## 3 Training Signal Optimization
### Worst-Case MSE Minimizing Training Structure  
Let \((\mathbf{P}^{\star},\tilde{\mathbf{T}}^{\star},\tilde{\mathbf{R}}^{\star},\tilde{ \mathbf{Q}}^{\star})\) be an optimal solution to the minimax problem (14). Then, the structures of the optimal training signal \(\mathbf{P}^{\star}\) and the worst-case correlation errors \((\tilde{\mathbf{T}}^{\star},\tilde{\mathbf{R}}^{\star},\tilde{\mathbf{Q}}^{ \star})\) are derived in the following theorem. **Theorem 1**. : _Let \(\mathbf{\hat{T}}=\mathbf{U}_{T}\mathbf{\Lambda}_{T}\mathbf{U}_{T}^{H}\) and \(\mathbf{\hat{Q}}=\mathbf{U}_{Q}\mathbf{\Lambda}_{Q}\mathbf{U}_{Q}^{H}\) be the eigenvalue decompositions (EVDs) of the matrices \(\mathbf{\hat{T}}\) and \(\mathbf{\hat{Q}}\), respectively, where_  
\[\mathbf{\Lambda}_{T}=\text{Diag}(\lambda_{T,1},\cdots,\lambda_{T,M_{t}})\]  
_and_  
\[\mathbf{\Lambda}_{Q}=\text{Diag}(\lambda_{Q,1},\cdots,\lambda_{Q,L})\]  
_are diagonal matrices whose diagonal entries are the eigenvalues of \(\mathbf{\hat{T}}\) and \(\mathbf{\hat{Q}}\), respectively. The columns of the unitary matrices \(\mathbf{U}_{T}\) and \(\mathbf{U}_{Q}\) consist of the eigenvectors of \(\mathbf{\hat{T}}\) and \(\mathbf{\hat{Q}}\), respectively. For massive MIMO systems, there are several efficient methods or low-complexity approximations to perform the EVDs of covariance matrices with large sizes. These include the power iteration, Lanczos algorithm, Arnoldi iteration, randomized EVD, Jacobi-Davidson, locally optimal block preconditioned conjugate gradient (LOBPCG), etc. Other matrix-free computation approaches based on matrix-vector multiplications can also be used. The optimal training signal \(\mathbf{P}^{\star}\), minimizing the worst-case MSE, has the following structure (this solution generally requires very low signaling overheads for the feedback transmission; specifically, only \(M_{t}\) real numbers and \(M_{t}\) integers need to be fed back to construct the training signal matrix at the transmitter [10]; accordingly, the limited feedback issue may not be a serious design concern in practice):_  
\[\mathbf{P}^{\star}=\mathbf{U}_{T}\mathbf{D}_{P}^{\star}\mathbf{U}_{Q}^{H}, \tag{15}\]  
_and the worst-case correlation error matrices \((\tilde{\mathbf{T}}^{\star},\tilde{\mathbf{R}}^{\star},\tilde{\mathbf{Q}}^{ \star})\), respectively, have the following structures:_  
\[\tilde{\mathbf{T}}^{\star}=\mathbf{U}_{T}\mathbf{D}_{T}^{\star}\mathbf{U}_{T}^ {H},\quad\tilde{\mathbf{R}}^{\star}=\sqrt{\frac{\epsilon_{R}}{M_{r}}}\mathbf{ I}_{M_{r}},\quad\tilde{\mathbf{Q}}^{\star}=\mathbf{U}_{Q}\mathbf{D}_{Q}^{ \star}\mathbf{U}_{Q}^{H}, \tag{16}\]  
_where \(\mathbf{D}_{P}^{\star}\in\mathbb{R}^{M_{t}\times L}\) is a rectangular diagonal matrix containing non-negative elements on its main diagonal, and \(\mathbf{D}_{T}^{\star}\in\mathbb{R}^{M_{t}\times M_{t}}\) and \(\mathbf{D}_{Q}^{\star}\in\mathbb{R}^{1\times L}\) are real diagonal matrices. It thus can be inferred that the worst case corresponds to a diagonal channel covariance matrix and the best case corresponds to a channel covariance matrix whose main diagonal values are close to zero and other elements are close to each other._  
Proof. : See Appendix A. The optimal training structure (15) means that the transmit directions of the training signal should be matched to the eigenvectors of the estimated correlation matrices at the transmitter side, whereas the training power has to be allocated according to the worst-case eigenvalues of the imperfect correlation matrices. This observation actually results from the fact that the uncertainty sets in (9)-(11) do not contain any directional information. Next, the worst-case value of the receiver correlation error seen by the transmitter is given by the scaled identity matrix, i.e., equal perturbation. This is due to the Kronecker model, in which the correlation information at the transmitter and receiver sides can be separable [25].