# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
Then we have  
\[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]0 \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]1 (37) subject to \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]2 \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]3 (37b) \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]4 (37c) \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]5 (37d) \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{tot_{2}}\mathbf{Q}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H}+\mathbf{\hat{H}}_{ tot_{3}}\mathbf{\hat{H}}_{tot_{3}}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+ \mathbf{\hat{H}}_{tot_{3}}\mathbf{\hat{Q}}_{r}\mathbf{\hat{H}}_{tot_{3}}^{H} \right|}, \tag{12}\]6  
Note that the two additional constraints (37d) and (37e) need to be satisfied due to the conditions of Lemma 1 (i.e., eigenvalues have to be in decreasing order). Interestingly,these two additional constraints are affine. The above optimization problem can further be simplified using the following lemma,  
**Lemma 2**. : _The objective function of the optimization problem (37) is optimized when the constraints (37a) and (37c) are satisfied with equality._  
Proof. : Intuitively, as the objective function is an increasing and decreasing function of each element of \(\gamma_{s}\) and \(\sigma_{r}^{2}\), respectively, at convergence, the constraints are met with equality. See Appendix C for the proof.