## 4 Analytical Expression for the DASR of RZFP in FDD Systems
Thus, using the \(P\)-DoF model, \(\mathbf{H}\) may be represented as  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]0  
where matrix \(\mathbf{Y}\) in Equation (32) is given as \(\mathbf{Y}\in\mathbb{C}^{\text{K}\times P}\), which can be demonstrated as \(\mathcal{CN}\big{(}0,1\big{)}\). Matrix \(\mathbf{B}\) in Equation (32) is denoted by \(\mathbf{B}\in\mathbb{C}^{\text{N}\times P}\), which can be obtained from \(P\leq N\) of an \(N\times N\) matrix to satisfy \(\mathbf{B}^{\text{H}}\mathbf{B}=\mathbf{I}_{P}\) with \(P/N=c\in(0,1]\), which represents the correlation degree [11; 77; 78; 79], and \(\mathbf{H}=\mathbb{E}[\mathbf{h}_{\text{z}}\mathbf{h}_{\text{z}}^{\text{H}}] =\frac{1}{c}\mathbf{B}\mathbf{B}^{\text{H}}\). Using the \(P\)-DoF model and EVD, the expression of MMSE CSI estimation in Equation (16) can be formulated as  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]1  
where \(m=\tau_{\text{tr}}\) if \(\tau_{\text{tr}}\leq P\) and is \(P\) if otherwise. The unitary matrix eigenvectors are given by \(\mathbf{U}_{m}\in\mathbb{C}^{N\times m}\). Now, recall the trace of the MMSE estimation in Equation (17), and, using the \(P\)-DoF model, the MMSE covariance matrix \((\mathbf{\Phi})\) can be simplified as  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]2  
The expression above can become a further simplified version of the MMSE estimation, which is defined as  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]3  
Now, we have obtained a simplified version of the MMSE estimation, and the transmit covariance matrix can also be simplified using the \(P\)-DoF model; hence, we can apply these simplifications to further simplify the \(\overline{\text{SINR}}\) with the RZFP. The main findings of the RZFP in MMIMO systems with the FDD mode and the \(P\)-DoF model are summed up in the following:  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]4  
where \(\tilde{\delta}\) denotes a simplified expression of the fixed-point expressions in Equations (23) and (24) and yields  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]5  
where the parameter \(\tilde{U}\) in Equation(37) is given as \(\tilde{U}=P\zeta\big{(}1+1/\rho_{\text{tr}}\big{)}\), and \(\tilde{\xi}\) and \(\tilde{\Psi}\) denote a simplified expression of Equations(28) and (29), which can be determined from Equations (38) and (39):  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]6  
where \(\tilde{D}\) is given as \(\tilde{D}=K+\tilde{U}+\tilde{\delta}\tilde{U}\), which is used to simplify the above equation:  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]7  
The SINR of the RZFP expression developed in this subsection allows the DASR of the FDD mode MMIMO system to be easily obtained without Monte Carlo simulations, and, hence, optimization with respect to power allocation becomes applicable. In the following section, we will use the closed-form analytical formula developed here for the SINR of the RZFP to investigate the DASR maximization over a wide range of scenarios.