## 3 Conventional Uplink Reception Strategies
### Conventional Inter-BBU Cooperation  
This subsection discusses the conventional inter-BBU cooperation scheme [7], where each BBU \(i\) sends a selected subset of the signals \(\hat{y}_{i,1},\hat{y}_{i,2},\ldots,\hat{y}_{i,M}\) to the other BBU \(\tilde{i}\). We note that it may not be optimal to send all the signals on the backhaul link, which has a limited capacity \(C_{B}\), particularly when the capacity \(C_{B}\) of backhaul links is much smaller than \(C_{F}\). We define \(\tilde{\mathcal{M}}_{i}\subseteq\mathcal{M}\) as the set of RUs' indices whose quantized signals are transferred to BBU \(\tilde{i}\) on the backhaul link. Without claim of optimality, following the policy proposed in [7], we fill each set \(\tilde{\mathcal{M}}_{i}\) with the indices of \(\tilde{M}\) RUs that have the largest channel gains from the users in the neighboring cluster \(\tilde{i}\), i.e.,  
\[\tilde{\mathcal{M}}_{i}=\big{\{}r_{i,1},r_{i,2},\ldots,r_{i,\tilde{M}}\big{\}}, \tag{8}\]  
where we have \(||\mathbf{g}_{i,r_{1}}||^{2}\geq||\mathbf{g}_{i,r_{2}}||^{2}\geq\ldots\geq|| \mathbf{g}_{i,r_{M}}||^{2}\) with \(\mathbf{g}_{i,r}\triangleq[g_{i,r,1}g_{i,r,2}\cdots g_{i,r,K}]^{T}\). The number \(\tilde{M}\in\{0,1,\ldots,M\}\) determines the level of inter-cluster cooperation, and setting \(\tilde{M}=0\) leads to the no cooperation scheme discussed in Section 3.1. In principle, increasing the level \(\tilde{M}\) enables stronger cooperation among the BBUs. However, when the backhaul links have small capacity, large \(\tilde{M}\) degrades the resolution of the quantized signals to satisfy the backhaul capacity constraints, and the fronthaul links are not fully utilized. Therefore, the optimal level \(\tilde{M}\) should be carefully chosen depending on the instantaneous channel states as well as the backhaul and fronthaul capacities. We denote the compression rate allocated to express the signal \(\hat{y}_{i,r}\) by \(R_{C,i,r}\in[0,C_{F}]\), and the rates \(\{R_{C,i,r}\}_{r\in\tilde{\mathcal{M}}_{i}}\) should satisfy the condition  
\[\sum_{r\in\tilde{\mathcal{M}}_{i}}R_{C,i,r}\leq C_{B}. \tag{9}\]  
In addition, with this approach, the upper-threshold \(C_{F}\) in the RHS of the fronthaul capacity constraint in Equation (3) is replaced with \(R_{C,i,r}\) for RUs \((i,r)\) with \(r\in\tilde{\mathcal{M}}_{i}\). This is because, if \(R_{C,i,r}<C_{F}\), we can use only partial, instead of full, capacity of those RUs' fronthaul links, since the quantized signals should be transferred on the fronthaul link to BBU \(i\) as well as on the backhaul link to the other BBU \(\tilde{i}\). We assume that BBU \(i\) decodes the messages \(W_{i,1},W_{i,2},\ldots,W_{i,K}\) by leveraging the signals \(\hat{\mathbf{y}}_{i}\) received on the fronthaul links from in-cluster RUs as well as the signals \(\hat{y}_{i,r_{j,1}},\hat{y}_{i,r_{i,2}},\ldots,\hat{y}_{i,r_{j,\tilde{M}}\tilde {\alpha}}\) received on the backhaul link from the other BBU \(\tilde{i}\). Then, the achievable sum-rate \(R_{\text{sum},i}\) of cluster \(i\) is given as  
\[R_{\text{sum},i} =I\left(\mathbf{x}_{i};\ \hat{\mathbf{y}}_{i,r_{j,r_{j,r_{j,r_{j}}}}} \right)_{r\in\tilde{\mathcal{M}}_{i}} \tag{10}\] \[=\log_{2}\det\left(\mathbf{I}+P\left(P\mathbf{\tilde{G}}_{i} \mathbf{\tilde{G}}_{i}^{H}+\sigma_{2}^{2}\mathbf{I}+\mathbf{\tilde{\Omega}}_ {i}\right)^{-1}\mathbf{\tilde{H}}_{i}\mathbf{\tilde{H}}_{i}^{H}\right),\]  
where we define the matrices \(\mathbf{\tilde{H}}_{i}\in\mathbb{C}^{(M+\tilde{M})\times K}\), \(\mathbf{\tilde{G}}_{i}\in\mathbb{C}^{(M+\tilde{M})\times K}\) and \(\mathbf{\tilde{\Omega}}_{i}\in\mathbb{C}^{(M+\tilde{M})\times(M+\tilde{M})}\) as  
\[\mathbf{\tilde{H}}_{i}=\left[ \begin{array}{c}\mathbf{H}_{i}\\ \left[\begin{array}{c}\mathbf{e}_{r_{i,1}}^{H}\\ \vdots\\ \mathbf{e}_{r_{j,M}}^{H}\end{array}\right]\end{array} \right],\ \mathbf{\tilde{G}}_{i}=\left[ \begin{array}{c}\mathbf{G}_{i}\\ \left[\begin{array}{c}\mathbf{e}_{r_{i,1}}^{H}\\ \vdots\\ \mathbf{e}_{r_{j,M}}^{H}\end{array}\right]\end{array}