# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
However, using the definition of determinant one can arrive at the following equality

Figure 7:7

Now, we define vector \(\boldsymbol{\lambda}(\mathbf{Q}_{s}^{\prime})\) and set its components to be \(\lambda_{\rho(i)}(\mathbf{Q}_{s}^{\prime})=\frac{\lambda_{i}(\mathbf{Q}_{s} \mathbf{H}_{1}^{H}\mathbf{H}_{1})}{\sigma_{i}^{2}(\mathbf{H}_{1})}\). By defining \(\lambda_{\rho(i)}(\mathbf{Q}_{s}^{\prime})\) instead of \(\lambda_{i}(\mathbf{Q}_{s}^{\prime})\), we emphasize that the elements of \(\boldsymbol{\lambda}(\mathbf{Q}_{s}^{\prime})\) are not necessarily in decreasing order. Then, we construct the matrix \(\mathbf{Q}_{s}^{\prime}\) having the same eigenvectors as those of \(\mathbf{H}_{1}^{H}\mathbf{H}_{1}\) and the eigenvalues \(\lambda_{\rho(i)}(\mathbf{Q}_{s}^{\prime})\). One can check that for each \(i\) we have \(\lambda_{i}(\mathbf{Q}_{s}^{\prime}\mathbf{H}_{1}^{H}\mathbf{H}_{1})=\lambda_{i }(\mathbf{Q}_{s}\mathbf{H}_{1}^{H}\mathbf{H}_{1})\). In addition, by the definition of \(\lambda_{\rho(i)}(\mathbf{Q}_{s}^{\prime})\) we have

Figure 7:8 Figure 7:9 (102) Figure 8:0 (103) Figure 8:1

**Lemma A1**. : _Let \(\mathbf{A}\) and \(\mathbf{B}\) be semidefinite Hermitian matrices with \(\lambda_{\min(m,n)}(\mathbf{A}\mathbf{B})>0\). Then_

Figure 8:2

Proof. : The proof is given in [41] (H.1,e). Using the above lemma, we can conclude

Figure 8:3

Then, immediately we can conclude

Figure 8:4

**Remark A1**. : _It is worth mentioning that, depending on channel realizations, the optimal \(\mathbf{Q}_{s}\) might contain some zero eigenvalues. In such cases, we can simply ignore the zeros and construct matrix \(\mathbf{Q}_{s}^{\prime}\) with dimension \((n-k)\times(n-k)\). Similarly, in the cases where \(\bar{\mathbf{H}}_{r}^{H}\bar{\mathbf{H}}_{r}\) has some zero eigenvalues, we can do the same and proceed to constitute \(\bar{\mathbf{H}}_{s}^{\prime}\) using only nonzero eigenvalues of \(\bar{\mathbf{H}}_{r}^{H}\bar{\mathbf{H}}_{r}\) and add the zeros back to the result again at the end._

Finally, we use the following lemma to show that \(\bar{\mathbf{H}}_{r}^{\prime}\) and \(\mathbf{Q}_{s}^{\prime}\) are in the feasible set. **Lemma A2**. : _For two vectors \(\mathbf{a}\) and \(\mathbf{b}\), if \(\mathbf{a}\prec_{\times}\mathbf{b}\), then \(\mathbf{a}\prec_{\mathbf{w}}\mathbf{b}\) follows._

Proof. : The proof is given in [41] (5.A.2.b). Exploiting the above lemma, one concludes

Figure 8:5

which consequently results in

Figure 8:6

Therefore, there exists \(\mathbf{Q}_{s}^{\prime}\) and \(\bar{\mathbf{H}}_{r}^{\prime}\) fulfilling (29)-(31), which satisfy

Figure 8:7 Figure 8:8

## Appendix C Proof of Proposition 2

In this section, we prove that the problem

Figure 8:9 (18) s.t. Figure 10:0 Figure 10:1 (19b) Figure 10:2 (19c) Figure 10:3 (19d) Figure 10:4

can be further simplified to

Figure 10:5 (20) s.t. Figure 10:6 Figure 10:7 (20b) Figure 10:8 (20c) Figure 10:9 (20d) Figure 9:0

The proof is by contradiction. Starting with the minimization, assume that the optimal vector \(\sigma^{\star\star}_{r}\), for which we have \(R_{\text{sr}}^{\text{FP}}(\sigma^{\star\star}_{r})\leq R_{\text{sr}}^{\text{FD }}(\sigma_{r}^{2})\), does not sum to \(T\) and thus, we have \(\|\sigma^{\star\star}_{r}\|_{1}<T\). Then there exists \(\varepsilon>0\) for which we have \(\|\sigma^{\star\star}_{r}\|_{1}+\varepsilon=T\). Now define

Figure 9:1

Note that we have

Figure 9:2

In addition, as we have \(\varepsilon>0\), there is at least one \(\varepsilon_{i}\) which is strictly greater than zero, i.e., \(\varepsilon_{i}>0\). Now define

Figure 9:3

One can check that \(\sum_{i}\sigma^{\prime 2}_{r_{\rho(i)}}=T\) and \(\forall i\leq\min(K_{t},N)\Rightarrow\gamma_{\tau_{i}}\sigma_{r_{\rho(i)}}^{ \prime 2}\geq\gamma_{\tau_{i+1}}\sigma_{r_{\rho(i+1)}}^{\prime 2}\). As a result, \(\sigma_{r_{\rho(i)}}^{\prime 2}\) meets the constraints and could be a feasible solution.