# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
The hexagon and the circle are the mapping output of the cube and the sphere respectively. Note that in (21), the inequalities hold with equality if and only if \(\mathbf{A}\) and \(\mathbf{B}\) are diagonalizable over a common basis. Using the result of Lemma 1, \(R_{\text{sr}}^{\text{FD}}\) can be lower-bounded as  
\[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]5 \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]6  
In addition, it holds that \(\lambda_{i}\!\left(\!\sigma_{t}^{2}\mathbf{I}_{\mathbf{K}_{r}}\!+\!\tilde{ \mathbf{H}}_{1}\mathbf{Q}_{s}\!\tilde{\mathbf{H}}_{1}^{H}\!+\!\mathbf{H}_{tot} \mathbf{Q}_{r}\mathbf{H}_{tot_{1}}^{H}\right)=\sigma_{t}^{2}+\lambda_{i}\! \left(\!\tilde{\mathbf{H}}_{1}\mathbf{Q}_{s}\!\tilde{\mathbf{H}}_{1}^{H}\!+ \!\mathbf{H}_{tot_{1}}\!\mathbf{Q}_{r}\!\mathbf{H}_{tot_{1}}^{H}\right)\). Hence, we obtain  
\[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]7 \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]8  
Note that the inequality holds with equality whenever \(\mathbf{H}_{1}\mathbf{Q}_{s}\mathbf{H}_{1}^{H}\) and \(\tilde{\mathbf{H}}_{1}\mathbf{Q}_{s}\!\tilde{\mathbf{H}}_{1}^{H}\!+\!\mathbf{ H}_{tot}\!\mathbf{Q}_{r}\!\mathbf{H}_{tot_{1}}^{H}\) share a common basis. Next, we use the following inequality  
\[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]9 \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1} \mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r} \mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+ \mathbf{\hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{ r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{11}\]0  
The above inequality holds true since \(\lambda_{i}\!\left(\tilde{\mathbf{H}}_{1}\mathbf{Q}_{s}\tilde{\mathbf{H}}_{1} ^{H}\right)\leq T_{1}P_{s}\). Now, instead of completing the minimization over the left-hand side (LHS) of Equation (26), we can first minimize the right-hand side (RHS) of Equation (27) to find an achievable rate. Similarly, for \(R_{\text{rd}}^{\text{FD}}\) we have  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1} \mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r} \mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+ \mathbf{\hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{ r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{11}\]1  
**Remark 1**.