## 2 System Model  
Our system model is illustrated in Figure 1. We consider a single-cell multi-user massive MIMO downlink system with limited feedback, where a BS with \(M\) transmit antennas serves \(K\) single-antenna users simultaneously. Let \(\mathbf{F}(\triangleq[\mathbf{f}_{1},\cdots,\mathbf{f}_{K}])\in\mathbb{C}^{M \times K}\) be a linear precoding matrix and \(\mathbf{d}(\triangleq[d_{1},...,d_{K}])\in\mathbb{C}^{K\times 1}\) be a data symbol vector for all \(k\in\{1,...,K\}\). Then, the transmit signal vector at the BS denoted by \(\mathbf{x}\in\mathbb{C}^{M\times 1}\) is obtained from \(\mathbf{x}=\mathbf{F}\mathbf{d}\). The received signal denoted by \(\mathbf{y}\in\mathbb{C}^{K\times 1}\) becomes  
\[\mathbf{y}=\mathbf{H}^{\dagger}\mathbf{x}+\mathbf{n}=\mathbf{H}^{\dagger} \mathbf{F}\mathbf{d}+\mathbf{n}, \tag{1}\]  
where \(\mathbf{H}\in\mathbb{C}^{M\times K}\) is a concatenated channel matrix such that \(\mathbf{H}\triangleq[\mathbf{h}_{1},...,\mathbf{h}_{K}]\), where \(\mathbf{h}_{k}\in\mathbb{C}^{M\times 1}\) is the user \(K\)'s channel vector, and \(\mathbf{n}(\triangleq[n_{1},...,n_{K}])\in\mathbb{C}^{K\times 1}\) is additive white Gaussian noise. In this paper, we assume that the BS only utilizes the channel direction information (CDI) for beamforming vector design to save the additional feedback overhead required for power allocation [6; 7]. Therefore, when the total transmit signal power is \(P\), the BS allocates equal powers to each user, i.e., \(\mathbb{E}[|d_{k}|^{2}]=P/K\). Meanwhile, the beamforming vector for user \(k\) should satisfy \(\mathbf{f}_{k}^{\dagger}\mathbf{f}_{k}=1\). In our channel model, we consider the Rayleigh correlated channel such that \(\mathbf{h}_{k}\sim\mathcal{CN}(\mathbf{0},\mathbf{R}_{k})\), where \(\mathbf{R}_{k}\in\mathbb{C}^{M\times M}\) is a positive semi-definite channel covariance matrix represented by \(\mathbf{R}_{k}\triangleq(\mathbf{U}_{k}^{\prime})(\mathbf{\Lambda}_{k}^{ \prime})(\mathbf{U}_{k}^{\prime})^{\dagger}\) with the singular value decomposition (SVD). We denote by \(r_{k}\) the number of non-zero singular values in \(\mathbf{\Lambda}_{k}^{\prime}\). Then, with the Karhunen-Loeve representation, \(\mathbf{h}_{k}\) can be represented by [9; 10; 19]  
\[\mathbf{h}_{k}\triangleq\mathbf{U}_{k}\mathbf{\Lambda}_{k}^{1/2}\mathbf{e}_{ k}, \tag{2}\]  
where \(\mathbf{U}_{k}\in\mathbb{C}^{M\times r_{k}}\) is a tall unitary matrix comprised of \(r_{k}\) column vectors in \(\mathbf{U}_{k}^{\prime}\) corresponding to non-zero singular values, \(\mathbf{\Lambda}_{k}\in\mathbb{R}^{r_{k}\times r_{k}}\) is a diagonal matrix with \(r_{k}\) non-zero positive eigenvalues, and \(\mathbf{e}_{k}\in\mathbb{C}^{r_{k}\times 1}\sim\mathcal{CN}(\mathbf{0}, \mathbf{I}_{r_{k}})\). Assuming the one-ring scattering model along with spatial correlation of the transmitter's uniform linear array (ULA) antennas (as shown in Figure 1), the channel covariance matrix of the \(g\)-th group is given by [9; 10; 19]  
\[[\mathbf{R}_{g}]_{P,q}=\frac{1}{2\Delta_{g}}\int_{\theta_{g}-\Delta_{g}}^{ \theta_{g}+\Delta_{g}}e^{-2\pi\frac{\theta_{g}}{\lambda_{c}}(p-q)\sin\theta} \,d\theta, \tag{3}\]  
where \(p\) and \(q\) denote \((p,q)\)-th element of covariance matrix \(\mathbf{R}_{g}\), \(\lambda_{c}\) is the carrier wavelength, \(\delta_{a}\) is the space between adjacent antennas, and \(\theta_{g}\) and \(\Delta_{g}\) represent the azimuth center angle and angular spread, respectively. With this model, total \(K\) users are divided into \(G\) groups according to the channel  
Figure 1: Illustration of our system model. covariance matrices \(\{\mathbf{R}_{g}\}_{g=1}^{G}\). We denote by \(K_{g}\) the number of users in the \(g\)-th group, so we have \(K=\sum_{g=1}^{G}K_{g}\). We assume that all users in the same group have the same channel covariance matrix, and the BS perfectly knows the covariance matrices of all user groups, i.e., \(\mathbf{R}_{1},\ldots,\mathbf{R}_{G}\). **3.