## 4 Uplink Reception With Proposed Cooperation
To investigate the convergence property of the proposed algorithm, Figure 2 plots the average sum-rate \(R_{\text{sum}}\) with respect to the number of iterations for a two-cluster C-RAN uplink system with \(M\in\{1,3,5\}\), \(K=6\), \(C_{B}=1\), \(C_{F}=2\) and \(20\) dB SNR. It is observed that, as the network size increases (i.e., the number \(M\) of RUs increases), more iterations are needed for convergence. However, for all simulated cases, the algorithm converges within a few tens of iterations. In the simulation of the remaining results, we limit the maximum number of iterations to \(N_{\text{max}}=30\), which means that Algorithm 1 stops if the updated sum-rate is sufficiently close to the previous sum-rate, or the number of iterations reaches \(N_{\text{max}}\). In Figure 3, we plot the average sum-rate \(R_{\text{sum}}\) versus the fronthaul capacity \(C_{F}\) for a two-cluster C-RAN uplink system with \(M=2\), \(K=6\), \(C_{B}=1\), and \(P/\sigma_{2}^{2}=20\) dB. From the figure, we can see that the performance of the conventional cooperation scheme with fixed \(\tilde{M}=M\) can be worse than that of the no cooperation scheme, particularly when the RUs-to-BBU fronthaul links have much larger  
\[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]8 Average sum-rate \(R_{\text{sum}}\) versus the number of iterations for a two-cluster C-RAN uplink system with \(M\in\{1,3,5\}\), \(K=6\), \(C_{B}=1\), \(C_{F}=2\), and \(20\) dB SNR. capacity than the inter-BBU backhaul links (i.e., \(C_{F}>C_{B}\)). This is because forwarding the quantized signals of many in-cluster RUs to the other BBU on low-capacity backhaul links limits the resolution of the quantized signals and makes the capacity of fronthaul links not fully utilized. In addition, we can achieve a notable gain by adopting the conventional inter-BBU cooperation scheme with the optimal \(\tilde{M}\) compared to the no cooperation scheme, and the gain increases with the fronthaul capacity \(C_{F}\). However, we should perform an MM algorithm for each possible value \(\tilde{M}\in\{0,1,\ldots,M\}\) to find the optimal \(\tilde{M}\). We note that the proposed scheme can achieve a further gain particularly at large \(C_{F}\) without resorting to a discrete search. Figure 4 plots the average sum-rate \(R_{\text{sum}}\) with respect to the SNR \(P/\sigma_{z}^{2}\) for a two-cluster C-RAN uplink system with \(M=2\), \(K=6\), \(C_{B}=1\), and \(C_{F}=4\). The figure shows that the performance gaps among the schemes increase with the SNR of the uplink channel. This suggests that the importance of inter-BBU cooperation on the backhaul links becomes more significant at high SNRs, since the overall performance will be more interference-limited in that regime. \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]9 Average sum-rate \(R_{\text{sum}}\) versus the fronthaul capacity \(C_{F}\) for a two-cluster C-RAN uplink system with \(M=2\), \(K=6\), \(C_{B}=1\), and \(P/\sigma_{z}^{2}=20\) dB. In Figure 5, we investigate the impact of capacity \(C_{B}\) of the backhaul links by plotting the average sum-rate with respect to \(C_{B}\) for a two-cluster C-RAN uplink system with \(M=3\), \(K=10\), \(C_{F}=2\), and \(20\) dB SNR. For the conventional cooperation scheme, we choose the cooperation level \(\bar{M}\) from \(\bar{M}\in\{0,M\}\). We can see in the figure that the proposed scheme outperforms the conventional cooperation scheme when the backhaul links do not have enough capacity. However, as the backhaul capacity \(C_{B}\) becomes sufficiently large, both the proposed and conventional inter-BBU cooperation schemes achieve the performance of the perfect backhaul scheme.