# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Overview
\begin{table}
\begin{tabular}{c l c l} \hline
**Notation** & **Definition** & **Notation** & **Definition** \\ \hline \(N_{t}\) & Number of antennas at the source & \(\mathbf{\Theta}\) & IRS phase profile \\ \hline \(N_{r}\) & Number of antennas at the destination & \(x_{s}\), \(x_{r}\) & Source and relay transmit signals \\ \hline \(K_{t}\) & Number of antennas at the relay’s & \(n_{r}\), \(n_{d}\) & Additive noise at the relay and destination \\ \hline \(K_{r}\) & Number of antennas at the relay’s receiver & \(y_{r}\), \(y_{d}\) & Received signals at the relay and destination, respectively \\ \hline \(M\) & Number of IRS elements & \(\kappa\) & FD/HD mode indicator \\ \hline \(\mathbf{H}_{r}\), \(\mathbf{\hat{H}}_{r}\), \(\mathbf{\hat{H}}_{r}\) & Self-interference actual channel, estimated channel and channel error, respectively & \(R_{\text{sr}}^{\text{FD}}\), \(R_{\text{rd}}^{\text{FD}}\), \(R^{\text{FD}}\) & Source–relay, relay–destination and the overall throughput in FD mode \\ \hline \(\mathbf{H}_{1}\), \(\mathbf{\hat{H}}_{1}\), \(\mathbf{\hat{H}}_{1}\) & Source–Relay actual channel, estimated channel and channel error, respectively & \(R_{\text{sr}}^{\text{HD}}\), \(R_{\text{rd}}^{\text{HD}}\), \(R^{\text{HD}}\) & Source–relay, relay–destination and the overall throughput in HD mode \\ \hline \(\mathbf{H}_{2}\), \(\mathbf{\hat{H}}_{2}\), \(\mathbf{\hat{H}}_{2}\) & Relay–Destination actual channel, estimated channel and channel error, respectively & \(\mathbf{Q}_{s}\), \(\mathbf{Q}_{r}\) & Source and relay covarriance matrices \\ \hline \(\mathbf{H}_{SI}\), \(\mathbf{\hat{H}}_{SI}\), \(\mathbf{\hat{H}}_{SI}\) & Source–IRS actual channel, estimated channel and channel error, respectively & \(\mathbf{H}_{tot_{1}}\) & \(\mathbf{H}_{tot_{1}}=(\mathbf{\hat{H}}_{r}+\mathbf{H}_{R}\mathbf{\Theta} \mathbf{\hat{H}}_{IR})\) \\ \hline \(\mathbf{H}_{IR}\), \(\mathbf{\hat{H}}_{IR}\), \(\mathbf{\hat{H}}_{IR}\), \(\mathbf{\hat{H}}_{IR}\) & IRS–Destination actual channel, estimated channel and channel error, respectively & \(\mathbf{\hat{H}}_{tot_{2}}\), \(\mathbf{\hat{H}}_{tot_{2}}\) & \(\mathbf{\hat{H}}_{tot_{2}}=(\mathbf{\hat{H}}_{1}+\mathbf{H}_{R}\mathbf{\Theta} \mathbf{\hat{H}}_{IR})\) \\ \hline \(\mathbf{H}_{RI}\), \(\mathbf{\hat{H}}_{RI}\), \(\mathbf{\hat{H}}_{RI}\) & Relay–IRS actual channel, estimated channel and channel error, respectively & \(\mathbf{\hat{H}}_{tot_{3}}\), \(\mathbf{\hat{H}}_{tot_{3}}=(\mathbf{\hat{H}}_{2}+\mathbf{H}_{R}\mathbf{\Theta }\mathbf{\hat{H}}_{IR})\) \\ \hline \(\mathbf{H}_{IR}\), \(\mathbf{\hat{H}}_{IR}\), \(\mathbf{\hat{H}}_{IR}\) & IRS–Relay actual channel, estimated channel and channel error, respectively & \(\lambda_{i}(X)\) & \(i\)’th largest eigenvalue of matrix \(X\) \\ \hline \(\gamma_{s_{i}}\) & \(\gamma_{s_{i}}=\lambda_{i}(\mathbf{Q}_{s})\) & \(\sigma_{1_{i}}^{2}=\lambda_{i}(\mathbf{H}_{1}\mathbf{H}_{1}^{H})\) \\ \hline \(\gamma_{r_{i}}\) & \(\gamma_{r_{i}}=\lambda_{i}(\mathbf{Q}_{r})\) & \(\sigma_{r_{i}}^{2}\) & \(\sigma_{r_{i}}^{2}=\lambda_{i}(\mathbf{\hat{H}}_{tot_{1}}\mathbf{\hat{H}}_{ tot_{1}}^{H})\) \\ \hline \(T_{r}\) & The RSI channel uncertainty bound & \(\sigma_{2_{i}}^{2}\) & \(\sigma_{2_{i}}^{2}=\lambda_{i}(\mathbf{H}_{2}\mathbf{H}_{2}^{H})\) \\ \hline \(T_{x}\) & Channel estimation error bound & \(T^{\prime}\) & The remaining RSI channel uncertainty \\ \(x\in\{1,2,SI,IR,RI,ID\}\) & & \(\mbox{after considering the impact of the IRS}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters.