# Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling
## 2 System Model
### Space-Frequency Analog Fronthaul Channel  
In the considered analog fronthaul architecture, the \(k\)-th EN forwards the signal \(\mathbf{y}_{k}(t)\) received by the UEs to the BBU over a wired-access link in a fully analog fashion. As depicted in Figure 2, we focus our attention on a multichannel link that is possibly affected by inter-channel interference. While the model considered here can apply also to wireless multichannel links, as in Figure 1, we adopt here the terminology of Analog Radio-over-Copper (A-RoC) as an important example in which A-RoX is affected by fronthauling inter-link interference. Accordingly, each of the cables employed for the fronthaul contains \(l_{S}\) twisted-pairs, i.e., \(l_{S}\) space-separated channels, indexed as \(c\in[1,l_{S}]\). Each pair carries a bandwidth equal to \(l_{F}\leq n_{F}\) frequency channels of the RAN, indexed as \(f^{\prime}\in[1,l_{F}]\), so that a total of \(l_{S}l_{F}\) space-frequency resource blocks are available over each cable, as shown in Figure 4. Furthermore, we assume that each analog fronthaul link has enough resources to accommodate the transmission of the whole radio signal at each EN, i.e., \(l_{S}l_{F}\geq n_{F}\). The fronthaul channel between each EN and the BBU is described by the matrix \(\mathbf{H}_{c}\in\mathbb{R}^{l_{S}\times l_{S}}\), which accounts for direct channel gains on each cable, given by the diagonal elements \([\mathbf{H}_{c}]_{ii}\), and for the intra-cable crosstalk, described by the off-diagonal elements \([\mathbf{H}_{c}]_{ij}\), with \(i\
eq j\). We assume here that the channel coefficients in \(\mathbf{H}_{c}\) do not depend on frequency \(f^{\prime}\). Furthermore, in keeping the modeling assumptions of the Wyner model, we posit that the direct channel gains for all the pairs are normalized to 1, while the crosstalk coefficients between any pair of twisted-pairs are given by a coupling parameter \(\gamma\geq 0\). It follows that the fronthaul channel matrix can be written as  
\[\mathbf{H}_{c}=\gamma\mathbf{1}_{l_{S}}\mathbf{1}_{l_{S}}^{T}+(1-\gamma) \mathbf{I}_{l_{S}}, \tag{6}\]  
where \(\mathbf{1}_{n}\) denotes a column vector of size \(n\) of all ones and \(\mathbf{I}_{n}\) is the identity matrix of size \(n\). We note that, in case of wireless fronthaul links such as in A-RoR, the coefficient \(\gamma\) accounts for the mutual interference between spatially separated radio links. As a result, one typically has \(\gamma>0\) when considering sub-6 GHz frequency bands, while the condition \(\gamma=0\) may be reasonable in the mmWave or THz bands, in which communication is mainly noise-limited due to the highly directive beams [36]. Figure 4: Space-frequency cable resource allocation. For a given time \(t\), the symbols \(\mathbf{y}_{k}(t)\) received at EN \(k\)-th over all the \(n_{F}\) radio frequency channels are transported to the BBU over the \(l_{S}l_{F}\) cable resource blocks, where the mapping between radio and cable resources is to be designed (see Section 3.1) and depends on the bandwidth \(l_{F}\) available at each twisted-pair. In this regards, we define the fraction \(\mu\in[1/l_{S},1]\) of the radio bandwidth \(n_{F}\) that can be carried by each pair, referred to as _normalized cable bandwidth_, as  
\[\mu=\frac{l_{F}}{n_{F}}. \tag{7}\]  
As a result, the quantity  
\[\eta=\mu\cdot l_{S} \tag{8}\]  
expresses the bandwidth amplification factor (or redundancy) over the cable fronthaul, as \(\eta\geq 1\). To simplify, we assume here that both \(1/\mu\) and \(\eta\) are integer numbers. The two extreme situations with \(\mu=1\), or \(\eta=l_{S}\), and \(\mu=1/l_{S}\), or \(\eta=1\), are shown in Figure 5 for \(l_{S}=4\) twisted-pairs and \(n_{F}=8\) subcarriers. For the first case, one replica of the whole radio signal \(\mathbf{y}_{k}(t)\) can be transmitted over all of the \(l_{S}\) pairs, and the bandwidth amplification over cable is \(\eta=l_{S}\). In the second case, disjoint fractions of the received bandwidth can be forwarded on each pair and the bandwidth amplification factor is \(\eta=1\).