# Impact of Multiple Beams and Mobility-Based Beam Alignment Error on Millimeter Wave Communication +
## 2 System Model  
We consider a single-tier outdoor mmwave network with BSs and UE distributed in \(\mathbb{R}^{2}\) according to Poisson point process (PPP) of density \(\lambda\) and \(\lambda_{U}\), respectively. Random-sized rectangular buildings are independently distributed according to another PPP. The analysis is conducted for a typical UE that randomly moves in a straight line with velocity \(v\) m/s. It is assumed that a UE always associates with the BS in LOS with minimum path loss. Universal frequency reuse is assumed across the network with \(\lambda_{U}>>\lambda\), and each BS has at least one active user to serve at any time. The blockages cause the path length between typical UE and serving BS to be either line-of-sight (LOS) (unblocked path) with probability \(P_{LOS}\) or non-LOS (blocked path) with probability \(P_{NLOS}\). Different path loss exponents \(\alpha_{L},\alpha_{N}\) and path loss intercepts, \(C_{L},C_{N}\), are applied to each LOS and NLOS links according to the following functions [12]:  
\[l(r)=\begin{cases}C_{L}r^{-\alpha_{L}}&\text{with probability }P_{LOS}(r)=e^{-\beta r}\\ C_{N}r^{-\alpha_{N}}&\text{with probability }P_{NLOS}(r)=1-e^{-\beta r} \end{cases} \tag{1}\]  
where \(\beta\) is the blockage parameter that depends on the length and width of the building.  
The channel fading follows independent Nakagami distribution with the parameters for LOS and NLOS links denoted by \(N_{L}\) and \(N_{N}\), respectively. The antenna gain is parameterized by the Gaussian main lobe profile [13], and both the BS and UE are capable of random directional beamforming with \(2^{n}\) and \(2^{k}\) beams, respectively, where \(\{n,k\}\in N\) are integers. The corresponding beamwidths for BS and UE are given as \(\theta_{n}=2\pi/2^{n}\) and \(\theta_{k}=2\pi/2^{k}\). The gain of the antenna main lobe and side lobes for BS and UE depends upon the number of beams and is given by following expressions:  
\[G_{BS,n}(\phi)=\begin{cases}G_{o,n}exp\left(-\eta_{\phi,n}.\phi_{n} ^{2}\right)&|\phi_{n}|\leq\theta_{n}\\ g_{n}&\text{otherwise}\end{cases} \tag{2}\]  
\[G_{UE,k}(\phi)=\begin{cases}G_{o,k}exp\left(-\eta_{\phi,k}\phi_{k}^{2}\right)& |\phi_{k}|\leq\theta_{k}\\ g_{k}&\text{otherwise}\end{cases} \tag{3}\]  
where \(\eta_{\phi,N}=2.028\ln(10)/\theta_{N}^{2}\), \(\phi_{N}\) represents the antenna's angle relative to its bore-sight direction \(\in[-\pi,\pi]\). The maximum main lobe gain and the side-lobe gain are given by \(G_{o,N}=\pi 10^{2.028}/(42.64\ \phi+\pi)\) and \(g=10^{-2.028}\ G_{o}\), respectively. The probability of beam alignment error at BS due to UE mobility is proposed in [6] as \(P_{bm}^{BS}=1-exp(-v\tau 2^{n}\sqrt{\lambda}/\pi)\). Similarly, the probability of beam misalignment at the UE's side is given by \(P_{bm}^{UE}=1-exp(-v\tau 2^{k}\sqrt{\lambda}/\pi)\). The total gain of the desired link (i.e., from typical UE to typical BS) is given by  
\[G_{o}=\begin{cases}G_{o,n}G_{o,k}&\text{with probability}\quad(1-P_{bm}^{BS})(1-P_{bm}^{UE})\\ G_{o,n}g_{k}&\text{with probability}\quad(1-P_{bm}^{BS})(P_{bm}^{UE})\\ g_{n}G_{o,k}&\text{with probability}\quad(P_{bm}^{BS})(1-P_{bm}^{UE})\\ g_{n}g_{k}&\text{with probability}\quad(P_{bm}^{BS})(P_{bm}^{UE})\end{cases} \tag{4}\]The interfering UEs have independent and uniformly distributed beam directions over [0, 2\(\pi\)]. The total gain on the interfering links is a discrete random variable whose distribution is given by  
\[G_{Z}=\begin{cases}G_{0,n}G_{0,k}&\text{with probability}\Big{(}\frac{\theta_{n }\theta_{k}}{4\pi^{2}}\Big{)}\\ G_{0,n}g_{k}&\text{with probability}\Big{(}\frac{\theta_{n}}{2\pi}\Big{)} \Big{(}1-\frac{\theta_{k}}{2\pi}\Big{)}\\ g_{n}G_{0,k}&\text{with probability}\Big{(}1-\frac{\theta_{n}}{2\pi} \Big{)}\Big{(}\frac{\theta_{k}}{2\pi}\Big{)}\\ g_{n}g_{k}&\text{with probability}\Big{(}1-\frac{\theta_{n}}{2\pi} \Big{)}\Big{(}1-\frac{\theta_{k}}{2\pi}\Big{)}\end{cases} \tag{5}\]