# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
In (14), first, the RSI does its worst damage on the performance of the best IRS configuration and after that performs another optimization to bring the worst power allocation against the best covariance matrices (This will be clearer later on when the geometrical representation of the problem is given). In what follows, we provide our proposed ways to deal with optimization problems (14) and (15), respectively. Theorem 1: _For the optimization problem (14), one can show that \(T^{\prime}(T,\Theta)\leq(\sqrt{T_{r}}-\sigma_{\min}(\mathbf{H}_{IR}*\mathbf{ H}_{RI}^{T}))^{2}\)._Proof. : We begin the proof with an intuitive example and then extend it to the more general case. Assume that \(K_{t}=1,K_{r}=2\) and \(M=3\). Then we have  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]1 (18) subject to \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]2 \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]3  
In addition, consider the following optimization problem  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]4 (19) subject to \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]5 \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]6  
Here, notice that \(\mathbf{H}_{IR}diag(\mathbf{H}_{RI}^{T})\) is a linear map from a three-dimensional into a two-dimensional space.