## 4 Uplink Reception With Proposed Cooperation  
In this section, we propose an improved inter-BBU cooperation scheme based on layered compression, or successive refinement quantization, strategy [16; 17] (see also [13]). In this approach, each RU \((i,r)\) is equipped with two compression encoders, where the \(j\)th encoder generates a description \(U_{i,r,j}\) of compression rate \(R_{C,i,r,j}\) bit/symbol by quantizing and compressing the received signal \(y_{i,r}\) of RU \((i,r)\). The description \(U_{i,r,1}\) is referred to as _basement layer_ from which a quantized signal \(\hat{y}_{i,r,1}\) can be recovered. The other description \(U_{i,r,2}\) is referred to as _enhancement layer_, since a better reconstruction \(\hat{y}_{i,r,2}\) can be obtained by using both descriptions \(U_{i,r,1}\) and \(U_{i,r,2}\), i.e., \(y_{i,r}\leftrightarrow\hat{y}_{i,r,2}\leftrightarrow\hat{y}_{i,r,1}\). Following [12], we assume the Gaussian test channel for both \(\hat{y}_{i,r,1}\) and \(\hat{y}_{i,r,2}\), i.e.,  
\[\hat{y}_{i,r,j}=y_{i,r}+q_{i,r,j}, \tag{12}\]  
with the quantization noise \(\hat{q}_{i,r,j}\sim\mathcal{CN}(0,\omega_{i,r,j})\) being independent of \(y_{i,r}\). In order for the descriptions \(\hat{y}_{i,r,1}\) and \(\hat{y}_{i,r,2}\) to be successfully recovered, the following conditions should be satisfied. \[I\left(y_{i,r};\hat{y}_{i,r,j}\right)=\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2} }{\omega_{i,r,j}}\right)\leq\sum_{m=1}^{j}R_{C,i,r,m}, \tag{13}\]  
for \(j\in\{1,2\}\). To enable a flexible inter-BBU cooperation, we assume that the basement layer \(U_{i,r,1}\) of each RU \((i,r)\) is transferred to both the serving BBU \(i\) and the neighboring BBU \(\bar{i}\), while the enhancement layer \(U_{i,r,2}\) is delivered only to the serving BBU \(i\). Therefore, BBU \(i\) can recover the quantized signal \(\hat{y}_{i,r,2}\), which better represents the received signal \(y_{i,r}\) of RU \((i,r)\) than the other quantized signal \(\hat{y}_{i,r,1}\) reconstructed at BBU \(\bar{i}\) does. Under the assumption that we use the El Gamal-Cover coding scheme [18] for the compression encoders and decoders, the described process can be made possible if the compression rates \(R_{C,i,r,j}\) satisfy the following constraints. \[R_{C,i,r,1}+R_{C,i,r,2}\leq C_{F},\text{ for }i\in\{1,2\},r\in\mathcal{M}, \tag{14}\]  
\[\sum_{r\in\mathcal{M}}R_{C,i,r,1}\leq C_{B},\text{ for }i\in\{1,2\}. \tag{15}\]  
We refer to ([13], Section 13.5) for the detailed proof. We assume that each BBU \(i\) decodes the messages \(W_{i,1},W_{i,2},\ldots,W_{i,K}\) from the high-resolution quantized signals \(\hat{y}_{i,2}\triangleq[\hat{y}_{i,1,2}\hat{y}_{i,2,2}\cdots\hat{y}_{i,M,2}]^ {T}\) associated with in-cluster RUs and the low-resolution quantized signals \(\hat{y}_{\bar{i},1}\triangleq[\hat{y}_{\bar{i},1,\bar{i}}\hat{y}_{i,2,1} \cdots\hat{y}_{i,M,1}]^{T}\) corresponding to the other-cluster RUs. The achievable sum-rate \(R_{\text{sum},i}\) of cluster \(i\) is given as  
\[R_{\text{sum},i} =I\left(\mathbf{x}_{i};\ \hat{y}_{i,2},\hat{y}_{\bar{i},1}\right) \tag{16}\] \[=\log_{2}\det\left(\mathbf{I}+P\left(P\mathbf{\hat{G}}_{i}\mathbf{ \hat{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\mathbf{\tilde{\Omega}}_{i}\right)^ {-1}\mathbf{\tilde{H}}_{i}\mathbf{\tilde{H}}_{i}^{H}\right),\]  
where we define the matrices \(\mathbf{\tilde{H}}_{i}\in\mathbb{C}^{2M\times K}\), \(\mathbf{\tilde{G}}_{i}\in\mathbb{C}^{2M\times K}\) and \(\mathbf{\tilde{\Omega}}_{i}\in\mathbb{C}^{2M\times 2M}\) as  
\[\mathbf{\tilde{H}}_{i}=\left[ \begin{array}{c}\mathbf{H}_{i}\\ \mathbf{G}_{\bar{i}}\end{array} \right],\ \mathbf{\tilde{G}}_{i}=\left[ \begin{array}{c} \mathbf{G}_{i}\\ \mathbf{H}_{\bar{i}}\end{array} \right],\ \mathbf{\tilde{\Omega}}_{i}=\text{diag}\left(\{ \omega_{i,r,2}\}_{r\in\mathcal{M}},\{\omega_{\bar{i},r,1}\}_{r\in\mathcal{M}} \right). \tag{17}\]  
We now discuss the optimization of the proposed layered inter-cluster cooperation strategy.