## 5 Low-Complexity Hybrid Beamforming
### Hybrid Beamforming Based on Spectral Efficiency Maximization
\begin{array}{c}V^{\prime H}_{\rm{kc}}F_{RF}F_{BB,k}F_{RF}{}^{H}F_{BB,k}{}^{ H}V^{\prime}_{\rm{kc}}=\begin{bmatrix}V^{\prime H}_{\rm{kc},1}F_{RF}F_{BB,k}F_{RF}{}^{H}F_{BB,k} {}^{H}V^{\prime}_{\rm{kc},1}&V^{\prime H}_{\rm{kc},1}F_{RF}F_{BB,k}F_{RF}{}^{H}F _{BB,k}V^{\prime}_{\rm{kc},2}\\ V^{\prime H}_{\rm{kc}}F_{RF}F_{BB,k}F_{RF}{}^{H}F_{BB,k}V^{\prime}_{\rm{kc},1}&V^ {\prime H}_{\rm{kc},2}F_{RF}F_{BB,k}F_{RF}{}^{H}F_{BB,k}V^{\prime}_{\rm{kc},2} \end{bmatrix}\\ \approx\begin{bmatrix}\overline{\mathbb{Q}}&0\\ 0&0\end{bmatrix}\end{array}  \tag{74}\]  
The spectral efficiency can then be approximated by deducing as follows:\[R =\log_{2}\biggl{(}\biggl{|}I_{kN_{k},k}+\frac{\dot{p^{{}^{\prime}}}}{ kN_{k}\omega_{n}^{2}}\Bigl{(}\Sigma^{\prime}{}_{kc}^{2}\Bigr{)}V^{H}{}_{kc}F_{RF}F_{BB,k}F_{ RF}{}^{H}F_{BB,k}HV^{\prime}{}_{kc}\biggr{|}\biggr{)} \tag{75}\] \[\qquad\approx\log_{2}\biggl{(}\biggl{|}I_{kN_{k},k}+\frac{\dot{p^{ {}^{\prime}}}}{kN_{k}\omega_{n}^{2}}\Bigl{[}\Sigma^{\prime}{}_{kc,1}^{2}\Bigr{]} 0\biggr{.}\biggl{.}\biggl{. }\biggl{|} \begin{matrix}\overline{Q}&0\\ 0&0\end{matrix} \biggr{|}\biggr{)}\] \[=\log_{2}\biggl{(}\biggl{|}I_{kN_{k},k}+\frac{\dot{p^{{}^{\prime} }}}{kN_{k}\omega_{n}^{2}}\Sigma^{\prime}{}_{kc,1}^{2}\overline{Q}\biggr{|} \biggr{)}\] \[=\log_{2}\biggl{(}\biggl{|}I_{kN_{k},k}+\frac{\dot{p^{{}^{\prime} }}}{kN_{k}\omega_{n}^{2}}\Sigma^{\prime}{}_{kc,1}^{2}\overline{Q}\biggr{|} \biggr{)}\] \[=\log_{2}\biggl{(}\biggl{|}I_{kN_{k},k}+\frac{\dot{p^{{}^{\prime} }}}{kN_{k}\omega_{n}^{2}}\Sigma^{\prime}{}_{kc,1}\biggr{|}\biggr{)}\] \[\qquad+\log_{2}\Biggl{(}\biggl{|}I_{kN_{k},k}-\Bigl{(}I_{kN_{k},k} +\frac{\dot{p^{{}^{\prime}}}}{kN_{k}\omega_{n}^{2}}\Sigma^{\prime}{}_{kc,1}^{2} \biggr{)}^{-1}\frac{\dot{p^{{}^{\prime}}}}{kN_{k}\omega_{n}^{2}}\Sigma^{\prime }{}_{kc,1}^{2}\Bigl{(}I_{kN_{k},k}-\overline{Q}\biggr{)}\biggr{|}\biggr{)}\] \[\overset{a}{\approx}\log_{2}\biggl{(}\biggl{|}I_{kN_{k},k}+\frac{ \dot{p^{{}^{\prime}}}}{kN_{k}\omega_{n}^{2}}\Sigma^{\prime}{}_{kc,1}^{2} \biggr{|}\biggr{)}-tr\Bigl{(}I_{kN_{k},k}-V^{H}{}_{kc,1}F_{RF}F_{BB,k}F_{RF}F_ {RF}{}^{H}F_{BB,k}HV^{\prime}{}_{kc,1}\Bigr{)}\] \[=\log_{2}\biggl{(}\biggl{|}I_{kN_{k},k}+\frac{\dot{p^{{}^{\prime} }}}{kN_{k}\omega_{n}^{2}}\Sigma^{\prime}{}_{kc,1}^{2}\biggr{|}\biggr{)}-\biggl{(} kN_{s,k}-\Bigl{\|}V^{H}{}_{kc,1}F_{RF}F_{BB,k}\Bigr{\|}_{F}^{2}\biggr{)}\]  
where \(\overset{a}{\approx}\) denotes the two approximations \(\biggl{(}I_{kN_{s,k}}+\frac{\dot{p^{{}^{\prime}}}}{kN_{k}\omega_{n}^{2}}\Sigma ^{\prime}{}_{kc,1}^{2}\biggr{)}^{-1}\frac{\dot{p^{{}^{\prime}}}}{kN_{k}\omega _{n}^{2}}\Sigma^{\prime}{}_{kc,1}^{2}\approx I_{kN_{s,k}}\). When the \(\tilde{Y}\) values are not very large, \(\log_{2}\bigl{|}I_{kN_{s}}-\tilde{Y}\bigr{|}\approx\log_{2}\bigl{(}1-tr\Bigl{(} \tilde{Y}\Bigr{)}\bigr{)}\approx-tr\Bigl{(}\tilde{Y}\Bigr{)}\), \(\overline{Q}=V^{\prime H}_{kc,1}F_{RF}F_{BB,k}F_{RF}{}^{H}F_{BB,k}HV^{\prime}{}_ {kc,1}\), and \(\tilde{Y}=\log_{2}\bigl{|}I_{kN_{s,k}}-\overline{Q}\bigr{|}\). The system spectral efficiency is maximizing \(\bigl{\|}V^{H}{}_{kc,1}F_{RF}F_{BB,k}\bigr{\|}_{F}\), which represents the optimal beamforming matrix \(F_{opt}=V^{\prime}{}_{kc,1}\) and \(F_{RF}F_{BB,k}\) the chord distance in the Grassmann manifold. The problem can be expressed as \(\bigl{\|}F_{opt}-F_{RF}F_{BB,k}\bigr{\|}_{F}\). The beamforming optimization problem for a millimeter-wave massive MIMO UAV is expressed as follows:  
\[ \begin{split}\Bigl{(}F_{RF}^{opt},F_{BB}^{opt}\Bigr{)}& =\arg\min_{F_{RF},F_{BB}}\bigl{\|}F_{opt}-F_{RF}F_{BB,k}\bigr{\|}_{F}\\ & s.t.\ F_{RF}F_{RF}F_{RF}\\ &\bigl{\|}F_{RF}F_{BB,k}\bigr{\|}_{F}^{2}=kN_{s}\end{split}  \tag{76}\]  
Because the constraint \(F_{RF}\epsilon\mathcal{F}_{RF}\), that is \(F_{RF}\) the constant mode constraint on the column elements makes the problem non-convex and difficult to find an optimal solution, there is a need to improve the performance of the system while reducing the complexity and solving the non-convex optimization problem.