# Hybrid Beamforming for Reduction of Inter-Beam Interference in Millimeter-Wave Cellular Systems
## 2 Hybrid Beamforming Techniques for a Multicell Multiuser Environment
### Type-3 Precoder  
The Type-2 precoder requires a large network overhead because the precoder is operated on the network-level consisting of all neighboring \(BS\)s. To reduce the network overhead, the \(BS\) with a Type-3 precoder forms analog beams in the direction of not only the \(MS\)s in the serving cell but also the \(MS\)s in the neighboring cell, as shown in Figure 2c. In this case, the Type-3 precoder allows us to design the digital precoder of each \(BS\) independently. The \(RF\) precoder in the Type-3 precoder is given by  
\[\mathbf{F}_{RF}^{b_{n}}=\mathbf{A}_{BS}^{b_{n}}=\left[\mathbf{a}_{BS}^{u_{1,1} }\left(\phi_{BS}^{u_{1,1}},\theta_{BS}^{u_{1,1}}\right),\ \cdots,\ \mathbf{a}_{BS}^{u_{1,l}}\left(\phi_{BS}^{u_{1,l}},\theta_{BS}^{u_{1,l}} \right),\ \mathbf{a}_{BS}^{u_{2,1}}\left(\phi_{BS}^{u_{2,1}},\theta_{BS}^{u_{2,1}} \right),\ \cdots,\ \mathbf{a}_{BS}^{u_{1,l}}\left(\phi_{BS}^{u_{2,l}},\theta_{BS}^{u_{2,l}} \right)\right] \tag{13}\]  
The effective network channel for each \(BS\) with the Type-3 precoder is given by (6). However, unlike the Type-1 precoder, the effective network channel for the \(BS\)\(b_{n}\) with the Type-3 precoder, \(\widetilde{\mathbf{H}}^{b_{n}}\in\mathbb{C}^{BU\times BU}\), becomes a square matrix because of the increase in the number of \(RF\) precoder. In the Type-3 case, a ZF precoder for the effective network channel is obtained as follows:  
\[\mathbf{F}_{BB}^{b_{n}}=\left(\widetilde{\mathbf{H}}^{b_{n}}\right)^{-1} \tag{14}\]  
Unlike the Type-2 case, the \(BS\) with the Type-3 precoder can operate independently during the data transmission period. In addition, the number of beams that eventually formed at the \(BS\) is the same as the number of \(MS\)s in the serving cell, \(U\). Using (6) and (13), the effective network channel in the Type-3 case can be expressed as  
\[ \begin{split}&\widetilde{\mathbf{H}}^{b_{n}}=\mathbf{D}^{b_{n}} \mathbf{A}_{BS}^{b_{n}}\Big{(}\mathbf{A}_{BS}^{b_{n}}\Big{)}^{*}\\ &\Big{[}\mathbf{D}^{b_{n}}\Big{]}_{u_{l,S},u_{l,S}}=\sqrt{N_{BS} N_{MS}}\mathbf{a}_{u_{l,S}}\Big{(}\mathbf{w}_{RF}^{u_{l,S}}\Big{)}^{*}\mathbf{a}_{ MS}^{u_{l,S}}\Big{(}\phi_{MS}^{u_{l,S}}\theta_{MS}^{u_{l,S}}\Big{)}\end{split}  \tag{15}\]  
If \(l\) is equal to \(n\) and the \(RF\) combiner of the \(MS\)\(u_{n,S}\) is correctly steered to the direction of the \(BS\)\(b_{n}\), the equation \(\left(\mathbf{w}_{RF}^{u_{n,S}}\right)^{*}\mathbf{a}_{MS}^{u_{n,S}}\Big{(}\phi_ {MS}^{u_{n,S}},\theta_{MS}^{u_{n,S}}\Big{)}=1\) is always satisfied. In addition, the hybrid precoder satisfies the following power constraint:  
\[\left\|\mathbf{F}_{RF}^{b_{n}}\mathbf{\hat{e}}_{BB,u_{n,m}}^{b_{n}}\right\|^{2}=1 \tag{16}\]  
Then, the precoder in (14) can be rewritten as follows:  
\[ \begin{split}&\mathbf{F}_{BB}^{b_{n}}=\left(\widetilde{\mathbf{H}} ^{b_{n}}\right)^{-1}\mathbf{\Lambda}=\left(\mathbf{D}^{b_{n}}\mathbf{A}_{BS}^{b _{n}}\Big{(}\mathbf{A}_{BS}^{b_{n}}\Big{)}^{*}\right)^{-1}\mathbf{\Lambda}\\ &\mathbf{\hat{e}}_{BB,u_{n,m}}^{b_{n}}=\left[\mathbf{F}_{BB}^{b_{n }}\right]_{:,u_{n,m}}=\left(\mathbf{D}^{b_{n}}\mathbf{A}_{BS}^{b_{n}}\Big{(} \mathbf{A}_{BS}^{b_{n}}\Big{)}^{*}\right)^{-1}[\mathbf{\Lambda}]_{:,u_{n,m}} \end{split}  \tag{17}\]where \(\mathbf{\Lambda}\) represents the diagonal matrix inserted to satisfy the power constraint in (16). \([\mathbf{\Lambda}]_{:,u_{n,m}}\) denotes the \((U\)\({}^{*}(n-1)+m)\)-th row vector of \(\mathbf{\Lambda}\). Using (13) and (17), (16) can be rewritten as  
\[