## 2 System Model
### Two-Stage Precoder
Treating each group separately and exploiting only the CDI of the dimension-reduced effective channel of each group, the IGI should be cancelled out by the outer precoder only on the channel covariance matrices i.e., \(\mathbf{R}_{1},\ldots,\mathbf{R}_{G}\). Hence, the outer precoder for the \(g\)-th group is designed with the criterion given by  
\[\mathbf{H}_{c}^{\dagger}\mathbf{T}_{g}\approx 0\text{ for all }c\
eq g. \tag{6}\]  
According to the approximation (6), we adopt the block diagonalization (BD) method for the design structure of the outer precoder proposed in [10] in order to cancel out the IGI among the other groups, which constructs the precoder by the null-space of the channels of the other groups. Note that the BD method is a generalization of the zero-forcing channel-inversion for multi-user MIMO channels with linear processing [10; 28]. With a similar approach used in [10], the outer precoder for the \(g\)-th group is designed as follows. We define the matrix  
\[\mathbf{Y}_{g}\triangleq[\mathbf{U}_{1}^{\star},\ldots,\mathbf{U}_{g-1}^{ \star},\mathbf{U}_{g+1}^{\star},\ldots,\mathbf{U}_{G}^{\star}], \tag{7}\]  
where \(\mathbf{U}_{c}^{\star}\) is an \(M\times r_{c}^{\star}\) matrix comprised of the dominant eigenvectors of \(\mathbf{U}_{c}\), which is the eigenmatrix of the \(c\)th group covariance matrix such that \(\mathbf{R}_{c}=\mathbf{U}_{c}\mathbf{\Lambda}_{c}\mathbf{U}_{c}^{\dagger}\), i.e., \(r_{c}^{\star}\leq r_{c}\). The dimension of \(\mathbf{Y}_{g}\) is \(M\times\sum_{c\
eq g}^{G}r_{c}^{\star}\) with satisfying \(\sum_{c\
eq g}^{G}r_{c}^{\star}\leq M\). Note that, if \(\sum_{g=1}^{G}r_{g}\leq M\), we can choose \(r_{c}^{\star}=r_{c}\) to reflect the eigenvectors of the other groups exactly. Thus, we assume \(r_{c}^{\star}=r_{c}\), (\(\forall c\
eq g\)) in the definition (7) to construct \(\mathbf{Y}_{g}\) for group \(g\in\{1,2,...,G\}\). Using the SVD, \(\mathbf{Y}_{g}\) can be expressed as  
\[\mathbf{Y}_{g}=\mathbf{\Psi}_{g}\mathbf{\Sigma}_{g}[\mathbf{\Phi}_{g},\ \mathbf{\Phi}_{g}^{\perp}]^{\dagger}, \tag{8}\]  
where \(\mathbf{\Phi}_{g}^{\perp}\) is an \(M\times(M-\sum_{c\
eq g}^{G}r_{c})\) sub-unitary matrix ( \(\mathbf{\Phi}_{g}^{\perp}\mathbf{\Phi}_{g}^{\perp}=\mathbf{I}_{M-\sum_{c\
eq g }^{G}r_{c}}\) ) comprised of the orthogonal bases of the null-space of \(\mathbf{Y}_{g}\). After projecting onto the null-space, i.e., \(\text{Span}(\mathbf{\[PERSON]}_{g}^{\perp})\), the covariance matrix of the projected channel is obtained by  
\[\mathbf{R}_{g} =\left(\mathbf{\[PERSON]}_{g}^{\perp}\right)^{\dagger}\mathbf{R}_{g} \mathbf{\[PERSON]}_{g}^{\perp}\] \[=\mathbf{\tilde{U}}_{g}\mathbf{\tilde{\Lambda}}_{g}\mathbf{ \tilde{U}}_{g}^{\dagger}, \tag{9}\]  
where (9) is obtained from the SVD. Selecting the dominant \(s_{g}\) (\(\leq r_{g}\)) eigenmodes of \(\mathbf{\tilde{R}}_{g}\), we can construct the dimension-reduced effective channel for group \(g\) according to the BD method. Consequently, the outer precoder of the \(g\)-th group, i.e., \(\mathbf{T}_{g}\in\mathbb{C}^{M\times s_{g}}\), is given by  
\[\mathbf{T}_{g}=\mathbf{\Phi}_{g}^{\perp}\mathbf{\tilde{U}}_{g}^{\star}, \tag{10}\]  
where \(\mathbf{\tilde{U}}_{g}^{\star}\) contains the dominant \(s_{g}\) eigenvectors of \(\mathbf{\tilde{U}}_{g}\), where \(s_{g}\) is the design parameter to be optimized. Note that it is satisfied that \(\mathbf{T}_{g}^{\dagger}\mathbf{T}_{g}=\left(\mathbf{\tilde{U}}_{g}^{\star} \right)^{\dagger}\left(\mathbf{\Phi}_{g}^{\perp}\right)^{\dagger}\mathbf{ \Phi}_{g}^{\perp}\mathbf{\tilde{U}}_{g}^{\star}=\left(\mathbf{\tilde{U}}_{g}^{ \star}\right)^{\dagger}\mathbf{\tilde{U}}_{g}^{\star}=\mathbf{I}_{s_{g}}\). For the inner precoder, we adopt the zero-forcing (ZF) beamforming to mitigate multiuser interference among the users in each group (i.e., SGI). The beamforming vector of user \(k\) in the \(g\)-th group is constructed by null-space of the effective channel vectors of all the other users in the \(g\)-th group.