where $\overline{P}_m$ is the maximum average transmit power available at $m$. Using this notation, the signal received by $k$ can be expressed as $${y_d}_k=\sum_{m=1}^M {\boldsymbol}{h}_{mk}^T {\boldsymbol}{x}_m + {n_d}_k,
\label{eq:yk}$$ where ${n_d}_k \sim \mathcal{CN}(0,\sigma_d^2)$ is the Gaussian noise sample at $k$. The vector ${\boldsymbol}{y}_d = \left[{y_d}_1\, \ldots\, {y_d}_K\right]^T$ containing the signals received by the $K$ scheduled in the network can then be expressed as $${\boldsymbol}{y}_d=\sum_{m=1}^M {\boldsymbol}{H}_m^T {\boldsymbol}{x}_m + {\boldsymbol}{n}_d,$$ where ${\boldsymbol}{H}_m=\left[{\boldsymbol}{h}_{m1}\, \ldots\, {\boldsymbol}{h}_{mK}\right]$ and ${\boldsymbol}{n}_d=\left[{n_d}_1\, \ldots\, {n_d}_K\right]^T$.

The mathematical operations that symbol vector ${\boldsymbol}{s}_d$ undergoes before being transmitted, generically represented as ${\boldsymbol}{x}_m = \mathcal{P}_m({\boldsymbol}{s}_d)$, for all $m \in \{1,\ldots,M\}$, include, first, a baseband precoding task at the , second, a compressing process of all or part of the data that must be sent from the to the through the fronthaul links and, third, an precoding task at each of the . Let us denote by ${\mathcal{Q}_d}_m({\boldsymbol}{x})$ and ${\mathcal{Q}_d}_m^{-1}({\boldsymbol}{x})$ the quantization and unquantization mathematical operations performed by the -based - functional split on a vector of signal samples ${\boldsymbol}{x}$ 