## 3 Beamforming Design for BIOS-FD Systems
### SIC-WMMSE Approach
Thus, with other parameters fixed, the optimal \(\mathbf{F}\) can be obtained by utilizing the Karush-Kuhn-Tucker (KKT) conditions as  
\[\mathbf{F}^{\star}=\mathbf{\hat{H}}_{\text{D}}\mathbf{A}_{\text{D}}\mathbf{U}_ {\text{D}}, \tag{28}\]  
where \(\mathbf{\hat{F}}=(\mathbf{H}_{\text{SI}}^{H}\mathbf{U}_{\text{U}}\mathbf{A}_ {\text{U}}\mathbf{U}_{\text{U}}^{H}\mathbf{H}_{\text{SI}}+\mathbf{\hat{H}}_{ \text{D}}\mathbf{A}_{\text{D}}\mathbf{U}_{\text{D}}^{H}\mathbf{U}_{\text{D}} \mathbf{\hat{H}}_{\text{D}}^{H}+\frac{\rho}{N_{\text{R}}}\mathbf{H}_{\text{SI }}^{H}\mathbf{H}_{\text{SI}}+\delta\mathbf{I})^{-1}\). \(\delta\) is the KKT coefficient whose optimal solution can be obtained through a one-dimensional search based on the power constraint. The follow two subsections focus on the optimization of BIOS, i.e., \(\boldsymbol{\Phi}_{1}\), \(\boldsymbol{\Phi}_{2}\). Optimize \(\boldsymbol{\Phi}_{1}\) with Fixed \(\mathbf{F}\), \(\boldsymbol{\Phi}_{2}\), \(\mathbf{A}_{\text{D}}\), \(\mathbf{A}_{\text{U}}\), \(\mathbf{U}_{\text{D}}\), \(\mathbf{U}_{\text{U}}\)  
In this subsection, we will optimize \(\boldsymbol{\Phi}_{1}\) by fixing other variables. By further defining \(\mathbf{H}_{\text{D}}=\big{[}\mathbf{h}_{\text{D},1},\cdots,\mathbf{h}_{\text {D},K_{\text{D}}}\big{]}\), \(\mathbf{H}_{\text{U}}=\big{[}\mathbf{h}_{\text{U},1},\cdots,\mathbf{\hat{h}}_{ \text{U},K_{\text{U}}}\big{]}\), \(\mathbf{\overline{H}}=\mathbf{H}_{\text{U}}^{H}\boldsymbol{\Phi}_{2}^{H} \mathbf{G}^{H}\), \(\mathbf{\overline{F}}=\mathbf{H}_{\text{TB}}\mathbf{F}\), \(\mathbf{H}_{\text{fle}}=\mathbf{H}_{\text{BR}}\mathbf{U}_{\text{U}}\mathbf{A}_{ \text{U}}\mathbf{U}_{\text{U}}^{H}\mathbf{H}_{\text{BR}}^{H}\), and \(\mathbf{H}_{\text{fra}}=\mathbf{G}^{H}\boldsymbol{\Phi}_{2}^{H}\mathbf{H}_{ \text{D}}\mathbf{U}_{\text{D}}\mathbf{U}_{\text{D}}\mathbf{U}_{\text{D}} \mathbf{U}_{\text{D}}^{H}\mathbf{H}_{\text{D}}^{H}\boldsymbol{\Phi}_{2}\mathbf{G}\), the objective function in (18) related to \(\boldsymbol{\Phi}_{1}\) can be simplified as follows:  
\[\min_{\boldsymbol{\Phi}_{1}}f_{1}(\boldsymbol{\Phi}_{1})= 2\operatorname{Re}\Big{(}\operatorname{Tr}\Big{(}\boldsymbol{\Phi}_{1} ^{H}\Big{(}\sqrt{\xi_{\text{r}}}\mathbf{H}_{\text{BR}}\overline{\mathbf{U}}_{ \text{U}}\mathbf{H}_{\text{TR}}\mathbf{F}\overline{\mathbf{F}}^{H}-\sqrt{\xi_{ \text{r}}P_{\text{U}}}\mathbf{H}_{\text{BR}}\mathbf{U}_{\text{U}}\mathbf{A}_{ \text{U}}\overline{\mathbf{H}} \tag{29}\] \[-\sqrt{\xi_{\text{r}}}\mathbf{G}^{H}\boldsymbol{\Phi}_{2}^{H} \overline{\mathbf{U}}_{\text{D}}\overline{\mathbf{F}}^{H}+\frac{\rho}{N_{\text{R}}} \sqrt{\xi_{\text{r}}}\mathbf{H}_{\text{BR}}\mathbf{H}_{\text{TR}}\mathbf{H}_{ \text{TB}}^{H}\Big{)}\Big{)}\Big{)}\Big{)}\] \[+\operatorname{Tr}\Big{(}\xi_{\text{r}}P_{\text{U}}\boldsymbol{ \Phi}_{1}^{H}\mathbf{H}_{\text{fle}}\boldsymbol{\Phi}_{1}\overline{\mathbf{F}}^{H} \overline{\mathbf{H}}\Big{)}+\operatorname{Tr}\Big{(}\xi_{\text{r}} \boldsymbol{\Phi}_{1}^{H}\mathbf{H}_{\text{fle}}\boldsymbol{\Phi}_{1}\overline{ \mathbf{F}}\overline{\mathbf{F}}^{H}\Big{)}\] \[u_{\text{D},k}{}^{\star}=\Big{(}\hat{\mathbf{h}}_{\\[a_{\text{U},l}{}^{\star}=e_{\text{U},l}^{-1}=\Big{(}\mathbf{u}_{\text{U},l}^{H} \Big{(}P_{\text{U}}\mathbf{\hat{H}}_{\text{U}}\mathbf{\hat{H}}_{\text{U}}^{H}+ \mathbf{H}_{\text{SI}}\mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\text{SI}}^{H}+\sigma _{\text{U}}^{2}\mathbf{I}\Big{)}\mathbf{u}_{\text{U},l}-2\sqrt{P_{\text{U}}} \operatorname{Re}\Big{\{}\mathbf{u}_{\text{U},l}^{H}\mathbf{\hat{h}}_{\text{U}, l}\Big{\}}+1\Big{)}^{-1}, \tag{25}\]0 s.t.