# Hybrid Beamforming for Reduction of Inter-Beam Interference in Millimeter-Wave Cellular Systems
## 2 Hybrid Beamforming Techniques for a Multicell Multiuser Environment
### Type-1 Precoder  
In a Type-1 precoder, the previous digital precoder developed for a single-cell multiuser environment is directly applied to a multicell multiuser environment. In this case, analog beams are formed in the direction of _MSs_ in each cell, as shown in Figure 2a. Beam direction in this figure is obtained after completion of the first stage. Here, the solid line between the left beam of _BS_\(b_{1}\) and the _MS_\(u_{1,1}\) represents the beam direction between the serving _BS_ (\(b_{1}\)) and desired _MS_ (\(u_{1,1}\)). The solid line between the right beam of _BS_\(b_{1}\) and the _MS_\(u_{1,2}\) represents the beam direction between the serving _BS_ (\(b_{1}\)) and desired _MS_ (\(u_{1,2}\)). The broken line between the right beam of _BS_\(b_{1}\) and the _MS_\(u_{1,1}\) represents the direction of inter-beam interference in the serving cell. The inter-beam interference is mainly caused by the sidelobe of analog beam formed toward _MS_\(u_{1,2}\) in the serving cell. The dotted lines between two beams of neighboring _BS_\(b_{2}\) and the _MS_\(u_{1,1}\) represent the directions of inter-cell interference. The channel coefficients associated with these lines correspond to the elements of the effective network channel matrix. In a Type-1 precoder, the _RF_ precoder in each cell is given by  
\[\mathbf{F}_{RF}^{b_{n}}=\mathbf{A}_{BS}^{b_{n}}=\left[\mathbf{a}_{BS}^{u_{n,1} }\left(\phi_{BS}^{u_{n,1}},\phi_{BS}^{u_{n,1}}\right),\ \mathbf{a}_{BS}^{u_{n,2}}\left(\phi_{BS}^{u_{n,2}},\phi_{BS}^{u_{n,2}}\right),\cdots,\ \mathbf{a}_{BS}^{u_{n,1}}\left(\phi_{BS}^{u_{n,1}},\phi_{BS}^{u_{n,1}} \right)\right] \tag{5}\]  
where \(\mathbf{F}_{RF}^{b_{n}}\) is the _RF_ precoder of the \(b_{n}\)-th _BS_ and \(\mathbf{A}_{BS}^{b_{n}}\) is the array response matrix of the \(b_{n}\)-th _BS_. We assumed that the number of _MSs_ served by a _BS_ is \(U\). That is, a _BS_ transmits multiple streams simultaneously to _U MSs_s using the _RF_ precoder obtained in the first stage. Here, the _MSs_ served by the \(b_{n}\)-th _BS_ are denoted by \(u_{n,1},u_{n,2},\cdots,u_{n,U}\). However, in a multicell multiuser environment, a new channel model is needed to take into account the interference terms transmitted from neighboring _BSs_. Here, we define an effective network channel of \(b_{n}\), which can be viewed as the effective channel matrix between the _BS_\(b_{n}\) and all _MSs_ in neighboring cells (including the serving cell). The term \"effective\" is used to indicate that the _RF_ precoder of the _BS_ and the _RF_ combiner of the _MS_ are included in the channel model. From the perspective of the _BS_\(b_{n}\), an effective network channel can be expressed as  
\[\widetilde{\mathbf{H}}^{b_{n}}=\left[ \begin{array}{c}\left(\mathbf{w}_{RF}^ {u_{1,1}}\right)^{*}\mathbf{H}_{u_{1,1}}^{b_{n}}\mathbf{F}_{RF}^{b_{n}}\\ \left(\mathbf{w}_{RF}^{u_{1,2}}\right)^{*}\mathbf{H}_{u_{1,2}}^{b_{n}}\mathbf{ F}_{RF}^{b_{n}}\\ \vdots\\ \left(\mathbf{w}_{RF}^{u_{1,U}}\right)^{*}\mathbf{H}_{u_{1,U}}^{b_{n}}\mathbf{ F}_{RF}^{b_{n}}\\ \left(\mathbf{w}_{RF}^{u_{1,1}}\right)^{*}\mathbf{H}_{u_{2,1}}^{b_{n}}\mathbf{ F}_{RF}^{b_{n}}\\ \vdots\\ \left(\mathbf{w}_{RF}^{u_{B,U}}\right)^{*}\mathbf{H}_{u_{B,U}}^{b_{n}}\mathbf{ F}_{RF}^{b_{n}}\end{array} \right] \tag{6}\]  
The effective network channel of \(b_{n}\) includes not only the effective channels between the _BS_\(b_{n}\) and _MSs_ in the serving cell, but also the effective channels between the _BS_\(b_{n}\) and _MSs_ in the neighboring cells. Here, the total number of _MSs_, denoted by \(u_{1,1},\ u_{1,2},\ \cdots,\ u_{1,U},\ u_{2,1},\ \cdots,\ u_{B,U}\), is _BU_ because the number of neighboring _BSs_ is \(B\). A baseband precoder can be designed if we can obtain an inverse matrix of the effective network channel as follows:  
\[\mathbf{F}_{BB}^{b_{n}}=\left(\widetilde{\mathbf{H}}^{b_{n}}\right)^{\dagger} \tag{7}\]  
where \((\ )^{\dagger}\) denotes a pseudo inverse matrix operation.