# An Efficient Precoding Algorithm for mmWave Massive MIMO Systems
## 2 System Model  
The shared array architecture of the mmWave massive MIMO system is shown in Figure 1. The number of transmitting antennas at the transmitting end is \(N_{t}\), the number of receiving antennas at the receiving end is \(N_{r}\), the number of RF links at the transmitting end is \(N_{t}^{RF}\), the number of RF links at the receiving end is \(N_{r}^{RF}\), and the data streams of the transmitting end and the receiving end are both \(N_{s}\). In order to ensure multi-stream transmission, it is necessary to satisfy \(N_{s}\leq N_{t}^{RF}\leq N_{t}\) and \(N_{s}\leq N_{r}^{RF}\leq N_{r}\). Under this hardware structure, the signal is transmitted to the channel \(\mathbf{H}\in\mathbb{C}^{N_{t}\times N_{t}}\) by the processing of the baseband precoder \(\mathbf{F}_{\text{BB}}\in\mathbb{C}^{N_{t}^{RF}\times N_{t}}\) and the RF precoder \(\mathbf{F}_{\text{RF}}\in\mathbb{C}^{N_{t}\times N_{t}^{RF}}\), and the transmitting signal of the transmitting end is:  
\[\mathbf{x}=\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\mathbf{S} \tag{1}\]  
where \(\mathbf{S}=[s_{1},s_{2},\ldots,s_{N_{s}}]^{T}\) is the data stream of the signal, and \(E\big{[}\mathbf{S}\mathbf{S}^{H}\big{]}=\frac{1}{N_{s}}\mathbf{I}_{N_{s}}\); \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{N_{t}}]^{T}\) is the transmitting signal for the transmitting end. Figure 1: Proposed array architecture of mmWave massive MIMO systems. The received signal arriving at the receiving end antenna after channel transmission is:  
\[\mathbf{y}=\sqrt{\rho}\mathbf{H}\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\mathbf{S}+\mathbf{n} \tag{2}\]  
where \(\mathbf{n}\in\mathbb{C}^{N_{r}}\) represents a Gaussian white noise with zero mean and a covariance matrix of \(\sigma^{2}\mathbf{I}_{N_{r}}\), \(\mathbf{y}=\left[y_{1},y_{2},\ldots,y_{N_{r}}\right]^{T}\) is the signal received by the receiving antenna, \(\rho\) is the average received power, \(\mathbf{H}\) is the channel matrix, and \(E\left[\parallel\mathbf{H}\parallel_{\mathbf{F}}^{2}\right]=N_{t}N_{r}\). The received signal is further processed by the RF combiner \(\mathbf{W}_{\text{RF}}\in\mathbb{C}^{N_{r}\times N_{r}^{RF}}\) and the baseband combiner \(\mathbf{W}_{\text{BB}}\in\mathbb{C}^{N_{r}^{RF}\times N_{s}}\), and the received signal received by the receiver is:  
\[\hat{\mathbf{y}}=\sqrt{\rho}\ \mathbf{W}_{\text{BB}}^{H}\mathbf{W}_{\text{RF}}^{H}\mathbf{H} \mathbf{F}_{\text{RF}}\mathbf{S}+\mathbf{W}_{\text{BB}}^{H}\mathbf{W}_{\text{RF}}^{H}\bm {n} \tag{3}\]  
where \(\hat{\mathbf{y}}=\left[\hat{y}_{1},\hat{y}_{2},\ldots,\hat{y}_{N_{s}}\right]^{T}\) is the signal received at the receiving end. \(\mathbf{F}_{\text{RF}}\) is implemented using the analog network so that it satisfies the elements \(\left(\mathbf{F}_{\text{RF}}^{(i)}\mathbf{F}_{\text{RF}}^{(i)H}\right)_{l,l}=\frac{1}{ N_{t}}\), where \(\left(\cdot\right)_{l,l}\) represent the \(l\)th diagonal element of the matrix. Similarly, \(\left(\mathbf{W}_{\text{RF}}^{(i)}\mathbf{W}_{\text{RF}}^{(i)H}\right)_{l,l}=\frac{1} {N_{r}}\). The total power constraint at the transmitting end is \(\parallel\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\parallel=\mathbf{z}_{\text{F}} ^{2}=N_{s}\). Considering the high path loss of the mmWave channel, the sparse distribution in space, the close arrangement of the antenna array on the transceiver in the massive MIMO system, and the high correlation of the antenna elements, the traditional fading statistical channel model is not applicable. Therefore, the ray-tracing model is usually used for modeling.