# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
Hence, the average worst-case throughput rate is defined as the average of worst-case rates for \(L\) randomization, i.e., \(R_{\mathrm{av}}=\frac{1}{L}\sum_{l=1}^{L}R_{l}\). Notice that for each set of realizations, we solve the robust transceiver design as is elaborated in Algorithm 2. We run different sets of simulations as described in the following subsections. ### Antenna Array Increment with No IRS

In this part, first we assume that there is no IRS installed. Then we evaluate the performance of the system using different strategies. Thereafter, we examine how installing an IRS can help increase the throughput. We consider two cases where the source, relay and destination are equipped with (a) a small antenna array, and (b) a large antenna arrays. In order to see the impact of IRS, we first assume that there is no IRS installed. For these cases, we have

1. \(N_{t}=4,K_{r}+K_{t}=10,N_{r}=4\),
2. \(N_{t}=10,K_{r}+K_{t}=24,N_{r}=10\). These cases are considered to highlight the performance of full-duplex DF relaying as a function of the number of antennas with the worst-case RSI. Interestingly, as the number of antennas at the source, relay and destination increase, full-duplex relaying achieves a higher throughput rate even with strong RSI. This can be seen by comparing rates from Figure 5(a) to those from Figure 5(b). Furthermore, notice that the worst-case RSI casts strong interference on the strong streams from the source to the destination. With very low RSI power \(T_{r}\to 0\), full-duplex almost doubles the throughput rate compared to the half-duplex counterpart. This can be seen in Figure 6, where the curves have their intercept point with the vertical axis. Figure 6: Sum rate throughput of the FD mode as a function of normalized interference \(\frac{T}{T}\) with different number of antennas at the source, relay and destination. The transmit power budget at the source and the relay are assumed to be equal, i.e., \(P_{\rm s}=5\) and \(P_{\rm r}=1\). (**a**) \(\{N_{t},K_{r}+K_{t},N_{r}\}=\{4,10,4\}\). (**b**) \(\{N_{t},K_{r}+K_{t},N_{r}\}=\{10,24,10\}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Transmitter location & \((0\;\text{m},0\;\text{m})\) \\ \hline IRS location & \((4000\;\text{m},20\;\text{m})\) \\ \hline Relay location & \((4000\;\text{m},0\;\text{m})\) \\ \hline Receiver location & \((8000\;\text{m},0\;\text{m})\) \\ \hline Path-loss & \(32.6+36.7\text{log}(d)\) \\ \hline Transmission bandwidth \(B\) & 180 Mb \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. However, as \(T_{r}\) increases, the efficiency of full-duplex operation drops. It is worth noting that at low RSI power the DoF plays the most important role to have a higher sum rate. For instance, consider Figure 6a in which the cases FD = {4,5,5,4}, FD = {4,6,4} and FD = {4,6,4,4} have DoF\({}_{\text{total}}=4\) - DoF\({}_{\text{total}}\) is the minimum of the DoF of source-relay and relay-destination channels, i.e., DoF\({}_{\text{total}}=\min\left(\text{DoF}_{\text{sr}},\text{DoF}_{\text{rd}}\right)\) - while the cases FD = {4,7,3,4} and FD = {4,3,7,4} have DoF\({}_{\text{total}}=3\). At \(T_{r}=0\), there is a noticeable gap between the first three cases and the last two, while the difference of the first three cases from each other is small. The big gap is due to the difference in DoF\({}_{\text{total}}\), and the small one is due to the difference in SNR. Similarly, in Figure 6b, the three cases FD = {10,12,12,10}, FD = {10,10,14,10} and FD = {10,10,14,10} with DoF\({}_{\text{total}}=10\) have higher rates than the two cases FD = {10,6,18,10} and FD = {10,18,6,10} with DoF\({}_{\text{total}}=6\). Finally, it can be seen in both Figure 6a,b that at \(T_{r}=0\) there is no difference between cases that have the same DoF\({}_{\text{total}}\) but different DoF\({}_{\text{sr}}\) and DoF\({}_{\text{rd}}\). As it can be seen in both Figure 6a,b, for cases with \(K_{t}>K_{t}\) the sum rate drops quickly as RSI increases.