## 5 Simulation Results  
This section presents numerical results to illustrate the performance evaluation of the proposed BIOS-assisted FD system. For comparison, an IOS-assisted FD system is employed as a benchmark, which is characterized by identical phase matrices for reflecting and refracting with \(\xi_{\text{r}}=\xi_{\text{t}}=1/2\). Throughout simulations, the centers of the BS and BIOS are set at the origin of a three-dimensional coordinate system. The distance between two neighboring IOSs of the BIOS is set as \(d_{\text{II}}=0.05\) m. The DL and UL UEs are randomly distributed within two respective circles of 3m radius centered at \((20,-10,0)\) and \((10,10,0)\). Considering the NLoS paths, \(\mathbf{h}_{\text{D},k}\) and \(\mathbf{h}_{\text{U},l}\) are modeled as Rician channels with a Ricianfactor \(\kappa=4\). Moreover, \(\alpha_{0}=10^{-4}\) is the pass loss of the reference distance \(d_{0}=1\) m with the path loss exponent \(\beta=2.2\). Furthermore, \(\lambda=0.05\) m and \(\sigma_{\rm U}^{2}=\sigma_{\rm D}^{2}=-80\) dBm. Figure 2 shows the SR performance as a function of the BIOS/IOS size for different designs in a multi-user MISO system with \(N_{\rm T}=N_{\rm R}=3\) and \(K_{\rm D}=K_{\rm U}=2\). Specifically, the solid lines illustrate the SR of FD systems, and the dashed lines indicate the rate of HD systems, with blue for BIOS assistance and red for IOS assistance, respectively. Meanwhile, as the BIOS may be too small to enable effective SIC, we switch the system from the FD mode to the HD one with only the DL transmission when the SIC constraint cannot be satisfied, which is indicated by the shaded area. It is worth noting that as the channels are randomly generated, the shaded area means that in most of the simulation realizations, the system has to switch to the HD mode as the size of the BIOS/IOS is too small to achieve the SIC threshold. It can be observed that for all systems, the data rate monotonically increases with \(M_{y}\) for both BIOS and IOS due to the enhanced beamforming gains. For FD systems, as \(M_{y}\) increases, the SR of BIOS-assisted systems significantly outperforms that of HD systems. However, the IOS-assisted FD system demonstrates only a slight enhancement over HD systems. This is because the IOS beams for SIC and SRM are tightly coupled, limiting the system's efficiency. Specifically, if the IOS is designed to cancel the SI signal on the reflecting side, it cannot be efficiently optimized for SRM on the refracting side. Conversely, the BIOS-assisted FD system can provide flexible beamforming on both sides, thereby enhancing communication performance and achieving SIC simultaneously. To further investigate the beamforming performance of the BIOS and IOS in FD systems, Figure 3 shows the detailed UL and DL rates of the BIOS/IOS-assisted system with different \(M\). As can be observed, the system switches from the FD mode to the HD mode with \(M_{y}<9\). In this range, the data rate, i.e., the DL rate, consistently increases with \(M\). This is because the expansion of the array scale provides higher degrees of freedom, thus enhancing the beamforming gain. However, when \(M_{y}\) changes from 8 to 9, the DL rate unexpectedly decreases while the UL rate substantially increases. This is because the power of the cascaded signal generated by the BIOS/IOS is strong enough to facilitate effective SIC, thus leading to a trade-off between the SRM and SIC for beamforming. It is worth noting that, compared with the IOS-assisted system, the DL rate in the BIOS-assisted system is less affected due to the independent beamforming on both sides. Consequently, the SR performance of the BIOS-assisted FD system significantly surpasses that of the IOS-assisted system. Furthermore, Figure 4 shows the performance comparison between the proposed BIOS and an IOS with the same scattering elements.