## 4 Simulation Results
### Performance Comparison  
In Figures 3 and 4, we illustrate the worst-case MSE performance as a function of the effective training signal-to-interference ratio (SIR) for the systems with \(M_{t}=M_{r}=L=4\) and \(M_{t}=M_{r}=L=8\). The effective training SIR is defined as \(\text{SIR}=\mathcal{P}_{T}\mathbb{E}[\text{Tr}(\mathbf{\hat{T}})]/\mathbb{E}[ \text{Tr}(\mathbf{\hat{Q}})]=\mathcal{P}_{T}\text{Tr}(\mathbf{C}_{T})/\sum_{k=1 }^{3}\text{Tr}(\mathbf{C}_{s,k}\mathbf{C}_{l,k})\mathbf{C}_{r,k}\). In Figure 3, we set \(\Theta_{T}=10^{\circ}\), \(\Theta_{I,k}=10^{\circ}\) and \(\rho_{\tau,k}=\rho_{s,k}=0.9\), \(k=1,2,3\), which represents a strongly correlated environment. On the other hand, in Figure 4, we set \(\Theta_{T}=30^{\circ}\), \(\Theta_{I,k}=30^{\circ}\), and \(\rho_{\tau,k}=\rho_{s,k}=0.3\), \(k=1,2,3\), which represents a weakly correlated environment. The values of \(\alpha_{T}\), \(\alpha_{R}\), and \(\alpha_{Q}\) are set to \(\alpha_{T}=\alpha_{R}=\alpha_{Q}=0.3\). In the figures, the results are averaged over 500 realizations. It can be observed from Figures 3 and 4 that the proposed schemes outperform the other schemes. For example, in Figure 3, the proposed schemes have roughly 5 dB improvement in terms of the effective training SIR for the system with \(M_{t}=M_{r}=L=8\) when the effective training SIR is higher than 10 dB (similar trends can still be observed even for other performance metrics, e.g., the ratio of the standard deviation to the mean value). provides comparable performance to that of the optimal power allocation. The semi-robust scheme with the proposed suboptimal power allocation works well for the weakly correlated case, but it shows noticeable performance loss for the strongly correlated case. The worst-case MSE performance is compared in Figures 5 and 6 for various values of \(\alpha=\alpha_{T}=\alpha_{R}=\alpha_{Q}\) when \(\Theta_{T}=10^{\circ}\), \(\Theta_{I,k}=10^{\circ}\) and \(\rho_{\tau,k}=\rho_{\tau,k}=0.9\), \(k=1,2,3\), where the parameter \(\alpha\) is introduced to set \((\epsilon_{T},\epsilon_{R}.\epsilon_{Q})=(\alpha||\hat{\mathbf{T}}||_{F}^{2},\alpha||\hat{\mathbf{R}}||_{F}^{2},\alpha||\hat{\mathbf{Q}}||_{F}^{2})\). Figures 5 and 6 show the performance comparison for the systems with \(M_{t}=M_{r}=L=4\) and \(M_{t}=M_{r}=L=6\), respectively. When the value of \(\alpha\) decreases from 0.6 to 0.1, the performances of all the schemes are improved, and the gap between the proposed and semi-robust schemes increases, but the gap between the semi-robust and nonrobust schemes decreases. This means that the performance is dominated by the uncertainty in the interference covariance matrix when the value of \(\alpha\) equals 0.1, i.e., the error is small, but the performance is dominated by the uncertainty in the channel covariance matrix when the value of \(\alpha\) equals 0.6, i.e., the error is large. The suboptimal power allocation shows almost the same performance as that of the optimal one in the proposed scheme, but shows notable performance loss in the semi-robust scheme when \(\alpha=0.6\). Figure 4: Worst-case MSE performance comparison of various training schemes for weakly correlated environment when \(\alpha_{T}=\alpha_{R}=\alpha_{Q}=0.3\). The results are shown for the systems with \(M_{t}=M_{r}=L=4\) and \(M_{t}=M_{r}=L=8\). Figure 5: Worst-case MSE performance comparison of various training schemes for different values of the uncertainty parameter \(\alpha=\alpha_{T}=\alpha_{R}=\alpha_{Q}\) when \(M_{t}=M_{r}=L=4\). The results are shown for \(\alpha=0.1\) and \(\alpha=0.6\). Figure 6: Worst-case MSE performance comparison of various training schemes for different values of the uncertainty parameter \(\alpha=\alpha_{T}=\alpha_{R}=\alpha_{Q}\) when \(M_{t}=M_{r}=L=6\). The results are shown for \(\alpha=0.1\) and \(\alpha=0.6\).