# Joint Hybrid Beamforming Design for Millimeter Wave Amplify-and-Forward Relay Communication Systems
## 3 Hybrid Beamforming Design for HD AF Relay Systems
### MO-Based HBF Design  
It can be observed from the optimization problem (7) that there exist nine variables and four nonconvex constraints, which is tough to get the global optimal solution. In this subsection, an alternating optimization algorithm on the basis of the MO method is proposed. To overcome the difficulty of solving the original problem, it is decomposed into three HBF design sub-problems [25; 26]. _(1) The Source Node Design_: When \(\mathbf{G}\) and \(\mathbf{W}\) are fixed, the sub-problem can be further expressed as  
\[\underset{\gamma_{1},\mathbf{F}_{\text{RF}},\mathbf{F}_{\text{U}}}{ \text{min}} \text{IMSE}\] (8a) s.t. \[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\] \[\mathbb{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{\text{F}}^{2}\Big{\}} =P_{\text{B}}. \tag{8c}\]  
According to \(\mathbf{F}=\gamma_{1}\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{U}}\), (8c) can be rewritten as \(\gamma_{1}^{2}\operatorname{tr}\left(\overline{\mathbf{F}}\mathbf{F}^{H} \right)=P_{\text{B}}\), and we can obtain  
\[\gamma_{1}=\left[\frac{\operatorname{tr}\left(\mathbf{F}_{\text{RF}}\mathbf{ F}_{\text{U}}\mathbf{F}_{\text{RF}}^{H}\right)}{P_{\text{B}}}\right]^{-\frac{1}{2}}. \tag{9}\]  
To this end, the Lagrange function \(\operatorname{L1}(\lambda_{1},\mathbf{F}_{\text{U}},\gamma_{1})\) of problem (8) can be constructed. The minimum of \(\operatorname{L1}(\lambda_{1},\mathbf{F}_{\text{U}},\gamma_{1})\) is achieved when its partial derivatives equal to zero, which gives the optimal \(\mathbf{F}_{\text{U}}\) as  
\[\mathbf{F}_{\text{U}}=\Big{(}\mathbf{F}_{\text{RF}}^{H}\mathbf{ H}_{1}^{H}\overline{\mathbf{G}}^{H}\mathbf{H}_{2}^{H}\mathbf{W}\mathbf{W}^{H} \mathbf{H}_{2}\overline{\mathbf{G}}\mathbf{H}_{1}\mathbf{F}_{\text{RF}}+ \mathbf{B}_{11}\Big{)}^{-1}\mathbf{F}_{\text{RF}}^{H}\mathbf{H}_{1}^{H} \overline{\mathbf{G}}^{H}\mathbf{H}_{2}^{H}\mathbf{W}, \tag{10}\]  
with \(\mathbf{B}_{11}\)=\(b_{11}\big{(}\mathbf{F}_{\text{RF}}^{H}\mathbf{F}_{\text{RF}}\big{)}\) and \(b_{11}=\frac{\sigma_{\text{B}}^{2}}{\mathbf{B}}\text{tr}\Big{(}\mathbf{W}^{H} \mathbf{H}_{2}\overline{\mathbf{G}}\overline{\mathbf{G}}^{H}\ \ \mathbf{H}_{2}^{H} \mathbf{W}\Big{)}+\frac{\sigma_{\text{D}}^{2}}{\mathbf{B}}\gamma_{2}^{-2} \text{tr}\big{(}\mathbf{W}^{H}\mathbf{W}\big{)}.\)  
One more step, substituting (8a) with (10), the optimization sub-problem can be concisely re-formulated as  
\[\underset{\mathbf{F}_{\text{RF}}}{\text{min}} \text{T1}(\mathbf{F}_{\text{RF}})\] (11a) s.t. \[\text{s.t. }\text{s.t}\big{|}\big{|}\mathbf{F}_{\text{RF}}\big{|}_{a,b} \big{|}^{2}=1,\forall a,b, \tag{11b}\]  
where we denote  
\[\text{T1}(\mathbf{F}_{\text{RF}})=\text{tr}\bigg{\{}\Big{[}\mathbf{I}_{N_{ \text{S}}}+\Big{(}\mathbf{W}^{H}\mathbf{H}_{2}\overline{\mathbf{G}}\mathbf{H} _{1}\mathbf{F}_{\text{RF}}\Big{)}\,\mathbf{B}_{11}^{-1}\Big{(}\mathbf{W}^{H} \mathbf{H}_{2}\overline{\mathbf{G}}\mathbf{H}_{1}\mathbf{F}_{\text{RF}}\Big{)} ^{H}\Big{]}^{-1} \tag{12}\]  
We can see that (11) is an optimization problem with only a constant-modulus constraint. Om the grounds of [21; 27], we can tackle it by applying the MO method. However, it is difficult to use MO in the Euclidean space directly. Fortunately, if the gradient of objective function in Euclidean space is derived, the above problem can be easily solved [27].