# 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 [31], it is required to meet \(N_{\text{RF,T}}\geq N_{\text{S}}+N_{\text{RF,R}}\), with \(N_{\text{RF,T}}\) and \(N_{\text{RF,R}}\) being the number of the transmit and receive RF chains at the AF relay node, while [35] needs to satisfy \(N_{\text{RF,R}}\geq 2N_{\text{S}}\). The authors of [30] proposed an SI suppression algorithm suitable for the system with \(N_{\text{RF,T}}=N_{\text{RF,R}}=N_{\text{S}}\). Nevertheless, its performance remains to be improved. On the basis of the above literature, we further develop the SIC-NP algorithm. We first construct an equivalent SI channel matrix \(\overline{\mathbf{H}}_{\text{SI-EQ}}\in\mathbb{C}^{N_{\text{S}}\times N_{ \text{R}}}\), which can be written as  
\[\overline{\mathbf{H}}_{\text{SI-EQ}}=\mathbf{G}_{1}^{H}\overline{\mathbf{H}} _{\text{SI}}. \tag{55}\]  
It is obvious that the rank of \(\overline{\mathbf{H}}_{\text{SI-EQ}}\) is \(N_{\text{S}}(N_{\text{R}}\gg N_{\text{S}})\), so its null-space definitely exists. There are a number of ways to obtain the null-space of a matrix. We adopt the singular value decomposition (SVD) method in this paper, which yields  
\[\overline{\mathbf{H}}_{\text{SI-EQ}}=\overline{\mathbf{U}}_{\text{SI-EQ}} \overline{\mathbf{\Sigma}}_{\text{SI-EQ}}\overline{\mathbf{\
abla}}_{\text{ SI-EQ}}^{H}, \tag{56}\]  
where \(\overline{\mathbf{U}}_{\text{SI-EQ}}\in\mathbb{C}^{N_{\text{S}}\times N_{ \text{R}}}\) and \(\overline{\mathbf{\
abla}}_{\text{SI-EQ}}\in\mathbb{C}^{N_{\text{R}}\times N_{ \text{R}}}\) are unitary matrixes, \(\overline{\mathbf{\Sigma}}_{\text{SI-EQ}}\in\mathbb{C}^{N_{\text{S}}\times N_{ \text{R}}}\) is a diagonal matrix. \(\overline{\mathbf{\
abla}}_{\text{SI-EQ}}\) can be further decomposed as \(\overline{\mathbf{\
abla}}_{\text{SI-EQ}}=\left[\overline{\mathbf{\
abla}}_{ \text{SI-EQ}}^{1}\overline{\mathbf{\
abla}}_{\text{SI-EQ}}^{0}\right]\), where \(\overline{\mathbf{\
abla}}_{\text{SI-EQ}}^{1}\in\mathbb{C}^{N_{\text{R}}\times N _{\text{S}}}\) and \(\overline{\mathbf{\
abla}}_{\text{SI-EQ}}^{0}\in\mathbb{C}^{N_{\text{R}}\times( N_{\text{R}}-N_{\text{S}})}\) are the right singular matrix for non-zero singular value and zero singular value. And the latter satisfies  
\[\overline{\mathbf{H}}_{\text{SI-EQ}}\overline{\mathbf{\
abla}}_{\text{SI-EQ}}^ {0}=\mathbf{0}. \tag{57}\]  
In this correspondence, we can construct the null-space matrix \(\mathbf{P}\in\mathbb{C}^{N_{\text{B}}\times N_{\text{R}}}\) of \(\overline{\mathbf{H}}_{\text{SI-EQ}}\) using the column vectors of \(\overline{\mathbf{\
abla}}_{\text{SI-EQ}}^{0}\). Subsequently, \(\mathbf{G}_{2}\) is projected to \(\mathbf{P}\) to mitigate the SI. The projection matrix \(\mathbf{P}_{1}\) and SI suppression matrix \(\mathbf{G}_{2}^{\text{SIC}}\) can be written as  
\[\text{s.t.} \tag{50b}\]0  
\[\text{s.t.} \tag{50b}\]1  
Hence, the HBF matrix of FD AF relay node takes the form of  
\[\text{s.t.} \tag{50b}\]2  
For clarity, the above SI suppression procedure is summarized in Algorithm 2. ```
0:\(\mathbf{G}_{1}\), \(\mathbf{G}_{2}\), \(\overline{\mathbf{H}}_{\text{SI}}\). 0:\(\mathbf{G}_{2}^{\text{SIC}}\). 1:Calculate the equivalent SI channel matrix \(\overline{\mathbf{H}}_{\text{SI-EQ}}\) according to (55). 2:Calculate \(\overline{\mathbf{\
abla}}_{\text{SI-EQ}}^{0}\) by taking SVD of \(\overline{\mathbf{H}}_{\text{SI-EQ}}\). 3:Construct the null-space matrix \(\mathbf{P}\). 4:Calculate \(\mathbf{P}_{1}\) according to (58). 5:Compute \(\mathbf{G}_{2}^{\text{SIC}}\) according to (59). ```  
**Algorithm 2** SIC-NP Algorithm. _(3) The Destination Node Design_: Similar to that of Section II, the optimization subproblem of the destination node has only constant-modulus constraint. For any given source and relay HBF matrixes, the destination sub-problem can be formulated as  
\[\text{s.t.} \tag{50b}\]3 (61a) s.t. \[\text{s.t.} \tag{50b}\]4  
with \(\mathbf{W}_{\text{RF}}\) fixed, the problem (61) can be transformed into an unconstrained optimization problem.