# Hybrid Beamforming for Reduction of Inter-Beam Interference in Millimeter-Wave Cellular Systems
## 2 Hybrid Beamforming Techniques for a Multicell Multiuser Environment  
As shown in Figure 1, the signal received at an \(MS\) in an mm-wave system with a hybrid beamformer under a multicell multiuser environment can be expressed as follows:  
\[y_{u_{u,m}}=\left(\mathbf{w}_{RF}^{u_{u,m}}\right)^{*}\Bigg{\{}\sum_{l=1}^{B} \left(\mathbf{H}_{u_{u,m}}^{b_{l}}\mathbf{F}_{RF}^{b_{l}}\mathbf{F}_{BB}^{b_{l }}\mathbf{s}^{b_{l}}\right)+\mathbf{n}_{u_{u,m}}\Bigg{\}} \tag{1}\]  
where \(y_{u_{u,m}}\) is the signal received at the \(MS\)\(u_{u,m}\), which is given by a superposition of signals transmitted from neighboring \(BSs\)\(b_{1}\), \(b_{2}\), \(\cdots\), \(b_{B}\). In addition, \(u_{n,m}\) is the \(m\)-th \(MS\) served by the \(b_{n}\)-th \(BS\), \(B\) is the number of neighboring \(BSs\), \(\mathbf{w}_{RF}^{u_{u,m}}\) is the \(RF\) combiner of \(MS\)\(u_{n,m}\), \(\mathbf{H}_{u_{n,m}}^{b_{l}}\in\mathbb{C}^{N_{MS}\times N_{BS}}\) is the channel between the \(BS\)\(b_{l}\) and \(MS\)\(u_{n,m}\), and \(\mathbf{n}_{u_{n,m}}\) is the noise component. \(\mathbf{F}_{RF}^{b_{l}}\in\mathbb{C}^{N_{BS}\times L_{BS}}\), \(\mathbf{F}_{BB}^{b_{l}}\in\mathbb{C}^{L_{BS}\times M_{BS}}\), and \(\mathbf{s}^{b_{l}}\in\mathbb{C}^{M_{BS}\times 1}\) denote the \(RF\) precoder, baseband precoder, and symbol transmitted from \(BS\)\(b_{l}\), respectively. Here, \(N_{BS}\), \(L_{BS}\), and \(M_{BS}\) denote the number of antenna arrays, number of \(RF\) chains, and number of streams, respectively, all at the \(BS\). The analog precoder is called the \(RF\) precoder, while the digital precoder is called the baseband precoder. Moreover, the transmitted symbol \(\mathbf{s}^{b_{l}}\) is assumed to be circular [24]. Then, the signal transmitted from the \(BS\)\(b_{n}\) with a hybrid beamformer can be expressed as  
\[\mathbf{x}^{b_{n}}=\mathbf{F}_{RF}^{b_{n}}\mathbf{F}_{BB}^{b_{n}}\mathbf{s}^{ b_{n}} \tag{2}\]  
The \(RF\) precoder \(\mathbf{F}_{RF}^{b_{n}}\) is usually implemented by a phase shifter. Its entry is given by \(\frac{1}{\sqrt{N_{BS}}}e^{\phi\phi}\), where \(\phi\) is the quantized angle of the phase shifter. A ray-based channel model is normally used for mm-wave channels because of a strong line-of-sight (LOS) path component and limited scattering components. Here, we ignore the effect of carrier frequency offset (CFO) and time delays of multi-path  
Figure 1: An mm-wave system with a hybrid beamformer in a multicell multiuser environment. channel [25; 26]. The delay term for each multipath component is ignored for notational simplicity because, in the following sections, only dominant path (LOS) is considered for the design of proposed precoders. In this paper, it is assumed that each \(BS\) is equipped with multiple beamforming antennas and serves a number of \(MS\)s simultaneously. However, only one beam is formed in the direction of LOS path for each \(MS\). Then, the channel with \(V\) paths can be expressed as  
\[\mathbf{H}_{u_{u},m}^{b_{n}}=\sqrt{\frac{N_{BS}N_{MS}}{V}}\sum_{v=1}^{V}\alpha _{u_{u},u_{v},b_{n},v}\mathbf{a}_{MS}\left(\phi_{MS}^{u_{u},m},b_{n},\rho^{u_{ u},m},\theta_{MS}^{u_{u},m}\right)\mathbf{a}_{BS}{}^{s}\left(\phi_{BS}^{u_{u},m },\theta_{BS}^{u_{u},m}\right) \tag{3}\]  
where \(\mathbf{H}_{u_{u},m}^{b_{n}}\) is the channel matrix between the \(BS\)\(b_{n}\) and \(MS\)\(u_{u,m}\), and \(\alpha_{u_{u},m},b_{n},\rho\) is the gain of the \(v\)-th path between the \(BS\)\(b_{n}\) and \(MS\)\(u_{n,m}\). In addition, \(\mathbf{a}_{BS}\left(\phi_{BS}^{u_{u},m},\theta_{BS}^{u_{u},m},\theta_{BS}^{u_ {u},m}\right)\) and \(\mathbf{a}_{MS}\left(\phi_{MS}^{u_{u},m},\theta_{MS}^{u_{u},m},\right)\) denote the antenna array response vectors of the \(v\)-th path at the \(BS\)\(b_{n}\) and \(MS\)\(u_{n,m}\), respectively. When a uniform planar array (UPA) is used, the antenna response vector is given by  
\[\mathbf{a}_{BS}\left(\phi_{BS}^{u_{u},m},b_{BS}^{u_{u},m},\phi_{BS}^{u_{u},m} \right)=\frac{1}{\sqrt{N_{BS}}}\left[