## 2 System Model of Downlink FDD MMIMO  
In this study, a single-cell wireless communication system is considered in which the BS has an array of \(N\) antennas and connects with \(K\) single-antenna uncorrelated users in the downlink. The users' single-antenna assumption makes low-cost, straightforward hardware with effective power consumption achievable as discussed in [10]. All users obtain the downlink transmission concurrently via the same time frequency resources. The number of users \(K\) is assumed to be limited and less than \(N\), or \(N\gg K\), which is the typical assumption in MMIMO systems as described in [5]. We consider a single-frequency band FDD transmission mode in the downlink. In addition, this work also considers a block fading structure in which \(\tau_{\mathrm{c}}\in\mathbb{Z}^{+}\), where the channel with the LCT block is assumed as in the A-LTE mobile networks [45]. In the training phase, the downlink channel is employed per each coherence block, where the BS can transmit a training sequence in the downlink with a power of \(\rho_{\mathrm{tr}}\) and a length \(\tau_{\mathrm{tr}}\). As a result, the remaining amount of power and time is dedicated for allocating the useful transmission data. The channel should be estimated in the downlink to beamform the data symbols correctly to the targeted uses. The main aim of this research is to investigate maximizing the DASR when the power is optimized with respect to the downlink training sequence and the data phase. A block diagram of the MMIMO systems at the BS with digital precoding is shown in Figure 1. The \(k\)th user's received downlink data signal at the user's side, which is denoted by \(z_{k}\), is provided as follows:  
\[z_{k}=\sqrt{\rho_{\mathrm{d}}}\,\mathbf{h}_{k}^{\mathrm{H}}\mathbf{x}+n_{k}, \tag{1}\]  
where the transmit power that is transmitted in the data phase is given as \(\rho_{\mathrm{d}}\). The downlink transmit vector is represented by the parameter \(\mathbf{x}\). This transmit vector includes the transmitting data symbols and the precoding vector that will be defined later in Equation (4). The additive noise is represented by the parameter \(n_{k}\), which is defined here as a Gaussian distribution with a zero mean. The actual downlink channel vector is denoted by \(\mathbf{h}_{k}\in\mathbb{C}^{N}\). It can be modeled as  
\[\mathbf{h}_{k}=\mathbf{H}^{1/2}\mathbf{\hat{h}}_{k}, \tag{2}\]  
where \(\mathbf{\hat{h}}_{k}\) is the downlink channel vector, and the covariance matrix at the transmitter is given as \(\mathbf{H}=\mathbb{E}\left[\mathbf{h}_{k}\mathbf{h}_{k}^{\mathrm{H}}\right] \in\mathbb{C}^{N\times N}\), which can be decomposed as in [46]:  
\[\mathbf{H}=\mathbf{U}\mathbf{\Sigma}\mathbf{U}^{\mathrm{H}}, \tag{3}\]Therein, we use the eigenvalue decomposition (EVD) method with eigenvectors of \(\mathbf{U}=[\mathbf{u}_{1},\dots,\mathbf{u}_{N}]\in\mathbb{C}^{N\times N}\) and eigenvalues of \(\mathbf{\Sigma}\). Note that the eigenvalues are given as \(\sigma_{1}\geq\sigma_{2}\geq\dots\geq\sigma_{N}\). In practice, obtaining the channel covariance matrix involves either estimating it at the BS or determining the feedback of a quantized version of it from the users. Inaccuracies in estimating or defining the feedback of the covariance matrix can impact the system performance, thereby leading to suboptimal solutions and reduced overall system performance. Since the covariance matrix is represented by an expectation of the instantaneous channel vectors and relies on the scattering environment and the propagation geometry, it can be considered as locally stationary and varies much more slowly than the instantaneous channel of the coherence time [47; 48; 49]. Therefore, the channel covariance matrix can be accurately obtained in both the TDD and FDD transmission modes [50; 51; 52; 53]. To this end, several research works have investigated the system performance when the channel covariance matrix is assumed to be unknown.