# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
Consequently, we show that the problem (13) can be simplified to the following optimization problem  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]3 (14) subject to \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]4 \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]5 \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]6 \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]7 (14a) where \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]8 (15) subject to \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]9 (15a) \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]0 (15b) and where \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]1 denotes the vector of all non-zero elements of its input matrix.