# A New Design of Codebook for Hybrid Precoding in Millimeter-Wave Massive MIMO Systems
## 2 System Model
### Problem Formulation  
Based on [PERSON]'s theorem, the spectral efficiency of the system in Figure 1 can be formulated as follows:  
\[R=\log_{2}\left(\bigg{|}\mathbf{I}_{N_{\mathrm{s}}}+\frac{\rho}{N_{\mathrm{s}} }\mathbf{R}_{n}^{-1}\mathbf{W}_{\mathrm{BB}}^{H}\mathbf{W}_{\mathrm{RF}}^{H} \mathbf{H}\mathbf{F}_{\mathrm{RF}}\mathbf{F}_{\mathrm{BB}}^{H}\mathbf{F}_{ \mathrm{RF}}^{H}\mathbf{H}^{H}\mathbf{W}_{\mathrm{RF}}\mathbf{W}_{\mathrm{BB} }\bigg{|}\right), \tag{4}\]  
where \(\mathbf{R}_{n}=\sigma_{n}^{2}\mathbf{W}_{\mathrm{BB}}^{H}\mathbf{W}_{\mathrm{ RF}}^{H}\mathbf{W}_{\mathrm{BB}}\) denotes the combined noise covariance matrix.  
We need to jointly optimize the four variables (\(\mathbf{F}_{\mathrm{RF}}\), \(\mathbf{F}_{\mathrm{BB}}\),\(\mathbf{W}_{\mathrm{RF}}\), \(\mathbf{W}_{\mathrm{BB}}\)) in (4) to maximize spectral efficiency, but they are hard to optimize together in practice [18; 19]. By decoupling the joint optimization problem at the transmitter and receiver, the problem of maximizing spectral efficiency can be transformed into a problem of maximizing mutual information in [1], which can be expressed as follows:  
\[I(\mathbf{F}_{\mathrm{RF}},\mathbf{F}_{\mathrm{BB}})=\log_{2}\left(\bigg{|} \mathbf{I}_{N_{\mathrm{s}}}+\frac{\rho}{N_{\mathrm{s}}\sigma_{n}^{2}}\mathbf{ H}\mathbf{F}_{\mathrm{RF}}\mathbf{F}_{\mathrm{BB}}\mathbf{F}_{\mathrm{RF}}^{H} \mathbf{H}^{H}\bigg{|}\right), \tag{5}\]  
where \(I(\mathbf{F}_{\mathrm{RF}},\mathbf{F}_{\mathrm{BB}})\) denotes the amount of mutual information that can be maximized by optimizing the design of the hybrid precoder \((\mathbf{F}_{\mathrm{RF}},\mathbf{F}_{\mathrm{BB}})\); the optimization of the analog and digital precoding matrix is expressed as follows:  
\[\begin{split}&(\mathbf{F}_{\mathrm{RF}}^{\mathrm{opt}},\mathbf{F }_{\mathrm{BB}}^{\mathrm{opt}})=\arg\max_{\mathbf{F}_{\mathrm{RF}},\mathbf{F}_ {\mathrm{BB}}}I(\mathbf{F}_{\mathrm{RF}},\mathbf{F}_{\mathrm{BB}}),\\ &\mathrm{s.t.}\quad\mathbf{F}_{\mathrm{RF}}\in\mathcal{F}_{ \mathrm{RF}},\\ &\quad\left\|\mathbf{F}_{\mathrm{RF}}\mathbf{F}_{\mathrm{BB}} \right\|_{F}^{2}=N_{\mathrm{s}},\end{split} \tag{6}\]  
Figure 1: Simplified block diagram of the mmWave hybrid precoding system with the fully-connected transceiver architecture.  
where \(\mathcal{F}_{\text{RF}}\) is the feasible set whose elements are satisfied with the constraint of constant modulus and it is hard to find a general solution to work out (6) under the nonconvex constraint \(\mathbf{F}_{\text{RF}}\in\mathcal{F}_{\text{RF}}\). Refs. [1; 4] proved that maximizing the above amount of mutual information can be converted to the minimization problem, which aims to minimize the Euclidean distance between the designed hybrid precoder and the optimal unconstrained precoder. So, the design problem of hybrid precoders can be further reformulated as the follows:  
\[\begin{split}(\mathbf{F}_{\text{RF}}^{\text{opt}},\mathbf{F}_{ \text{BB}}^{\text{opt}})=&\text{arg}\min_{\mathbf{F}_{\text{RF}},\mathbf{F}_{\text{BB}}}\left\|\mathbf{F}_{\text{opt}}-\mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}\right\|_{F},\\ \text{s.t.}&\mathbf{F}_{\text{RF}}(m,n)\in\mathcal{F }_{\text{RF}},\forall m,n,\\ &\left\|\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\right\|_{F}^ {2}=N_{\text{s}},\end{split} \tag{7}\]  
where \(\mathbf{F}_{\text{opt}}\) is the optimal unconstrained precoder that can theoretically achieve the best spectral efficiency. Additionally, \(\mathbf{F}_{\text{opt}}=\mathbf{V}(\cdot,1:N_{\text{s}})\), where \(\mathbf{V}\) is the right singular matrix of the channel matrix \(\mathbf{H}=\mathbf{U}\mathbf{E}\mathbf{V}^{H}\) and can be obtained by performing singular value decomposition (SVD). In addition, when using a \(q\)bit resolution phase shifter, the phase shifter can achieve \(2^{q}\) phase angles, so \(\mathcal{F}_{\text{RF}}\) can be obtained by \(\mathcal{F}_{\text{RF}}=\left\{1,\omega,\omega^{2},\cdots,\omega^{2q-1}\right\}\), \(\omega=e^{2\pi/2^{q}}\).