# Massive MIMO Indoor Transmissions at 38 and 65 GHz Applying Novel HBF Techniques for 5G
## 3 Methodology
### Implemented Algorithms
#### 3.1.1 Massive MIMO Hybrid Beamforming (HBF)  
As shown in Figure 3, the transmitting HBF structure modified the \(N\)s data streams or layers in each of the \(N\) subcarriers passing through the digital precoder \(\mathbf{V}_{D}\). Then, the signals were sent over RF chains to the analog precoder \(\mathbf{V}_{RF}\), and finally, the output data were transmitted by the \(Nt\) antennas, and then were multiplied by the channel matrix \(H\). The received signal vector \(\mathbf{y}(k)\ \in\mathbb{C}^{N_{x}x1}\) at the \(k\)th subcarrier can be expressed as:  
\[\mathbf{y}(k)=\mathbf{H}(k)\mathbf{V}_{RF}\mathbf{V}_{D}(k)\mathbf{s}(k)+\mathbf{n}(k), \tag{1}\]  
where \(k=1,\ldots,N,\mathbf{s}(k)\ \in\mathbb{C}^{N_{x}x1}\), and \(\mathbf{n}(k)\ \sim\ \mathcal{CN}\big{(}0,\sigma^{2}\mathbf{I}_{Nt}\big{)}\) denote the transmitted symbol vector and the additive noise vector at the \(k\)th subcarrier, respectively. \(\mathbf{V}_{RF}\in\mathbb{C}^{N_{x}xN_{RF}}\) and \(\mathbf{V}_{D}(k)\in\mathbb{C}^{N_{RF}xN_{s}}\) denote the analog and digital precoder, respectively. The analog precoder was the same for all subcarriers. Finally, \(\mathbf{H}(k)\in\mathbb{C}^{N_{x}xN_{t}}\) was the channel matrix in the frequency domain at the \(k\)th subcarrier.  
Figure 3: Block diagram of the HBF technique.  
The \(\mathbf{V}_{RF}\) is first obtained as:  
\[\mathbf{V}_{RF}(i,j)=\left\{\begin{array}{ll}1,\;\;if\;\eta_{i,j}=0,\\ \frac{\eta_{i,j}}{|\eta_{i,j}|},\;\;otherwise\end{array}\right\} \tag{2}\]  
where  
\[\eta_{i,j}=\sum_{l\
eq i}\mathbf{G}_{j}(i,l)\mathbf{V}_{RF}(l,j), \tag{3}\]  
\[\mathbf{G}_{j}=\frac{\gamma^{2}}{\sigma^{2}}\mathbf{F}_{1}-\frac{\gamma^{4}}{\sigma^{ 4}}\mathbf{F}_{1}\mathbf{\overline{V}}_{RF}^{j}\mathbf{C}_{j}^{-1}\Big{(}\mathbf{\overline{V}} _{RF}^{j}\Big{)}^{H}\mathbf{F}_{1}, \tag{4}\]  
\[\mathbf{F}_{1}=\frac{1}{N}\sum_{k=1}^{N}(\mathbf{H}[k])^{H}\mathbf{H}[k]), \tag{5}\]  
\[\mathbf{C}_{j}=\mathbf{I}+\frac{\gamma^{2}}{\sigma^{2}}\Big{(}\mathbf{\overline{V}}_{RF}^ {j}\Big{)}^{H}\mathbf{F}_{1}\mathbf{\overline{V}}_{RF}^{j} \tag{6}\]  
and \(\gamma=\sqrt{P/(N_{t}N_{RF})}\), \(P\) being the total transmit power per subcarrier, \(\mathbf{\overline{V}}_{RF}^{j}\) is the sub \(-\) matrix of \(\mathbf{V}_{RF}\) with \(j^{\text{th}}\) column removed, \(\sigma^{2}\) is the noise power, and \((\cdot)^{\mathcal{H}}\) represents the conjugate transpose of \((\cdot)\).  
For a fixed \(\mathbf{V}_{RF}\), the design of \(\mathbf{V}_{\mathbf{D}}\) can be written as:  
\[\mathbf{V}_{\mathbf{D}}(k)=\mathbf{Q}^{-\frac{1}{2}}\mathbf{U}_{e}(k)\mathbf{\Gamma}_{e}(k), \tag{7}\]  
where \(\mathbf{Q}=\mathbf{V}_{RF}^{H}\mathbf{V}_{RF},\mathbf{U}_{e}[k]\) is the set of right singular vectors corresponding to the N\(s\) largest singular values of \(\mathbf{H}_{eff}[k]\mathbf{Q}^{-1/2}\), \(\mathbf{H}_{eff}[k]=\mathbf{H}[k]\mathbf{V}_{RF}\) being the effective channel of subcarrier \(k\). \(\mathbf{\Gamma}_{e}[k]\) is the diagonal matrix of allocated powers to each stream, i.e., assuming equal power allocation for all streams in each subcarrier, then \(\mathbf{\Gamma}_{e}[k]=\sqrt{P/N_{s}}\mathbf{I}\).  
In reception, as the number of receiving antennas was low (5 antennas), the linear decoding method was applied using Zero-Fencing (ZF) [26]. The estimation of the transmitted symbols, assuming \(\mathbf{H}_{new}(k)=\mathbf{H}(k)\mathbf{V}_{RF}\mathbf{V}_{\mathbf{D}}(k)\), is as follows:  
\[\hat{\mathbf{s}}(k)=\mathbf{H}_{new}(k)^{+}\mathbf{H}_{new}(k)\mathbf{s}(k)+\mathbf{H}_{new}(k)^{ +}\mathbf{n}(k) \tag{8}\]  
where \((\cdot)^{+}\) represents the the Penrose-Monroe pseudo inverse of \((\cdot)\).