# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
In fact, the more relative antennas at the relay transmitter compared to its receiver, the faster the sum rate drops with the rise in RSI. To understand this behaviour of the system better, again, consider case {10,18,6,10} and also suppose \(T_{r}\rightarrow\infty\). As discussed before, we have DoF\({}_{\text{sr}}=10\) and DoF\({}_{\text{rd}}=6\). Moreover, we have DoF\({}_{\text{t}}=6\) for the interference channel (\(\tilde{\mathbf{H}}_{\text{r}}\)). Unlike the case with no interference, in this case the bottleneck is no longer the relay-destination link. This is due to the fact that interference can act to the detriment of some six of the source-relay subchannels. As we have DoF\({}_{\text{t}}=6\), interference can choose at most six independent subchannels, and as we assumed \(T_{r}\rightarrow\infty\), for those subchannels we obtain SINR \(\rightarrow\) 0. Therefore, no information can be conveyed from those links, and the bottleneck becomes the source-relay link with 4 usable subchannels. It can be seen in Figure 6a,b that as \(T_{r}\) increases, the cases with the same sum-rate at \(T_{r}=0\) start to diverge because of the different characteristics of the interference they experience. We explain the effect of interference in the following subsection in more detail. ### The Impact of IRS

In this part, we evaluate the impact of IRS on the throughput rate when it is used to perform different tasks. Figure 7 shows the throughput for three different scenarios, namely, when IRS is used to help the transmitter-relay link, when it is applied to cancel RSI and when the IRS job is to help the relay-destination link. Then the results are compared with two cases where the system is working in HD with IRS and the case where the system is working in FD with no IRS. It is also assumed that \(\frac{T_{r}}{\left\langle\mathbf{H}_{\text{r}},\mathbf{H}_{\text{r}}^{\prime} \right\rangle}=75\%\), i.e., the system works at high RSI range. As it can be seen, the highest performance is achieved when the IRS is utilized to deal with the RSI. As a result, for the rest of the paper we use the IRS for this purpose. Figure 7: Comparison of five different scenarios: HD with IRS, FD with no IRS, FD with IRS as RSI cancelator; FD with IRS to help transmitter–relay link; FD with IRS to help relay–destination link. We considered the case where \(\left\{N_{t},K_{t},K_{r},N_{r},M\right\}=\left\{4,5,5,4,100\right\}\). Figure 8 shows the impact of IRS on the throughput. For Figure 8a, we considered the case \(\{N_{t},K_{t},K_{r},N_{r}\}=\{4,5,5,4\}\), and for Figure 8b, we considered \(\{N_{t},K_{t},K_{r},N_{r}\}=\{10,12,12,10\}\). As shown in the figure, the number of IRS elements has a great impact on RSI cancellation to the extend that having an IRS with \(M=100\) and \(M=300\) can cancel interference of \(\frac{T_{r}}{\hat{\mathbf{H}}_{r}(\hat{\mathbf{H}}_{r}\hat{\mathbf{H}}_{r})^{ \prime}}=0.75\) for \(\{N_{t},K_{t},K_{r},N_{r}\}=\{4,5,5,4\}\), and \(\{N_{t},K_{t},K_{r},N_{r}\}=\{10,12,12,10\}\), respectively. Further, it can also be seen in the figure that having IRS with \(20\) and \(100\) elements for the small and large antenna array cases, respectively, is not helpful at all. This is mainly due to the fact that unlike the average case, for the case of worst-case scenario, the number of IRS elements should be at least as large as the dimension of \(\hat{\mathbf{H}}_{r}\). Otherwise, the IRS feasible set cannot span into all dimensions of \(\hat{\mathbf{H}}_{r}\). Therefore, there is always at least one representation for \(\hat{\mathbf{H}}_{r}\) in which IRS cannot perform any RSI cancellation. In addition, comparing two figures Figure 8a,b one can conclude that, when the dimension of \(\hat{\mathbf{H}}_{r}\) increases, the effort that IRS has to make in order to cancel RSI remarkably increases which is consistent with the previous statement. ### Relay Tx/Rx Antenna allocation

Suppose that the relay has \(K_{t}+K_{r}=8\) in total.