## 7 Solution of The Digital Beamforming Matrix and The Analog Beamforming Matrix
### Solution of The Analog Beamforming Matrix Based on The MM Algorithm  
According to the solution process of the alternating optimization algorithm, we first solve the analog beamforming matrix \(\mathbf{F}_{RF}\) based on the digital beamforming matrix \(F_{BB}\). Thus, the problem \(P_{1}\) can be expressed as:  
\[P_{2}:\min_{\mathbf{F}_{RF}}\left\|\mathbf{F}-\mathbf{F}_{RF}\mathbf{F}_{BB} ^{(n)}\right\|_{F}^{2}, \tag{85}\]  
\[s.t.\ C_{1}:\mathbf{F}_{RF}\in\mathbb{F}, \tag{86}\]  
where \(\mathbf{F}_{BB}^{(n)}\) represents the estimated value of the digital beamforming matrix \(\mathbf{F}_{BB}\) at the \(n\)th iteration. Due to the unit module constraint of elements in \(F_{RF}\), the problem \(P_{2}\) is a nonconvex optimization problem. According to the MM framework theory, the key step is constructing a surrogate function of the objective function in the optimization problem [39]. To construct the surrogate function, we decompose the matrix \(F\) by rows. According to the equivalence of the F-norm and \(L_{2}\)-norm of the vector, the problem \(P_{2}\) can be rewritten as:  
\[P_{3}:\ \min_{\mathbf{F}_{RF}}\sum_{i=1}^{N}\mathbf{F}_{i}^{H}\mathbf{F}_{i}-2 \Re\left(\mathbf{F}_{i}^{H}\mathbf{F}_{BB}^{(n)}\mathbf{F}_{RF,i}\right)+ \mathbf{F}_{RF,i}^{H}\mathbf{F}_{BB}^{(n)}\mathbf{F}_{BB}^{(n)H}\mathbf{F}_{ RF,i}, \tag{87}\]  
\[s.t.\ C_{1}:\mathbf{F}_{RF}\in\mathbb{F}, \tag{88}\]  
where \(\mathbf{F}_{i}^{H}\) represents the \(i\)th row vector of the matrix \(\mathbf{F}\), and \(\mathbf{F}_{RF,i}^{H}\) represents the \(i\)th row vector of the matrix \(\mathbf{F}_{RF}\). It should be noted that the third term \(\textbf{F}_{RF,i}^{H}\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{(n)H}\textbf{F}_{RF,i}\) in (86) is a convex function term, which needs further conversion. According to the first-order Taylor expansion, \(\textbf{F}_{RF,i}^{H}\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{(n)H}\textbf{F}_{RF,i}\) can be converted into:  
\[ \begin{array}{ll}\textbf{F}_{RF,i}^{H}\textbf{F}_{BB}^{(n)H}\textbf{F}_{RF,i }&=\textbf{F}_{RF,i}^{(q)H}\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{(q)H} \textbf{F}_{RF,i}^{(q)}+2\Re\Big{(}\textbf{F}_{RF,i}^{(q)H}\textbf{F}_{BB}^{(n) H}\Big{(}\textbf{F}_{RF,i}-\textbf{F}_{RF,i}^{(q)}\Big{)}\Big{)}\\ &+\Big{(}\textbf{F}_{RF,i}-\textbf{F}_{RF,i}^{(q)}\Big{)}^{H}\textbf{F}_{BB}^{ (n)H}\Big{(}\textbf{F}_{RF,i}-\textbf{F}_{RF,i}^{(q)}\Big{)}\end{array}  \tag{89}\]  
where \(\textbf{F}_{RF,i}^{(q)}\) represents the estimated value of \(\textbf{F}_{RF,i}\) at the \(q\)th iteration. According to the MM algorithm, the surrogate of \(\textbf{F}_{RF,i}^{H}\textbf{F}_{BB}^{(n)H}\textbf{F}_{RF,i}\) can be expressed as follows:  
\[ \begin{array}{ll}\textbf{F}_{RF,i}^{H}\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{ (n)H}\textbf{F}_{RF,i}&\leq\textbf{F}_{RF,i}^{(q)H}\textbf{F}_{BB}^{(n)H} \textbf{F}_{RF,i}^{(q)}+2\Re\Big{(}\textbf{F}_{RF,i}^{(q)H}\textbf{F}_{BB}^{ (n)}\textbf{F}_{BB}^{(n)H}\Big{(}\textbf{F}_{RF,i}-\textbf{F}_{RF,i}^{(q)} \Big{)}\Big{)}\\ &+\Big{(}\textbf{F}_{RF,i}-\textbf{F}_{RF,i}^{(q)}\Big{)}^{H}\textbf{X}^{(n)} \Big{(}\textbf{F}_{RF,i}-\textbf{F}_{RF,i}^{(q)}\Big{)}\end{array}  \tag{90}\]  
where \(\textbf{X}^{(n)}\) is a positive semidefinite matrix and satisfies the constraint \(\textbf{X}^{(n)}\)\(\succeq\)\(\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{(n)H}\), here, we let \(\textbf{X}^{(n)}=\lambda_{\max}\Big{(}\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{( n)H}\Big{)}\textbf{I}\) and \(\lambda_{\max}\Big{(}\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{(n)H}\Big{)}\) represents the maximum eigenvalue of the matrix \(\textbf{F}_{BB}^{(n)}\textbf{F}_{BB}^{(n)H}\). In conclusion, (89) can be further expressed as:  
\[