# A New Design of Codebook for Hybrid Precoding in Millimeter-Wave Massive MIMO Systems
## 2 System Model
### Channel Model  
The overall system architecture is shown in Figure 1. As shown in the single-user mmWave massive MIMO system, we consider a fully-connected structure in the system in which each radio frequency (RF) chain will be connected to all antennas through phase shifters. In addition, the antenna array will adopt a uniform linear array (ULA) configuration. In this system, the transmitter and receiver are equipped with \(N_{\rm t}\) and \(N_{\rm r}\) antennas, and \(N_{\rm t}^{\rm RF}\) and \(N_{\rm r}^{\rm RF}\) RF chains, where \(N_{\rm t}\) and \(N_{\rm r}\) denote the number of antennas for the transmitter and receiver, respectively, and \(N_{\rm t}^{\rm RF}\) and \(N_{\rm t}^{\rm RF}\) denote the number of RF chains for the transmitter and receiver, respectively. For the hybrid precoding system, the number of data streams is \(N_{\rm s}\), where \(N_{\rm s}\leq N_{\rm t}^{\rm RF}\leq N_{\rm t}\,,N_{\rm s}\leq N_{\rm r}^{\rm RF} \leq N_{\rm r}\). Unlike traditional digital precoding, the hybrid precoder is composed of a digital precoder \(\mathbf{F_{\rm{BB}}}\) and an analog precoder \(\mathbf{F_{\rm{RF}}}\), where \(\mathbf{F_{\rm{BB}}}\) denotes the digital precoding matrix of size \(N_{\rm t}^{\rm RF}\times N_{\rm s}\) and \(\mathbf{F_{\rm{RF}}}\) denotes the analog precoding matrix of size \(N_{\rm t}\times N_{\rm t}^{\rm RF}\). So, after passing the hybrid precoder, the transmitted signal is \(\mathbf{x}=\mathbf{F_{\rm{RF}}}\mathbf{F_{\rm{BB}}}\), where \(\mathbf{s}\) denotes the data stream's vector satisfying \(\mathbb{E}[\mathbf{s}\mathbf{s}^{H}]=\frac{1}{N_{\rm s}}\mathbf{I}_{N_{\rm s}}\), where \(\mathbf{I}_{N_{\rm s}}\) denotes the identity matrix of size \(N_{\rm s}\times N_{\rm s}\). With the limitation of the total power in the transmitter, the hybrid precoder power needs to satisfy \(\|\mathbf{F_{\rm{RF}}}\mathbf{F_{\rm{BB}}}\|_{F}^{2}=N_{\rm s}\). At the receiver, there are also a \(N_{\rm r}^{\rm RF}\times N_{\rm s}\) digital combining matrix \(\mathbf{W_{\rm{BB}}}\) and a \(N_{\rm r}\times N_{\rm r}^{\rm RF}\) analog combining matrix \(\mathbf{W_{\rm{RF}}}\), so the received signal vector \(\mathbf{y}\in\mathbb{C}^{N_{\rm t}\times 1}\) can be expressed as [14]  
\[\mathbf{y}=\sqrt{\rho}\mathbf{W_{\rm{BB}}^{H}}\mathbf{W_{\rm{RF}}^{H}}\mathbf{ H}\mathbf{F_{\rm{RF}}}\mathbf{F_{\rm{BB}}}\mathbf{s}+\mathbf{W_{\rm{BB}}^{H}} \mathbf{W_{\rm{RF}}^{H}}\mathbf{n}, \tag{1}\]  
where \(\rho\) is the average received power, \(\mathbf{H}\) of size \(N_{\rm r}\times N_{\rm t}\) is the channel matrix, and \(\mathbf{n}\) denotes the noise that follows the distribution \(\mathcal{CN}(0,\sigma_{n}^{2})\). The features of mmWave transmission are the high free-space path loss and the large-scale antenna arrays, resulting in limited spatial selectivity or scattering [14] and high correlation of antennas [15]. For mmWave channel modeling, many of the classical statistical fading distributions used in traditional MIMO scenarios are obviously not applicable to the characteristic of mmWave transmission environments. In order to better utilize the feature in mmWave channels, we present a narrowband clustered channel model, according to the extended Saleh-Valenzuela model [14; 15]. Therefore, the discrete-time narrowband clustered channel model can be expressed as [14; 15; 16]  
\[\mathbf{H}=\sqrt{\frac{N_{\rm t}N_{\rm r}}{N_{\rm{cl}}N_{\rm{ray}}}}\sum_{i=1} ^{N_{\rm{cl}}}\sum_{l=1}^{N_{\rm{ray}}}\alpha_{il}\mathbf{a}_{\rm{r}}(\phi_{il} ^{\rm{r}})\mathbf{a}_{\rm{t}}(\phi_{il}^{\rm{t}})^{H}, \tag{2}\]where \(\mathbf{H}\) is the channel matrix, \(\alpha_{il}\) is the complex gain of the \(l\)-th path in the \(i\)-th cluster that is subjected to the complex Gaussian distribution \(\mathcal{CN}(0,\sigma_{\alpha,i}^{2})\), where \(\sum_{i=1}^{N_{\mathrm{cl}}}\sigma_{\alpha,i}^{2}=\sqrt{\frac{N_{\mathrm{cl}}N _{\mathrm{ray}}}{N_{\mathrm{cl}}N_{\mathrm{ray}}}}\) and \(N_{\mathrm{cl}}\) and \(N_{\mathrm{ray}}\) denote the number of clusters and the number of paths in each cluster, respectively.