## 2 System Model of Downlink FDD MMIMO
For example, the work of [54] proposed a resampling method for obtaining the channel covariance matrix using cubic splines. In addition, a dictionary of known uplink/downlink covariance matrices was proposed in [55]. In this approach, upon encountering a new uplink covariance matrix, the associated downlink covariance matrix was approximated by interpolating across a Riemannian space, thereby utilizing the elements present in the dictionary. The works of [56; 57; 43] proposed a potential approach for estimating a high-dimensional covariance matrix with limited observations involving the application of regularization to the sample covariance matrix. The work of [58] proposed a spatial spectrum method for the quantization and the feedback of the covariance matrix. The work of [59] proposed a codebook-based approach to feedback quantized information with respect to the effective eigenvectors and eigenvalues of the channel covariance matrix. The work of [60] proposed an angle estimation with a power angular spectrum (PAS) that can be extracted from the instantaneous uplink channel estimate to obtain the uplink channel covariance matrix. Then, the downlink channel covariance matrix is reconstructed using the inference of the uplink one. Recently, a limited channel covariance feedback approach was proposed in [61], where the user feedback provided only partial information regarding the covariance matrix of the BS. The investigations above have shown that the system performance using partial channel covariance knowledge suffers from minor losses compared to perfect channel covariance knowledge. It is worth noting that the proposed approach in this paper does not necessitate full knowledge of the channel covariance matrix. Instead, only the effective eigenvectors need to be obtained. Indeed, obtaining this partial information about the channel covariance matrix is more straightforward compared to dealing with the entire (complete) covariance matrix. Figure 1: Block diagram of MMIMO systems with precoder at BS. Therefore, a more precise evaluation of the performance degradation under different levels of covariance matrix estimation errors can be investigated in the future. The transmitted signal is given by the vector \(\mathbf{x}\in\mathbb{C}^{N}\), which is placed in Equation (1) and defined here as follows:  
\[\mathbf{x}=\sqrt{\sigma}\mathbf{V}\mathbf{a}, \tag{4}\]  
where \(\mathbf{a}=[a_{1},\dots,a_{K}]^{\mathrm{T}}\in\mathbb{C}^{K}\) denotes the useful data vector, which is modeled with a zero mean and a variable unit to satisfy \(\big{[}\mathbf{a}\mathbf{a}^{\mathrm{H}}\big{]}=\mathbf{I}_{K}\), and \(\mathbf{V}=[\mathbf{V}_{1},\dots,\mathbf{V}_{K}]\in\mathbb{C}^{N\times K}\) represents the precoder at the BS, where the type of precoder used in this paper is defined later in (6). This precoder depends on the channel estimation. We need to normalize the power in order to satisfy \(\mathbb{E}\big{[}\|\mathbf{x}\|^{2}\big{]}=K\) and ensure the power used for the transmit data as \(\rho_{\mathrm{d}}\). As such, the parameter \(\alpha\) given in Equation (4) is defined here, which represents the normalization coefficient to meet the above requirement as described in [11] so that  
\[\alpha=\frac{K}{\mathbb{E}\big{[}\mathrm{tr}\big{(}\mathbf{V}\mathbf{V}^{ \mathrm{H}}\big{)}\big{]}}. \tag{5}\]  
In this paper, a RZFP technique is used. To this end, the precoder matrix can be formulated as in [11]:  
\[\mathbf{V}=\mathbf{\hat{H}}^{\mathrm{H}}\big{(}\mathbf{\hat{H}}\mathbf{\hat{ H}}^{\mathrm{H}}+N\varrho\mathbf{I}_{K}\big{)}^{-1}, \tag{6}\]  
where \(\mathbf{\hat{H}}\in\mathbb{C}^{K\times N}\) represents the estimated channel with \(\mathbf{\hat{H}}=[\mathbf{\hat{h}}_{1},\mathbf{\hat{h}}_{2},\dots,\mathbf{ \hat{h}}_{K}]^{\mathrm{H}}\in\mathbb{C}^{K\times N}\), and \(\varrho\) denotes the regularization parameter of the RZFP, which is considered to be \(1/\rho_{\mathrm{d}}\), as given in [11].