## 2 Related Work
### Antenna Models  
The antenna model is one of the critical elements for time-dependent analysis. Usually, three types of models are utilized in the literature for directional mmWave/sub-THz systems. The first type is the so-called theoretical geometric cone or cone-plus-circle modelsthat capture only the essentials of directional propagation [13]. Specifically, the code antenna model captures the main beam only and accepts only one parameter--half-power beamwidth (HPBW). Cone-plus-sphere extends this simple model to account for the loss of power to the side and back lobes. These models can be parameterized theoretically as discussed in detail in [17] or by utilizing the array parameters, including the power loss to the side and back lobes, \(k\), the average gain over the main lobe, [29]  
\[G_{A}(\theta_{3 db}^{\pm})=\frac{1}{\theta_{3 db}^{+}-\theta_{3 db}^{-}}\int_{ \theta_{3 db}^{-}}^{\theta_{3 db}^{+}}\frac{\sin(N\pi\cos(\theta)/2)}{\sin(\pi \cos(\theta)/2)}d\theta. \tag{1}\]  
where \(N\) is the number of antenna elements and \(\theta_{3 db}\) is the angle and where the gain of the radiation pattern decreases by 3 dB as compared to the array maximum. The HPBW \(\alpha\) is given by  
\[\alpha=2|\theta_{m}-\theta_{3 db}|, \tag{2}\]  
where \(\theta_{m}\) is the location of the array maximum. The location \(\theta_{m}\) can be calculated as \(\theta_{m}=\arccos(-\beta/\pi)\), where \(\beta\) is the phase excitation difference affecting the physical orientation of the array. In our case, \(\theta_{m}=\pi/2\) for \(\beta=0\). The 3 dB point is provided by  
\[\theta_{3 db}^{\pm}=\arccos[-\beta\pm 2.782/(N\pi)]. \tag{3}\]  
Note that this type of model does not capture the fine-grain changes in the antenna transmit and receive gains, reporting abrupt changes even in the case of cone-plus-sphere models. To account for smooth changes in the gains, the so-called pattern-based models can be used [30]. These models are defined by the actual antenna radiation pattern, expressing the gain in the transmit or receive direction by using a function \(g(\theta,\phi)\), where \(\theta\) is the elevation angle and \(\phi\) is the azimuth angle. Since realistic antenna radiation patterns are not readily available, complex superposition-based models can be used to approximate them. One such model is specified in 3 GPP TR 39.901.  
The topic considered in our paper is also closely related to modeling wave propagation under the micro-mobility of antennas. In this context, ray-tracing models offer high accuracy. Specifically, in [31], the authors consider the extended Walfisch-Ikegami model (EWM) that is characterized by lower computational time with tolerable accuracy when compared to the deterministic ray optical models. Wave-guiding effects are not considered in this model, which generally occurs in urban canyons. This accuracy of the model is higher when the transmitters are located above the rooftop levels of the buildings. While it considers only the path loss, it still discriminates between line-of-sight (LoS) and non-line-of-sight (NLoS) cases.  
The authors in [32] characterize the channel of a point-to-point communication link between unstable transmitters and receivers by taking into account the effects of transceiver vibrations, three-dimensional antenna pattern, path loss, and small-scale fading. They proposed an approximate Gaussian model based on the number of antenna elements for the standard array antenna pattern. By using this model, they derived the probability density function (pdf) and cumulative distribution functions (CDF) of the pointing error in the presence of unstable transmitters and receivers as a function of the antenna pattern.  
The study in [33] provides a detailed overview of antenna arrays based on feeding techniques and considers reconfigurable arrays, while the study in [34] highlights the impact of misalignment and narrow beam bias on the quality of service in the sub-terahertz band.