# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
Notice that assuming that the RSI remains uncanceled, a robust transceiver against the worst-case RSI channel is required which is formulated as an optimization problem as follows  
\[\max_{\mathbf{Q}_{s},\mathbf{Q}_{r},\mathbf{\Theta}}\min_{ \mathbf{H},} \min\left(R^{\text{FD}}_{\text{sr}},R^{\text{FD}}_{\text{rd}}\right)\] (13) subject to \[\text{Tr}(\mathbf{Q}_{s})\leq P_{s}, \tag{13a}\] \[\text{Tr}(\mathbf{Q}_{r})\leq P_{r},\] (13b) \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]0 \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]1 (13c) \[R^{\text{FD}}_{\text{sr}} =\log_{2}\frac{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{ \hat{H}}_{1}\mathbf{Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{\hat{H}}_{1}\mathbf{ Q}_{s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{H} \right|}{\left|\sigma_{t}^{2}\mathbf{I}_{K_{r}}+\mathbf{\hat{H}}_{1}\mathbf{Q}_ {s}\mathbf{\hat{H}}_{1}^{H}+\mathbf{H}_{tot1}\mathbf{Q}_{r}\mathbf{H}_{tot1}^{ H}\right|}, \tag{7}\]2  
in which the throughput rate with respect to the worst-case RSI channel is maximized. Two constraints, \(P_{s}\) and \(P_{r}\), represent the transmit power budgets at the source and the relay, respectively. In constraint (13c), \(T_{x}\) represents the RSI or the channel estimation error bound corresponding to \(\mathbf{H}_{x}\). Notice that \(\text{Tr}(\mathbf{\hat{H}}_{x}\mathbf{\hat{H}}_{x}^{H})\) represents the sum of the squared singular values of \(\mathbf{H}_{x}\). It should be noted that using a bounded matrix norm is the most common way for modeling the uncertainty of a matrix [38; 39]. In practice, \(T_{x}\) can be found using stochastic methods when the distribution of the channel error is known. Otherwise, one may find it using a sample average approximation method. Finally, constraints (13d) are due to the unit modulus limitation of the IRS elements. The problem (13) is non-convex and hard to solve. As a result, for each of the above-mentioned scenarios, we propose a simplified version of the optimization problem and try to solve it instead. Note that as we are interested in finding the throughput corresponding to the worst-case RSI, any simplification in the optimization problem should be in favor of the RSI and interference. First, we analyse the performance of the system when the IRS is helping the relay cancel the RSI.