# Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling
## 3 Analog Fronthaul Signal Processing
### Signal Combining at the Fronthaul Output  
As discussed, depending on the fronthaul bandwidth, a number \(\eta\) of noisy replicas of the radio signals received at each EN are relayed to the BBU over \(\eta\) different twisted-pairs. Hence, in order to maximize the SNRs for all signals, Maximum Ratio Combining (MRC) [39] is applied at the cable output as  
\[\mathbf{R}_{k}=\tilde{\mathbf{R}}_{k}\mathbf{G}, \tag{19}\]  
where \(\mathbf{R}_{k}\in\mathbb{C}^{l_{F}\times\frac{1}{\eta}}\) is the signal received at BBU from the \(k\)-th EN after the combiner and \(\mathbf{G}\in\mathbb{R}^{l_{S}\times\frac{1}{\eta}}\) is the MRC matrix. Under the assumptions here, MRC coincides with equal ratio combining and hence matrix \(\mathbf{G}\) can be written as  
\[\mathbf{G}=\frac{1}{\eta}\left(\mathbf{I}_{\frac{1}{\eta}}\otimes\mathbf{1}_ {\eta}\right). \tag{20}\]  
As an example, in the case of maximum redundancy, i.e., \(\eta=l_{S}\), the MRC matrix \(\mathbf{G}=l_{S}^{-1}\mathbf{1}_{l_{S}}\) combines the analog signals received over all pairs, since they carry the same information signal. On the contrary, in the case of minimal normalized bandwidth \(\mu=1/l_{S}\) or \(\eta=1\), the matrix \(\mathbf{G}\) equals the identity matrix as \(\mathbf{G}=\mathbf{I}_{l_{S}}\), since no combining is possible. The signal \(\mathbf{r}_{k}\in\mathbb{C}^{n_{F}}\) received at the BBU from EN\({}_{k}\) across the \(n_{F}\) subcarriers is thus obtained by vectorizing matrix \(\mathbf{R}_{k}\) in (19) as  
\[\mathbf{r}_{k}=\text{vec}(\mathbf{R}_{k}). \tag{21}\]  
The relationship between the signal \(\mathbf{r}_{k}\) (21) obtained at the output of the combiner and the radio received signal \(\mathbf{y}_{k}\) in (15) is summarized by the block-scheme in Figure 6, and it is  
\[\mathbf{r}_{k}=\text{vec}\left[\left(\left(\text{vec}_{\frac{1}{\eta}}^{-1} \left(\mathbf{y}_{k}\right)\otimes\mathbf{1}_{\eta}^{T}\right)\mathbf{H}_{c} +\tilde{\mathbf{W}}_{k}\right)\mathbf{G}_{k}\right]. \tag{22}\]  
Finally, we collect the overall signal \(\mathbf{R}\in\mathbb{C}^{n_{F}\times M}\) received at the BBU from all ENs across all frequencies in the matrix  
\[\mathbf{R}=[\mathbf{r}_{1},\mathbf{r}_{2},\;...,\;\mathbf{r}_{M}]. \tag{23}\]  
After some algebraic manipulations, it is possible to express Equation (23) in a more compact form, which is reported in the following Lemma 1. Figure 6: Relationship between the signal \(\mathbf{r}_{k}\) (21) obtained at the output of the combiner and the radio received signal \(\mathbf{y}_{k}\) in (15). **Lemma 1**: _In the given C-RAN architecture with analog fronthaul links, for a given bandwidth amplification factor \(\eta\geq 1\), the signal \(\mathbf{R}\in\mathbb{C}^{n_{F}\times M}\) received at the BBU from all ENs across all radio frequencies after MRC can be written as_  
\[\mathbf{R}=\left(\mathbf{H}_{\mathbf{c}}^{\eta}\otimes\mathbf{I}_{I_{F}}\right) \mathbf{Y}+\mathbf{W}, \tag{24}\]  
_where_  
\[\mathbf{H}_{\mathbf{c}}^{\eta}=\gamma\eta\mathbf{1}_{\frac{1}{\eta}}\mathbf{ \frac{1}{\eta}}+(1-\gamma)\mathbf{I}_{\frac{1}{\eta}} \tag{25}\]  
_is the equivalent fronthaul channel matrix; \(\mathbf{Y}\) is the signal received at all ENs across all frequency radio channels in (5); and \(\mathbf{W}=[\mathbf{w_{1}},\mathbf{w_{2}},...,\mathbf{w_{M}}]\) is the equivalent cable noise at the BBU after MRC, with the \(k\)-th column distributed as \(\mathbf{w}_{k}\sim\mathcal{CN}\left(\mathbf{0},\frac{1}{\eta}\mathbf{I}_{n_{ F}}\right)\) for all \(k=1,2,...,M\)._  
see Appendix A. To gain some insights, it is useful again to consider the two extreme cases of maximum redundancy, i.e., \(\eta=l_{S}\), and no redundancy, i.e., \(\eta=1\). In the former case, the equivalent channel (25) equals the scalar \(\mathbf{H}_{\mathbf{c}}^{\eta}=1+\gamma(l_{S}-1)\). This demonstrates the effect of transmitting a replica of the whole radio signal over all pairs.