# RIS-Assisted Cooperative Time-Division Multiple Access
## 2 RIS-Assisted Cooperative Time-Division Multiple Access
### Beamforming Vector Design  
In this section, for a given random phase shift matrix of RIS, \(\mathbf{\Phi}\), we determine and derive optimal transmit weight vectors, \(\mathbf{w}_{n}\), \(\mathbf{w}_{c,s}\), and \(\mathbf{w}_{c,w}\) to maximize \(\text{SINR}_{n}\), \(\text{SINR}_{c,s}\) and \(\text{SINR}_{c,w}\), respectively. Based on these analyses, the RC-TDMA algorithm is proposed to guarantee the required capacity of the weak-channel-conditioned UE as much as possible. To maximize the capacity of a weak-channel-conditioned UE in a non-cooperative time slot, i.e., Equation (3), the maximum ratio transmission (MRT) beamforming is adopted, which is given as  
\[\mathbf{w}_{n}=\frac{\mathbf{h}^{H}}{\|\mathbf{h}\|}. \tag{9}\]  
For a cooperative time slot, the transmit beamforming vector for the strong-channel-conditioned UE should be determined to maximize the SINR of Equation (8), which can be rewritten as  
\[\max_{\mathbf{w}_{c,s}}\text{SINR}_{c,s}=\max_{\mathbf{w}_{c,s}}\rho_{s} \mathbf{w}_{c,s}^{H}(\mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{s})^{H}( \mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{s})\mathbf{w}_{c,s}\quad\text{s.t. }\|\mathbf{w}_{c,s}\|^{2}=1. \tag{10}\]  
By applying the eigenvalue decomposition to Equation (10), the optimal \(\mathbf{w}_{c,s}\) for maximizing \(\text{SINR}_{c,s}\) is obtained by  
\[\mathbf{w}_{c,s}^{*}=\text{max. eigenvector of }(\mathbf{g}_{s}\mathbf{\Phi H}+ \mathbf{h}_{s})^{H}(\mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{s}). \tag{11}\]  
Note that the optimal \(\mathbf{w}_{c,s}\) is determined regardless of the value of \(\rho_{s}\) in Equation (11). Based on the optimal \(\mathbf{w}_{c,s}\), the minimum transmit SNR of \(\rho_{s}\) to satisfy the required SINR can be calculated as  
\[\rho_{s}^{*}=\frac{\gamma_{req}}{(\mathbf{w}_{c,s}^{*})^{H}(\mathbf{g}_{s} \mathbf{\Phi H}+\mathbf{h}_{s})^{H}(\mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{ s})\mathbf{w}_{c,s}^{*}}, \tag{12}\]where \(\gamma_{req}\) is the required SINR. The required capacity for each UE can then be given as \(\mathbf{C}_{req}=\text{log}_{2}(1+\gamma_{req})\). The reason for using the minimum \(\rho_{s}\) to meet the required SINR is to maximize the SINR of Equation (5) by allocating the transmit power to the weak-channel-conditioned UE as much as possible. Thus, the transmit SNR for the weak-channel-conditioned UE is given as  
\[\rho_{w}^{*}=\rho-\rho_{s}^{*}. \tag{13}\]  
In a cooperative time slot, the transmit beamforming vector for the weak-channel-conditioned UE should be designed to maximize the SINR of Equation (5) while guaranteeing \((\mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{s})\mathbf{w}_{c,w}=0\). Thus, we first design the transmit beamforming vector that maximizes the SINR of Equation (5), and the optimal transmit beamforming vector \(\mathbf{w}_{c,w}^{*}\) is then determined by projecting it onto the null space of \(\mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{s}\) so that \(\mathbf{w}_{c,w}^{*}\) can satisfy \((\mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{s})\mathbf{w}_{c,w}^{*}=0\). By applying \(\mathbf{w}_{c,s}^{*}\) determined in Equation (11), the SINR of Equation (5) can be rewritten as  
\[\text{SINR}_{c,w}=\frac{\rho_{w}\mathbf{w}_{c,w}^{H}(\mathbf{H}^{H}\mathbf{ \Phi}^{H}\mathbf{g}_{w}^{H}\mathbf{g}_{w}\mathbf{\Phi H}+\mathbf{h}_{w}^{H} \mathbf{h}_{w})\mathbf{w}_{c,w}}{1+\rho_{s}\|(\mathbf{g}_{w}\mathbf{\Phi H}+ \mathbf{h}_{w})\mathbf{w}_{c,s}^{*}\|^{2}}. \tag{14}\]  
Note that \(\frac{\rho_{w}}{1+\rho_{s}\|\mathbf{g}_{w}\mathbf{\Phi H}\mathbf{w}_{c,w}^{2} \|+\rho_{s}\|\mathbf{h}_{w}\mathbf{w}_{c,s}\|^{2}}\) is the determined value.