# Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling
## 5 Non-Orthogonal Multiple Access (NOMA)
### eMBB Rate by Puncturing  
To carry out joint decoding at the BBU of the eMBB signals under NOMA, the standard approach is to simply discard at the eMBB decoder those signals that are interfered by URLLC. As shown in Figure 8, this technique, referred to as puncturing, is based on the detection of URLLC transmissions at the BBU: if a URLLC transmission is detected in the signal received from EN\({}_{k}\), such signal is discarded. Otherwise, the interference-free eMBB signals are jointly decoded at the BBU.  
Considering the aforementioned assumptions, the signal model for puncturing can be equivalently described by assuming that if the signal \(Y_{k}^{f}\) received at the EN\({}_{k}\) on frequency \(f\) is interfered by an URLLC transmission, then such signal is discarded, and the BBU receives only noise. This is mathematical described by  
\[Y_{k}^{f}=B_{k}(X_{k}^{f}+\alpha X_{k+1}^{f}+\alpha X_{k-1}^{f})+Z_{k}^{f}, \tag{36}\]  
where the Bernoulli variable \(B_{k}=1-A_{k}\sim\mathcal{B}(1-q)\) indicates the absence (\(B_{k}=1\)) or presence (\(B_{k}=0\)) of URLLC transmissions in the given minislot. The signal in (36) received across all ENs and frequencies can be written in matrix form as  
\[\mathbf{Y}=\mathbf{X}\mathbf{H}\mathbf{B}+\mathbf{Z}, \tag{37}\]  
with definitions given in Section 2 and with \(\mathbf{B}=\text{diag}(B_{1},B_{2},...,B_{M})\). The rate for the eMBB UE under NOMA by puncturing is reported in Lemma 3 and can be derived by substituting signal (37) in Equation (24).  
**Lemma 3**.: _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 puncturing yields_  
\[R_{B}=\frac{\mu}{M}\mathbb{E}_{\mathbf{B}}\left[log\left(det\left(\mathbf{I}+ P_{B}\mathbb{R}_{z_{eq}}^{-1}\mathbf{H}_{B,eq}\mathbf{H}_{B,eq}^{T}\right) \right)\right], \tag{38}\]  
_where \(\mathbf{R}_{z_{eq}}\) is defined as in Lemma 2; \(\mathbf{H}_{B,eq}=\mathbf{B}\mathbf{H}\otimes\mathbf{H}_{c}^{\eta}\) is the equivalent wireless plus cable channel in case of puncturing; and we have \(\mathbf{B}=\text{diag}(B_{1},B_{2},...,B_{M})\), with \(B_{k}\) being i.i.d. \(\mathcal{B}(1-q)\) variables._  
Figure 8: Block diagram of the operation of the ENs and BBU for Non-Orthogonal Multiple Access (NOMA) by puncturing and Treating Interference as Noise (TIN). A/A stands for Analog-to-Analog.  
Proof.: Lemma 3 can be proved by following similar steps as for the proof of Lemma 2 with two minor differences: _(i)_ the radio channel matrix \(\mathbf{H}\) is right multiplied by the random matrix \(\mathbf{B}\) and _(ii)_ the capacity is computed by averaging over the distribution of \(\mathbf{B}\).  
Differently from the rate (32) achieved by OMA, under NOMA, the eMBB transmission spreads over all the minislots, so that there is no scaling factor \(1-L_{U}^{-1}\) in front of the rate expression (38) to account for the resulting loss in spectral efficiency. In case of NOMA by puncturing, the noise covariance is exactly as the one in the eMBB OMA rate in Equation (32), since the eMBB signal, if not discarded at the BBU, is guaranteed to be URLLC interference-free. The overall rate is computed by averaging over all the possible realizations of the random matrix \(\mathbf{B}\), which left-multiplies the radio channel matrix \(\mathbf{H}\) and accounts for the probability that the entire signal is discarded due to an incoming URLLC packet.  
In the case of C-RAN with digital limited-capacity fronthaul links, as discussed in [32], it is advantageous to carry out the operation of detecting and, eventually, discarding the eMBB signal at the ENs. In fact, with a digital fronthaul, only the undiscarded minislots can be quantized, hence devoting the limited fronthaul resources to increase the resolution of interference-free eMBB samples [32]. The same does not apply to the analog fronthaul considered here, as signals are directly relayed to the BBU without any digitization.