# Joint Hybrid Beamforming Design for Millimeter Wave Amplify-and-Forward Relay Communication Systems
## 4 Hybrid Beamforming Design for FD AF Relay Systems
### Proposed HBF-MO-FD-R Algorithm
\begin{split}\text{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2} \Big{\}}=P_{\text{B}},\end{split} \] (36f) \[ \begin{split}\text{E}_{\text{AH}_{1}}\Big{\{}\|\mathbf{x}_{ \text{R}}\|_{F}^{2}\Big{\}}=P_{\text{R}},\end{split} \] (36g) \[ \begin{split}\text{G}_{1}^{H}\overline{\mathbf{H}}_{ \text{SI}}\text{G}_{2}=\mathbf{0},\end{split}  \tag{36h}\]  
where (36f) and (36g) denote the source and AF relay power constraints. Specifically, the former has the same form as that in Section III, while the latter can be given by \(\text{E}_{\text{AH}_{1}}\Big{\{}\|\mathbf{x}_{\text{R}}\|_{F}^{2}\Big{\}}=\gamma_{ 2}^{2}\text{tr}\Big{[}\gamma_{1}^{2}\overline{\mathbf{G}}\overline{\mathbf{H}}_{1} \overline{\mathbf{F}}\overline{\mathbf{F}}^{H}\overline{\mathbf{H}}_{1}^{H} \overline{\mathbf{G}}^{H}\ +\gamma_{1}^{2}\text{tr}\Big{(}\overline{\mathbf{F}}^{H}\Theta_{1}\Big{)} \overline{\mathbf{G}}\Phi_{1}\overline{\mathbf{G}}^{H}+\sigma_{\text{R}}^{2} \overline{\mathbf{G}}\overline{\mathbf{G}}\overline{\mathbf{G}}^{H}\Big{]}\). _(1) The Source Node Design:_ While fixing \(\mathbf{G}\) and \(\mathbf{W}\), the HBF design sub-problem of the source node can be formulated as  
\[\min_{\gamma_{1},\text{F}_{\text{RF}},\text{F}_{\text{U}}} \text{IMSE} \tag{37a}\] \[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\] (37b) \[\text{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2}\Big{\}}=P_{\text{B}}. \tag{37c}\]  
with \(\mathbf{F}=\gamma_{1}\mathbf{F}_{\text{RF}}\text{F}_{\text{U}}\), we have the power constraint as \(\gamma_{1}^{2}\text{tr}\Big{(}\overline{\mathbf{F}}\overline{\mathbf{F}}^{H} \Big{)}=P_{\text{B}}\). Therefore, \(\gamma_{1}\) can be expressed as  
\[\gamma_{1}=\Bigg{[}\frac{\text{tr}\big{(}\mathbf{F}_{\text{RF}}\text{F}_{ \text{U}}\mathbf{F}_{\text{U}}^{H}\mathbf{F}_{\text{RF}}^{H}\big{)}}{P_{\text{B}}} \Bigg{]}^{-\frac{1}{2}}. \tag{38}\]Upon having fixed \(\mathbf{F}_{\text{RF}}\), we can obtain \(\mathbf{F}_{\text{U}}\) by setting the partial derivatives of the Lagrange function \(\text{L1}(\lambda_{1},\mathbf{F}_{\text{U}},\gamma_{1})\) to zero:  
\[\mathbf{F}_{\text{U}}=\Big{(}\mathbf{F}_{\text{RF}}^{H}\overline{\mathbf{H}}_{ 1}^{H}\overline{\mathbf{G}}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{W}\mathbf{ W}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{G}}\overline{\mathbf{H}}_{1} \mathbf{F}_{\text{RF}}+\mathbf{C}_{11}\Big{)}^{-1}\mathbf{F}_{\text{RF}}^{H} \overline{\mathbf{H}}_{1}^{H}\overline{\mathbf{G}}^{H}\overline{\mathbf{H}}_{2 }^{H}\mathbf{W}, \tag{39}\]  
where \(\mathbf{C}_{11}=\)\(c_{11}\mathbf{F}_{\text{RF}}^{H}\overline{\mathbf{H}}_{1}^{H}\overline{\mathbf{G}}^{H} \mathbf{\Theta}_{2}\overline{\mathbf{G}}\overline{\mathbf{H}}_{1}\mathbf{F}_{ \text{RF}}+c_{12}\mathbf{F}_{\text{RF}}^{H}\mathbf{\Theta}_{1}\mathbf{F}_{ \text{RF}}+c_{13}\mathbf{F}_{\text{RF}}^{H}\mathbf{F}_{\text{RF}}\), \(c_{11}=\text{tr}\big{(}\mathbf{W}^{H}\mathbf{\Phi}_{2}\mathbf{W}\big{)}\), \(c_{12}=\text{tr}\big{(}\mathbf{W}^{H}\overline{\mathbf{H}}_{2}\overline{ \mathbf{G}}\mathbf{\Phi}_{1}\overline{\mathbf{G}}^{H}\overline{\mathbf{H}}_{2 }^{H}\mathbf{W}\Big{)}+\text{tr}\big{(}\mathbf{W}^{H}\mathbf{\Phi}_{2}\mathbf{ W}\big{)}\text{tr}\big{(}\overline{\mathbf{G}}\mathbf{\Phi}_{1}\overline{ \mathbf{G}}^{H}\mathbf{\Theta}_{2}\big{)}\) and \(c_{13}=\gamma_{2}^{-2}\frac{\sigma_{\text{R}}^{2}}{P_{\text{R}}}\text{tr} \big{(}\mathbf{W}^{H}\mathbf{W}\big{)}\)\(+\)\(\frac{\sigma_{\text{R}}^{2}}{P_{\text{R}}}\text{tr}\big{(}\mathbf{W}^{H}\overline{ \mathbf{H}}_{2}\overline{\mathbf{G}}\overline{\mathbf{G}}^{H}\overline{ \mathbf{H}}_{2}^{H}\mathbf{W}\big{)}+\frac{\sigma_{\text{R}}^{2}}{P_{\text{R}} }\text{tr}\big{(}\overline{\mathbf{G}}\overline{\mathbf{G}}^{H}\mathbf{\Theta }_{2}\big{)}\text{tr}\big{(}\mathbf{W}^{H}\mathbf{\Phi}_{2}\mathbf{W}\big{)}\).