## 4 Analytical Expression for the DASR of RZFP in FDD Systems  
The SINR in the downlink is provided analytically here and is based on the theory of the random matrix method [11; 75; 76]. Let us assume that the users have a common correlation at the BS so that the covariance matrix is \(\mathbf{R}_{k}=\mathbf{R}\ \forall k=1,\ldots,K\), and, hence, \(\text{SINR}_{k}=\text{SINR}\ \forall k=1,\ldots,K\), as considered in [20; 21; 22; 24]. Specifically, \(\overline{\text{SINR}}\), an analytical expression of the SINR is calculated as \(\text{SINR}-\overline{\text{SINR}}\)\(\xrightarrow[N\to\infty]{}0\). A straightforward system evaluation is obtained by replacing the SINR SINR term in Equation (9) with the approximation provided in this section thanks to the theory of the random matrix. Therefore, the results can be easily reproducible. The expression for SINR of the RZFP in the MMIMO system with the FDD mode is written as  
\[\overline{\text{SINR}}=\frac{KN\,\bar{\omega}\,\delta^{2}}{\frac{(1+\delta)^{ 2}}{\rho_{4}}+K^{2}\bar{\omega}\overline{\Psi}}, \tag{22}\]  
where the fixed-point technique is used to find the parameter \(\delta\), and the variable \(\bar{\omega}\) is the analytical counterpart of the RZFP with  
\[\delta^{(t)}=\frac{1}{N}\text{tr}\Biggl{(}\Theta\left(\frac{K\,\Theta}{N\bigl{(} 1+\delta^{(t-1)}\bigr{)}}+\varrho\mathds{I}_{N}\right)^{-1}\Biggr{)}, \tag{23}\]where the integer \(t=1,2,\dots\), has the starting value \(\delta^{(0)}=1/\varrho\), and \(\delta\) is determined after the fixed-point technique in Equation (24) is converged. Thus, we have the following:  
\[\delta=\lim_{t\to\infty}\delta^{(t)}. \tag{24}\]  
Once we obtain the value of \(\delta\) in Equations (23) and (24), it is substituted into  
\[\mathbf{T}=\left(\frac{K\mathbf{\Theta}}{N(1+\delta)}+\varrho\mathbf{I}_{N} \right)^{-1}, \tag{25}\]  
with  
\[\bar{\mathbf{T}}=\mathbf{T}\Bigg{(}\mathbf{I}_{N}+\frac{\mathbf{\Theta} \bar{\delta}K}{(1+\delta)^{2}N}\Bigg{)}\mathbf{T}, \tag{26}\]  
with \(\bar{\delta}\) being obtained as  
\[\bar{\delta}=\frac{1}{N}\Bigg{(}1-\frac{\text{tr}\big{(}\mathbf{\Theta} \mathbf{T}\mathbf{\Theta}\mathbf{T}\big{)}}{\big{(}N(1+\delta)\big{)}^{2}} \Bigg{)}^{-1}\text{tr}\big{(}\mathbf{T}\mathbf{\Theta}\mathbf{T}\big{)}. \tag{27}\]  
The RZFP normalization \(\bar{\omega}\) is obtained as  
\[\bar{\omega}=\big{(}\text{tr}(\mathbf{T})-\varrho\text{tr}(\bar{\mathbf{T}}) \big{)}^{-1}, \tag{28}\]  
with \(\bar{\Psi}\) being obtained in Equations (29)-(31):  
\[\mathbf{\Psi}=\text{tr}\big{(}\mathbf{H}\mathbf{T}^{\prime}\big{)}\frac{1}{N} -\frac{2\big{(}\text{tr}\big{(}\mathbf{\Theta}\mathbf{T}\mathbf{\Theta} \mathbf{T}^{\prime}\big{)}\big{)}\big{(}1+\delta\big{)}-\text{tr}\big{(} \mathbf{\[PERSON]}\mathbf{T}\big{)}^{2}\delta^{\prime}}{\big{(}(1+\delta)N\big{)} ^{2}}, \tag{29}\]  
\[\mathbf{T}^{\prime}=\mathbf{T}\Bigg{(}\mathbf{\[PERSON]}+\frac{\mathbf{\[PERSON]}K \delta^{\prime}}{(1+\delta)^{2}N}\Bigg{)}\mathbf{T}, \tag{30}\]  
\[\delta^{\prime}=\frac{1}{N}\Bigg{(}1-\frac{\text{tr}\big{(}\mathbf{\[PERSON]} \mathbf{T}\mathbf{\Theta}\mathbf{T}\big{)}}{N^{2}\big{(}(1+\delta)\big{)}^{2} }\Bigg{)}^{-1}\text{tr}\big{(}\mathbf{T}\mathbf{\Theta}\mathbf{T}\mathbf{ \Theta}\big{)}. \tag{31}\]  
The SINR equation given in Equation (22) represents the analytical expression of the SINR, which applies to all models of channel correlation. Simplified Analytical Expression for the DASR of RZFPWe use the \(P\)-DoF model to further simplify the analysis of the SINR and, consequently, the DASR. The correlations in the channel depend on how many degrees of freedom there are in this channel. Nevertheless, in MMIMO, the channel's degrees of freedom may be substantially less than \(N\). The \(P\)-DoF model as defined in [11; 77; 78; 79] is presented in this subsection.