# 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
\begin{split}&\mathbf{W}_{BB}^{u_{\text{tr},m}}\Big{(}\mathbf{W}_{ RF}^{u_{\text{tr},m}}\Big{)}^{*}\mathbf{H}_{u_{\text{tr},m}}^{b_{\text{tr}}} \mathbf{R}_{RF}^{b_{\text{tr}}}\mathbf{g}_{BB,u_{\text{tr},m}}^{b_{\text{tr},m}} \\ &=\mathbf{W}_{BB}^{u_{\text{tr},m}}\Big{(}\mathbf{W}_{RF}^{u_{ \text{tr},m}}\Big{)}^{*}\mathbf{a}_{MS}^{b_{\text{tr},m}}\Big{(}\phi_{BS}^{b_{ \text{tr},m}}\,,\,\theta_{BS}^{b_{\text{tr},m}}\Big{)}\Big{[}\mathbf{A}_{BS}^{ b_{\text{tr},m}}\Big{]}\Big{[}\mathbf{A}_{BS}^{b_{\text{tr},m}}\Big{]}_{u_{ \text{tr},m},u_{\text{tr},m}}\\ &=\frac{1}{\sqrt{N_{BS}N_{MS}}\Big{\{}}\Big{(}\Big{(}\mathbf{a}_{ BS}^{b_{\text{tr},m}}\Big{)}^{*}\mathbf{a}_{BS}^{b_{\text{tr},m}}\Big{)}^{-1} \Big{(}\mathbf{a}_{MS}^{u_{\text{tr},m}}\Big{)}^{*}\mathbf{a}_{MS}^{u_{\text{tr },m}}}\Bigg{)}^{*}\mathbf{a}_{b_{\text{tr},m},u_{\text{tr},m}}}\Bigg{[}\mathbf{ A}_{MS}^{u_{\text{tr},m}}\Bigg{]}_{b_{\text{tr},m},b_{\text{tr}}}\Big{[} \mathbf{A}_{BS}^{b_{\text{tr}}}\Bigg{]}_{u_{\text{tr},m},u_{\text{tr},m}}\\ &=\sqrt{\Bigg{(}\Big{(}\mathbf{a}_{BS}^{b_{\text{tr}}}\Big{)}^{*} \mathbf{a}_{BS}^{b_{\text{tr},m}}\Big{)}^{-1}\Bigg{(}\mathbf{a}_{MS}^{u_{\text{ tr},m}}\Big{)}^{*}\mathbf{a}_{MS}^{u_{\text{tr},m}}}\Bigg{)}^{-1}_{u_{\text{tr},m},u_{ \text{tr},m}}}\Bigg{|}\end{split}  \tag{31}\]  
Then, the averaged achievable rate is given by  
\[R_{u_{\text{tr},m}}^{b_{\text{tr}}}=\log_{2}\Bigg{(}1+\frac{p}{U_{1}\sigma^{2} }\frac{N_{BS}N_{MS}\Big{|}\alpha_{b_{\text{tr},m}}\Big{|}^{2}}{\bigg{(}\big{(} \mathbf{a}_{BS}^{b_{\text{tr}}}\big{)}^{*}\mathbf{a}_{BS}^{b_{\text{tr}}} \Big{)}^{-1}_{u_{\text{tr},m},u_{\text{tr},m}}\bigg{(}\big{(}\mathbf{a}_{MS}^{ u_{\text{tr},m}}\big{)}^{*}\mathbf{A}_{MS}^{u_{\text{tr},m}}\Big{)}^{-1}_{b_{ \text{tr},b_{\text{tr}}}}\Bigg{)}} \tag{32}\]  
From (32), one can see that the averaged achievable rate in the Type-4 precoder is affected by not only the array response matrix of the \(BS\), but also the array-response matrix of the \(MS\). In summary, in this section, we propose four different precoding techniques to reduce the IBI in mm-wave cellular systems with a hybrid beamformer. The Type-1 precoder is simple because the precoding technique developed for a single-cell multiuser environment is applied to each cell. However, in this case, we cannot reduce the interference from neighboring cells. The Type-2 precoder can reduce interference from both the serving cell and neighboring cells because the precoder is designed for the combined network consisting of all neighboring cells. However, in this case, all \(BSs\) should share data streams, increasing the network overhead. In the Type-3 precoder, the \(RF\) precoder is extended such that the digital precoder can be designed independently by each \(BS\). However, in this case, the number of \(RF\) chains that are required at the \(BS\) increases as the number of neighboring cells increases. The Type-4 precoder can be used when a hybrid beamformer is available at the \(MS\) as well as at the \(BS\). In this case, the number of \(RF\) chains required at the \(BS\) is reduced to a single-cell case; however, the number of \(RF\) chains required at the \(MS\) increases by the number of neighboring cells.