# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries  
Considering a FD DF relay, the following rates are achievable [37],  
\[R^{\text{FD}}=\min(R^{\text{FD}}_{\text{sr}},R^{\text{FD}}_{\text{rd}}), \tag{6}\]  
in which, depending on how the IRS is applied to the system, the three following sets of rates are possible. First,  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\] \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]  
where \(\mathbf{H}_{tot_{1}}=(\mathbf{\hat{H}}_{r}+\mathbf{H}_{RI}\mathbf{\Theta} \mathbf{H}_{IR})\) when the IRS is used to cancel the self interference. Second,  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\] \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]  
where \(\mathbf{\hat{H}}_{tot_{2}}=(\mathbf{\hat{H}}_{1}+\mathbf{\hat{H}}_{RI}\mathbf{ \Theta}\mathbf{\hat{H}}_{IR})\) and \(\mathbf{\hat{H}}_{tot_{2}}=(\mathbf{\hat{H}}_{1}+\mathbf{\hat{H}}_{RI}\mathbf{ \Theta}\mathbf{\hat{H}}_{IR})\) if the IRS is established to help the source-relay channel and finally  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1} \mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r} \mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+ \mathbf{\hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{ r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{11}\] \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]  
where \(\mathbf{\hat{H}}_{tot_{3}}=\left(\mathbf{\hat{H}}_{2}+\mathbf{\hat{H}}_{RI} \mathbf{\Theta}\mathbf{\hat{H}}_{IR}\right)\) and \(\mathbf{\hat{H}}_{tot_{3}}=(\mathbf{\hat{H}}_{2}+\mathbf{\hat{H}}_{RI}\mathbf{ \Theta}\mathbf{\hat{H}}_{IR})\) if the IRS is utilized to enhance the rate of the relay-destination channel.