# Joint Hybrid Beamforming Design for Millimeter Wave Amplify-and-Forward Relay Communication Systems
## 2 System Model  
As shown in Figure 1, an AF mmWave relay communication system is adopted in this paper, including a source node with \(N_{\text{T}}\) antennas, an HD AF relay node with \(N_{\text{R}}\) antennas and a destination node with \(N_{\text{D}}\) antennas. And all nodes are equipped with \(N_{\text{RF}}\) RF chains. In our scenario, the blockage between the source-destination link prevents direct communication. Consequently, it is necessary for the two nodes to communicate through the AF relay node.  
We assume that \(\mathbf{s}\in\mathbb{C}^{N_{\text{R}}\times 1}\) denotes the original signal of the source node characterized by \(\mathbb{E}\big{\{}\mathbf{s}\mathbf{s}^{H}\big{\}}=\mathbf{I}_{N_{\text{R}}}\), with \(N_{\text{S}}\) representing the number of data streams. The signal is processed by the digital beamformer \(\mathbf{F}_{\text{B}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{S}}}\) and then the analog beamformer \(\mathbf{F}_{\text{RF}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{RF}}}\), which subjects to the constant-modulus constraint, i.e., \(\big{|}\big{|}\mathbf{F}_{\text{RF}}\big{|}_{a,b}\big{|}^{2}=1\). Therefore, the source transmit signal can be given by \(\mathbf{x}_{\text{B}}=\mathbf{F}\mathbf{s}\), where \(\mathbf{F}=\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{B}}\) is the HBF matrix of the source node. The source transmit power is stated as \(\mathbb{E}\big{\{}\|\mathbf{x}_{\text{B}}\|_{F}^{2}\big{\}}=\mathbf{p}_{\text{ B}}\).  
Through the mmWave channel \(\mathbf{H}_{1}\in\mathbb{C}^{N_{\text{R}}\times N_{\text{T}}}\), the signal is received by the AF relay node, the transmit signal of which can be given by \(\mathbf{x}_{\text{R}}=\mathbf{G}(\mathbf{H}_{1}\mathbf{x}_{\text{B}}+\mathbf{ n}_{\text{R}})\), where \(\mathbf{G}=\mathbf{G}_{\text{RF}}\mathbf{G}_{\text{B}}\mathbf{G}_{\text{RF}}^ {H}\) is the relay node HBF matrix, with \(\mathbf{G}_{\text{RF}_{1}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{RF}}}\), \(\mathbf{G}_{\text{RF}_{2}}\in\mathbb{C}^{N_{\text{R}}\times N_{\text{RF}}}\) and \(\mathbf{G}_{\text{B}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{RF}}}\) being the receive ABF matrix, transmit ABF matrix and digital beamforming (DBF) matrix, respectively. \(\mathbf{n}_{\text{R}}\in\mathbb{C}^{N_{\text{R}}\times 1}\) denotes the addictive complex Gaussian noise with zero mean and covariance matrix \(\mathbb{E}\big{\{}\mathbf{n}_{\text{R}}\mathbf{n}_{\text{R}}^{H}\big{\}}= \sigma_{\text{R}}^{2}\mathbf{I}_{N_{\text{R}}}\) at the relay node. Similarly, the analog beamformers should subject to the constant-modulus constraints, while the transmit signal \(\mathbf{x}_{\text{R}}\) should satisfy the power constraint \(\mathbb{E}\big{\{}\|\mathbf{x}_{\text{R}}\|_{\text{F}}^{2}\big{\}}=\mathbf{p}_{ \text{R}}\).  
With an analog beamformer \(\mathbf{W}_{\text{RF}}\in\mathbb{C}^{N_{\text{D}}\times N_{\text{RF}}}\) followed by a digital beamformer \(\mathbf{W}_{\text{B}}\in\mathbb{C}^{N_{\text{RF}}\times N_{\text{S}}}\), the received signal at the destination node takes the form of  
\[\mathbf{y}_{\text{D}}=\mathbf{W}^{H}\mathbf{H}_{2}\mathbf{G}\mathbf{H}_{1} \mathbf{F}\mathbf{s}+\mathbf{W}^{H}\mathbf{H}_{2}\mathbf{G}\mathbf{n}_{\text{ R}}+\mathbf{W}^{H}\mathbf{n}_{\text{D}}, \tag{1}\]  
where \(\mathbf{H}_{2}\in\mathbb{C}^{N_{\text{D}}\times N_{\text{R}}}\) denotes the mmWave channel of the relay-to-destination link, \(\mathbf{n}_{\text{D}}\in\mathbb{C}^{N_{\text{D}}\times 1}\) is the zero-mean addictive complex Gaussian noise vector with covariance matrix \(\mathbb{E}\big{\{}\mathbf{n}_{\text{D}}\mathbf{n}_{\text{D}}^{H}\big{\}}= \sigma_{\text{D}}^{2}\mathbf{I}_{N_{\text{D}}}\) at the destination node. For simplicity, we define \(\mathbf{W}=\mathbf{W}_{\text{RF}}\mathbf{W}_{\text{B}}\) as the HBF matrix of the destination node.