# Robust Hybrid Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks
## 2 System Model  
### System Specifications  
The system framework is shown in Figure 1. Consider a base station configured with \(N_{t}^{RF}\) transmit antennas and \(N_{r}^{RF}\) radio links using \(N_{s}\) data streams, data communication with users configuring \(N_{r}\) receiver antennas, and \(N_{r}^{RF}\) links. The number of data streams, antennas, and RF links at the transmitter and receiver satisfies \(N_{s}\leq N_{t}^{RF}\leq N_{t}\), respectively.  
Where: uppercase letter \(A\) represents the matrix, and lowercase letter \(x\) represents the column vector. \(A^{-1}\) represents the inverse of matrix \(A\); \(A^{*}\) indicates the conjugate transpose of the matrix \(A\); \(\text{diag}(\cdot)\) denotes a diagonal matrix; \(A\)[:\([1\): \(a\)\(]\)] takes the first \(a\) column of matrix \(A\); \(\|A\|_{\text{F}}^{2}\) represents the Frobenius norm of matrix \(A\); \(A^{*}\) indicates the pseudoinverse of \(A\); \(\text{Tr}(A)\) represents the trace of matrix \(A\)The modulated symbol is first processed by digital beamforming matrix \(\mathbf{F}_{\text{BB}}\in\mathbb{C}^{N_{t}^{RF}\times N_{s}}\) for the baseband signal, and the processed signal is then subjected to analog beamforming matrix \(\mathbf{F}_{\text{RF}}\in\mathbb{C}^{N_{t}\times N_{t}^{RF}}\) to form the final transmitted signal:  
\[\mathbf{x}=\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\mathbf{s} \tag{1}\]  
where \(\mathbf{s}=[s_{1},\dots,s_{N_{s}}]^{T}\in\mathbb{C}^{N_{s}}\).  
In this paper, using the Saleh-Valenzuela channel model [12; 13; 14; 15; 16], \(\mathbf{H}\in\mathbb{C}^{N_{r}\times N_{t}}\) represents the downlink channel matrix, then received signal \(\mathbf{y}\) is expressed as:  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}=\mathbf{H}\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\mathbf{s}+\bm {n} \tag{2}\]  
Among them, noise vector \(\mathbf{n}\) obeys a Gaussian distribution with a mean of 0 and a variance of \(\sigma^{2}\mathbf{I}\) that is, \(\mathbf{n}\sim\mathbb{C}\mathbb{N}\big{(}0,\sigma^{2}\mathbf{I}\big{)}\).  
\(\mathbf{W}_{\text{RF}}\in\mathbb{C}^{N_{r}\times N_{r}^{RF}}\) and \(\mathbf{W}_{\text{BB}}\in\mathbb{C}^{N_{RF}^{RF}\times N_{s}}\) represent, respectively, the analog merge matrix and the digital merge matrix. Assuming that the transceiver knows the state information of the channel, received signal \(\mathbf{y}\) is processed by the signal in the analog domain and the digital domain. The received symbol vector can be expressed as:  
\[\mathbf{\bar{s}}=\mathbf{W}_{\text{RF}}^{*}\mathbf{W}_{\text{BB}}^{*}\mathbf{H}\mathbf{F}_{\text{ RF}}\mathbf{F}_{\text{BB}}\mathbf{s}+\mathbf{W}_{\text{RF}}^{*}\mathbf{W}_{\text{BB}}^{*}\mathbf{n} \tag{3}\]  
Analog combining matrix \(\mathbf{W}_{\text{RF}}\) and analog beamforming matrix \(\mathbf{F}_{\text{RF}}\) are implemented by using a phase shifter with phase modulation at the transceiver end. Because of limited amplitude modulation capability, the analog beamforming matrix is used, and the combining matrix elements generally satisfy constant modulus constraint \(\big{|}(\mathbf{W}_{\text{RF}})_{m,n}=1\big{|}_{r}\big{|}(\mathbf{F}_{\text{RF}})_{m,n}= 1\big{|}\). These constraints mean that the matrix elements are unit-normalized by taking the modulus that allows to represent them with unit amplitude. In the case where the channel state information (CSI) is known, the spectral efficiency [23]\(R\) of the system can be expressed as:  
\[\mathbf{R}=\log_{2}\left(\det[\mathbf{I}_{N_{s}}+\beta(\mathbf{W}_{\text{RF}}\;\mathbf{W}_{ \text{BB}})^{\dagger}\mathbf{H}\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\times\mathbf{F}_{ \text{BB}}^{*}\mathbf{W}_{\text{RF}}^{*}\mathbf{H}^{*}\mathbf{W}_{\text{RF}}\;\mathbf{W}_{ \text{BB}}]\right) \tag{4}\]  
where \(\beta=\frac{p}{N_{s}}\), \(P\) represents the average transmit power of the sender.