# Robust Hybrid Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks
## 3 Proposed Algorithm
### Algorithm Design  
The purpose of the proposed algorithm is to reduce the minimum mean square error (MMSE) of the transmitted and received signals. On the basis of the initial digital matrix, the orthogonal function of the digital matrix was used to further simplify the objective function in Equation (10). Multiple iterations were performed to finally obtain a hybrid beamforming matrix. Expanding the objective function in Equation (10), we get [21; 28; 29]:  
\[\begin{array}{rl}E[\|\overline{\mathbf{s}}-\mathbf{s}\|^{2}]=&\|\overline{ \mathbf{H}}\mathbf{F_{\mathrm{RF}}}\mathbf{F_{\mathrm{BB}}}-\mathbf{I}\|_{\mathrm{F}}^{2}+ \sigma^{2}=\mathrm{Tr}\mathbf{(}\overline{\mathbf{H}}\mathbf{F_{\mathrm{RF}}}\mathbf{F_{ \mathrm{BB}}}-\mathbf{I}\mathbf{)(}\mathbf{I}-\mathbf{F_{\mathrm{BB}}}\mathbf{F_{\mathrm{RF}}}\mathbf{ (}\overline{\mathbf{H}}\mathbf{)}^{*}\mathbf{)}\|+\sigma^{2}\\ &\quad=\mathrm{Tr}\mathbf{(}\overline{\mathbf{H}}\mathbf{F_{\mathrm{RF}}}\mathbf{F_{\mathrm{BB }}}-\mathrm{Tr}\mathbf{(}\overline{\mathbf{H}}\mathbf{F_{\mathrm{RF}}}\mathbf{F_{\mathrm{BB}}} \mathbf{F_{\mathrm{RF}}}\mathbf{(}\overline{\mathbf{H}}\mathbf{)}^{*}\mathbf{)}+\mathrm{Tr}\mathbf{(} \mathbf{F_{\mathrm{BB}}}\mathbf{F_{\mathrm{RF}}}\mathbf{(}\overline{\mathbf{H}}\mathbf{)}^{*}\bm {)}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad  
Utilizing the orthogonality of the digital beamforming matrix, namely, \(\mathbf{F_{\text{BB}}}\mathbf{F_{\text{BB}}}=\mathbf{I}\), Equation (16), assuming \(\mathbf{F_{\text{RF}}}\) is known, can be further transformed to:  
\[\min_{\mathbf{F_{\text{RF}}}}Tr(\mathbf{F_{\text{e}}^{*}F_{\text{RF}}F_{\text{BB}}}),\]  
subject to  
\[\mathbf{F_{\text{BB}}}\mathbf{F_{\text{BB}}}=\mathbf{I}. \tag{17}\]  
The MMSE of the transmitted and received signals can be approximated by Equation (18), which is known by [PERSON]'s inequality:  
\[\text{Tr}(\mathbf{F_{e}^{*}F_{\text{RF}}F_{\text{BB}}})\leq\|\mathbf{F_{\text{BB}}}^{*} \|_{\infty}\mathbf{F_{\text{e}}}\|\mathbf{F_{\text{RF}}}\|_{1},\]  
subject to  
\[\|\mathbf{U_{\text{et}}}\mathbf{S_{\text{et}}}\mathbf{V_{\text{et}}}\|_{1}=\|\mathbf{F_{e}^{*} F_{\text{RF}}}\|_{1} \tag{18}\]  
The equation holds only when \(\mathbf{F_{\text{BB}}}\) satisfies Equation (19):  
\[\mathbf{F_{\text{BB}}}=\mathbf{U}\mathbf{V}^{*} \tag{19}\]  
where \(\mathbf{F_{e}^{*}F_{\text{RF}}}=\mathbf{U}\mathbf{S}\mathbf{V}\). Since each element of the analog matrix satisfies the constant-modulus requirement, the analog wave obtained according to Equation (15), the phase information of each element value of the beam-shaping matrix can further obtain analog matrix \(\mathbf{F_{\text{RF}}}\), \(\text{arg}(\mathbf{F_{\text{RF}}})=\text{arg}\Big{(}\mathbf{F_{\text{opt}}}\Big{(}F_{ \text{BB}_{\text{int}}}\Big{)}^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!and compared with pure digital beamforming, the OMP hybrid beamforming algorithm [20], and other references [18; 19; 24]. The specific simulation parameters are shown in Table 1.