## 2 System Model and Problem Formulation
### System Model  
Consider a narrowband multi-input single-output (MISO) FD system comprising a BS, a BIOS, \(K_{\text{D}}\) DL UEs, and \(K_{\text{U}}\) UL UEs in Figure 1. We assume that all the UEs are single-antenna, \(\text{BS}_{\text{Tx}}\) employs \(N_{\text{T}}\) transmit antennas, and \(\text{BS}_{\text{Rx}}\) has \(N_{\text{R}}\) receive antennas. The BIOS is deployed significantly close to the BS to ensure the reflected signals are strong enough to efficiently suppress the SI signals, so we assume that there is no line-of-sight (LoS) path between the BS and UEs [28].  
For this FD system, the DL signal from the BS is denoted as \(\mathbf{s_{\text{D}}}=\mathbf{F}\mathbf{x_{\text{D}}}\) with \(\mathbb{E}\{|\mathbf{s_{\text{D}}}|^{2}\}=P_{\text{s}}\). Similarly, \(\mathbf{F}=[\mathbf{f_{1}},\cdots,\mathbf{f_{K_{\text{D}}}}]\in\mathbb{C}^{N_{\text{T}} \times K_{\text{D}}}\) denotes the precoder at the BS, \(\mathbf{x_{\text{D}}}=[x_{\text{D},1},\cdots,\)\(x_{\text{D},K_{\text{D}}}]^{T}\in\mathbb{C}^{K_{\text{D}}\times 1}\) denotes the original symbol vector, and \(x_{\text{U},l}\) is the symbol transmitted by the \(l\)-th UL UE with the power constraint \(\mathbb{E}\{|\mathbf{x_{\text{U},l}}|^{2}\}=P_{\text{U}}\). The channels between the \(\text{BS}_{\text{Tx}}\)-\(\text{IOS}_{1}\), \(\text{BS}_{\text{Tx}}\)-\(\text{BS}_{\text{Rx}}\), \(\text{BS}_{\text{Rx}}\)-\(\text{IOS}_{1}\), \(\text{IOS}_{2}\)-\(\text{UE}_{\text{D},k}\), and \(\text{IOS}_{2}\)-\(\text{UE}_{\text{U},l}\) are denoted, respectively, as \(\mathbf{H_{\text{TB}}}\in\mathbb{C}^{M\times N_{\text{T}}}\), \(\mathbf{H_{\text{TR}}}\in\mathbb{C}^{N_{\text{R}}\times N_{\text{T}}}\), \(\mathbf{H_{\text{BR}}}\in\mathbb{C}^{M\times N_{\text{R}}}\), \(\mathbf{h_{\text{D},k}}\in\mathbb{C}^{M\times 1}\), and \(\mathbf{h_{\text{U}}}\in\mathbb{C}^{M\times 1}\). Therefore, the DL received signal at the \(k\)-th DL UE and the UL received signal at the \(\text{BS}_{\text{Rx}}\) can be respectively represented as follows:  
Figure 1: System model of the BIOS-assisted FD system.  
\[y_{\text{D},k}=\sqrt{\xi_{\text{t}}}\mathbf{h}_{\text{D},k}^{H}\mathbf{\Phi}_{2} \mathbf{G}\mathbf{\Phi}_{1}\mathbf{H}_{\text{TB}}\mathbf{f}_{k}x_{\text{D},k}+ \underbrace{\sum_{i=1,i\
eq k}^{K_{\text{D}}}\sqrt{\xi_{\text{t}}}\mathbf{h}_{ \text{D},k}^{H}\mathbf{\Phi}_{2}\mathbf{G}\mathbf{\Phi}_{1}\mathbf{H}_{\text{TB}} \mathbf{f}_{i}x_{\text{D},i}}_{n_{\text{D},k}^{\text{MU}}}+n_{\text{D},k}, \tag{2}\]  
\[\mathbf{y}_{\text{U}}=\sqrt{\xi_{\text{t}}}\mathbf{H}_{\text{BR}}^{H}\mathbf{\Phi} _{1}\mathbf{G}^{H}\mathbf{\Phi}_{2}\mathbf{h}_{\text{U},l}x_{\text{U},l}+ \underbrace{\sum_{i=1,i\
eq l}^{K_{\text{U}}}\sqrt{\xi_{\text{t}}}\mathbf{H}_{ \text{BR}}^{H}\mathbf{\Phi}_{1}\mathbf{G}^{H}\mathbf{\Phi}_{2}\mathbf{h}_{\text{U},l}x _{\text{U},l}}_{n_{\text{U},l}^{\text{MU}}}+\underbrace{\left(\mathbf{H}_{ \text{TR}}+\sqrt{\xi_{\text{r}}}\mathbf{H}_{\text{BR}}^{H}\mathbf{\Phi}_{1} \mathbf{H}_{\text{TB}}\right)\sum_{i=1}^{K_{\text{D}}}\mathbf{f}_{i}x_{\text{ D},i}}_{n_{\text{U},l}^{\text{MU}}}+\mathbf{n}_{\text{U}}, \tag{3}\]  
where \(n_{\text{D},k}^{\text{MU}}\) represents the interference from other DL UEs for the \(k\)-th DL UE and \(\mathbf{n}_{\text{U},l}^{\text{MU}}\) represents the interference from other UL UEs for the \(l\)-th UL UE. \(\mathbf{n}_{\text{U},l}^{\text{SI}}\) is the SI at the BS. \(n_{\text{D},k}\) denotes the Gaussian noise at the \(k\)-th DL UE with mean zero and variance \(\sigma_{\text{D}}^{2}\), and \(\mathbf{n}_{\text{U}}\) denotes the noise at the BS satisfying the distribution of \(\mathbf{n}_{\text{U}}\sim\mathcal{CN}\big{(}\mathbf{0},\sigma_{\text{U}}^{2} \mathbf{I}\big{)}\).