# Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling
## 5 Non-Orthogonal Multiple Access (NOMA)
### eMBB Rate by Successive Interference Cancellation  
Finally, a more complex receiver architecture can be considered at the BBU, whereby interference is cancelled out from the useful signal. This technique, referred to as Successive Interference Cancellation (SIC), is based on the idea that, if an URLLC signal is successfully decoded at the EN\({}_{k}\), it can be cancelled from the overall received signal \(\mathbf{y}_{k}\) prior to the relaying over the cable, so that an ideally interference-free eMBB signal is forwarded to the BBU. We also assume that, if the URLLC signal is not successfully decoded, signal \(\mathbf{y}_{k}\) is discarded. As a practical note, SIC must be performed in the analog domain, thus complicating the system design. Practical complications are not considered in the analysis here. As shown in Figure 9, if the URLLC signal is successfully decoded at EN\({}_{k}\), this needs first to be Digital-to-Analog Converted (DAC) and then cancelled from the analog signal \(\mathbf{y}_{k}\). Therefore, signal \(\mathbf{y}_{k}\) needs to be suitably delayed in order to wait for the cascade of ADC, decoding, and DAC operations to be completed at the URLLC decoder. Being latency not an issue for eMBB traffic, in this work we assume to employ ideal ADC/DAC, so that the delay in Figure 9 is assumed as ideally zero. To account for imperfect SIC, the amplitude of the residual URLLC interference on eMBB signal is assumed to be proportional to a factor \(\rho\in[0,1]\). Accordingly, perfect SIC corresponds to \(\rho=0\), and no SIC to \(\rho=1\). The signal received at the BBU from EN\({}_{k}\) can be thus written as  
\[\gamma_{k}^{f}=(1-A_{k}E_{k})(X_{k}^{f}+\alpha X_{k+1}^{f}+\alpha X_{k-1}^{f})+ \rho\beta A_{k}(1-E_{k})U_{k}^{f}+Z_{k}^{f}, \tag{40}\]  
where the Bernoulli variable \(E_{k}\sim\mathcal{B}(q\varepsilon_{U}^{D})\) indicates whether there has been an error in decoding the URLLC packet (i.e., \(E_{k}=1\)), or it has been successfully decoded (i.e., \(E_{k}=0\)), and \(A_{k}\) is the same as above. It is easy to show that the factor \((1-A_{k}E_{k})\) multiplying the eMBB signal indicates that the eMBB signal (40) is discarded only when the two following events simultaneously happen: (i) there is a URLLC transmission (i.e., \(A_{k}=1\), whose probability is \(q\)), and (ii) such URLLC transmission is not successfully decoded (\(E_{k}=1\), whose probability is \(\varepsilon_{U}^{D}\)). In turn, the factor \(A_{k}(1-E_{k})\) multiplying the URLLC signal implies that, if there is a URLLC transmission (i.e., \(A_{k}=1\)) and such transmission is successfully decoded at the EN\({}_{k}\) (i.e., \(E_{k}=0\)), then the URLLC signal is mitigated by analog SIC so that only a \(\rho\)-fraction of it is forwarded to the BBU and impairs the eMBB transmission. The signal in (40) received across all ENs and frequencies in the case of NOMA by SIC can be equivalently written in matrix form as  
\[\mathbf{Y}=\mathbf{X}\mathbf{H}\left(\mathbf{I}-\mathbf{A}\mathbf{E}\right)+ \rho\mathbf{\theta}\mathbf{U}\mathbf{A}\left(\mathbf{I}-\mathbf{E}\right)+ \mathbf{Z}, \tag{41}\]  
where \(\mathbf{E}=\text{diag}(E_{1},E_{2},...,E_{M})\). The eMBB UE rate for NOMA by SIC can thus be computed by substituting signal (41) in (24) and the final result is in Lemma 5. Lemma 5: _In the given C-RAN architecture with analog fronthaul links, for a given bandwidth amplification factor \(\eta\geq 1\), the eMBB user rate under NOMA by SIC yields_  
\[R_{B}=\frac{\mu}{M}\mathbb{E}_{\mathbf{A},\mathbf{E}}\left[\log\left(det \left(\mathbf{I}+P_{B}\mathbf{R}_{AE,\varepsilon_{q}}^{-1}\mathbf{H}_{AE} \mathbf{H}_{AE}^{T}\right)\right)\right], \tag{42}\]  
Figure 9: Block diagram of the operation of the ENs and BBU for Non-Orthogonal Multiple Access (NOMA) by Successive Interference Cancellation (SIC). A/A stands for Analog-to-Analog.