# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
where \(\alpha\) is the time-sharing parameter. Note that in half-duplex relaying, the source and relay transmissions are conducted in separate channel uses. Hence, the transmit covariance matrices \(\mathbf{Q}_{s}\in\mathbb{H}^{N_{t}\times N_{t}}\) and \(\mathbf{Q}_{r}\in\mathbb{H}^{K_{t}\times K_{t}}\) are optimized by maximizing the achievable rate from the source to the destination. Here, the convex cone of Hermitian positive semidefinite matrices of dimensions \(N_{t}\times N_{t}\) and \(K_{t}\times K_{t}\) are represented by \(\mathbb{H}^{N_{t}\times N_{t}}\) and \(\mathbb{H}^{K_{t}\times K_{t}}\), respectively. Importantly, for maximizing this achievable rate, the time-sharing parameter, i.e., \(\alpha\), needs to be optimized alongside the system parameters, e.g., power allocation. Readily, optimal \(\alpha\) occurs at \(\alpha R_{\mathrm{sr}}^{\mathrm{HD}}=(1-\alpha)R_{\mathrm{rd}}^{\mathrm{HD}}\). Therefore, the achievable rate becomes as follows

\[R^{\mathrm{HD}}=\frac{R_{\mathrm{sr}}^{\mathrm{HD}}R_{\mathrm{rd}}^{\mathrm{HD} }}{R_{\mathrm{sr}}^{\mathrm{HD}}+R_{\mathrm{rd}}^{\mathrm{HD}}}. \tag{45}\]

Notice that as the objective function of the above optimization problem is a monotonically increasing function of both \(R_{\mathrm{sr}}^{\mathrm{HD}}\) and \(R_{\mathrm{rd}}^{\mathrm{HD}}\), the problem can be simplified to maximizing \(R_{\mathrm{sr}}^{\mathrm{HD}}\) and \(R_{\mathrm{rd}}^{\mathrm{HD}}\) separately. Next, we provide the solution to the rate optimization problem when IRS is assisting the source-relay link. We have

\[\max_{\mathbf{Q}_{s}} \min_{\hat{\mathbf{H}}_{S|}\hat{\mathbf{H}}_{1},\hat{\mathbf{H}} _{1R}} R_{\mathrm{sr}}^{\mathrm{HD}}\] (46) subject to \[\mathrm{Tr}(\mathbf{Q}_{s})\leq P_{s}, \tag{46a}\] \[\mathrm{Tr}(\hat{\mathbf{H}}_{1}\hat{\mathbf{H}}_{1}^{H})\leq T_ {1},\] (46b) \[\mathrm{Tr}(\hat{\mathbf{H}}_{S|}\hat{\mathbf{H}}_{S|}^{H})\leq T_ {SI},\] (46c) \[\mathrm{Tr}(\hat{\mathbf{H}}_{IR}\hat{\mathbf{H}}_{IR}^{H})\leq T _{IR}, \tag{46d}\]

The above optimization problem follows the same approach applied for the optimization of the relay-destination link in the FD scenario. As a result, the same method could be applied to find it. In other words, the well-known water-filling algorithm can be used to find the optimal covariance matrices along with the algorithm introduced in [8] to find the best IRS pattern. This process continues iteratively until it finally converges. The solution to \(R_{rd}^{HD}\) is the same as well, and the same procedure can be applied to find \(\mathbf{Q}_{r}\). The overall procedure of finding the solution for the HD mode is summarized in Algorithm 4. ```
1:\(\hat{\mathbf{H}}_{1 tot_{2}}=(\hat{\mathbf{H}}_{1}+\hat{\mathbf{H}}_{R1}\Theta \hat{\mathbf{H}}_{IR})\)
2:Find the optimum \(\Theta\), using Algorithm 1 in [8]
3:\(\hat{\mathbf{H}}_{1 tot_{3}}=(\hat{\mathbf{H}}_{2}+\hat{\mathbf{H}}_{R1}\Theta \hat{\mathbf{H}}_{IR})\)
4:Find the optimum \(\Theta\), using Algorithm 1 in [8]
5:Find \(R_{\mathrm{sr}}^{\mathrm{HD}}\), using Equation (42)
6:Find \(R_{\mathrm{rd}}^{\mathrm{HD}}\), using Equation (43)
7:Find \(R^{\mathrm{HD}}\), using Equation (45)
```

**Algorithm 4** Robust Transceiver Design for HD scenario

## 5 Numerical Results

We assume the transmit power budgets at the source and at the relay are \(P_{s}=5\) and \(P_{r}=1\), respectively. Moreover, the AWGN spectral density is assumed to be \(-175\) dBm and the bandwidth is \(BW=180\) MHz. In this section, we investigate the performance of IRS-assisted full-duplex relaying with RSI channel uncertainty bound \(T_{r}\), i.e., \(\mathbf{Tr}(\hat{\mathbf{H}}_{r}\hat{\mathbf{H}}_{r}^{H})\leq T_{r}\). We consider all the channels to follow the Rician distribution with the factor \(\epsilon=0.1\) and the specificaiton given in Table 3. We also assume \(T_{x}=0.001\), \(x\in\{1,2,SI,IR,RI,ID\}\). We perform Monte Carlo simulations with \(L=10^{3}\) realizations from random channels and noise vectors.