# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
We can equivalently write \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]2 (16) subject to \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]3 (16a) \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]4 (16b) where \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]5 denotes a column-wise Khatri-Rao product defined as below \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]6 (17) and where \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]7 is the \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]8 th column of \[R^{\text{FD}}_{\text{rd}} =\log_{2}\frac{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{\hat{H} }_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}+\mathbf{\hat{H}}_{2}\mathbf{Q}_ {r}\mathbf{\hat{H}}_{2}^{H}\right|}{\left|\sigma_{d}^{2}\mathbf{I}_{N}+\mathbf{ \hat{H}}_{2}\mathbf{Q}_{r}\mathbf{\hat{H}}_{2}^{H}\right|}, \tag{8}\]9, and \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}\!+\!\mathbf{ \hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{tot_{2}}^{H}+\mathbf{\hat{H} }_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}{\left|\sigma_{t}^{2} \mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{tot_{2}}\mathbf{Q}_{s}\mathbf{\hat{H}}_{ tot_{2}}^{H}+\mathbf{\hat{H}}_{r}\mathbf{Q}_{r}\mathbf{\hat{H}}_{r}^{H}\right|}, \tag{9}\]0 denotes the Kronecker product. See Appendix A for proof. One can show that \(T^{\prime}\leq(\sqrt{T_{r}}-\sigma_{\min}(\mathbf{H}_{IR}*\mathbf{H}_{RI}^{T}) )^{2}\). As mentioned before, problem (14) is a simplification of problem (13). This means every achievable rate which is inside the feasible set of (14) is also inside the feasible set of (13) (Notice that the reverse is not necessarily true, i.e., every achievable rate which is a feasible solution of (13) is not necessarily a feasible solution for (14) as well. However, as we look for achievable rates, we can still use this method). The reason is that in problem (13), the minimization over RSI happens only one time, and the worst-case RSI simultaneously tries to cancel the effect of the best configuration of IRS and the best covariance matrices.