# Robust Hybrid Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks
## 2 System Model
### Channel Model  
A notable feature of mmWave communication is that the number of scatterers in the transmission path is relatively small, so the existing channel model [27] is not suitable for mmWave communication systems. The existing channel model is only intended for MIMO systems that do not address the sparse characteristics of channel and are also not applicable for mmWave bands. The channel model  
Figure 1: Proposed framework for millimeter-wave (mmWave) MIMO system.  
in [27] is also only for MIMO systems and not for massive MIMO systems, which motivated us to deploy the Saleh-Valenzuela Channel Model that has sparse characteristics, is suitable for low-rank channel estimation, and embodies the spatial correlation characteristics of mmWave communications. In response to this situation, the widely used Saleh-Valenzuela model was used [12; 13; 14; 15; 16], and mmWave channel matrix \(\mathbf{H}\) under this model can be expressed as:  
\[\mathbf{H}=\sqrt{\frac{N_{l}N_{r}}{N_{d}N_{my}}}\sum\limits_{i=1}^{N_{d}}\sum \limits_{l=1}^{N_{my}}\beta_{i,l}a_{r}(\alpha^{r}_{i,l^{\prime}},\beta^{r}_{i, l})a_{t}(\alpha^{t}_{i,l^{\prime}}\beta^{t}_{i,l})^{*} \tag{5}\]  
where \(N_{cl}\) and \(N_{my}\) represent the number of clusters and scatterers in each cluster, respectively. \(\beta_{i,l}\) represents the gain coefficient of the \(l\)th path in the \(i\)th cluster. The gain coefficient satisfies the mean of 0, variance is \(\sigma_{\beta,i^{\prime}}^{2}\), and \(\sum\limits_{i=1}^{N_{c}}\sigma_{\beta,i}^{2}=\gamma\). Where \(\gamma\) is a normalization factor that causes channel matrix \(\mathbf{H}\) to satisfy condition \(E[\|\mathbf{H}\|_{\rm F}^{2}]=N_{t}N_{r}\), the matrix representation of the channel matrix is  
\[\mathbf{H}=\mathbf{A}_{r}\mathbf{\Lambda}\mathbf{\Lambda}_{t}^{*}, \tag{6}\]  
where  
\[\left\{\begin{array}{ll}\mathbf{A}_{r}=\left[a_{r}(\alpha^{r}_{i,l^{\prime}}, \beta^{r}_{i,l}),\ldots,\ a_{r}(\alpha^{r}_{i,l_{my}},\beta^{r}_{i,l_{my}}) \ldots\right]&i=1,\ldots,\ N_{cl}\\ \mathbf{A}_{t}=\left[a_{t}(\alpha^{t}_{i,l^{\prime}},\beta^{t}_{i,l}),\ldots,\ a_{t}(\alpha^{t}_{i,l_{my}},\beta^{t}_{i,l_{my}}) \ldots\right]&i=1,\ldots,\ N_{cl}\\ \mathbf{\Lambda}=\mathrm{diag}\left(\sqrt{\frac{N_{t}N_{t}}{N_{t}N_{my}}}\beta_{t,1},\ldots,\sqrt{\frac{N_{t}N_{t}}{N_{t}N_{my}}}\beta_{t,N_{my}}\right)&i=1, \ldots,\ N_{cl}\end{array}\right., \tag{7}\]  
where \(\alpha^{r}_{i,l}(\alpha^{t}_{i,l})\) and \(\beta^{r}_{i,l}(\beta^{t}_{i,l})\) represent the azimuth and elevation angles of the angle of arrival (AoA) and the angle of departure (AoD), respectively. \(a_{r}(\alpha^{r}_{i,l^{\prime}},\beta^{r}_{i,l})\), \(a_{t}(\alpha^{t}_{i,l^{\prime}},\beta^{t}_{i,l})\) are the antenna array response sequences of receiver and transmitter respectively, and the transmitting and receiving antenna arrays are arranged in a uniform planar array (UPA), so the corresponding antenna-array vector is:  
\[\begin{array}{ll}a\Big{(}\alpha_{i,1},\beta_{i,l}\Big{)}=\\ \frac{1}{\sqrt{N_{t}}}\Big{[}1,\ldots,\ e^{j\frac{2n}{\lambda}\ d((m\sin\alpha_{i,l }\sin\beta_{i,l}+n\cos\beta_{i,l})},\ldots,e^{j\frac{2n}{\lambda}\ d((\sqrt{N_ {t}}-1)\sin\alpha_{i,l}\sin\beta_{i,l}+(\sqrt{N_{t}}-1)\cos\beta_{i,l})}\ \Big{]},\end{array} \tag{8}\]  
where \(m\) and \(n\) are index values of the two-dimensional planar array, respectively; for the antenna array of the transceiver end, values of \(m\) and \(n\) are, respectively, \(m=\Big{[}1,\ldots,\ \sqrt{N_{t}}\Big{]}\Big{(}m=\Big{[}1,\ldots,\ \sqrt{N_{t}}\Big{]}\Big{)}\), \(n=\Big{[}1,\ldots,\ \sqrt{N_{t}}\Big{]}\Big{(}n=\Big{[}1,\ldots,\ \sqrt{N_{t}}\Big{]}\Big{)}\), where \(d\) is the antenna spacing and \(\lambda\) is the signal wavelength.