# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
Furthermore, following cases in which the number of antenna at the source and destination are \(\{N_{t},N_{r}\}=\{4,4\}\). The question is, from eight antennas at the relay, how many should be used for reception for the robust design? Figure 9 shows the sum rate as a function of \(K_{r}\) for different values of \(T\) where there is no IRS and there is an IRS with \(M=60\) elements, respectively. As it can be seen, by using more antennas for reception than for transmission, i.e., \(K_{r}>K_{t}\), at the relay, the throughput rate is maximized. This is due to the fact that increasing the signal-to-noise ratio (SNR) of the source-relay streams enhances the overall throughput rate more than increasing the number of antennas for transmission in order to enhance the DoF of the relay-destination link. Furthermore, notice that in this setup the overall DoF from the source to destination is limited by the DoF of the source-relay link, i.e., the bottleneck is in the first hop. By comparing two scenarios, we see that having an IRS not only improves the rates in all cases, but also it may change the best antenna allocation. For instance, for the case of \(T=15\%\), it is best to have six antennas at the relay receiver and four4 antennas at the relay transmitter. However, after establishing the IRS, the best antenna allocation changes to five antennas at each end. For instance, the results show that although the DoF\({}_{\text{total}}\) for both \(\{N_{t},K_{t},K_{r},N_{r}\}=\{4,3,5,4\}\) and \(\{N_{t},K_{t},K_{r},N_{r}\}=\{4,5,3,4\}\) is three, the sum rate capacity of the latter is much better than that of the former at high interference. This is because of the fact when DoF\({}_{\text{sr}}>\) DoF\({}_{\text{rd}}\), the source-relay link enjoys DoF\({}_{\text{sr}}-\) DoF\({}_{\text{rd}}\) subchannels with no interference. Therefore, the source can manage to obtain a higher sum rate by choosing its power allocation wisely. However, in the case of DoF\({}_{\text{sr}}\leq\) DoF\({}_{\text{rd}}\), no matter how well the power allocation is performed, all sub channels suffer from interference at the source-relay end. Figure 8: Sum rate throughput as a function of IRS elements. The transmit power budget at the source and the relay are assumed to be equal, i.e., \(P_{s}=5\) and \(P_{\text{r}}=1\): (**a**) \(\{N_{t},K_{t},K_{r},N_{r}\}=\{4,5,5,4\}\); (**b**) \(\{N_{t},K_{t},K_{r},N_{r}\}=\{10,12,12,10\}\). ### Full-Duplex vs. Half-Duplex

In this subsection, we determine the thresholds where the HD relaying outperforms the FD relaying. This threshold provides a mode-switching threshold in hybrid HD/FD relay systems. As it can be seen in Figure 10, for each case of \(K_{r}\), there is a maximum value of \(\frac{T}{P}\) above which the HD mode outperforms the FD mode in terms of sum rate maximization. Furthermore, Figure 10 shows the threshold for different IRS configurations. For this part, we continued with the case of \(N_{t}=4,K_{r}=5,K_{t}=5,N_{r}=4\). As it can be seen, by increasing the number of antennas, the threshold occurs at higher RSI. This is in fact a direct result of obtaining better performance by having more antennas at the relay's receiver. It is worth noting that the IRS has a great impact on the performance of FD relaying. For instance, by having an IRS consisting of only 60 elements, the FD mode outperforms the HD mode in almost all cases. ## 6 Conclusions

In this paper, we investigated a multi-antenna source communicating with a multi-antenna destination through a multi-antenna relay. The relay is assumed to exploit a decode-and-forward (DF) strategy. An IRS is installed to help the relay cope with the RSI

Figure 10: Thresholds for different \(K_{r}\) and \(M\). The region above each curve indicates values of \(\frac{T}{P}\) for which HD outperforms FD. In contrast, points below the curve belong to cases where FD performs better than HD. Figure 9: Sum rate throughput as a function of relay receiver antennas \(K_{r}\) with and without RSI.