# 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  
In this subsection, the IMSE criterion is still adopted in the HBF design for each node in the FD AF relay communication system, where the optimization objective can be written as \[ \begin{split}&\text{IMSE}=\text{E}_{\text{AH}_{1}\text{AH}_{2}}\Big{\{} \text{tr}\Big{(}\mathbf{W}^{\text{H}}\text{H}_{2}\overline{\mathbf{G}}\text{H}_ {1}\overline{\mathbf{F}}\overline{\mathbf{F}}^{\text{H}}\text{H}_{1}^{\text{H }}\overline{\mathbf{G}}^{\text{H}}\text{H}_{2}^{\text{H}}\mathbf{W}\ -\mathbf{W}^{\text{H}}\text{H}_{2}\overline{\mathbf{G}}\text{H}_{1}\overline{ \mathbf{F}}\\ &-\overline{\mathbf{F}}^{H}\text{H}_{1}^{H}\overline{\mathbf{G}}^ {H}\text{H}_{2}^{H}\mathbf{W}+\gamma_{1}^{-2}\sigma_{\text{R}}^{2}\mathbf{W}^ {H}\text{H}_{2}\overline{\mathbf{G}}\overline{\mathbf{G}}^{H}\text{H}_{2}^{H} \mathbf{W}+\gamma_{1}^{-2}\gamma_{2}^{-2}\sigma_{\text{D}}^{2}\mathbf{W}^{H} \mathbf{W}+\mathbf{I}_{\text{N}_{\text{S}}}\Big{)}\Big{\}}.\end{split}  \tag{32}\]  
When a matrix \(\mathbf{X}\) satisfies \(\mathbf{X}\sim\mathcal{CN}\big{(}\overline{\mathbf{X}},\mathbf{\Sigma}\otimes \overline{\mathbf{Y}}\big{)}\), we have [36]  
\[\text{E}\Big{\{}\mathbf{X}\mathbf{C}\mathbf{X}^{H}\Big{\}}=\overline{\mathbf{X }}\mathbf{C}\overline{\mathbf{X}}^{H}+\text{tr}\Big{(}\mathbf{C}\mathbf{ \Sigma}^{T}\Big{)}\overline{\mathbf{Y}}, \tag{33}\]  
\[\text{E}\Big{\{}\mathbf{X}^{H}\mathbf{C}\mathbf{X}\Big{\}}=\overline{\mathbf{ X}}^{H}\mathbf{C}\overline{\mathbf{X}}+\text{tr}(\mathbf{\Psi}\mathbf{C}) \mathbf{\Sigma}^{T}. \tag{34}\]  
Since the error matrixes \(\Delta\text{H}_{1}\) and \(\Delta\text{H}_{2}\) are independent from each other, we can rewrite IMSE as  
\[ \begin{split}&\text{IMSE}=\text{tr}\Big{[}\mathbf{W}^{H}\overline{ \mathbf{H}}_{2}\overline{\mathbf{G}}\overline{\mathbf{H}}_{1}\overline{ \mathbf{F}}\overline{\mathbf{F}}^{H}\overline{\mathbf{H}}_{1}^{H}\overline{ \mathbf{G}}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{W}+\text{tr}\Big{(} \overline{\mathbf{G}}\overline{\mathbf{H}}_{1}\overline{\mathbf{F}}^{H} \overline{\mathbf{H}}_{1}^{H}\overline{\mathbf{G}}^{H}\Theta_{2}\Big{)} \mathbf{W}^{H}\Phi_{2}\mathbf{W}\\ &+\text{tr}\Big{(}\overline{\mathbf{F}}\overline{\mathbf{F}}^{H} \Theta_{1}\Big{)}\mathbf{W}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{G}} \Phi_{1}\overline{\mathbf{G}}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{W}+ \text{tr}\Big{(}\overline{\mathbf{F}}\overline{\mathbf{F}}^{H}\Theta_{1} \Big{)}\text{tr}\Big{(}\overline{\mathbf{G}}\Phi_{1}\overline{\mathbf{G}}^ {H}\Theta_{2}\Big{)}\mathbf{W}^{H}\Phi_{2}\mathbf{W}\\ &-\mathbf{W}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{G}} \overline{\mathbf{H}}_{1}\overline{\mathbf{F}}-\overline{\mathbf{F}}^{H} \overline{\mathbf{H}}_{1}^{H}\overline{\mathbf{G}}^{H}\overline{\mathbf{H}}_ {2}^{H}\mathbf{W}+\gamma_{1}^{-2}\sigma_{\text{R}}^{2}\mathbf{W}^{H} \overline{\mathbf{H}}_{2}\overline{\mathbf{G}}\overline{\mathbf{G}}^{H}\overline {\mathbf{H}}_{2}^{H}\mathbf{W}\\ &+\gamma_{1}^{-2}\sigma_{\text{R}}^{2}\text{tr}\Big{(}\overline {\mathbf{G}}\overline{\mathbf{G}}^{H}\Theta_{2}\Big{)}\mathbf{W}^{H}\Phi_{2} \mathbf{W}+\gamma_{1}^{-2}\gamma_{2}^{-2}\sigma_{\text{D}}^{2}\mathbf{W}^{H} \mathbf{W}+\mathbf{I}_{\text{N}_{\text{S}}}\Big{)}\end{split}  \tag{35}\]  
in view of (33) and (34). Therefore, the IMSE problem can be formulated as  
\[ \begin{split}\min_{\gamma_{1}\gamma_{2},\text{F}_{\text{RF}}, \text{F}_{\text{U}},\text{G}_{\text{RF}_{1}},\text{G}_{\text{RF}_{2}},\text{G} _{\text{U}_{2}},\text{W}_{\text{RF}},\text{W}_{\text{B}}}\text{IMSE}\end{split}  \tag{36a}\] \[ \begin{split}\text{s.t.}&\left|\left[\text{F}_{ \text{RF}}\right]_{a,b}\right|^{2}=1,\forall a,b,\end{split} \] (36b) \[ \begin{split}\left|\left[\text{G}_{\text{RF}_{1}}\right]_{c,d }\right|^{2}=1,\forall c,d,\end{split} \] (36c) \[ \begin{split}\left|\left[\text{G}_{\text{RF}_{2}}\right]_{c,f }\right|^{2}=1,\forall e,f,\end{split} \] (36d) \[ \begin{split}\left|\left[\text{W}_{\text{RF}}\right]_{g,h} \right|^{2}=1,\forall g,h,\end{split} \] (36e) \[