## 3 Conventional Uplink Reception Strategies
### No Inter-BBU Cooperation  
In this subsection, we review the conventional scheme without inter-BBU cooperation, in which each BBU \(i\) decodes the messages \(W_{i,1},W_{i,2},\ldots,W_{i,K}\) sent by in-cluster users by exploiting only the signals \(\hat{y}_{i,1},\hat{y}_{i,2},\ldots,\hat{y}_{i,M}\) received on the fronthaul links. With this approach, the backhaul links connecting the BBUs are not utilized at all.  
If we assume that BBU \(i\) decodes the messages \(W_{i,1},W_{i,2},\ldots,W_{i,K}\) while treating the interference signals \(x_{i,1},x_{i,2},\ldots,x_{i,K}\) from the other cluster \(\overline{i}\) as noise, the achievable sum-rate \(R_{\text{sum}j}\triangleq\sum_{k\in\mathcal{K}}R_{i,k}\) of cluster \(i\) is given as  
\[R_{\text{sum},i}=I\left(\mathbf{x}_{i};\hat{y}_{i}\right)=\log_{2}\det\left( \mathbf{I}+P\left(P\mathbf{G}_{i}\mathbf{G}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+ \mathbf{\Omega}_{i}\right)^{-1}\mathbf{H}_{i}\mathbf{H}_{i}^{H}\right), \tag{5}\]  
where we define the notations \(\mathbf{x}_{i}\triangleq[x_{i,1}x_{i,2}\cdots x_{i,K}]^{T}\), \(\mathbf{\hat{y}}_{i}\triangleq[\hat{y}_{i,1}\hat{y}_{i,2}\cdots\hat{y}_{i,M}] ^{T}\), \(\mathbf{\Omega}_{i}\triangleq\text{diag}(\{\omega_{i,r}\}_{r\in\mathcal{M}})\), and the \((r,k)\)th elements of \(\mathbf{H}_{i}\in\mathbb{C}^{M\times K}\) and \(\mathbf{G}_{i}\in\mathbb{C}^{M\times K}\) are given as \(h_{i,r,k}\) and \(g_{i,r,k}\), respectively.  
Note that, in this approach that does not employ inter-BBU cooperation, the optimal quantization power \(\omega_{i,r}\) is simply the minimum value that satisfies the condition in Equation (3), since there is no overhead of the backhaul links. Such minimum value of \(\omega_{i,r}\) is given as  
\[\omega_{i,r}=\frac{\sigma_{y,i,r}^{2}}{2^{C_{F}}-1}. \tag{6}\]  
**Remark 1**.: _Suppose that BBU \(i\) decodes and cancels the interference signals \(x_{i,k}\), \(k\in\tilde{K}_{i}\), from the neighboring cluster \(\overline{i}\) prior to decoding the desired in-cluster signals. Then, the sum-rate \(R_{\text{sum},i}\) of cluster \(i\) is bounded as_  
\[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(\sum_{k\in\mathcal{K }\setminus\mathcal{K}_{i}}P\mathbf{g}_{i,k}\mathbf{g}_{i,k}^{H}+\sigma_{z}^{2 }\mathbf{I}+\mathbf{\Omega}_{i}\right)^{-1}\mathbf{H}_{i}\mathbf{H}_{i}^{H} \right), \tag{7}\]  
_where \(\mathbf{g}_{i,k}\in\mathbb{C}^{M\times 1}\) denotes the \(k\)th column vector of \(\mathbf{G}_{i}\). Note that, as long as \(\tilde{\mathcal{K}}_{\overline{i}}\) is not empty, the RHS of Equation (7) is strictly larger than that of Equation (5), since the interference covariance terms have been reduced. However, unlike Equation (5), the condition in Equation (7) may not be satisfied with equality, since the rates \(R_{i,k}\) with \(k\in\tilde{\mathcal{K}}_{i}\) are subject to additional constraints for successful interference decoding at the neighboring BBU \(\overline{i}\). This suggests that the sets \(\tilde{\mathcal{K}}_{1}\) and \(\tilde{\mathcal{K}}_{2}\) of decoded interference signals need to be carefully chosen depending on the SNR \(P/\sigma_{z}^{2}\) as well as the instantaneous CSI. An exhaustive search for finding the optimal sets \(\tilde{\mathcal{K}}_{1}\) and \(\tilde{\mathcal{K}}_{2}\)requires a search size exponentially increasing with \(K\). This calls for the development of an efficient selection algorithm that achieves a good trade-off between the performance and complexity._