# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
Notice that one can readily extend this interpretation into the complex domain, as the constraint (19b) will still be a subset of constraints (18b). It remains to show one can generalize the geometrical proof for arbitrary large dimensions. This means that the channel dimensions and the number of IRS elements can be any natural numbers. Interestingly, it is enough to show that the geometrical proof based on \(\ell_{2}\) norms and Euclidean distance exists for higher dimensions. This proof is given in Appendix A where it is shown \(||\mathbf{\bar{H}}_{r}+\mathbf{H}_{R\mathcal{I}}\mathbf{\Theta}\mathbf{H}_{IR}||_ {F}^{2}=||Vec(\mathbf{\bar{H}}_{r})+(\mathbf{H}_{IR}*\mathbf{H}_{R\mathcal{I}}^ {T})Vec(\mathbf{\Theta})||_{2}^{2}\). Next we solve problem (14). Solving this problem is hard in general as it is non-convex. Hence, we first use the following lemma and theorem to solve it. There, it is shown that for every possible choice of \(\mathbf{H}_{1}\) and \(\mathbf{H}_{2}\), there exists at least one set of simultaneously diagonalizable matrices \(\mathbf{H}_{tot}\), \(\mathbf{Q}_{s}\) and \(\mathbf{Q}_{r}\) that are the solutions to the problem (14). **Lemma 1**. : _For two positive semi-definite and positive definite matrices \(\mathbf{A}\) and \(\mathbf{B}\) with eigenvalues \(\lambda_{1}(\mathbf{A})\geq\lambda_{2}(\mathbf{A})\geq...\geq\lambda_{N}( \mathbf{A})\) and \(\lambda_{1}(\mathbf{B})\geq\lambda_{2}(\mathbf{B})\geq...\geq\lambda_{N}( \mathbf{B})\), respectively, the following inequalities hold,_  
\[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]9  
Proof. : Consider [PERSON]'s inequality given by [40],  
\[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]0  
Furthermore, given \(\mathbf{B}\) as a positive definite matrix, the following are true,  
\[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]1 \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]2  
Now, dividing the sides of (22) by \(\bigl{|}\mathbf{B}\bigr{|}\), one can readily obtain (21). \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]3 Geometrical representation of optimization problem (18). \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_{r }\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{10}\]4 An example of mapping from three-dimensional to two-dimensional space.