# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
Note that as \(\gamma_{s}^{\star}\) is the optimal source power allocation based on all other parameters, by changing \(\sigma^{\star\star 2}_{r}\) to \(\sigma^{\prime 2}_{r}\), \(\gamma_{s}^{\star}\) might also change. However, we created each \(\sigma^{\prime 2}_{r_{\rho(i)}}\) in a special way to avoid this change. To show this, first notice that we have

Figure 9:4where \(\lambda\) is water level and can be found based on power constraints. Substituting the new power allocation for interference, we obtain new power allocation for input power as follows

Figure 9:5 Figure 9:6

where \((a)\) comes from the fact that \(\
icefrac{{\epsilon}}{{\sum_{j=1}^{N}\frac{\sigma^{2}_{1_{j}}}{\tau_{j}}}}\) is a constant independent of \(i\). So we can define \(\lambda^{\prime}=\lambda+\
icefrac{{\epsilon}}{{\sum_{j=1}^{N}\frac{\sigma^{2 }_{1_{j}}}{\tau_{j}}}}\). This shows, for \(\sigma^{\prime 2}_{r}\), all the optimal variables and parameters remain the same as those of \(\sigma^{\star 2}_{r}\). Now we compare \(R_{\text{sr}}^{\text{FP}}\) for both cases. First, notice that we have \(\forall i,\varepsilon_{i}\geq 0\) and among them there is at least one index \(i^{\prime}\), for which we have \(\varepsilon_{i^{\prime}}>0\). This means \(\forall i,\sigma^{\prime 2}_{r_{p(i)}}\geq\sigma^{\star 2}_{r_{p(i)}}\) and \(\sigma^{\prime 2}_{r_{p(^{\prime})}}>\sigma^{\star 2}_{r_{p(^{\prime})}}\). Now, notice that \(f_{i}(x)=\text{log}_{2}\Bigg{(}1+\frac{\sigma^{2}_{1_{i}}\gamma^{\star}_{s_{p( i)}}}{1+\gamma_{r_{i}}x}\Bigg{)}\) is a monotonically decreasing function of \(x\). Thus, we have \(f_{i}(\sigma^{\prime 2}_{r_{p(i)}})\leq f_{i}(\sigma^{\star 2}_{r_{p(i)}})\) and \(f_{i^{\prime}}(\sigma^{\prime 2}_{r_{p(^{\prime})}})<f_{i^{\prime}}(\sigma^{ \star 2}_{r_{p(^{\prime})}})\). Adding all above inequalities, we obtain

Figure 9:7

The above equation indicates \(R_{\text{sr}}^{\text{FD}}(\sigma^{\star 2}_{r})>R_{\text{sr}}^{\text{FD}}( \sigma^{2}_{r})\) which contradicts the first assumption \(R_{\text{sr}}^{\text{FD}}(\sigma^{\star 2}_{r})\leq R_{\text{sr}}^{\text{FD}}( \sigma^{2}_{r})\). This completes the proof of the minimization part. For the maximization part, the general idea is the same. Again, the proof is by contradiction. We assume the optimal vector \(\gamma^{\star}_{s}\), for which we have \(R_{\text{sr}}^{\text{FD}}(\gamma^{\star}_{s})\geq R_{\text{sr}}^{\text{FD}}( \gamma_{s})\), does not sum to \(P_{s}\). Therefore, we have \(\|\gamma_{s}\|_{1}<P_{s}\). Then there exists \(\varepsilon>0\) for which we have \(\|\gamma_{s}\|_{1}+\varepsilon=P_{s}\). Now we define

Figure 9:8

where, \(\eta=\sum_{i}\bigg{(}\frac{1+\sigma^{\star 2}_{r_{i}}\gamma_{r_{i}}}{\sigma^{2}_{1_{i}}}+ \gamma^{\star}_{s_{i}}\bigg{)}\). Now we define the new source power allocation as below

Figure 9:9

One can check that \(\sum_{i}\gamma^{\prime}_{s_{p(i)}}=P_{s}\) and \(\sigma^{2}_{1_{i}}\gamma^{\prime}_{s_{p(i)}}\geq\sigma^{2}_{1_{i+1}}\gamma^{ \prime}_{s_{p(i+1)}}\). Thus, the new source power allocation is in the feasible set. Now the remaining is to make sure the new allocation does not change the corresponding \(\sigma^{2}_{r}\).