# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
Therefore, the received signals at the relay and the destination can, respectively, be expressed as

\[\mathbf{y}_{\mathrm{r}} =(\hat{\mathbf{H}}_{1}\!+\!\hat{\mathbf{H}}_{IR}\mathbf{\Theta} \hat{\mathbf{H}}_{\mathrm{SI}})\mathbf{x}_{\mathrm{r}}\!+\!(\hat{\mathbf{H}}_ {1}\!+\!\hat{\mathbf{H}}_{IR}\mathbf{\Theta}\hat{\mathbf{H}}_{\mathrm{SI}}) \mathbf{x}_{\mathrm{r}}+\mathbf{n}_{\mathrm{t}}, \tag{40}\] \[\mathbf{y}_{\mathrm{d}} =(\hat{\mathbf{H}}_{2}\!+\!\hat{\mathbf{H}}_{ID}\mathbf{\Theta} \hat{\mathbf{H}}_{RI})\mathbf{x}_{\mathrm{r}}\!+\!(\hat{\mathbf{H}}_{2}\!+\! \hat{\mathbf{H}}_{ID}\mathbf{\Theta}\hat{\mathbf{H}}_{RI})\mathbf{x}_{\mathrm{ r}}+\mathbf{n}_{\mathrm{d}}. \tag{41}\]

Consequently, the achievable rates for the transmitter-relay and relay-destination links can be expressed as below

\[R_{\mathrm{sr}}^{\mathrm{HD}} =\!\log\left|\mathbf{I}_{K_{r}}\!+\!\left(\hat{\mathbf{H}}_{1}\! +\!\hat{\mathbf{H}}_{IR}\mathbf{\Theta}\hat{\mathbf{H}}_{\mathrm{SI}}\right) \mathbf{Q}_{\mathrm{s}}\!\left(\hat{\mathbf{H}}_{1}\!+\!\hat{\mathbf{H}}_{IR} \mathbf{\Theta}\hat{\mathbf{H}}_{\mathrm{SI}}\right)^{H}\right.\] \[\left.\left(\sigma_{t}^{2}\mathbf{I}_{K_{r}}\!+\!(\hat{\mathbf{H}} _{1}\!+\!\hat{\mathbf{H}}_{IR}\mathbf{\Theta}\hat{\mathbf{H}}_{\mathrm{SI}}) \mathbf{Q}_{\mathrm{s}}\!\left(\hat{\mathbf{H}}_{1}\!+\!\hat{\mathbf{H}}_{IR} \mathbf{\Theta}\hat{\mathbf{H}}_{\mathrm{SI}}\right)^{H}\!\right)^{-1}\!\right|\] \[\geq\sum_{i=1}^{\min\left(M,K_{r}\right)}\log_{2}\!\!\left(\!1+ \!\frac{\lambda_{i}\!\left(\mathbf{H}_{1}^{\prime}\mathbf{Q}_{\mathrm{s}}\mathbf{ H}_{1}^{\prime}\right)}{\sigma_{t}^{2}+T_{2}P_{r}}\right), \tag{42}\] \[R_{\mathrm{rd}}^{\mathrm{HD}} =\!\log\left|\mathbf{I}_{\mathrm{N}}\!+\!\left(\hat{\mathbf{H}}_{ 2}\!+\!\hat{\mathbf{H}}_{ID}\mathbf{\[PERSON]}\hat{\mathbf{H}}_{\mathrm{RI}} \right)\mathbf{Q}_{r}\!\left(\hat{\mathbf{H}}_{2}\!+\!\hat{\mathbf{H}}_{ID} \mathbf{\[PERSON]}\hat{\mathbf{H}}_{\mathrm{RI}}\right)^{H}\right.\] \[\sigma_{r_{i}}^{2}=\left[\frac{\sqrt{\left(\sigma_{1_{i}}^{2}\gamma_{s_{i}} \right)^{2}+\frac{4\sigma_{1_{i}}^{2}\gamma_{s_{i}}\gamma_{t_{i}}}{\lambda}} -\sigma_{1_{i}}^{2}\gamma_{s_{i}}-2(\sigma_{t}^{2}+T_{1}P_{s})}{2\gamma_{r_{i }}}\right]^{+}, \tag{39}\]0 \[\sigma_{r_{i}}^{2}=\left[\frac{\sqrt{\left(\sigma_{1_{i}}^{2}\gamma_{s_{i}} \right)^{2}+\frac{4\sigma_{1_{i}}^{2}\gamma_{s_{i}}\gamma_{t_{i}}}{\lambda}} -\sigma_{1_{i}}^{2}\gamma_{s_{i}}-2(\sigma_{t}^{2}+T_{1}P_{s})}{2\gamma_{r_{i }}}\right]^{+}, \tag{39}\]1

where \(\mathbf{H}_{1}^{\prime}=\hat{\mathbf{H}}_{1}\!+\!\hat{\mathbf{H}}_{IR}\mathbf{ \Theta}\hat{\mathbf{H}}_{\mathrm{SI}}\) and \(\mathbf{H}_{2}^{\prime}=\hat{\mathbf{H}}_{2}\!+\!\hat{\mathbf{H}}_{ID}\mathbf{ \Theta}\hat{\mathbf{H}}_{\mathrm{RI}}\). In addition, \(R_{\mathrm{sr}}^{\mathrm{HD}}\) and \(R_{\mathrm{rd}}^{\mathrm{HD}}\) are the achievable rates on the source-relay and relay-destination links, respectively. Using time sharing, the achievable rate between the source and destination nodes is given by

\[\sigma_{r_{i}}^{2}=\left[\frac{\sqrt{\left(\sigma_{1_{i}}^{2}\gamma_{s_{i}} \right)^{2}+\frac{4\sigma_{1_{i}}^{2}\gamma_{s_{i}}\gamma_{t_{i}}}{\lambda}} -\sigma_{1_{i}}^{2}\gamma_{s_{i}}-2(\sigma_{t}^{2}+T_{1}P_{s})}{2\gamma_{r_{i }}}\right]^{+}, \tag{39}\]2 \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Complexity** & **Robust Design \({}^{*}\)** & **Solution** & **System Model** \\ \hline This work & \(\mathcal{O}(MK_{r}K_{t}\min(M,K_{r}K_{t}))\) & Yes & Analytical bounds & Relay and IRS \\ \hline [PERSON] et al. [8] & \(\mathcal{O}(I_{O}M^{3})\) & No & Practical solution & Only IRS \\ \hline [PERSON] et al. [34] & \(\mathcal{O}(I_{t}(I_{w}\min(N_{t},K_{r})))\) & Yes & Practical solution & Only relay \\ \hline [PERSON] et al. [9] & \(\mathcal{O}(I_{1}I_{O}I_{w}M^{3})\) & No & Practical solution & Relay and IRS \\ \hline \hline \end{tabular} \({}^{*}\) This means that the method is robust against channel uncertainties and/or RSI.

\end{table} 
Table 2: Comparison of the proposed method with previous studies.