# Joint Hybrid Beamforming Design for Millimeter Wave Amplify-and-Forward Relay Communication Systems
## 4 Hybrid Beamforming Design for FD AF Relay Systems
### System Model  
_(1) System Model_: It is inevitable that the AF relay system will lead to severe SI in FD mode. As shown in Figure 2, in order to mitigate the SI, we decompose the DBF matrix \(\mathbf{G}_{\text{B}}\) of the relay node into \(\mathbf{G}_{\text{B}_{1}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{S}}}\) and \(\mathbf{G}_{\text{B}_{2}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{S}}}\) to design individually, which are the relay digital receive and transmit beamforming matrixes, respectively. The FD relay node can receive its own transmit signal and that of the source node simultaneously. Hence, the received signal of the relay node takes the form of [30; 31]  
\[\mathbf{y}_{\text{R}}=\mathbf{G}_{1}^{H}\mathbf{H}_{1}\mathbf{F}\mathbf{s}_{ \text{B}}+\mathbf{G}_{1}^{H}\mathbf{H}_{\text{SI}}\mathbf{G}_{2}\mathbf{s}_{ \text{R}}+\mathbf{G}_{1}^{H}\mathbf{n}_{\text{R}}, \tag{24}\]  
where \(\mathbf{G}_{1}=\mathbf{G}_{\text{RF}_{1}}\mathbf{G}_{\text{B}_{1}}\in\mathbb{ C}^{N_{\text{RF}}\times N_{\text{S}}}\) and \(\mathbf{G}_{2}=\mathbf{G}_{\text{RF}_{2}}\mathbf{G}_{\text{B}_{2}}\in\mathbb{ C}^{N_{\text{RF}}\times N_{\text{S}}}\) are the relay hybrid receive and transmit beamforming matrixes. \(\mathbf{s}_{\text{B}}\) and \(\mathbf{s}_{\text{B}}\) denote the transmit signals of the source and relay nodes, which are subject to \(\mathbb{E}\big{\{}\mathbf{s}_{\text{BS}}\mathbf{s}_{\text{B}}^{H}\big{\}}= \mathbf{I}_{N_{\text{S}}}\) and \(\mathbb{E}\big{\{}\mathbf{s}_{\text{R}}\mathbf{s}_{\text{R}}^{H}\big{\}}= \mathbf{P}_{\text{th}}\mathbf{I}_{N_{\text{S}}}\), with \(P_{\text{th}}\) denoting the power of interference signals. And \(\mathbf{H}_{\text{SI}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{R}}}\) stands for the SI channel matrix. The received signal at the destination node can be expressed as  
\[\mathbf{y}_{\text{D}} =\underbrace{\mathbf{W}^{H}\mathbf{H}_{2}\mathbf{G}_{2}\mathbf{G }_{1}^{H}\mathbf{H}_{1}\mathbf{F}\mathbf{s}_{\text{B}}}_{\text{desired signal}}+\underbrace{\mathbf{W}^{H}\mathbf{H}_{2} \mathbf{G}_{2}\mathbf{G}_{1}^{H}\mathbf{H}_{\text{SI}}\mathbf{G}_{2}\mathbf{s }_{\text{R}}}_{\text{SI}} \tag{25}\] \[+\underbrace{\mathbf{W}^{H}\mathbf{H}_{2}\mathbf{G}_{2}\mathbf{G }_{1}^{H}\mathbf{n}_{\text{R}}}_{\text{noise at the relay node}}+\underbrace{ \mathbf{W}^{H}\mathbf{n}_{\text{D}}}_{\text{noise at the destination node}}.\]  
Figure 2: FD AF relay communication system. With the SI constraint \(\mathbf{G}_{1}^{H}\mathbf{H}_{\text{SI}}\mathbf{G}_{2}=\mathbf{0}\) set in the HBF design, \(\mathbf{y}_{\text{D}}\) can be further expressed as  
\[\mathbf{y}_{\text{D}}=\mathbf{W}^{H}\mathbf{H}_{2}\mathbf{G}_{2}\mathbf{G}_{1}^{ H}\mathbf{H}_{1}\mathbf{F}\mathbf{s}_{\text{B}}+\mathbf{W}^{H}\mathbf{H}_{2} \mathbf{G}_{2}\mathbf{G}_{1}^{H}\mathbf{n}_{\text{R}}+\mathbf{W}^{H}\mathbf{n} _{\text{D}}. \tag{26}\]  
_(2) Imperfect CSI Model:_ In practice, the CSI is usually not available accurately at every node because of user mobility and algorithm processing delay. The channel model with estimation errors is modelled as  
\[\mathbf{H}=\overline{\mathbf{H}}+\Delta\mathbf{H}, \tag{27}\]  
where \(\mathbf{H}\) and \(\overline{\mathbf{H}}\) are the true and estimated channel matrixes, respectively. \(\Delta\mathbf{H}\) denotes the channel estimation error, which is given by [14; 15]  
\[\Delta\mathbf{H}=\mathbf{\Phi}^{\frac{1}{2}}\Delta\mathbf{\Theta}^{\frac{1}{2}}, \tag{28}\]  
where \(\mathbf{\Theta}\) and \(\mathbf{\Phi}\) are the covariance matrices of estimation error at the receiver and transmitter sides. \(\Delta\) denotes the unknown part of CSI, which is a complex Gaussian matrix with identically distributed zero mean and unit variance entries. Therefore, the true channel matrix satisfies \(\mathbf{H}\sim\mathcal{CN}\big{(}\overline{\mathbf{H}},\mathbf{\Theta}\otimes \mathbf{\Phi}\big{)}\).