# RIS-Assisted Cooperative Time-Division Multiple Access
## 2 RIS-Assisted Cooperative Time-Division Multiple Access
### System Model
In a non-cooperative time slot, the received signal of the UE is given as  
\[\mathbf{y}_{n}=\sqrt{P}\mathbf{h}\mathbf{w}_{n}x+n, \tag{1}\]  
where \(P\) is the total transmit power, \(x\) denotes the transmit symbol of the UE, \(\mathbb{E}\left[|x|^{2}\right]=1\), \(\mathbf{h}(\in\mathbb{C}^{1\times M})\) is the channel vector from BS to the UE, \(\mathbf{w}_{n}(\in\mathbb{C}^{M\times 1})\) is the transmit beamforming vector for transmitting \(x\), and \(n\) is the additive white Gaussian noise with zero mean and variance \(N_{0}\). The received SINR and capacity in the non-cooperative time slot are then given as  
\[\mathrm{SINR}_{n}=\rho\|\mathbf{h}\mathbf{w}_{n}\|^{2}, \tag{2}\] \[\mathbf{C}_{n}=log_{2}(1+\mathrm{SINR}_{n}), \tag{3}\]  
where \(\rho\) is the transmit SNR, i.e., \(\rho=\frac{P}{N_{0}}\). In a cooperative time slot, the received signal of the weak-channel-conditioned UE is given as  
\[\mathbf{y}_{c,w}=\underbrace{\sqrt{P_{w}}\mathbf{g}_{w}\boldsymbol{\Phi} \mathbf{H}\mathbf{w}_{c,w}x_{w}+\sqrt{P_{w}}\mathbf{h}_{w}\mathbf{w}_{c,w}x_{ w}}_{desired\ signal\ term}+\underbrace{\sqrt{P_{s}}\mathbf{g}_{w}\boldsymbol{\Phi} \mathbf{H}\mathbf{w}_{c,s}x_{s}+\sqrt{P_{s}}\mathbf{h}_{w}\mathbf{w}_{c,s}x_{ s}}_{inter\,f\,rence\,signal\ term}  
for transmitting \(x_{s}\) in the cooperative time slot. The received SINR of the weak-channel-conditioned UE in the cooperative time slot can be expressed as  
\[\text{SINR}_{c,w}=\frac{\rho_{w}\|(\mathbf{g}_{w}\mathbf{\Phi H}+\mathbf{h}_{w}) \mathbf{w}_{c,w}\|^{2}}{1+\rho_{s}\|(\mathbf{g}_{w}\mathbf{\Phi H}+\mathbf{h}_{ w})\mathbf{w}_{c,s}\|^{2}}, \tag{5}\]  
where \(\rho_{w}\) and \(\rho_{s}\) are, respectively, the transmit SNRs of the weak-channel-conditioned and strong-channel-conditioned UEs in the cooperative time slot, i.e., \(\rho_{w}=\frac{P_{w}}{N_{0}}\), \(\rho_{s}=\frac{P_{w}}{N_{0}}\), and \(\rho=\rho_{w}+\rho_{s}\). In a cooperative time slot, the received signal of the strong-channel-conditioned UE is given as  
\[\mathbf{y}_{c,s} = \underbrace{\sqrt{P_{s}}\mathbf{g}_{s}\mathbf{\Phi Hw}_{c,s}x_{s }+\sqrt{P_{s}}\mathbf{h}_{s}\mathbf{w}_{c,s}x_{s}}_{\text{{desired signal term}}}+\underbrace{\sqrt{P_{w}}\mathbf{g}_{s}\mathbf{\Phi Hw}_{c,w}x_{w}+ \sqrt{P_{w}}\mathbf{h}_{s}\mathbf{w}_{c,w}x_{w}}_{\text{{interference signal term}}}+n, \tag{6}\] \[= \sqrt{P_{s}}\mathbf{g}_{s}\mathbf{\Phi Hw}_{c,s}x_{s}+\sqrt{P_{s} }\mathbf{h}_{s}\mathbf{w}_{c,s}x_{s}, \tag{7}\]  
where \(\mathbf{g}_{s}(\in\mathbb{C}^{1\times N})\) is the channel vector from the RIS to the strong-channel-conditioned UE and \(\mathbf{h}_{s}(\in\mathbb{C}^{1\times M})\) is the channel vector from BS to the strong-channel-conditioned UE. As mentioned above, in the cooperative slots, data transmission for the weak-channel-conditioned UE must not interfere with the strong-channel-conditioned UE. Thus, the beamforming vector of \(\mathbf{w}_{c,w}\) should be designed to satisfy \((\mathbf{g}_{s}\mathbf{\Phi H}+\mathbf{h}_{s})\mathbf{w}_{c,w}=0\). Based on \(\mathbf{w}_{c,w}\) nullifying the interference signal term in Equation (6), Equation (7) is derived. The received SINR of the strong-channel-conditioned UE in the cooperative time slot can be then expressed as  
Figure 2.0