# An Efficient Precoding Algorithm for mmWave Massive MIMO Systems
## 2 System Model
If the mmWave channel contains \(N_{\text{cl}}\) scattering clusters, with each cluster containing \(N_{\text{ray}}\) strip propagation paths, the channel \(\mathbf{H}\) of the system can be described as:  
\[\mathbf{H}=\sqrt{\frac{N_{t}N_{r}}{N_{\text{cl}}N_{\text{ray}}}}\sum_{i,l}\mathbf{ \alpha}_{i}\mathbf{\alpha}_{r}\!\left(\varphi_{il}^{r}\right)\!\mathbf{\alpha}_{i}\! \left(\varphi_{il}^{t}\right)^{H} \tag{4}\]  
where \(\mathbf{\alpha}_{i,l}\) represents the gain factor of the \(l\)th propagation path in the \(i\)th scattering cluster that follows a complex Gaussian distribution with zero mean and a variance of \(\sigma_{a_{i}}^{2}\), and it satisfies \(\sum_{i=1}^{N_{\text{cl}}}\sigma_{a_{i}}^{2}=\gamma\), and \(\gamma\) must satisfy \(E\left[\parallel\mathbf{H}\parallel_{\mathbf{F}}^{2}\right]=N_{t}N_{r}\) with \(\varphi_{il}^{r}\) as the angle of arrival (AoA). For the \(i\)th scattering cluster, \(\varphi_{il}^{r}\) is randomly distributed on \(\varphi_{i}^{r}\) with \(\varphi_{il}^{t}\) as the angle of departure (AoD), and for the \(i\)th scattering cluster, \(\varphi_{il}^{t}\) is randomly distributed on \(\varphi_{i}^{t}\). Generally, we choose the Laplacian distribution as a random distribution; \(\mathbf{\alpha}_{t}\!\left(\varphi_{il}^{t}\right)\) and \(\mathbf{\alpha}_{r}\!\left(\varphi_{il}^{r}\right)\) represent the array response vectors of the transmitter and receiver, respectively. The types of antenna arrays can be combined into various configurations depending on the arrangement of the antenna in the array. In mmWave massive MIMO systems, a uniform antenna array is generally selected to design the antennas at both ends of the transceiver. Common uniform antenna arrays have a uniform linear array (ULA) and uniform planar array (UPA). In a ULA, either the elevation or the azimuth perspective is considered, since it is a one-dimensional antenna array. The UPA is a two-dimensional antenna array, and this type of antenna arrays is preferable for mmWaves since it can accommodate more antenna elements within a small area at both the user equipment and the base station (BS). It also facilitates beamforming in an extra dimension, which results in the 3D-beamforming. For the convenience of analysis, a ULA is used in this paper. For a ULA, assuming that there are \(N\) antennas on the y-axis, the array response vector can be expressed as:  
\[\mathbf{\alpha}(\varphi)=\frac{1}{\sqrt{N}}\Big{[}1,\ e^{j\text{i}\text{i}\text{d }\sin\left(\varphi\right)},\ e^{j\text{i}\text{d}\text{i}\text{d}\sin\left( \varphi\right)},\ \ldots,\ e^{j\left(N-1\right)\text{d}\text{i}\text{d}\text{i}\text{d}\text{i} \text{d}}\Big{]}^{T} \tag{5}\]  
where \(\varphi\in[0,2\pi]\), \(k=\frac{2n}{\lambda}\), \(\lambda\) is the wavelength of the signal, and \(d\) is the spacing between antenna elements. In an actual system, channel state information (CSI) can be known by channel estimation. Inorder to focus only on precoding research, assuming that the transceiver knows the CSI, the spectral efficiency of the system is:  
\[R=\log_{2}\left|I_{N_{s}}+\frac{\rho}{N_{s}}\mathbf{R}_{n}^{-1}\mathbf{W}_{\text{BB}}^{H} \mathbf{W}_{\text{RF}}^{H}\mathbf{H}\mathbf{F}_{\text{RF}}F_{\text{BB}}\mathbf{F}_{\text{BB}}^{ H}\mathbf{F}_{\text{RF}}^{H}\mathbf{H}^{H}\mathbf{W}_{\text{RF}}\mathbf{W}_{\text{BB}}\right| \tag{6}\]  
where \(\mathbf{R}_{\text{n}}=\sigma_{n}^{2}\mathbf{W}_{\text{BB}}^{H}\mathbf{W}_{\text{RF}}^{H} \mathbf{W}_{\text{RF}}\mathbf{W}_{\text{BB}}\) is the noise covariance matrix processed by the receiver and \(\mathbf{I}_{N_{s}}\) is the identity matrix of the noise. Reference [9] approximates the spectral efficiency to minimize the Euclidian distance of the hybrid precoding matrix and the all-digital precoding matrix, whereby the precoding design problem can be written as:  
\[\left\{