# 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
The gradient of \(\text{T1}(\mathbf{F}_{\text{RF}})\) with respect to \(\mathbf{F}_{\text{RF}}\) is defined as \(\
abla\text{T1}(\mathbf{F}_{\text{RF}})\)=\(\frac{\partial\text{T1}(\mathbf{F}_{\text{RF}})}{\partial\mathbf{F}_{ \text{RF}}^{H}}\)[28], which is given by  
\[\
abla\text{T1}(\mathbf{F}_{\text{RF}})=\Big{(}b_{11}\mathbf{F}_{\text{RF}} \mathbf{B}_{11}^{-1}\mathbf{F}_{\text{RF}}^{H}-\mathbf{I}_{N_{\text{T}}} \Big{)}\Big{(}\mathbf{W}^{H}\mathbf{H}_{2}\overline{\mathbf{G}}\mathbf{H}_{1} \Big{)}^{H}\mathbf{B}_{12}^{-2}\mathbf{W}^{H}\mathbf{H}_{2}\overline{\mathbf{G }}\mathbf{H}_{1}\mathbf{F}_{\text{RF}}\mathbf{B}_{11}^{-1}, \tag{13}\]  
where \(\mathbf{B}_{12}\overset{\Delta}{=}\mathbf{I}_{N_{\text{S}}}+\big{(}\mathbf{W}^{H }\mathbf{H}_{2}\overline{\mathbf{G}}\mathbf{H}_{1}\mathbf{F}_{\text{RF}}\big{)} \mathbf{B}_{11}^{-1}\big{(}\mathbf{W}^{H}\mathbf{H}_{2}\overline{\mathbf{G}} \mathbf{H}_{1}\mathbf{F}_{\text{RF}}\big{)}^{H}\) for brevity. The detailed derivation is provided in Appendix A. _(2) The Relay Node Design_: In a similar manner, it is assumed that \(\mathbf{F}\) and \(\mathbf{W}\) are fixed. The optimization sub-problem of the relay node can be formulated as  
\[\min_{\mathbf{G}_{\text{RF}_{1}},\mathbf{G}_{\text{RF}_{2}}} \text{IMSE}\] (14a) s.t. \[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]0 \[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]1 (14c) \[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]2  
For any given \(\mathbf{G}_{\text{RF}_{1}}\) and \(\mathbf{G}_{\text{RF}_{2}}\), we can obtain  
\[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]3  
This leads to the optimization problem with only power constraint. Based on the Lagrange function, we have  
\[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]4 \[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]5  
By substituting (15) into (14a), the objective function is reduced to  
\[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]6  
where \(\mathbf{B}_{21}=\mathbf{G}_{\text{RF}_{2}}^{H}\mathbf{H}_{2}^{H}\mathbf{W} \mathbf{W}^{H}\mathbf{H}_{2}\mathbf{G}_{\text{RF}_{2}}+\frac{\sigma_{\text{D} }^{2}}{P_{\text{R}}^{2}}\text{tr}(\mathbf{W}^{H}\mathbf{W})\ \mathbf{G}_{\text{RF}_{2}}^{H}\mathbf{G}_{\text{RF}_{2}}\), \(\mathbf{B}_{22}=\mathbf{G}_{\text{RF}_{2}}^{H}\mathbf{H}_{2}^{H}\mathbf{W} \overline{\mathbf{F}}^{H}\mathbf{H}_{1}^{H}\mathbf{G}_{\text{RF}_{1}}\) and \(\mathbf{B}_{23}=\mathbf{G}_{\text{RF}_{1}}^{H}\mathbf{H}_{1}\overline{ \mathbf{F}}^{H}\mathbf{H}_{1}^{H}\mathbf{G}_{\text{RF}_{1}}+\gamma_{1}^{-2} \sigma_{\text{R}}^{2}\mathbf{G}_{\text{RF}_{1}}^{H}\mathbf{G}_{\text{RF}_{1}}\). The partial derivative of \(\text{T2}\big{(}\mathbf{G}_{\text{RF}_{1}},\mathbf{G}_{\text{RF}_{2}}\big{)}\) for \(\mathbf{G}_{\text{RF}_{1}}^{*}\) and \(\mathbf{G}_{\text{RF}_{2}}^{*}\) can be obtained as  
\[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]7 \[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]8 \[\left|\left|\mathbf{F}_{\text{RF}}\right|_{a,b}\right|^{2}=1, \forall a,b, \tag{8b}\]9  
_(3) The Destination Node Design_: With \(\mathbf{F}\) and \(\mathbf{G}\) fixed, the optimization sub-problem of the destination node can be expressed as  
\[\mathbb{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{\text{F}}^{2}\Big{\}} =P_{\text{B}}. \tag{8c}\]0 (20a) s.t. \[\mathbb{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{\text{F}}^{2}\Big{\}} =P_{\text{B}}. \tag{8c}\]1  
Compared with the source and relay nodes, the destination node only transmits signals without receiving signals. Hence, there is no power constraint. When \(\mathbf{W}_{\text{RF}}\) is constant, (20) can be converted to an unconstrained optimization problem. By taking the derivative, the optimal \(\mathbf{W}_{\text{B}}\) is given by \[