# 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
For any given \(\mathbf{F}\), \(\mathbf{G}_{1}\) and \(\mathbf{W}\), we have the optimization sub-problem as follows by ignoring the SI constraint:  
\[\text{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2}\Big{\}}=P_{\text{B}}. \tag{37c}\]2 (48a) s.t. \[\text{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2}\Big{\}}=P_{\text{B}}. \tag{37c}\]3 \[\text{E}\Big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2}\Big{\}}=P_{\text{B}}. \tag{37c}\]4  
Likewise, we can obtain \(\gamma_{2}\) by taking transposition of (48b), which is given as  
\[ \begin{split}\gamma_{2}&=\Big{[}\text{tr}\Big{(} \gamma_{1}^{2}\overline{\mathbf{G}}\overline{\mathbf{H}}_{1}\overline{ \mathbf{F}}^{H}\overline{\mathbf{H}}_{1}^{H}\overline{\mathbf{G}}^{H}\\ +\gamma_{1}^{2}\text{tr}\Big{(}\overline{\mathbf{FF}}^{H} \mathbf{\Theta}_{1}\Big{)}\overline{\mathbf{G}}\mathbf{\Phi}_{1}\overline{ \mathbf{G}}^{H}+\sigma_{\text{R}}^{2}\overline{\mathbf{G}}\overline{\mathbf{G }}^{H}\Big{)}\Big{/}P_{\text{R}}\Big{]}^{-\frac{1}{2}}\end{split}  \tag{49}\]  
To this end, the optimization problem (48) can be equivalently recast as an optimization problem (50) with only power constraint by fixing \(\mathbf{G}_{\text{RF}_{2}}\). Based on the Lagrange function, the closed form of \(\mathbf{G}_{\text{U}_{2}}\) is expressed as (51). \[\min_{\gamma_{2},\mathbf{G}_{\text{U}_{2}}} \text{MSE}\] (50a) s.t. \[\text{s.t.} \tag{50b}\] \[\mathbf{G}_{\text{U}_{2}} =\Big{[}\mathbf{G}_{\text{RF}_{2}}^{H}\overline{\mathbf{H}}_{2}^{ H}\mathbf{W}\mathbf{W}^{H}\overline{\mathbf{H}}_{2}\mathbf{G}_{\text{RF}_{2}}+ \text{tr}\Big{(}\mathbf{W}^{H}\mathbf{\Phi}_{2}\mathbf{W}\Big{)}\mathbf{G}_{ \text{RF}_{2}}^{H}\mathbf{\Theta}_{2}\mathbf{G}_{\text{RF}_{2}}\] (51) \[+\frac{\sigma_{\text{D}}^{2}}{P_{\text{R}}}\text{tr}\Big{(} \mathbf{W}^{H}\mathbf{W}\Big{)}\mathbf{G}_{\text{RF}_{2}}^{H}\mathbf{G}_{\text{ RF}_{2}}\Big{]}^{-1}\Big{(}\mathbf{G}_{\text{RF}_{2}}^{H}\overline{ \mathbf{H}}_{2}^{H}\mathbf{W}\overline{\mathbf{F}}^{H}\overline{\mathbf{H}}_{ 1}^{H}\mathbf{G}_{1}\Big{)}\] \[\times\Big{[}\mathbf{G}_{1}^{H}\overline{\mathbf{H}}_{1} \overline{\mathbf{F}}^{H}\overline{\mathbf{H}}_{1}^{H}\mathbf{G}_{1}+\text{tr} \Big{(}\overline{\mathbf{FF}}^{H}\mathbf{\Theta}_{1}\Big{)}\mathbf{G}_{1}^{H} \mathbf{\Phi}_{1}\mathbf{G}_{1}+\gamma_{1}^{-2}\sigma_{\text{R}}^{2}\mathbf{G}_ {1}^{H}\mathbf{G}_{1}\Big{]}^{-1},\]  
where we define \(\mathbf{D}_{23}=\mathbf{G}_{1}^{H}\overline{\mathbf{H}}_{1}\overline{\mathbf{F}} ^{H}\overline{\mathbf{H}}_{1}^{H}\mathbf{G}_{1}+\text{tr}\Big{(}\overline{ \mathbf{FF}}^{H}\mathbf{\Theta}_{1}\Big{)}\mathbf{G}_{1}^{H}\mathbf{\Phi}_{1} \mathbf{G}_{1}+\gamma_{1}^{-2}\sigma_{\text{R}}^{2}\mathbf{G}_{1}^{H}\mathbf{G}_ {1}\), \(\mathbf{D}_{21}=\mathbf{G}_{\text{RF}_{2}}^{H}\overline{\mathbf{H}}_{2}^{H} \mathbf{W}\mathbf{W}^{H}\overline{\mathbf{H}}_{2}\mathbf{G}_{\text{RF}_{2}}+ \text{tr}\big{(}\mathbf{W}^{H}\mathbf{\Phi}_{2}\mathbf\[\
abla\text{Q2}\big{(}\mathbf{G}_{\text{RF}_{2}}\big{)} =\Big{[}\Big{[}\overline{\mathbf{H}}_{2}^{H}\mathbf{W}\mathbf{W}^{H }\overline{\mathbf{H}}_{2}\,\mathbf{G}_{\text{RF}_{2}}\mathbf{D}_{21}^{-1} \mathbf{D}_{22}+\text{tr}\Big{(}\mathbf{W}^{H}\mathbf{\Phi}_{2}\mathbf{W}\Big{)} \mathbf{\Theta}_{2}\mathbf{G}_{\text{RF}_{2}}\mathbf{D}_{21}^{-1}\mathbf{D}_{22} \tag{54}\] \[+\frac{\sigma_{\text{D}}^{2}}{P_{\text{R}}}\text{tr}\Big{(} \mathbf{W}^{H}\mathbf{W}\Big{)}\mathbf{G}_{\text{RF}_{2}}\mathbf{D}_{21}^{-1} \mathbf{D}_{22}-\overline{\mathbf{H}}_{2}^{H}\mathbf{W}\overline{\mathbf{F}}^ {H}\mathbf{H}_{1}^{H}\mathbf{G}_{1}\Big{]}\mathbf{D}_{23}^{-1}\mathbf{D}_{22}^ {H}\mathbf{D}_{21}^{-1}.\]  
It's worth noting that we temporarily neglect the SI constraint in the HBF design of the relay node. Consequently, we need to design the SI suppression matrix to satisfy \(\mathbf{G}_{1}^{H}\overline{\mathbf{H}}_{\text{SI}}\mathbf{G}_{2}=\mathbf{0}\). Severe SI acts as a key factor to affect the performance of FD communication systems. There are strict restrictions on the number of RF chains in traditional SI suppression algorithm based on NP.