## 5 Low-Complexity Hybrid Beamforming
### Digital and Analog Beamforming Matrix Solutions  
This section proposes an alternating minimization method to solve the above problem. Such methods have been successful in solving optimization problems on two variables. The optimization Equation (76) is decoupled by keeping one variable fixed and optimizing iteratively the other variable. The optimization Equation (76) is rewritten as the following equation:  
\[ \begin{split}&\min_{F_{BB}}\bigl{\|}F_{opt}-F_{RF}F_{BB,k}\bigr{\|} _{F}\\ &\min_{F_{RF}}\bigl{\|}F_{opt}-F_{RF}F_{BB,k}\bigr{\|}_{F}^{2}\\ & s.t.\ \bigl{|}\bigl{(}F_{RF}\bigr{)}_{ij}\Bigr{|}=1,\forall i,j \end{split}  \tag{77}\]  
Two sets of alternating minimizations (AltMin) methods are defined in the literature [33]. One of them is a method for high complexity manifold optimization. In this case, the above problem can be solved iteratively by defining a Riemannian manifold. Wefirst use the conjugate line-step gradient method to determine the analog beamformer under the constraints, after which we use the least-squares method to find the digital beamformer. The former uses the conjugate line-step gradient method to determine the analog beamformer under constraints, and the latter uses the least-squares method to find the digital beamformer. The least-squares method is used to find the digital beamformer. The Manifold Optimization Alternating Minimization (MO-AltMin) method offers better performance, as it optimizes both variables simultaneously. However, the complexity of the MO-AltMin method is relatively high in that the update of the simulated beamformer in each iteration involves a line search method, the conjugate gradient method. Therefore, the nested loops in the MO-AltMin method slow down the overall solution process. This requires the study of a hybrid beamforming method with low computational complexity and low performance loss. A low-complexity Phase Extraction Alternating Minimization is designed on this basis. In millimeter-wave massive MIMO of a UAV, the number of antennas at the transmitting end would be large, where \(F_{DD,k}\)\(\in\)\(\mathbb{C}^{N_{RF}^{T}\times N_{S_{k}}}\), \(F_{RF}\)\(\in\)\(\mathbb{C}^{N_{T}\times N_{RF}^{T}}\), \(F_{opt}\)\(\in\)\(\mathbb{C}^{N_{T}\times N_{S_{k}}}\). The time consumed and the corresponding complexity of the matrix \(F_{opt}F_{RF}\) Performing Singular Value Decomposition gradually increases. So, the projection approximation subspace tracking method with lower complexity and better real-time is designed to estimate only \(F_{opt}F_{RF}\) of the right singular matrix \(\overbrace{S}\). This avoids the need for Singular Value Decomposition to obtain the entire left and right singular matrices and the computation of the singular value matrix. In the projection approximation subspace tracking method, the \(F_{opt}F_{RF}\) is the data sample. \(F_{opt}F_{RF}\) is selected from a row in the data sample vector \(\overbrace{x}\). When inputting the \(\overbrace{j}\left(1\leq\overbrace{j}\leq N_{T}\right)\) row \(\overbrace{x}\), it is possible to obtain \(F_{opt}F_{RF}\) of \(\overbrace{s}\) (\(1\leq\overbrace{s}\leq\overbrace{s}\)) column \(\overbrace{B}\)\(\left(\cdot,\overbrace{s}\right)\). \(\overbrace{B}\)\(\left(\cdot,\overbrace{s}\right)\) can be expressed as the following equation:  
\[\overbrace{B} \tag{78}\]  
where \(\overbrace{y}=\overbrace{B}\)\(\overbrace{j}\)\(-1\)\(\left(\cdot,\overbrace{s}\right)\)\(\overbrace{s}\)\(\overbrace{s}\) is the intermediate variable, and \(\overbrace{y}^{*}\) denotes the \(\overbrace{y}\) the conjugate. \(d^{\prime}\)\(\overbrace{j}\)\(\left(\overbrace{s}\right)=d^{\prime}\)\(\overbrace{j}\)\(-1\)\(\left(\overbrace{s}\right)+\left|\overbrace{y}\right|^{2}\) is the step size used for the correction. The method's steps can be described in Algorithm 1. ```
1.Input:\(F_{opt}\), \(F_{RF}\), \(\overbrace{B}\)\(\left(\cdot,\overbrace{B}\right)\)\(\left(\cdot,\overbrace{B}\right)\)\(\left(\cdot,\overbrace{s}\right)\).