# 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
Substituting (38) and (39) into (37a), the optimization objective can be further expressed as  
\[\text{J1}(\mathbf{F}_{\text{RF}})=\text{tr}\Bigg{\{}\Big{[}\mathbf{I}_{\text{ N}_{\text{S}}}+\Big{(}\mathbf{W}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{G}} \overline{\mathbf{H}}_{1}\mathbf{F}_{\text{RF}}\Big{)}\ \mathbf{C}_{11}^{-1}\Big{(}\mathbf{W}^{H}\overline{\mathbf{H}}_{2}\overline{ \mathbf{G}}\overline{\mathbf{H}}_{1}\mathbf{F}_{\text{RF}}\Big{)}^{H}\Big{]}^{ -1}\Bigg{\}}. \tag{40}\]  
The sub-problem with only constant-modulus constraint can be re-formulated as  
\[\min_{\mathbf{F}_{\text{RF}}} \quad\text{J1}(\mathbf{F}_{\text{RF}})\] (41a) s.t. \[\quad\Big{|}[\mathbf{F}_{\text{RF}}]_{d,b}\Big{|}^{2}=1,\forall a,b. \tag{41b}\]  
By mathematical derivation, we can obtain the gradient of \(\text{J1}(\mathbf{F}_{\text{RF}})\) with respect to \(\mathbf{F}_{\text{RF}}\):  
\[\
abla\text{J1}(\mathbf{F}_{\text{RF}}) =\Big{(}c_{11}\overline{\mathbf{H}}_{1}^{H}\overline{\mathbf{G}} ^{H}\mathbf{\Theta}_{2}\overline{\mathbf{G}}\overline{\mathbf{H}}_{1}\mathbf{ F}_{\text{RF}}\mathbf{C}_{11}^{-1}\mathbf{F}_{\text{RF}}^{H} \tag{42}\] \[+c_{12}\mathbf{\Theta}_{1}\mathbf{F}_{\text{RF}}\mathbf{C}_{11}^ {-1}\mathbf{F}_{\text{RF}}^{H}-\mathbf{I}_{N_{\text{T}}}+c_{13}\mathbf{F}_{ \text{RF}}\mathbf{C}_{11}^{-1}\mathbf{F}_{\text{RF}}^{H}\Big{)}\] \[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]0  
_(2) The Relay Node Design_: First, we consider the HBF design for the receiver of the relay node. It is assumed that \(\mathbf{F}\), \(\mathbf{G}_{2}\) and \(\mathbf{W}\) are fixed. Taking SI constraint out of consideration, the optimization problem can be formulated as  
\[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]1 (43a) s.t. \[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]2  
with \(\mathbf{G}_{\text{RF}_{1}}\) fixed, the optimization problem (43) can be recast as an unconstrained optimization problem. By taking derivative, we have the closed-form solution of \(\mathbf{G}_{\text{B}_{1}}\) as  
\[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]3 \[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]4  
Substituting the optimization problem (43) with Equation (44), we have  
\[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]5 (45a) s.t. \[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]6 \[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]7with \(\mathbf{D}_{11}\)=\(\mathbf{G}_{\text{RF}_{1}}^{H}\overline{\mathbf{H}}_{1}\overline{\mathbf{FF}}^{H} \overline{\mathbf{H}}_{1}^{H}\mathbf{G}_{\text{RF}_{1}}+\text{tr}\Big{(} \overline{\mathbf{FF}}^{H}\mathbf{\Theta}_{1}\Big{)}\mathbf{G}_{\text{RF}_{1}}^{ H}\mathbf{\Phi}_{1}\mathbf{G}_{\text{RF}_{1}}+\gamma_{1}^{-2}\sigma_{\text{R}}^{2} \mathbf{G}_{\text{RF}_{1}}^{H}\mathbf{G}_{\text{RF}_{1}}\), \(\mathbf{D}_{12}\)=\(\mathbf{G}_{\text{RF}_{1}}^{H}\)\(\overline{\mathbf{H}}_{1}\overline{\mathbf{F}}\mathbf{W}^{H}\overline{\mathbf{H}}_{2} \overline{\mathbf{G}}_{2}\) and \(\mathbf{D}_{13}\)=\(\overline{\mathbf{G}}_{2}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{W}\mathbf{W}^{H} \overline{\mathbf{H}}_{2}\overline{\mathbf{G}}_{2}+\text{tr}\big{(}\mathbf{W} ^{H}\mathbf{\Phi}_{2}\mathbf{W}\big{)}\overline{\mathbf{G}}_{2}^{H}\mathbf{ \Theta}_{2}\overline{\mathbf{G}}_{2}\). The Euclidean gradient of \(\text{Q}1\big{(}\mathbf{G}_{\text{RF}_{1}}\big{)}\) is given as  
\[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]8 \[\text{s.t.} \left|\left[\mathbf{F}_{\text{RF}}\right]_{a,b}\right|^{2}=1, \forall a,b,\]9 \[\text{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2}\Big{\}}=P_{\text{B}}. \tag{37c}\]0 \[\text{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2}\Big{\}}=P_{\text{B}}. \tag{37c}\]1  
After finishing the design of the receiver, we turn to the transmitter.