## 5 Our Proposed Novel Resource Allocation Scheme
### Optimize Compression Noise \(\mathbf{\Phi}\)  
For the fixed assignment variables \(\underline{x}\) and backhaul resource allocation vector \(\underline{r}\), let \(F(\underline{x},\mathbf{\Gamma},\mathbf{\Phi})\) presents objective function. For any given \((x_{i,j,k},r_{k})\), the signals received by BS A and BS B are:  
\[y_{A,k}=h_{i,A}s_{i}+h_{j,A}s_{j}+n_{A} \tag{16}\]  
\[y_{B,k}=h_{i,B}s_{i}+h_{j,B}s_{j}+n_{B} \tag{17}\]  
Because of the assumption of orthogonality between subcarriers, \(\mathbf{\Phi}\) is a diagonal matrix, _i.e.,_\(\mathbf{\Phi}=diag(q_{k})\), \(q_{k}\sim\mathcal{NC}\left(0,\eta_{k}\right)\) is the compression noise of compression code used by subcarrier \(k\):  
\[\hat{y}_{B,k}=y_{B,k}+q_{k} \tag{18}\]  
The sum rate of user pair \(i\) and user \(j\) on subcarrier \(k\) is:  
\[f_{i,j,k}=I(s_{i}s_{j};y_{A,k}\hat{y}_{B,k}) \tag{19}\]  
And the sub-problem of maximizing achievable pair rate under given fixed backhaul rate is:  
\[\begin{array}{ll}\max_{\eta_{k}}&I(s_{i}s_{j};y_{A,k}\hat{y}_{B,k})\\ s.t.&I(y_{A,k};\hat{y}_{B,k})\leqslant r_{k}\end{array} \tag{20}\]  
The optimal solution \(q_{k}\) to the above problem is given in [13]:  
\[q_{k}^{*}=e_{i,j,k}/(2^{r_{k}}-1) \tag{21}\]  
where \(e_{i,j,k}\) is the eigenvalue of conditional covariance associated with user \(i\) and user \(j\) on subcarrier \(k\). Given the knowledge of \(y_{A,k},y_{B,k}\) is Gaussian distributed and the conditional covariance denoted as \(R_{B|A}^{i,j,k}\). It can be computed as follows:  
\[R_{B|A}^{i,j,k}=R_{B}^{i,j,k}-R_{B,A}^{i,j,k}{(R_{A}^{i,j,k})}^{-1}R_{A,B}^{i,j,k} \tag{22}\]  
and:  
\[R_{A}^{i,j,k}=P(|h_{i,A}^{k}|^{2}+|h_{j,A}^{k}|^{2})+\sigma^{2} \tag{23}\]\[R_{A,B}^{i,j,k} =P(h_{i,A}^{k}h_{i,B}^{k*}+h_{j,A}^{k}h_{j,B}^{k*}) \tag{24}\] \[R_{B}^{i,j,k} =P(|h_{i,B}^{k}|^{2}+|h_{j,B}^{k}|^{2})+\sigma^{2}\] (25) \[R_{B,A}^{i,j,k} =P(h_{i,B}^{k}h_{i,A}^{k*}+h_{j,B}^{k}h_{j,A}^{k*}) \tag{26}\]  
where \(R_{A}^{i,j,k}\) and \(R_{B}^{i,j,k}\) denote the covariance of the received signal on subcarrier \(k\) at BS A and BS B, while \(R_{A,B}^{i,j,k}(R_{B,A}^{i,j,k})\) denotes the cross-correlation between the BS A(B) and BS B(A) observations.  
As \(R_{B|A}^{i,j,k}\) is a scalar, the eigenvalue \(e_{i,j,k}\) of conditional covariance equals to itself. Substituting Equation (21) into Equation (19), the optimized pair rate \(f_{i,j,k}\) is:  
\[\begin{array}{ll}f_{i,j,k}^{*}=&\log_{2}\left(1+P(|h_{i,A}^{k}|^{2}+|h_{j,A}^ {k}|^{2})/\sigma^{2}\right)\\ &+\log_{2}\left(e_{i,j,k}2^{\tau_{k}}/((2^{\tau_{k}}-1)\sigma^{2}+e_{i,j,k}) \right)\end{array} \tag{27}\]  
This equation indicates that the achievable rate of the pair can be decomposed into two parts. The first one is attributed to \(y_{A,k}\), the second one is related to two factors: (1) the additional information offered by \(y_{B,k}\); (2) the backhaul rate allocated to subcarrier \(k\).  
It can be easily verified that \(f_{i,j,k}^{*}\) is a concave function in \(r_{k}\). Based on above formulations, \(F(\mathbf{x},\mathbf{r},\mathbf{\Phi})\) could be reduces to \(F(\mathbf{x},\mathbf{r},\mathbf{\Phi}(\mathbf{r}))\), which is now related to \(\mathbf{x}\) and \(\mathbf{r}\).