## 5 Low-Complexity Hybrid Beamforming  
### Hybrid Beamforming Based on Spectral Efficiency Maximization  
When the computation of the RIS reflection matrix \(\Phi\) is complete, we will next solve the optimization problem of \((F_{RF},F_{BB,k},W_{RF,k}W_{BB,k})\). The computational complexity becomes high if the joint optimization is performed at both \((F_{RF},F_{BB,k},W_{RF,k}W_{BB,k})\) transceivers at the same time. Because of the constant mode constraints of \(F_{RF,k}\) and \(W_{RF,k}\), this optimization problem will be nonconvex, further increasing the complexity of the problem. To facilitate the solution, we decouple the transceiver and the transmitter while considering the design of the hybrid beamforming matrix at the transmitter side only. The problem in this section is translated into designing the optimal hybrid beamforming matrix \((F_{RF},F_{BB,k})\). This maximizes the system spectral efficiency. After designing the RIS reflection matrix \(\Phi\), the channel \(H_{\rm{kc}}=\hat{M}_{k}\Phi\tilde{R}_{k}\). The channel \(H_{\rm{kc}}\) performs a singular value decomposition as the following equation:  
\[H_{\rm{kc}}=G^{\prime}_{\rm{kc}}\Sigma^{\prime}_{\rm{kc}}V^{\prime H}_{\rm{kc}} \tag{71}\]  
where \(\Sigma^{\prime}_{\rm{kc}}= \begin{bmatrix}\Sigma^{\prime}_{\rm{kc},1}&0\\ 0&\Sigma^{\prime}_{\rm{kc},2}\end{bmatrix} \), \(V^{\prime}_{\rm{kc}}= \begin{bmatrix}V^{\prime}_{\rm{kc},1}&V^{\prime}_{\rm{kc },2}\end{bmatrix} \), \(\Sigma^{\prime}_{\rm{kc},1}\) represents a stream of information to be processed, and \(V^{\prime}_{\rm{kc},1}\epsilon^{\rm{CN}_{\rm{T}}\times\mathbb{N}_{k,k}}\) represents the product between the information stream to be processed and the number of transmitting antennas. It is assumed that the beamforming matrix received at the receiving end is perfect, that is, \(G^{\prime}_{\rm{kc}}=W_{RF,k}W_{BB,k}\). By referring to the noise covariance matrix of the receiving user above, we can express \(R_{n}\) as:  
\[ \begin{array}{c}R_{n}=\sigma_{n}^{2}W_{BB,k}^{H}W_{RF,k}^{H}W_{RF,k}W_{BB,k} \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\sigma_{n}^{2}G^{\prime}_{\rm {kc}}H^{\prime}_{\rm{kc}}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad= \sigma_{n}^{2}I_{kN_{\rm{s},k}}\end{array}  \tag{72}\]  
Combined with the hybrid channel \(H_{\rm{kc}}\), the spectral efficiency of the system can be expressed through Equation (63) as:  
\[ \begin{array}{c}R=log_{2}\Bigg{(}\Bigg{[}I_{kN_{\rm{s},k}}+\frac{\rho^{ \prime}}{kN_{\rm{s},k}}R_{n}^{-1}W_{BB,k}^{H}W_{RF,k}^{H}\Big{(}G^{\prime}_{ \rm{kc}}\Sigma^{\prime}_{\rm{kc}}V^{\prime H}_{\rm{kc}}\Big{)}\\ \times F_{RF}F_{BB,k}\Big{(}W_{BB,k}^{H}W_{RF,k}^{H}\Big{(}G^{\prime}_{\rm{kc}} \Sigma^{\prime}_{\rm{kc}}V^{\prime H}_{\rm{kc}}\Big{)}F_{RF}F_{BB,k}\Big{)}^{H }\Bigg{]}\\ =log_{2}\Bigg{(}\Bigg{[}I_{kN_{\rm{s},k}}+\frac{\rho^{\prime}}{kN_{\rm{s},k} \sigma_{n}^{2}}\Big{(}\Sigma^{\prime}_{\rm{kc}}{}^{2}\Big{)}V^{\prime H}_{\rm {kc}}F_{RF}F_{BB,k}F_{RF}F_{HR}F_{BB,k}H^{\prime}V^{\prime}_{\rm{kc}}\Bigg{]} \Bigg{)}\end{array}  \tag{73}\]  
From the above, it is clear that \(V^{\prime}_{\rm{kc},1}\epsilon^{\rm{CN}_{\rm{T}}\times\mathbb{N}_{k}}\). Then, the best unconstrained digital beamforming matrix in the system can be \(F_{opt}=V^{\prime}_{\rm{kc},1}\). Because of the constant mode constraint, \(F_{opt}\) is not possible to use \(F_{RF,k}F_{BB,k}\) be represented. Assuming that \(V^{\prime H}_{\rm{1c}}F_{RF,k}F_{BB,k}\approx I_{\rm{kN}}\), \(V^{\prime H}_{\rm{2c}}F_{RF,k}F_{BB,k}\approx 0\), the precoding matrix is expressed as:  
\[