## 2 System Model and Problem Formulation  
### BIOS Structure  
As mentioned in Section 1, hardware limitations prevent IOS from independently controlling the phases of refracted and reflected signals, resulting in symmetric beams on both sides. However, in the proposed FD system, we aim to utilize the reflected signals to cancel SI and for the refracted signals to communicate. Thus, the symmetric beams will degrade the system performance. Specifically, the cascaded reflected signal, which is emitted from the \(\text{BS}_{\text{Tx}}\) into the IOS and then reflected to the \(\text{BS}_{\text{Rx}}\), serves exclusively for SIC while the refracted signals aim to maximize SR. Moreover, refraction beamforming must align DL signals with the DL UEs and enhance UL reception, complicating the optimization problem due to increased coupling and non-convexity. Consequently, performance trade-offs are necessary to find an optimal balance.  
To mitigate the constraints, we propose a BIOS-assisted FD system as shown in Figure 1, which can achieve simultaneous SIC and SRM by flexibly controlling the reflection and refraction beams on both sides of the BIOS [33; 34]. The BIOS consists of two neighboring IOSs (referred to as \(\text{IOS}_{1}\) and \(\text{IOS}_{2}\)). Situated proximately to the BS, \(\text{IOS}_{1}\) functions like a traditional IOS, managing both reflection and refraction, while \(\text{IOS}_{2}\) is set to the full-transmitting mode. With this arrangement, the reflected cascaded signals can be smartly designed by controlling the \(\text{IOS}_{1}\) to suppress SI signals. Furthermore, the SR can be maximized by independently controlling the reflection and refraction beams on both sides of the BIOS. Thus, the BIOS can achieve the dual-function target of SIC and SRM in FD systems better than the conventional RIS designs. Specifically, compared to traditional IRS,BIOS allows the BS to achieve greater service coverage by reflecting and refracting signals into the full space. Moreover, the capability of independent beamforming on both sides enables separate and effective management for SIC and SRM. Compared to single-layer IOS, the addition of \(\text{IOS}_{2}\) enables independent beams on both sides, allowing for more flexible adjustment. Additionally, compared to multilayer refractive RIS investigated in the previous studies [35; 36], the BIOS is distinguished by its specific dual function of simultaneous and decoupled reflection and refraction, which enhances its performance in FD systems.  
Without loss of generality, we assume that both \(\text{IOS}_{1}\) and \(\text{IOS}_{2}\) are uniform square planar arrays consisting of \(M=M_{y}\times M_{z}\) elements, where \(M_{y}=M_{z}\). Thus, the effective reflecting and refracting coefficient matrices of BIOS can be respectively presented as follows:  
\[\hat{\mathbf{\Phi}}_{\text{r}}=\sqrt{\xi_{\text{r}}}\mathbf{\Phi_{1}},\quad\hat{\mathbf{ \Phi}}_{\text{t}}=\sqrt{\epsilon_{\text{t}}\xi_{\text{t}}}\mathbf{\Phi_{2}}\mathbf{G }\mathbf{\Phi_{1}}, \tag{1}\]  
where \(\mathbf{\Phi_{1}}\in\mathbb{C}^{M\times M}\) and \(\mathbf{\Phi_{2}}\in\mathbb{C}^{M\times M}\) denote the diagonal coefficient matrices of \(\text{IOS}_{1}\) and \(\text{IOS}_{2}\). \(\xi_{\text{r}}\) and \(\xi_{\text{t}}\) are the respective fractions of reflection and refraction signal power relative to the total power of the incoming signal for \(\text{IOS}_{1}\), satisfying the power constraint \(\xi_{\text{r}}+\xi_{\text{t}}=1\). (Following the assumption in [32], we presume negligible penetration loss for IOS and BIOS. Nevertheless, the penetration loss will be an interesting and important issue that deserves further investigation.) Similarly, \(\epsilon_{\text{r}}\) and \(\epsilon_{\text{t}}\) denote those for \(\text{IOS}_{2}\) with \(\epsilon_{\text{r}}+\epsilon_{\text{t}}=1\). According to the setup, \(\epsilon_{\text{r}}=0\) and \(\epsilon_{\text{t}}=1\). \(\mathbf{G}\in\mathbb{C}^{M\times M}\) denotes the channel between \(\text{IOS}_{1}\) and \(\text{IOS}_{2}\).