## 3 Beamforming Design for BIOS-FD Systems
### SIC-WMMSE Approach  
Due to the multiple coupled variables and non-convex constraints, problem (13) is difficult to solve directly. However, by leveraging the relationship between MSE and data rate, we convert the primal problem (13) into an equivalent WMMSE problem as follows:  
\[ \begin{array}{ll}\min\limits_{\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{F},}&f=\sum_{k=1}^{K_{\text{D}}}\big{(}a_{\text{D},k}e_{\text{D},k}-\log_{2}|a_{ \text{D},k}|\big{)}+\sum_{l=1}^{K_{\text{U}}}\big{(}a_{\text{U},l}e_{\text{U}, l}-\log_{2}|a_{\text{U},l}|\big{)}\\ \text{s.t.}&\|\mathbf{H}_{\text{SI}}\mathbf{F}\|_{F}^{2}/N_{\text{R}}\leq \gamma,\\ &|\mathbf{\Phi}_{1}|_{ii}=|\mathbf{\Phi}_{2}|_{jj}=1,\forall i,j,\\ &\text{Tr}\big{(}\mathbf{F}\mathbf{F}^{H}\big{)}\leq P_{\text{s}},\end{array}  \tag{14}\]  
where \(a_{\text{D},k}\) and \(a_{\text{U},l}\) are the weighting coefficients for the \(k\)-th DL UE and the \(l\)-th UL UE, respectively. \(u_{\text{D},k}^{*}\) is the receive coefficient at the \(k\)-th DL UE and \(\mathbf{u}_{\text{U},l}^{H}\) is the receive combiner for the \(l\)-th UL UE at the \(\text{BS}_{\text{R}}\), respectively. \(e_{\text{D},k}\) and \(e_{\text{U},l}\), respectively, denote the MSEs of the \(k\)-th DL UE and the \(l\)-th UL UE as follows:  
\[ \begin{array}{ll}e_{\text{D},k}&=\mathbb{E}\bigg{\{}\Big{\|}u_{\text{D},k}^{ *}y_{\text{D},k}-x_{\text{D},k}\Big{\|}^{2}\bigg{\}}\\ &=\Big{(}u_{\text{D},k}^{*}\hat{\mathbf{H}}_{\text{D},k}^{H}t_{k}-1\Big{)} \Big{(}u_{\text{D},k}^{*}\hat{\mathbf{H}}_{\text{D},k}^{H}t_{k}-1\Big{)}^{H}+ \sum_{\begin{subarray}{c}i=1\\ i\
eq k\end{subarray}}^{K_{\text{U}}}u_{\text{D},k}^{*}u_{\text{D},k}\hat{ \mathbf{H}}_{\text{D},k}^{H}t_{i}\hat{\mathbf{H}}_{\text{D},k}+\sigma_{\text{ D},k}^{2}u_{\text{D},k}^{*}\mu_{\text{D},k}\\ &=u_{\text{D},k}^{*}\mu_{\text{D},k}\hat{\mathbf{H}}_{\text{D},k}^{H} \mathbf{F}^{H}\hat{\mathbf{H}}_{\text{D},k}-2\operatorname{Re}\Big{\{}u_{ \text{D},k}^{*}\hat{\mathbf{H}}_{\text{D},k}^{H}t_{k}\Big{\}}+\sigma_{\text{ D},k}^{2}u_{\text{D},k}^{*}u_{\text{D},k}+1,\end{array}  \tag{15}\]  
\[\begin{array}{ll}e_{\text{U},l}&=\mathbb{E}\bigg{\{}\Big{|}\mathbf{u}_{\text{ U},l}^{H}\mathbf{y}-x_{\text{U},l}\Big{|}^{2}\bigg{\}}\\ &=P_{\text{U}}\Big{(}\mathbf{u}_{\text{U},l}^{H}\hat{\mathbf{h}}_{\text{U},l} -1\Big{)}\Big{(}\mathbf{u}_{\text{U},l}^{H}\hat{\mathbf{h}}_{\text{U},l}-1 \Big{)}^{H}+\sum_{ \begin{subarray}{c}i=1\\ i\
eq l\end{subarray} }^{K_{\text{U}}}P_{\text{U}}\mathbf{u}_{\text{U},l}^{H} \hat{\mathbf{h}}_{\text{U},l}^{H}\mathbf{h}_{\text{U},l}^{H}\mathbf{u}_{\text{ U},l}+\mathbf{u}_{\text{U},l}^{H}\mathbf{H}_{\text{SI}}\mathbf{F} ^{H}\mathbf{H}_{\text{SI}}^{H}\mathbf{u}_{\text{U},l}\\ &\quad+\sigma_{\text{U}}^{2}\mathbf{u}_{\text{U},l}^{H}\mathbf{I} \mathbf{u}_{\text{U},l}\\ &=\mathbf{u}_{\text{U},l}^{H}\Big{(}P_{\text{U}}\hat{\mathbf{h}}_{\text{U}} \hat{\mathbf{H}}_{\text{U}}^{H}+\mathbf{H}_{\text{SI}}\mathbf{F}^{H}\mathbf{ H}_{\text{SI}}^{H}+\sigma_{\text{U}}^{2}\mathbf{I}\Big{)}\mathbf{u}_{\text{U},l}-2 \sqrt{P_{\text{U}}}\operatorname{Re}\Big{\{}\mathbf{u}_{\text{U},l}^{H}\hat{ \mathbf{I}}_{\text{U},l}\Big{\}}+1,\\ &\quad\text{where}\ \hat{\mathbf{h}}_{\text{D},k}^{H}=\sqrt{\zeta_{\text{t}}} \mathbf{h}_{\text{D},k}^{H}\mathbf{\Phi}_{2}\mathbf{G}\mathbf{\Phi}_{1}\mathbf{ H}_{\text{TB}}\text{ represents the equivalent channel of the $k$-th DL UE for simplification and $\hat{\mathbf{H}}_{\text{D}}^{H}=\Big{[}\hat{\mathbf{h}}_{\text{D},1},\cdots, \hat{\mathbf{h}}_{\text{D},K_{\text{D}}}\Big{]}^{H}$. Similarly, $\hat{\mathbf{h}}_{\text{U},l}=\sqrt{\zeta_{\text{t}}}\mathbf{H}_{\text{BR}}^ {H}\mathbf{\Phi}_{1}\mathbf{G}^{H}\mathbf{\Phi}_{2}\mathbf{h}_{\text{U},l}$ represents the equivalent channel of the $l$-th UL UE for simplification and $\hat{\mathbf{H}}_{\text{U}}=\Big{[}\hat{\mathbf{h}}_{\text{U},1},\cdots, \hat{\mathbf{h}}_{\text{U},K_{\text{U}}}\Big{]}$. The proof of the equivalence between problems (13) and (14) can be found in Appendix A.