# Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling
## Appendix B Proof of Lemma 2  
By substituting signal \(\mathbf{Y}\) in (31) received by all ENs over all radio channels in case of OMA in Equation (24), we obtain  
\[\mathbf{R}=\left(\mathbf{H}_{c}^{\eta}\otimes\mathbf{I}_{I_{F}}\right) \mathbf{X}\mathbf{H}+\left(\mathbf{H}_{c}^{\eta}\otimes\mathbf{I}_{I_{F}} \right)\mathbf{Z}+\mathbf{W}. \tag{15}\]To compute the per-UE eMBB rate, a further vectorization is needed, leading to  
\[\begin{split}\mathbf{r}&=\text{vec}(\mathbf{R})\\ &=\text{vec}\left(\left(\mathbf{H}_{\text{c}}^{\eta}\otimes\mathbf{ I}_{\text{I}_{\text{F}}}\right)\mathbf{X}\mathbf{H}\right)+\text{vec}\left(\left( \mathbf{H}_{\text{c}}^{\eta}\otimes\mathbf{I}_{\text{I}_{\text{F}}}\right) \mathbf{Z}\right)+\text{vec}(\mathbf{W})\\ &=\left(\mathbf{H}^{T}\otimes\mathbf{H}_{\text{c}}^{\eta}\otimes \mathbf{I}_{\text{I}_{\text{F}}}\right)\mathbf{x}+\left(\mathbf{I}_{M}\otimes \mathbf{H}_{\text{c}}^{\eta}\otimes\mathbf{I}_{\text{I}_{\text{F}}}\right) \mathbf{z}+\mathbf{w}\\ &=\tilde{\mathbf{H}}_{\text{eq}}\mathbf{x}+\tilde{\mathbf{z}}_{ \text{eq}},\end{split} \tag{10}\]  
where \(\tilde{\mathbf{H}}_{\text{eq}}=\mathbf{H}^{T}\otimes\mathbf{H}_{\text{c}}^{ \eta}\otimes\mathbf{I}_{\text{I}_{\text{F}}}\) is the overall equivalent channel comprising both cable and radio channels over all ENs, and \(\tilde{\mathbf{z}}_{\text{eq}}=\left(\mathbf{I}_{M}\otimes\mathbf{H}_{\text{ c}}^{\eta}\otimes\mathbf{I}_{\text{I}_{\text{F}}}\right)\mathbf{z}+\mathbf{w}\) is the overall noise vector comprising both the vectorized radio noise \(\mathbf{z}=\text{vec}(\mathbf{Z})\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{ \text{n}_{F}M})\) and the vectorized cable noise \(\mathbf{w}=\text{vec}(\mathbf{W})\sim\mathcal{CN}(\mathbf{0},\frac{1}{\lambda^ {2}\eta}\mathbf{I}_{\text{n}_{F}M})\), where we recall that the scaling \(\lambda\) is due to the cable power constraints. Hence, the covariance of the equivalent noise \(\mathbf{z}_{\text{eq}}\) yields  
\[\begin{split}\tilde{\mathbf{R}}_{\text{zeq}}&= \mathbb{E}\left[\mathbf{z}_{\text{eq}}\mathbf{z}_{\text{eq}}^{H}\right]\\ &=\left(\mathbf{I}_{M}\otimes\mathbf{H}_{\text{c}}^{\eta}\otimes \mathbf{I}_{\text{I}_{\text{F}}}\right)\left(\mathbf{I}_{M}\otimes\mathbf{H}_ {\text{c}}^{\eta}\otimes\mathbf{I}_{\text{I}_{\text{F}}}\right)^{H}+\frac{1}{ \lambda^{2}\eta}\mathbf{I}_{\text{n}_{F}M}\\ &=\mathbf{I}_{M}\otimes\mathbf{H}_{\text{c}}^{\eta}\mathbf{H}_{ \text{c}}^{\eta}\otimes\mathbf{I}_{\text{I}_{\text{F}}}+\frac{1}{\lambda^{2} \eta}\mathbf{I}_{\text{n}_{F}M}.\end{split} \tag{11}\]  
The eMBB per-UE rate under OMA is computed by  
\[\begin{split} R_{B}&=\frac{(1-L_{U}^{-1})}{n_{F}M}I (\mathbf{r},\mathbf{x})\\ &=\frac{(1-L_{U}^{-1})}{n_{F}M}\text{log}\left(\text{det}\left( \mathbf{I}+\bar{P}_{B}\mathbf{R}_{\text{zeq}}^{-1}\mathbf{H}_{\text{eq}} \mathbf{H}_{\text{eq}}^{T}\right)\right),\end{split} \tag{12}\]  
where \(\tilde{\mathbf{H}}_{\text{eq}}\tilde{\mathbf{H}}_{\text{eq}}^{T}=\mathbf{H} \mathbf{H}\otimes\mathbf{H}_{\text{c}}^{\eta}\mathbf{H}_{\text{c}}^{\eta} \otimes\mathbf{I}_{\text{I}_{\text{F}}}\). Finally, using the determinant property of the Kronecker product \(\left|\mathbf{A}\otimes\mathbf{B}\right|=\left|\mathbf{A}\right|^{m}\left| \mathbf{B}\right|^{n},\) Equation (12) simplifies to  
\[\begin{split} R_{B}&=\frac{l_{F}(1-L_{U}^{-1})}{n_{F}M }\text{log}\left(\text{det}\left(\mathbf{I}+\bar{P}_{B}\mathbf{R}_{\text{zeq }}^{-1}\mathbf{H}_{\text{eq}}\mathbf{H}_{\text{eq}}^{T}\right)\right)\\ &=\mu\frac{1-L_{U}^{-1}}{M}\text{log}\left(\text{det}\left( \mathbf{I}+\bar{P}_{B}\mathbf{R}_{\text{zeq}}^{-1}\mathbf{H}_{\text{eq}} \mathbf{H}_{\text{eq}}^{T}\right)\right),\end{split} \tag{13}\]  
where \(\mathbf{H}_{\text{eq}}=\mathbf{H}\otimes\mathbf{H}_{\text{c}}^{\eta}\) and \(\mathbf{R}_{\text{zeq}}=\mathbf{I}_{M}\otimes\mathbf{H}_{\text{c}}^{\eta} \mathbf{H}_{\text{c}}^{\eta}+\frac{1}{\lambda^{2}\eta}\mathbf{I}_{\frac{M}{ \eta}}\), thus concluding the proof.