# Hybrid Beamforming for Reduction of Inter-Beam Interference in Millimeter-Wave Cellular Systems
## 2 Hybrid Beamforming Techniques for a Multicell Multiuser Environment
### Type-4 Precoder  
A Type-4 precoder is proposed for the case where a hybrid beamformer is available at the \(MS\) as well as the \(BS\). In the Type-4 case, the precoder at the \(BS\) removes only the interference from adjacent beams in the serving cell, while the combiner at the \(MS\) is designed to remove interference from neighboring cells. As shown in Figure 2d, the \(BS\) forms beams in the directions of \(MS\)s in the serving cell, while the \(MS\)s form beams in the direction of not only the serving \(BS\), but also neighboring \(BS\)s that generate interference to \(MS\)s in the serving cell. For example, in Figure 2d, the \(MS\)\(u_{1,1}\) forms beams in the directions of \(BS\)s \(b_{1}\) and \(b_{2}\). The precoder at the \(BS\)\(b_{1}\) removes the inter-beam interference in the serving cell, while the combiner at the \(MS\)\(u_{1,1}\) removes interference from neighboring \(BS\)\(b_{2}\). In the Type-4 case, the \(RF\) precoder at the \(BS\) is given by  
\[\mathbf{F}_{RF}^{b_{n}}=\mathbf{A}_{BS}^{b_{n}}=\Big{[}\mathbf{a}_{BS}^{u_{n,1 }}\Big{(}\phi_{BS}^{u_{n,1}},\phi_{BS}^{u_{n,1}}\Big{)},\;\mathbf{a}_{BS}^{u_{ n,2}}\Big{(}\phi_{BS}^{u_{n,2}},\phi_{BS}^{u_{n,2}}\Big{)},\;\cdots,\;\mathbf{a}_{BS}^{u_{ n,1}l}\Big{(}\phi_{BS}^{u_{n,1}l},\theta_{BS}^{u_{n,1}l}\Big{)}\Big{]} \tag{21}\]  
A digital precoding technique at the \(BS\) in the Type-4 case is the same as that in the Type-1 case. The \(RF\) combiner at the \(MS\) is given by  
\[\mathbf{W}_{RF}^{u_{n,m}}=\Big{[}\mathbf{a}_{MS}^{b_{1},u_{n,m}}\Big{(}\phi_{ MS}^{b_{1},u_{n,m}},\theta_{MS}^{b_{1},u_{n,m}}\Big{)},\;\mathbf{a}_{MS}^{b_{2},u_{ n,m}}\Big{(}\phi_{MS}^{b_{2},u_{n,m}},\theta_{MS}^{b_{2},u_{n,m}}\Big{)},\;\cdots,\; \mathbf{a}_{MS}^{b_{2},u_{n,m}}\Big{(}\phi_{MS}^{b_{2},u_{n,m}},\theta_{MS}^{b_{ 2},u_{n,m}}\Big{)}\Big{]} \tag{22}\]  
where \(\mathbf{a}_{MS}^{b_{1},u_{n,m}}\Big{(}\phi_{MS}^{b_{1},u_{n,m}},\theta_{MS}^{b_{ 1},u_{n,m}}\Big{)}/\) denotes the antenna-array response vector at the \(MS\)\(u_{n,m}\) formed toward the \(BS\)\(b_{l}\). The effective network channel of the \(BS\)\(b_{n}\) in the Type-4 case is given by  
\[\widetilde{\mathbf{H}}^{b_{n}}=\left[ \begin{array}{c}\Big{(}\mathbf{w}_{RF}^ {u_{n,1}}\Big{)}^{*}\mathbf{H}_{u_{1}}^{b_{n}}\mathbf{F}_{RF}^{b_{n}}\\ \Big{(}\mathbf{w}_{RF}^{u_{n,2}}\Big{)}^{*}\mathbf{H}_{u_{1}}^{b_{n}}\mathbf{F }_{RF}^{b_{n}}\\ \vdots\\ \Big{(}\mathbf{w}_{RF}^{u_{n,1}l}\Big{)}^{*}\mathbf{H}_{u_{2}l}^{b_{n}}\mathbf{F }_{RF}^{b_{n}}\end{array} \right] \tag{23}\]Unlike (6) in the Type-1 case, only \(MS\)s in the serving cell are considered in the effective network channel in (23). In the Type-4 case, the \(MS\) also forms an effective network channel as follows:  
\[\widetilde{\mathbf{H}}_{u_{n,m}}=\left[ \begin{array}{cccc}\left(\mathbf{W}_{ RF}^{u_{n,m}}\right)^{*}\sum\limits_{g=1}^{\frac{U}{2}}\left(\mathbf{H}_{u_{n,m}}^{b_{1 }}\mathbf{f}_{RF,\mu_{1,g}}^{b_{1}}\right)&\cdots&\left(\mathbf{W}_{RF}^{u_{n,m }}\right)^{*}\sum\limits_{g=1}^{U}\left(\mathbf{H}_{u_{n,m}}^{b_{B}}\mathbf{f}_ {RF,\mu_{B,g}}^{b_{B}}\right)\end{array} \right] \tag{24}\]  
where \(\widetilde{\mathbf{H}}_{u_{n,m}}\) denotes an effective network channel of the \(MS\)\(u_{n,m}\). The baseband combiner at the \(MS\) is designed as  
\[\mathbf{W}_{BB}^{u_{n,m}}=\left(\widetilde{\mathbf{H}}_{u_{n,m}}\right)^{-1} \tag{25}\]  
The number of analog beams required at the \(BS\) in the Type-4 case is \(U\), which is much smaller than that in the Type-3 case. Thus, a smaller number of \(RF\) chains is required at the \(BS\) in the Type-4 case. However, the number of \(RF\) chains required at the \(MS\) in the Type-4 case increases from 1 to \(B\).