## 3 Beamforming Design for BIOS-FD Systems
### SIC-WMMSE Approach
\begin{split} f_{\text{SI}}&=\frac{\rho}{N_{\text{R}}} \Big{\lVert}\mathbf{H}_{\text{TR}}+\sqrt{\zeta_{\text{F}}}\mathbf{H}_{\text{ BR}}^{H}\mathbf{\Phi}_{1}\mathbf{H}_{\text{TB}}\Big{\rVert}_{F}^{2}\\ &=\frac{\rho}{N_{\text{R}}}\text{Tr}\Big{(}\Big{(}\mathbf{H}_{ \text{TR}}+\sqrt{\zeta_{\text{F}}}\mathbf{H}_{\text{BR}}^{H}\mathbf{\Phi}_{1} \mathbf{H}_{\text{TB}}\Big{)}\Big{(}\mathbf{H}_{\text{TR}}+\sqrt{\zeta_{\text{F}}} \mathbf{H}_{\text{BR}}^{H}\mathbf{\Phi}_{1}\mathbf{H}_{\text{TB}}\Big{)}^{H} \Big{)}\\ &=\frac{\rho}{N_{\text{R}}}\text{Tr}\Big{(}\mathbf{H}_{\text{TR}} \mathbf{H}_{\text{TR}}^{H}+\sqrt{\zeta_{\text{F}}}\mathbf{H}_{\text{TR}} \mathbf{H}_{\text{TB}}^{H}\mathbf{\Phi}_{1}^{H}\mathbf{H}_{\text{BR}}\] \[\quad+\sqrt{\zeta_{\text{F}}}\mathbf{H}_{\text{BR}}^{H}\mathbf{ \Phi}_{1}\mathbf{H}_{\text{TB}}\mathbf{H}_{\text{TR}}^{H}+\zeta_{\text{F}} \mathbf{H}_{\text{BR}}^{H}\mathbf{\Phi}_{1}\mathbf{H}_{\text{TB}}\mathbf{H}_{ \text{TB}}^{H}\mathbf{\Phi}_{1}^{H}\mathbf{H}_{\text{BR}}\Big{)}.\end{split}  \tag{22}\]  
Optimize \(\mathbf{A}_{\text{D}}\), \(\mathbf{A}_{\text{U}}\), \(\mathbf{U}_{\text{D}}\), \(\mathbf{U}_{\text{U}}\) with Fixed \(\mathbf{F}\), \(\mathbf{\Phi}_{1}\), \(\mathbf{\Phi}_{2}\)  
According to the WMMSE design approach [28] and the derivation in Appendix A, \(\mathbf{A}_{\text{D}}\), \(\mathbf{A}_{\text{U}}\), \(\mathbf{U}_{\text{D}}\), and \(\mathbf{U}_{\text{U}}\) all have closed-form solutions when other variables are fixed, which can be expressed as follows:  
\[a_{\text{D},k}{}^{\star}=e_{\text{D},k}^{-1}=\Big{(}u_{\text{D},k}^{*}u_{ \text{D},k}\hat{\mathbf{H}}_{\text{D},k}^{H}\mathbf{F}\mathbf{F}^{H}\hat{ \mathbf{h}}_{\text{D},k}-2\operatorname{Re}\Big{\{}u_{\text{D},k}^{*} \mathbf{\Phi}_{\text{D},k}^{H}\mathbf{f}_{k}\Big{\}}+\sigma_{\text{D},k}^{2}u_{ \text{D},k}^{*}u_{\text{D},k}+1\Big{)}^{-1}, \tag{23}\]  
\[u_{\text{D},k}{}^{\star}=\Big{(}\hat{\mathbf{h}}_{\\[a_{\text{U},l}{}^{\star}=e_{\text{U},l}^{-1}=\Big{(}\mathbf{u}_{\text{U},l}^{H} \Big{(}P_{\text{U}}\mathbf{\hat{H}}_{\text{U}}\mathbf{\hat{H}}_{\text{U}}^{H}+ \mathbf{H}_{\text{SI}}\mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\text{SI}}^{H}+\sigma _{\text{U}}^{2}\mathbf{I}\Big{)}\mathbf{u}_{\text{U},l}-2\sqrt{P_{\text{U}}} \operatorname{Re}\Big{\{}\mathbf{u}_{\text{U},l}^{H}\mathbf{\hat{h}}_{\text{U}, l}\Big{\}}+1\Big{)}^{-1}, \tag{25}\]  
\[\mathbf{u}_{\text{U},l}{}^{\star}=\sqrt{P_{\text{U}}}\Big{(}P_{\text{U}} \mathbf{\hat{H}}_{\text{U}}\mathbf{\hat{H}}_{\text{U}}^{H}+\mathbf{H}_{\text {SI}}\mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\text{SI}}^{H}+\sigma_{\text{U}}^{ 2}\mathbf{I}\Big{)}^{-1}\mathbf{\hat{h}}_{\text{U},l}. \tag{26}\]  
Optimize \(\mathbf{F}\) with Fixed \(\mathbf{A}_{\text{D}}\), \(\mathbf{A}_{\text{U}}\), \(\mathbf{U}_{\text{D}}\), \(\mathbf{U}_{\text{U}}\), \(\boldsymbol{\Phi}_{1}\), \(\boldsymbol{\Phi}_{2}\)  
When \(\mathbf{A}_{\text{D}}\), \(\mathbf{A}_{\text{U}}\), \(\mathbf{U}_{\text{D}}\), \(\mathbf{U}_{\text{U}}\), \(\boldsymbol{\Phi}_{1}\), and \(\boldsymbol{\Phi}_{2}\) are fixed, problem (18) is reduced into a subproblem for the optimization of \(\mathbf{F}\) with the power constraint, which can be rewritten as follows by ignoring constant terms:  
\[\min_{\mathbf{F}} f(\mathbf{F})=\operatorname{Tr}\Big{(}\mathbf{U}_{\text{U}}\mathbf{A}_{ \text{U}}\mathbf{U}_{\text{U}}^{H}\mathbf{H}_{\text{SI}}\mathbf{F}\mathbf{F} ^{H}\mathbf{H}_{\text{SI}}^{H}\Big{)}+\operatorname{Tr}\Big{(}\mathbf{F}^{H }\mathbf{\hat{H}}_{\text{D}}\mathbf{A}_{\text{D}}\mathbf{U}_{\text{D}}^{H} \mathbf{U}_{\text{D}}\mathbf{\hat{H}}_{\text{D}}^{H}\mathbf{F}\Big{)}\] \[-2\operatorname{Re}\Big{(}\operatorname{Tr}\Big{(}\mathbf{A}_{ \text{D}}\mathbf{U}_{\text{D}}^{H}\mathbf{H}_{\text{D}}^{H}\mathbf{F}\Big{)} \Big{)}+\frac{\rho}{N_{\text{R}}}\operatorname{Tr}\Big{(}\mathbf{H}_{\text{SI }}\mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\text{SI}}^{H}\Big{)}\] (27) s.t. \[\operatorname{Tr}\Big{(}\mathbf{F}^{H}\mathbf{F}\Big{)}\leq P_{ \text{s}}.\]  
It can be observed that sub-problem (27) is convex.