## 4 Analysis of BIOS Size  
To further guide the practical deployment of BIOS-assisted FD systems, a lower bound of BIOS size required for efficient SIC is derived here for the single-input single-output (SISO) scenario. Specifically, the channels between the \(\text{BS}_{\text{TX}}\text{-BS}_{\text{Rx}}\), \(\text{BS}_{\text{TX}}\text{-IOS}_{1}\), and \(\text{BS}_{\text{Rx}}\text{-IOS}_{1}\) are denoted as \(h_{\text{TR}}\), \(\mathbf{h}_{\text{TB}}\), and \(\mathbf{h}_{\text{BR}}\), respectively. Theorem 4.1: _To achieve efficient SIC, the number of elements in one layer of the BIOS must exceed a threshold \(M\) that satisfies the following inequality:_  
\[\sum_{m=1}^{M}\left(\frac{1}{d_{m}^{\text{BR}}d_{m}^{\text{TB}}}\right)^{\frac{ 3}{2}}\geq\frac{2\sqrt{\pi}}{\sqrt{\xi_{\rm r}}A_{\text{BB}}d_{\text{TR}}}, \tag{37}\]  
_where \(d_{m}^{\text{BR}},d_{m}^{\text{TB}}\), respectively, denote the distances from the \(m\)-th element of the \(\text{IOS}_{1}\) to the \(\text{BS}_{\text{Rx}}\) and \(\text{BS}_{\text{Tx}}\), and \(\text{d}_{\text{BB}}\) and \(\text{d}_{\text{TR}}\) denote the distance between \(\text{BS}\text{-BIOS}\) and \(\text{BS}_{\text{Rx}}\text{-BS}_{\text{Tx}}\)._  
Proof of Theorem 4.1.: To ensure effective SIC in the SISO scenario, the residual SI should satisfy the following power constraint:  
\[\|(h_{\text{TR}}+\sqrt{\xi_{\rm r}}\mathbf{h}_{\text{BR}}^{H}\mathbf{\Phi}_{1}\mathbf{h}_{ \text{TB}})_{S}\|_{F}^{2}\leq\gamma. \tag{38}\]Notably, for effective SIC, the threshold \(\gamma\) should be minimized, and the power constraint (38) should be universally valid for any \(s_{\text{D}}\). This requires that the residual interference approaches zero, which can be expressed as  
\[\left(h_{\text{TR}}+\sqrt{\xi_{\text{r}}}\mathbf{h}_{\text{BR}}^{H}\mathbf{\phi}_{1} \mathbf{h}_{\text{TB}}\right)\Rightarrow 0. \tag{39}\]  
To clarify, the necessary condition for efficient SIC is that the modulus of the cascaded channel \(\sqrt{\xi_{\text{r}}}\mathbf{h}_{\text{BR}}^{H}\mathbf{\phi}_{1}\mathbf{h}_{\text{TB}}\) should exceed that of the SI channel \(h_{\text{TR}}\), which can be formulated as follows:  
\[\left|\sqrt{\xi_{\text{r}}}\mathbf{h}_{\text{BR}}^{H}\mathbf{\phi}\mathbf{h}_{ \text{TB}}\right|^{2}\geq\left|h_{\text{TR}}\right|^{2}. \tag{40}\]  
We define \(g(M)\) as the maximum gain of the cascade channel generated by BIOS with \(M\) scattering elements in one layer, which is given by  
\[g(M)=\max_{\mathbf{\phi}}\left|\xi_{\text{r}}\mathbf{h}_{\text{BR}}^{H}\mathbf{\phi} \mathbf{h}_{\text{TB}}\right|^{2}=\max_{\mathbf{\phi}}\xi_{\text{r}}\left|\sum_{ m=1}^{M}\left[\mathbf{h}_{\text{BR}}^{H}\right]_{m}[\mathbf{\phi}]_{m}[\mathbf{\eta}]_{m }[\mathbf{\eta}]_{m}\right|^{2} \tag{41}\]  
It can be observed that maximizing the cascade channel gain occurs when the phase shift element \([\mathbf{\phi}]_{m}\) is aligned in phase with \(\left[\mathbf{h}_{\text{BR}}^{H}\right]_{m}[\mathbf{h}_{\text{TB}}]_{m}\). Thus, by setting \([\mathbf{\phi}_{1}]_{m}=\frac{\left|\mathbf{h}_{\text{BR}}\right|_{m}\mathbf{h}_{ \text{TB}}^{H}}{\left|\left|\mathbf{h}_{\text{BR}}^{H}\right|_{m}\right|^{2}}\), \(g(M)\) has an evident closed-form solution as  
\[g(M)=\xi_{\text{r}}\|\mathbf{h}_{\text{BR}}\circ\mathbf{h}_{\text{TB}}\|_{1}^ {2}=\xi_{\text{r}}\left(\sum_{m=1}^{M}\left|[\mathbf{h}_{\text{BR}}^{*}]_{m}[ \mathbf{h}_{\text{TB}}]_{m}\right|\right)^{2}. \tag{42}\]  
By substituting the channel settings, we have  
\[g(M)=\frac{\lambda^{2}d_{\text{BB}}}{8\pi}\sum_{m=1}^{M}(\frac{1}{d_{m}^{ \text{BR}}d_{m}^{\text{TB}}})^{\frac{3}{2}}. \tag{43}\]  
\[\left|h_{\text{TR}}\right|=\frac{\lambda}{4\sqrt{\pi}d_{\text{TR}}}. \tag{44}\]  
According to the derivations mentioned above, the lower bound of the element number can be derived through the inequality constraint  
\[\frac{\lambda^{2}d_{\text{BB}}}{8\pi}\sum_{m=1}^{M}(\frac{1}{d_{m}^{\text{BR} }d_{m}^{\text{TB}}})^{\frac{3}{2}}\geq\frac{\lambda}{4\sqrt{\pi}d_{\text{TR}}}. \tag{45}\]  
The proof is thus completed.