# Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling
## Appendix A Proof of Lemma 1
### Maximum Ratio Combining at the Cable Output  
The signal \(\mathbf{R}_{k}\in\mathbb{C}^{l_{F}\times\frac{1}{\eta}}\) after the MRC at the cable output is  
\[\begin{split}\mathbf{R}_{k}&=\tilde{\mathbf{R}}_{k} \mathbf{G}\\ &=\left(\tilde{\mathbf{Y}}_{k}\mathbf{H}_{c}+\tilde{\mathbf{W}}_ {k}\right)\mathbf{G},\end{split} \tag{10}\]  
where the definitions of \(\tilde{\mathbf{R}}_{k}\), \(\mathbf{G}\), \(\tilde{\mathbf{Y}}_{k}\) and \(\mathbf{H}_{c}\) follow from and (10), (20), (18) and (6), respectively. Thus, Equation (10) can be rewritten as  
\[\begin{split}\mathbf{R}_{k}&=\left(\mathbf{Y}_{k} \otimes\mathbf{1}_{\eta}^{T}\right)\left(\gamma\mathbf{1}_{I_{S}}\mathbf{1}_{I _{S}}^{T}+(1-\gamma)\mathbf{I}_{I_{S}}\right)\mathbf{G}+\tilde{\mathbf{W}}_{ k}\mathbf{G}\\ &=\underbrace{\gamma\left(\mathbf{Y}_{k}\otimes\mathbf{1}_{\eta }^{T}\right)\mathbf{1}_{I_{S}}\mathbf{1}_{I_{S}}^{T}\mathbf{G}}_{\mathbf{R}_ {k}^{{}^{\prime\prime}}}+\underbrace{(1-\gamma)\left(\mathbf{Y}_{k}\otimes \mathbf{1}_{\eta}^{T}\right)\mathbf{G}}_{\mathbf{R}_{k}^{{}^{\prime\prime}}}+ \underbrace{\tilde{\mathbf{W}}_{k}\mathbf{G}}_{\mathbf{W}_{k}},\end{split} \tag{11}\]  
where \(\mathbf{W}_{k}=\tilde{\mathbf{W}}_{k}\mathbf{G}\) is the noise post MRC. The first term in (11) can be rewritten as  
\[\begin{split}\mathbf{R}_{k}^{{}^{\prime}}&=\frac{ \gamma}{\eta}\left(\mathbf{Y}_{k}\otimes\mathbf{1}_{\eta}^{T}\right)\mathbf{1} _{I_{S}}\mathbf{1}_{I_{S}}^{T}\left(\mathbf{1}_{\frac{1}{\tilde{\eta}}}\otimes \mathbf{1}_{\eta}\right)\\ &\overset{(a)}{=}\gamma\left(\mathbf{Y}_{k}\otimes\mathbf{1}_{ \eta}^{T}\right)\mathbf{1}_{I_{S}}\mathbf{1}_{\frac{1}{\tilde{\eta}}}^{T}\\ &\overset{(b)}{=}\gamma\eta\mathbf{Y}_{k}\mathbf{1}_{\frac{1}{ \tilde{\eta}}}\mathbf{1}_{\frac{1}{\tilde{\eta}}}^{T},\end{split} \tag{12}\]  
where \(\overset{(a)}{=}\) comes from the fact that \(\mathbf{1}_{I_{S}}^{T}\left(\mathbf{1}_{\frac{1}{\tilde{\eta}}}\otimes \mathbf{1}_{\eta}\right)=\left(\mathbf{1}_{\frac{1}{\tilde{\eta}}}^{T} \otimes\mathbf{1}_{\eta}^{T}\right)\left(\mathbf{1}_{\frac{1}{\tilde{\eta}}} \otimes\mathbf{1}_{\eta}\right)=\eta\mathbf{1}_{\frac{1}{\tilde{\eta}}}^{T}\) due to the mixed-product property of Kronecker product operator \(\left(\mathbf{A}\otimes\mathbf{B}\right)\left(\mathbf{C}\otimes\mathbf{D} \right)=\mathbf{AC}\otimes\mathbf{BD}\) (see [38], Chapter 2), and, similarly, \(\overset{(b)}{=}\) is obtained by \(\left(\mathbf{Y}_{k}\otimes\mathbf{1}_{\eta}^{T}\right)\mathbf{1}_{I_{S}} \mathbf{1}_{\frac{1}{\tilde{\eta}}}^{T}=\left(\mathbf{Y}_{k}\otimes\mathbf{1} _{\eta}^{T}\right)\left(\mathbf{1}_{\frac{1}{\tilde{\eta}}}\mathbf{1}_{\frac{ 1}{\tilde{\eta}}}^{T}\otimes\mathbf{1}_{\eta}\right)=\eta\mathbf{Y}_{k} \mathbf{1}_{\frac{1}{\tilde{\eta}}}\mathbf{1}_{\frac{1}{\tilde{\eta}}}^{T}\). Using similar arguments, the second term in (11) simplifies to  
\[\begin{split}\mathbf{R}_{k}^{{}^{\prime\prime}}&= \frac{1-\gamma}{\eta}\left(\mathbf{Y}_{k}\otimes\mathbf{1}_{\eta}^{T}\right) \left(\mathbf{1}_{\frac{1}{\tilde{\eta}}}\otimes\mathbf{1}_{\eta}\right)\\ &=(1-\gamma)\mathbf{Y}_{k}.\end{split} \tag{13}\]  
Finally, substituting (12) and (13) in (11) we obtain  
\[\mathbf{R}_{k}=\mathbf{Y}_{k}\mathbf{H}_{c}^{\eta}+\mathbf{W}_{k}, \tag{14}\]  
where  
\[\mathbf{H}_{c}^{\eta}=\gamma\eta\mathbf{1}_{\frac{1}{\tilde{\eta}}}\mathbf{1}_ {\frac{1}{\tilde{\eta}}}^{T}+(1-\gamma)\mathbf{I}_{\frac{1}{\tilde{\eta}}} \tag{15}\]  
is the equivalent cable channel matrix accounting for the bandwidth amplification factor over cable \(\eta\), and \(\mathbf{Y}_{k}\) is the radio signal reorganized in matrix form as in (16).