## 3 Power Allocation Optimization and Problem Formulations
### Power Allocation Formulation  
Let \(\mathcal{E}\) represent the total amount of energy at the base station. This is found by multiplying the available power at the base station \(\rho\)--here, it is SNR--by the overall coherence time \(\tau_{\text{c}}\), as provided in Equation (18):  
\[\mathcal{E}=\rho\,\tau_{\text{c}}. \tag{18}\]  
To perform downlink channel estimation in the training phase, the base station sends a \(\tau_{\text{tr}}\) training sequence length to the user per each transmission block. The remaining duration is dedicated to data transmission, which can be used to send useful data. Furthermore, the amount of the total transmit energy at the base station can be freely distributed with respect to the data phase and channel estimation phase. Therefore, the energy allocated to the training sequence \(\mathcal{E}_{\text{tr}}\) can be expressed as  
\[\mathcal{E}_{\text{tr}}=\epsilon\mathcal{E}, \tag{19}\]  
where \(\epsilon\in(0,1]\) represents the fraction of energy used for the channel estimation phase. As a result, the amount of energy that is still available for the data phase \(\mathcal{E}_{\text{d}}\) can be written as  
\[\mathcal{E}_{\text{d}}=(1-\epsilon)\mathcal{E}, \tag{20}\]  
where \(\rho_{\text{tr}}=\mathcal{E}_{\text{tr}}/\tau_{\text{tr}}\), and \(\rho_{\text{d}}=\mathcal{E}_{\text{d}}/\tau_{\text{d}}\). In this case, by using the fraction of energy dedicated to the training \(\epsilon\), we are controlling the amount of energy dedicated to the channel estimation and data transmission. Hence, for example, we can allocate more energy to the channel estimation to improve the quality of estimation while perceiving some time duration to send useful data to the user. The connection between optimizing the energy allocation for the data transmission and channel estimation phase and maximizing the DASR in MMIMO systems with the FDD mode and LCT is illustrated clearly. In particular, improving training sequence energy becomes imperative to enhance CSI estimation accuracy in the downlink. This signifies that allocating an increased power to the training sequence during the training phase leads to an enhancement in CSI estimation accuracy. The obtained training sequence matrix must always satisfy \(\text{tr}\big{(}\mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{S}_{\text{tr}}\big{)} =\rho_{\text{tr}}\tau_{\text{tr}}\) to ensure constant power as transmitted by the base station. Therefore, the total energy allocated to the channel estimation is given as \(\mathcal{E}_{\text{tr}}=\rho_{\text{tr}}\tau_{\text{tr}}\), which is equivalent to the energy expended during the downlink training period. The data transmission would use the remaining energy as explained clearly above. Note that in the realistic LCT scenarios, which are considered in this paper, increasing the training sequence length \(\tau_{\text{tr}}\) would reduce the DASR, since it will affect the remaining time for sending useful data to users. The main contribution in this present paper is derived from the maximization of the DASR using an optimized power allocation procedure. Specifically, by thoroughly going over every conceivable value for \(\epsilon\), the optimal power allocation that maximizes the DASR with each \(N\) is obtained. In the FDD transmission mode, the DASR is maximized for each value of \(N\) in relation to a suitable weighted energy value. Increasing the power allocation with respect to the training sequence phase can lead to an improvement in the DASR, as we will see later in the results section in Section 5. Note that machine learning (ML) algorithms can be integrated into the modeling framework of power allocation [64; 71; 72; 63]. However, a recognizable challenge could be the potential lack of interpretability in the decision-making process, thus making it challenging to understand and validate the underlying reasoning behind power allocation choices. Therefore, this paper provides a feasible and tractable solution for optimizing the power allocation in the FDD system with straightforward implications.