## 3 Beamforming Design for BIOS-FD Systems
### SIC-WMMSE Approach
Extract the main diagonal elements of \(\mathbf{\Phi}_{2}\) to form vector \(\mathbf{\phi}_{2}\); the objective \(f_{2}\!\left(\mathbf{\Phi}_{2}\right)\) can be transformed into \(f_{2}\!\left(\mathbf{\phi}_{2}\right)\) as follows:  
\[\mathbf{u}_{\text{U},l}{}^{\star}=\sqrt{P_{\text{U}}}\Big{(}P_{\text{U}} \mathbf{\hat{H}}_{\text{U}}\mathbf{\hat{H}}_{\text{U}}^{H}+\mathbf{H}_{\text {SI}}\mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\text{SI}}^{H}+\sigma_{\text{U}}^{ 2}\mathbf{I}\Big{)}^{-1}\mathbf{\hat{h}}_{\text{U},l}. \tag{26}\]4  
where \(\mathbf{\phi}_{2}=\operatorname{diag}\!\left(\mathbf{\Phi}_{2}\right)\), and \(\mathbf{C}\) and \(\mathbf{D}\) are equivalent matrices as  
\[\mathbf{u}_{\text{U},l}{}^{\star}=\sqrt{P_{\text{U}}}\Big{(}P_{\text{U}} \mathbf{\hat{H}}_{\text{U}}\mathbf{\hat{H}}_{\text{U}}^{H}+\mathbf{H}_{\text {SI}}\mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\text{SI}}^{H}+\sigma_{\text{U}}^{ 2}\mathbf{I}\Big{)}^{-1}\mathbf{\hat{h}}_{\text{U},l}. \tag{26}\]5  
It is worth noting that, despite the similar form, the optimization of \(\mathbf{\Phi}_{1}\) and \(\mathbf{\Phi}_{2}\) has different meanings for this system. As indicated in the UL rate expression (12), the SI is tied to \(\mathbf{\Phi}_{1}\) while not directly related to the optimization of \(\mathbf{\Phi}_{2}\). Therefore, we can ignore the penalty item related to SIC in (19) during the \(\mathbf{\Phi}_{2}\) optimization. Furthermore, as the BIOS can independently control the reflection and refraction beams on both sides, the additional freedom enables \(\mathbf{\Phi}_{2}\) to focus on the beamforming for SRM without affecting the SIC quality. The proposed SIC-WMMSE algorithm for solving the sub-problem is summarized in Algorithm 1. It is worth noting that the application of the KKT conditions and the MO algorithm enables a reduction in the objective function for each variable within each iteration. When the decrement in the weighted MSE between two iterations is below a predefined threshold, the iteration is terminated. Since the objective function decreases after each step of updating, Algorithm 1 can converge to a locally optimal solution. To analyze its computational complexity, we count the number of complex multiplications of the key steps in one iteration turn. Assuming that \(N_{\rm T}=N_{\rm T}=N\), \(K_{\rm D}=K_{\rm U}=K\), and \(M>N>K\), the complexity can be summarized as follows: updating \(\mathbf{F}^{(i)}\): \(O(KM^{2}+N^{3})\); updating \(\mathbf{\Phi}_{1}{}^{(i)}\): \(O\big{(}M^{3}\big{)}\); updating \(\mathbf{\Phi}_{2}{}^{(i)}\): \(O\big{(}NM^{2}\big{)}\); updating \(\mathbf{A}_{\rm D}{}^{(i)}\): \(O\big{(}KM^{2}\big{)}\); updating \(\mathbf{A}_{\rm U}{}^{(i)}\): \(O\big{(}KM^{2}+MN^{2}\big{)}\); updating \(\mathbf{U}_{\rm D}{}^{(i)}\): \(O\big{(}NK^{2}\big{)}\); updating \(\mathbf{U}_{\rm U}{}^{(i)}\): \(O\big{(}N^{3}\big{)}\). ```
1:Randomly initialize \(\mathbf{\Phi}_{1}^{(0)}\), \(\mathbf{\Phi}_{2}^{(0)}\) and \(\mathbf{F}^{(0)}\) and set \(i=0\). 2:Calculate \(\mathbf{A}_{\rm D}^{(0)}\), \(\mathbf{A}_{\rm U}^{(0)}\), \(\mathbf{U}_{\rm D}^{(0)}\), and \(\mathbf{U}_{\rm U}^{(0)}\) according to (23)-(26). 3:repeat
4: Update \(\mathbf{F}^{(i)}\) according to (28). 5: Update \(\mathbf{\Phi}_{1}^{(i)}\), \(\mathbf{\Phi}_{2}^{(i)}\) according to (31), (36). 6: Update \(\mathbf{U}_{\rm D}^{(i)}\) and \(\mathbf{U}_{\rm U}^{(i)}\) according to (24), (26). 7: Update \(\mathbf{A}_{\rm D}^{(i)}\), \(\mathbf{A}_{\rm U}^{(i)}\) according to (23), (25). 8:\(i\gets i+1\). 9:until the stopping condition is satisfied. ```  
**Algorithm 1** Proposed SIC-WMMSE Algorithm