## 2 System Model and Problem Formulation
### Problem Formulation  
In practice, the matrices \(\mathbf{T}\), \(\mathbf{R}\), and \(\mathbf{Q}\) are imperfect at the transmitter side due to feedback-related issues [15; 16; 17]. Considering such imperfection, we mathematically model the correlation uncertainties as follows:  
\[\mathbf{T}=\mathbf{\hat{T}}+\mathbf{\hat{T}}\in\mathcal{T},\quad\mathbf{R}= \mathbf{\hat{R}}+\mathbf{\hat{R}}\in\mathcal{R}\quad\text{and}\quad\mathbf{Q} =\mathbf{\hat{Q}}+\mathbf{\hat{Q}}\in\mathcal{Q}, \tag{8}\]  
respectively, where \(\mathbf{\hat{T}}\), \(\mathbf{\hat{R}}\), and \(\mathbf{\hat{Q}}\) are imperfect estimates of \(\mathbf{T}\), \(\mathbf{R}\), and \(\mathbf{Q}\), respectively, and \(\mathbf{\hat{T}}\), \(\mathbf{\hat{R}}\), and \(\mathbf{\hat{Q}}\) represent the corresponding correlation error matrices. The uncertainty sets are denoted by \(\mathcal{T}\), \(\mathcal{R}\), and \(\mathcal{Q}\), and they are, respectively, defined as  
\[\mathcal{T} =\Big{\{}\mathbf{\hat{T}}:\|\mathbf{\hat{T}}\|_{F}^{2}\leq \epsilon_{T},\ \ \mathbf{\hat{T}}+\mathbf{\hat{T}}\succeq\mathbf{0}\Big{\}}, \tag{9}\] \[\mathcal{R} =\Big{\{}\mathbf{\hat{R}}:\|\mathbf{\hat{R}}\|_{F}^{2}\leq \epsilon_{R},\ \ \mathbf{\hat{R}}+\mathbf{\hat{R}}\succeq\mathbf{0}\Big{\}}, \tag{10}\]and  
\[\mathcal{Q}=\left\{\tilde{\mathbf{Q}}:\|\tilde{\mathbf{Q}}\|_{F}^{2}\leq e_{T},\ \ \tilde{\mathbf{Q}}+\tilde{\mathbf{Q}}\succeq \mathbf{0}\right\}, \tag{11}\]  
where the parameters \(e_{T}\), \(e_{R}\), and \(e_{Q}\) denote the spherical radii of \(\mathcal{T}\), \(\mathcal{R}\), and \(\mathcal{Q}\), respectively, and they are related to the quantization step size and equal error contour [15; 16; 17]. To guarantee certain performance of the channel estimation for any possible uncertainty on the covariance information, in this paper, we adopt the spherical uncertainty model for the covariance errors as this model is the most uncertain among the uncertainty models. Even when other uncertainty models (e.g., bounded spectral norms or element-wise errors) are adopted, similar results or conclusions to those for the spherical uncertainty model in this paper can still be drawn. Using (8), we can rewrite (7) as  
\[J(\mathbf{P},\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}})=\text{ Tr}\big{(}\tilde{\mathbf{R}}+\tilde{\mathbf{R}}\big{)}\ \cdot\ \text{Tr}\bigg{[}\left\{(\tilde{\mathbf{T}}+\tilde{\mathbf{T}})^{-1}+\mathbf{P} (\tilde{\mathbf{Q}}+\tilde{\mathbf{Q}})^{-1}\mathbf{P}^{\text{H}}\right\}^{- 1}\bigg{]}. \tag{12}\]  
Note from (12) that the estimation MSE is a function of the known parameter \(\mathbf{P}\) as well as the unknown parameters \((\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}})\). To deal with the unknown parameters, we follow the widely used concept of the worst-case robustness [15; 16; 17]. Specifically, we use the worst-case MSE,  
\[J^{\star}(\mathbf{P})=\max_{(\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{ \mathbf{Q}})\in\mathcal{T}\times\mathcal{R}\times\mathcal{Q}}J(\mathbf{P}, \tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}}), \tag{13}\]  
as a design criterion. Considering the training power constraint, an optimization problem of interest can then be formulated as follows:  
\[\min_{\mathbf{P}} \max_{\mathbf{T},\mathbf{R},\mathbf{Q}} J(\mathbf{P},\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}})\] (14a) s.t. \[\text{Tr}(\mathbf{P}\mathbf{P}^{H})\leq\mathcal{P}_{\mathcal{T}} \tag{14b}\] \[\tilde{\mathbf{T}}\in\mathcal{T},\quad\tilde{\mathbf{R}}\in \mathcal{R},\quad\tilde{\mathbf{Q}}\in\mathcal{Q}, \tag{14c}\]  
where \(\mathcal{P}_{T}\) denotes the total transmit power of the training signal.