## 4 Characterizing Spectral Efficiency Dynamics
### Accounting for UE Macro-Mobility  
Recall that we represent user mobility by two independent diffusions over the \(x\)- and \(y\)-axes confined within the deployment boundaries, \(L\times L\). Assume that the BS is located at \((0,0)\). By aligning the \(x\)-axis with the direction of the user, we observe that the initial coordinates of UE are \((x_{0},0)\). The illustration of the considered scenario is shown in Figure 3. Here, \((x_{0},0)\) are the initial coordinates of the user at time \(t_{0}=0\), \((x,y)\) are the new coordinates at time \(t\), \(t>0\), the angles \(\beta_{H}\) and \(\beta_{V}\) show the horizontal and vertical misalignments of the antenna radiation patterns induced by the macro-mobility of the user, and \(d_{1}\) and \(d_{2}\) are the distances to the BS at times \(t_{0}=0\) and \(t\), respectively. To determine \(d_{2}\) and \(\beta_{V}\), consider, first, the side view of the scenario; see Figure 4. Recall that statistical properties of a user diffusing in one-dimension and confined within the \((0,L)\) compartment are described by the Green's function \(p(x,t|x_{0},t_{0})\), representing the  
Figure 3: Illustration of the beam misalignment due to macro-mobility. Figure 2: Antenna radiation pattern example [46]. probability of finding a diffusing point at \(x\) at time \(t\) together with the initial conditions \(p(x,t|x_{0},t_{0})=\delta(x-x_{0})\) and boundary conditions  
\[\frac{\partial p(0,t|x_{0},t_{0})}{\partial x}=\frac{\partial p(L,t|x_{0},t_{0}) }{\partial x}=0. \tag{12}\]  
The solution to the problem is obtained by separation of the variables and reported in different sources [47; 48; 49]. It can be written in terms of the infinite series as  
\[p(x,t|x_{0},t_{0}) =\frac{1}{L}+\frac{2}{L}\sum_{n=1}^{\infty}\exp\left[-\left(\frac {n\pi}{2}\right)^{2}\frac{t-t_{0}}{\tau_{x}}\right]\times\] \[\times\,\cos\left(\frac{n\pi\pi\pi}{L}\right)\cos\left(\frac{n\pi x _{0}\pi}{L}\right), \tag{13}\]  
where \(\tau_{x}=L^{2}/D_{x}\) is the relaxation time. The behavior of the density \(p(x,t|x_{0},t_{0})\) for different values of \(x_{0}\), \(t\) and \(L=10\), \(D_{M}=D_{x}=D_{y}=1\) is shown in Figure 5. As one may observe, \(\lim_{t\to\infty}p(x,t|x_{0},t_{0})=1/L\), implying that, eventually, the user is uniformly distributed over \(L\). However, the behavior at shorter timescales heavily depends upon the initial position \(x_{0}\) and on the diffusion constant \(D\). Figure 4: Side view of the considered scenario. The first raw moment of (13) utilized further can be calculated as  
\[\mu(t,x;L,t_{0},x_{0})=\int_{0}^{L}x\left[\frac{1}{L}+\frac{2}{L} \sum_{n=1}^{\infty}\exp\left[-\left(\frac{n\pi}{2}\right)^{2}\frac{t-t_{0}}{ \tau_{x}}\right]\times\right.\] Figure 4.0 Figure 4.1  
We note that (14) provides the first moment of the location of the diffusing particle (in our interpretation, a moving user) as an explicit function of time. This moment is available from the literature for the mobility model we consider (see, e.g., [50]) and is instrumental for the proposed framework. Specifically, it facilitates further calculation of the time-dependent SNR, rate, and spectral efficiency metrics. Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5  
Let \(X(t)\) be a random variable, representing the location of the user at time \(t\), given that it started at \(x_{0}\) and time \(t_{0}=0\). By utilizing the geometry in Figure 4, we see that the distance to the BS, \(l_{2}(t)\), and the angle, \(\beta_{V}(t)\), at time \(t\) can be expressed as  
\[ \begin{cases}l_{2}(t)=\sqrt{[X(t)]^{2}+(h_{A}-h_{U})^{2}}\\ \beta_{V}(t)=\arccos[X(t)/l_{2}(t)]\end{cases}