## 4 Uplink Reception With Proposed Cooperation
** Set \(\omega_{i,r,j}^{\prime}\) to \(\omega_{i,r,j}^{\prime}\leftarrow\sigma_{y,i,r}^{2}/\left(2\Sigma_{m=1}^{j} R_{C,i,r,m}-1\right)\), i.e., the minimum value that satisfies the constraint in Equation (18c), for \(i\in\{1,2\}\), \(r\in\mathcal{M}\), \(j\in\{1,2\}\). **3. ** Update \(\omega\) as a solution of the convex problem:  
\[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]8 (25a) s.t. \[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]9 \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]0 \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]1 \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]2 \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]3 (25c) \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]4 (25d) \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]5 (25e) \[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]6  
**4. ** Stop if a convergence criterion is satisfied. Otherwise, go back to Step 3 with \(\omega^{\prime}\leftarrow\omega\). **5. Numerical Results**  
In this section, we demonstrate numerical results that validate the efficiency of the proposed inter-cluster cooperation scheme. In the simulation, we assume that the channel coefficients \(h_{i,r,k}\) and \(g_{i,r,k}\) follow independent and identically distributed (i.i.d.) Rayleigh fading distribution, i.e.,\(h_{i,r,k}\sim\mathcal{CN}(0,1)\) and \(g_{i,r,k}\sim\mathcal{CN}(0,1)\). We compare the performance of the proposed cooperation scheme (Section 4) with the following benchmark schemes. * Perfect backhaul: Two BBUs can perfectly cooperate without any constraint, and each BBU \(i\) decodes in-cluster messages \(W_{i,k},k\in\mathcal{K}\), while treating the other-cluster signals as noise. * No backhaul (Section 3.1): There are no backhaul links, and hence the BBUs do not exchange any information. * Conventional cooperation (Section 3.2) with fixed \(\tilde{M}\). * Conventional cooperation (Section 3.2) with optimal \(\tilde{M}\). The sum-rate that is achieved with the perfect backhaul links is given as \(R_{\text{sum}}=\sum_{i\in\{1,2\}}R_{\text{sum},i}\), where the sum-rate \(R_{\text{sum},i}\) of cluster \(i\) is given as  
\[\log_{2}\left(1+\frac{\sigma_{y,i,r}^{2}}{\omega_{i,r,j}}\right) \leq\sum_{m=1}^{j}R_{\text{C},i,r,m},\ \forall i\in\{1,2\},\,r\in\mathcal{M},\,j\in\{1,2\},\]7  
Here, we define the matrices \(\mathbf{A}_{1}=[\mathbf{H}_{1};\mathbf{G}_{2}]\), \(\mathbf{A}_{2}=[\mathbf{G}_{1};\mathbf{H}_{2}]\), and \(\mathbf{\tilde{\Omega}}\triangleq\text{diag}(\{\omega_{1,r}\}_{r\in\mathcal{M} },\{\omega_{2,r}\}_{r\in\mathcal{M}})\in\mathbb{C}^{2M\times 2M}\) with \(\omega_{i,r}\) given as Equation (6). To find the optimal \(\tilde{M}\) of the last scheme, we perform an exhaustive search over \(\tilde{M}\in\{0,1,\ldots,M\}\).