# 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
In the Type-4 case, the achievable rate of the \(MS\)\(u_{n,m}\) in the serving the \(BS\)\(b_{n}\) is given by  
\[R_{u_{n,m}}^{b_{n,m}}=\log_{2}\left(1+\frac{\frac{\ell_{1}^{B}\left|\mathbf{W }_{BS}^{u_{n,m}}\left(\mathbf{W}_{RF}^{u_{n,m}}\right)^{*}\mathbf{H}_{m,m}^{b_ {n,m}}\mathbf{F}_{RF}^{b_{n,m}}\mathbf{G}_{BS,\mu_{n,m}}^{b_{n,m}}\right|^{2} }{\frac{\ell_{1}^{B}\sum_{g=1}^{U}\left|\mathbf{W}_{BS}^{u_{n,m}}\left(\mathbf{ W}_{RF}^{u_{n,m}}\right)^{*}\mathbf{H}_{m,m}^{b_{n,m}}\mathbf{F}_{RF}^{b_{n,m}} \mathbf{G}_{BS,\mu_{n,g}}^{b_{n,m}}\right|^{2}+\sum_{g=1}^{U}\left(\frac{\ell_{1 }^{B}\sum_{g=1}^{U}\left|\mathbf{W}_{BS}^{u_{n,m}}\left(\mathbf{W}_{RF}^{u_{n,m }}\right)^{*}\mathbf{H}_{m,m}^{b_{n,m}}\mathbf{F}_{RF}^{b_{n,m}}\mathbf{G}_{BS,\mu_{n,g}}^{b_{n,m}}\right|^{2}+\sigma^{2}\right)}}}\right) \tag{26}\]  
Unlike the Type-3 case, the baseband combiner of the \(MS\) is reflected in the achievable rate in the Type-4 case. As in (15), the effective network channel of the \(BS\)\(b_{n}\) can be written as  
\[ \begin{array}{l}\widetilde{\mathbf{H}}^{b_{n}}=\mathbf{D}^{b_{n}}\mathbf{A} _{BS}^{b_{n}}\left(\mathbf{A}_{BS}^{b_{n}}\right)^{*}\\ \mathbf{A}_{BS}^{b_{n}}=\left[\mathbf{a}_{BS}^{u_{n,1}}\left(\mathbf{g}_{BS}^{ u_{n,1}},\mathbf{g}_{BS}^{u_{n,1}}\right),\,\mathbf{a}_{BS}^{u_{n,2}}\left( \mathbf{g}_{BS}^{u_{n,2}},\mathbf{g}_{BS}^{u_{n,2}}\right),\cdots,\,\mathbf{a}_ {BS}^{u_{n,U}}\left(\mathbf{g}_{BS}^{u_{n,U}},\mathbf{g}_{BS}^{u_{n,U}}\right) \right]\end{array}  \tag{27}\]  
Note that the effective network channel in (27) considers only \(MS\)s in the serving cell. Using the result in (19), the diagonal matrix for the power constraint in the \(BS\) precoder can be expressed as  
\[\left[\mathbf{\Lambda}_{BS}^{b_{n}}\right]_{u_{n,m},u_{m}}=\sqrt{\frac{N_{BS} N_{MS}}{\left(\left(\mathbf{A}_{BS}^{b_{n}}\right)^{*}\mathbf{A}_{BS}^{b_{n}} \right)^{-1}_{u_{n,m},u_{n}}}}\left|\mathbf{a}_{u_{n,m}}\right| \tag{28}\]  
From (24), the effective network channel of the \(MS\) can be rewritten as  
\[ \begin{array}{l}\widetilde{\mathbf{H}}_{u_{n,m}}=\left(\mathbf{A}_{MS}^{u_ {n,m}}\right)^{*}\mathbf{A}_{MS}^{b_{n},n}\mathbf{D}_{u_{n,m}}\\ \mathbf{A}_{MS}^{u_{n,m}}=\left[\mathbf{a}_{MS}^{b_{1},u_{n,m}}\left(\mathbf{ g}_{BS}^{b_{1},u_{n,m}},\mathbf{g}_{BS}^{b_{1},u_{n,m}}\right)\mathbf{a}_{MS}^{b_{2},u_{n,m}} \left(\mathbf{\phi}_{BS}^{b_{2},u_{n,m}},\mathbf{\phi}_{BS}^{b_{2},u_{n,m}} \right)\,\,\cdots\,\mathbf{a}_{MS}^{b_{B},u_{n,m}}\left(\mathbf{\phi}_{BS}^{ b_{B},u_{n,m}},\mathbf{\theta}_{BS}^{b_{B},u_{n,m}}\right)\right]\\ \left[\mathbf{D}_{u_{n,m}}\right]_{b_{n},b_{n}}=\sqrt{N_{BS}N_{MS}}\mathbf{a }_{b_{n},u_{n,m}}\sum\limits_{g=1}^{U}\left\{\left(\mathbf{a}_{BS}^{b_{n},u_{n, m}}\left(\mathbf{\phi}_{BS}^{b_{n},u_{n,m}},\mathbf{\theta}_{BS}^{b_{n},u_{n,m}} \right)\right)^{*}\mathbf{f}_{RF,\mu_{n,g}}^{b_{n}}\right\}\end{array}  \tag{29}\]  
The diagonal matrix for the power constraint in the baseband combiner of the \(MS\) can be obtained in a similar way, as follows:  
\[\left[\mathbf{\Lambda}_{MS}^{u_{n,m}}\right]_{b_{n},b_{n}}=\sqrt{\frac{N_{BS}N_{MS }}{\left(\left(\mathbf{A}_{MS}^{u_{n,m}}\right)^{*}\mathbf{A}_{MS}^{u_{n,m}} \right)^{-1}_{b_{n},b_{n},b_{n}}}}\left|\mathbf{a}_{b_{n,n,m}}\sum\limits_{g=1}^ {U}\left\{\left(\mathbf{a}_{BS}^{b_{n},u_{n,m}}\left(\mathbf{\phi}_{BS}^{b_{n},u_{ n,m}},\mathbf{\theta}_{BS}^{b_{n},\mu_{n,m}}\right)\right)^{*}\mathbf{f}_{RF,\mu_{n,g}}^{b_{n }}\right\}\right| \tag{30}\]  
The effective network channels in (27) and (29) can be estimated independently at the \(BS\) and \(MS\), respectively. Thus, the digital precoder at the \(BS\) and combiner at the \(MS\) can be designed independently in the Type-4 case. To obtain an achievable rate in the Type-4 precoder, the numerator in (26) is rewritten as  
\[