## 2 System Model
### Limited Feedback Method with a Two-Stage Precoder  
In the previous section, the dimension of the effective channel of each user in group \(g\) is reduced to \(s_{g}\) by the outer precoder. Hence, we focus on the the limited feedback system to acquire the CDI of the dimension-reduced effective channel of each group, i.e., \(\mathbf{H}_{g}^{\text{eff}}\). We define \(\mathbf{h}_{g,k}^{\text{eff}}\in\mathbb{C}^{\mathbf{s}_{g}\times 1}\) the dimension-reduced effective channel of user \(k\) in the \(g\)-th group as  
\[\mathbf{h}_{g,k}^{\text{eff}}\triangleq\mathbf{T}_{g}^{\dagger}\mathbf{h}_{g,k}. \tag{13}\]  
Given the covariance matrix \(\mathbf{R}_{g}\) and the outer precoder \(\mathbf{T}_{g}\), each user only needs to quantize its effective channel (\(\mathbf{h}_{g,k}^{\text{eff}}\)) and feeds the quantized channel back to the BS. For quantization of the effective channel, we adopt the random vector quantizer (RVQ), which is widely used to analyze the effects of quantization error and performs close to optimal quantization with a Rayleigh fading channel environment [6; 7; 9]. Using the RVQ, the Rayleigh component of the effective channel should be quantized according to (13). By the Karhunen-Loeve representation, the effective channel in (13) can be decomposed by the SVD [13], given as  
\[\mathbf{h}_{g,k}^{\text{eff}}=\mathbf{T}_{g}^{\dagger}\mathbf{U}_{g}\mathbf{ \Lambda}_{g}^{1/2}\mathbf{e}_{g,k}=\mathbf{\Omega}_{g}\bar{\mathbf{\Sigma}}_{ g}\mathbf{T}_{g}^{\dagger}\mathbf{e}_{g,k}=\mathbf{\Omega}_{g}\mathbf{\Sigma}_{ g}\mathbf{w}_{g,k}, \tag{14}\]  
where \(\mathbf{\Omega}_{g}\bar{\mathbf{\Sigma}}_{g}\mathbf{T}_{g}^{\dagger}\left( \mathbf{\Omega}_{g}\in\mathbb{C}^{\mathbf{s}_{g}\times\mathbf{s}_{g}}, \bar{\mathbf{\Sigma}}_{g}\in\mathbb{C}^{\mathbf{s}_{g}\times\mathbf{r}_{g}}, \mathbf{\Gamma}_{g}\in\mathbb{C}^{\mathbf{r}_{g}\times\mathbf{r}_{g}}\right)\) is SVD of \(\mathbf{T}_{g}^{\dagger}\mathbf{U}_{g}\mathbf{\Lambda}_{g}^{1/2}\), and \(\mathbf{\Sigma}_{g}\in\mathbb{C}^{\mathbf{s}_{g}\times\mathbf{s}_{g}}\) is the matrix comprised of the first \(s_{g}\) columns of \(\bar{\mathbf{\Sigma}}_{g}\), i.e., the diagonal matrix with \(s_{g}\) non-zero positive eigenvalues, and \(\mathbf{w}_{g,k}\in\mathbb{C}^{\mathbf{s}_{g}\times 1}\) is the vector with the first \(s_{g}\) elements of \(\mathbf{Y}_{g}^{\dagger}\mathbf{e}_{g,k}\). Note that \(\mathbf{w}_{g,k}\) follows the distribution of \(\mathcal{CN}(\mathbf{0},\mathbf{I}_{s_{g}})\), and we have  
\[\mathbf{T}_{g}^{\dagger}\mathbf{R}_{g}\mathbf{T}_{g}=\left(\mathbf{\Omega}_{g} \mathbf{\Sigma}_{g}\right)\left(\mathbf{\Omega}_{g}\mathbf{\Sigma}_{g}\right)^ {\dagger} \tag{15}\]  
due to the fact that \(\mathbb{E}\left[\mathbf{h}_{g,k}^{\text{eff}}\left(\mathbf{h}_{g,k}^{\text{ eff}}\right)^{\dagger}\right]=\mathbf{T}_{g}^{\dagger}\mathbb{E}\left[\mathbf{h}_{g,k} \mathbf{h}_{g,k}^{\text{H}}\right]\mathbf{T}_{g}=\mathbf{T}_{g}^{\dagger} \mathbf{R}_{g}\mathbf{T}_{g}\) from (13), and \(\mathbb{E}\left[\mathbf{h}_{g,k}^{\text{eff}}\left(\mathbf{h}_{g,k}^{\text{ eff}}\right)^{\dagger}\right]=\left(\mathbf{\Omega}_{g}\mathbf{\Sigma}_{g} \right)\mathbb{E}\left[\mathbf{w}_{g,k}\mathbf{w}_{g,k}^{\text{H}}\right] \left(\mathbf{\Omega}_{g}\mathbf{\Sigma}_{g}\right)^{\dagger}=\left(\mathbf{ \Omega}_{g}\mathbf{\Sigma}_{g}\right)\left(\mathbf{\Omega}_{g}\mathbf{\Sigma}_{ g}\right)^{\dagger}\) from (14) and \(\mathbf{w}_{g,k}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{s_{g}})\), respectively. Allocating \(B\)-bits to each user's feedback size and, using the RVQ, a quantization codebook for user \(k\) in the \(g\)-th group, i.e., \(\mathcal{C}_{g,k}=\{\mathbf{c}_{g,k,1},...,\mathbf{c}_{g,k,2^{B}}\}\), consists of \(2^{B}\) randomly chosen isotropic \(s_{g}\)-dimensional unit norm vectors.