esa-sceva/satcom-chunk-collection huggingface dataset:
# Multi-Wavelength Digital-Phase-Shifting Moire Based on Moire Wavelength
## 2 Method
### 3D Surface Reconstruction by Moire-Wavelength and Height Refinement  
In the 3D surface reconstruction, pixelwise moire-wavelength refinement and pixelwise surface height refinement [17] can be applied using the moire-wavelength to height relationship, determined in Section 2.2. Following the computation of height \(h(x,y)\) by Equation (9) after two-wavelength digital-moire temporal phase-unwrapping, or by Equation (16) after three-wavelength phase-unwrapping, the surface height is refined by: (a) Determining the maximum height \(h_{max}\) over all pixels, (b) computing \(h^{\prime}(x,y)=\lambda\langle h_{max}/2\rangle\varphi(x,y)/2\pi\), and (c) computing final pixelwise surface height refinement by \(h^{\prime\prime}(x,y)=\lambda\langle h^{\prime}(x,y)\rangle\varphi(x,y)/2\pi\), where \(\lambda\langle\rangle\) denotes the value of \(\lambda\) as a function of height for different heights.
## 5 Results and Discussions  
The results of the measured physicochemical values with the cation and anion concentrations were given as descriptive statistics in Table 1. Additionally, based on the obtained results, spatial distribution diagrams were prepared to study the contamination movement in the study region in relation to each result.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**S. No**} & \multirow{2}{*}{**Analytical**} & \multicolumn{3}{c}{**Jan-20**} & \multicolumn{3}{c}{**Jan-21**} & \multicolumn{3}{c}{**WHO \& BIS**} & \multirow{2}{*}{**Detection Limit**} & \multirow{2}{*}{**Quantification Limit**} \\ \cline{3-3} \cline{5-10}  & & & & & & & & & & & \\ \cline{3-10}  & \multirow{2}{*}{**Parameters**} & \multicolumn{1}{c}{**Min-Max**} & \multicolumn{1}{c}{**Med**} & \multicolumn{1}{c}{**St.dev**} & \multicolumn{1}{c}{**Min-Max**} & \multicolumn{1}{c}{**Med**} & \multicolumn{1}{c}{**St.dev**} & \multicolumn{1}{c}{**Guidelines**} & \multicolumn{1}{c}{**Limit**} \\ \hline
1 & pH & 7.14\(\pm\)8.15 & 7.4 & 0.3 & 7.5\(\pm\)6.6 & 8.1 & 0.2 & 6.5\(\pm\)8.5 & 0.01 & 14 \\
2 & EC (\(\mu\)S/Cm) & 266-1040 & 1156.0 & 2167.6 & 834-6857 & 2526.4 & 1111.4 & Nil & 199 & 20,000 \\
3 & TDS (mg/L) & 186-7280 & 809.2 & 1517.3 & 503.6-6011 & 1627.0 & 1490.4 & 500–2000 & 99 & 10,000 \\
4 & Ca2+ (mg/L) & 16–4528 & 46.4 & 97.8 & 48.7–700 & 122.0 & 152.8 & 75–200 & Nil & Nil \\
5 & Mg2+ (mg/L) & 1–9256.3 & 19.7 & 56.6 & 15–450 & 64.5 & 986.6 & 30–100 & Nil & Nil \\
6 & Na2+ (mg/L) & 31-100 & 139.0 & 261.9 & 20–600 & 150.0 & 181.1 & Nil & 1 & 1000 \\
7 & K* (mg/L) & 3–300 & 18.0 & 74.1 & 2–98 & 20.0 & 23.8 & Nil & 1 & 1000 \\
8 & Cl- (mg/L) & 12–2880 & 158.0 & 599.7 & 50–1674.8 & 315.5 & 371.0 & 250–1000 & Nil & Nil \\
9 & HCO2+ (mg/L) & 48-696 & 190.0 & 146.6 & 10–661.4 & 87.9 & 133.3 & Nil & Nil & Nil \\
10 & SO4-2 (mg/L) & 12–750 & 73.0 & 175.5 & 29-8493 & 143.3 & 114.6 & 200–400 & 3 & 500 \\
11 & NO- (mg/L) & 3-74 & 13.5 & 15.3 & 1–140 & 47.0 & 30.7 & 45.0 & 1 & 100 \\
12 & F- (mg/L) & BDL-1.4 & 0.2 & 0.3 & 0.3–1.3 & 0.5 & 0.3 & 1–1.5 & 0.01 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Analytical results descriptive statistics.
# Angle-Only Cooperative Orbit Determination Considering Attitude Uncertainty
## 2 Attitude and Orbit Determination Model
### Observation Model  
As shown in Figure 1, the spacecraft \(S_{2}\) is assumed to be observed by the spacecraft \(S_{1}\). The inertial inter-spacecraft angle measurements can be obtained if the observer \(S_{1}\) is an attitude-reference spacecraft. The inertial inter-spacecraft angle measurements can be represented by two angulars \(\alpha\) and \(\beta\), given as:  
\[\left\{\begin{array}{l}\alpha=\tan^{-1}(\Delta y/\Delta x)+\varepsilon_{\alpha} \\ \beta=\sin^{-1}\left(\Delta z/\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}} \right)+\varepsilon_{\beta}\end{array}\right. \tag{15}\]  
where \(\Delta x=x_{2}-x_{1}\), \(\Delta y=y_{2}-y_{1}\) and \(\Delta z=z_{2}-z_{1}\); \(\varepsilon_{\alpha}\) and \(\varepsilon_{\beta}\), respectively, denote the randomly distributed noise for the two angulars. To be convenient, the angular measurement equations in Equation (15) can be further replaced by the line-of-sight model, given by:  
\[\mathbf{y}^{\prime}=\frac{\mathbf{r}_{2}-\mathbf{r}_{1}}{\|\mathbf{r}_{2}-\mathbf{r}\|}+\mathbf{\epsilon} ^{\prime} \tag{16}\]  
where \(\mathbf{r}_{1}=[x_{1},y_{1},z_{1}]^{T}\) and \(\mathbf{r}_{2}=[x_{2},y_{2},z_{2}]^{T}\); \(\mathbf{\epsilon}^{\prime}\) denotes the randomly distributed noise for vector measurement \(\mathbf{y}^{\prime}\).  
Note that in this paper, the attitude of spacecraft \(S_{1}\) also needs to be estimated. Therefore, \(S_{1}\) can only measure the line-of-sight angle in the spacecraft body coordinate system (i.e., coordinate system \(O_{B}-X_{B}Y_{B}Z_{B}\)), and the real measurement \(\mathbf{y}\) is given by the following:  
\[\mathbf{y}=\mathbf{R}_{E}^{B}\mathbf{y}^{\prime}+\mathbf{\varepsilon}=\mathbf{R}_{E}^{B} \frac{\mathbf{r}_{2}-\mathbf{r}_{1}}{\|\mathbf{r}_{2}-\mathbf{r}_{1}\|}+\mathbf{\varepsilon} \tag{17}\]  
where \(\mathbf{\varepsilon}\) is the corresponding measurement noise vector characterized by a normal distribution with zero mean and covariance \(\mathbf{R}\in\mathbb{R}^{3\times 3}\) and \(\mathbf{R}_{E}^{B}\) is the rotation matrix from ECI (coordinate system \(O_{E}-X_{E}Y_{E}Z_{E}\)) to the spacecraft body coordinate system \(O_{B}-X_{B}Y_{B}Z_{B}\):  
\[\mathbf{R}_{E}^{B}=\left[\begin{array}{ll}q_{0}^{2}+q_{1}^{2}-q_{2}^{2}-q_{3 }^{2}&2(q_{0}q_{3}+q_{1}q_{2})&2(q_{1}q_{3}-q_{0}q_{2})\\ 2(-q_{0}q_{3}+q_{1}q_{2})&q_{1}^{2}-q_{1}^{2}+q_{2}^{2}-q_{3}^{2}&2(q_{2}q_{3} +q_{1}q_{1})\\ 2(q_{1}q_{2}+q_{1}q_{3})&2(q_{2}q_{3}-q_{0}q_{1})&q_{0}^{2}-q_{1}^{2}-q_{2}^{ 2}+q_{3}^{2}\end{array}\right] \tag{18}\]  
Hence, the navigation problem is given by:  
\[\left\{\begin{array}{l}\dot{\mathbf{X}}=\left[\dot{\mathbf{X}}_{1}^{T},\mathbf{X}_{2}^{ T},\dot{\mathbf{q}}^{T}\right]^{T}=\left[\mathbf{f}(\mathbf{X}_{1})^{T},\mathbf{f}(\mathbf{X}_{1})^{T}, \mathbf{g}(\mathbf{q})^{T}\right]^{T}\\ \mathbf{y}=\mathbf{R}_{E}^{B}\frac{\mathbf{r}_{2}-\mathbf{r}_{1}}{\|\mathbf{r}_{2}-\mathbf{r}_{1} \|}+\mathbf{\varepsilon}\end{array}\right. \tag{19}\]
# Infrared and Visible Image Fusion via Feature-Oriented Dual-Module Complementary
## 3 The Proposed Method
### Infrared Brightness Supplement Module  
As illustrated in Figure 2, the predicted image has better visual information. Compared with the infrared source image in Figure 2a,b, however, it is observed from Figure 2c,d that the infrared target is dim, which means that the infrared thermal information is not sufficiently extracted by the previous module. To address this, we propose an IBSM focused on image intensity.
# A PINN-Based Nonlinear PMSM Electromagnetic Model Using Differential Inductance Theory
## 4 Examples of PINN Surrogate Modeling
### PMSM Torque Calculation Based on PINN Modeling  
In order to verify the validity of the PMSM nonlinear mathematical model and the PINN Surrogate model, the torque calculations are performed under SE conditions and USE conditions, respectively, and compared with the FEA data and the OLAM results in order to verify the superiority of the PINN model. Theoretically, the farther the selected calculation operating point deviates from the FEA operating point, the larger the error between the parameter surfaces of the OLAM and the PINN model and the actual parameters are. In this paper, the typical field weakening operating point (\(I_{\text{AMP}}=450\) A, \(\beta=70\) deg) and the operating point (\(I_{\text{AMP}}=490\) A, \(\beta=62\) deg) are computed and verified, where \(I_{\text{AMP}}\) is the current amplitude and \(\beta\) is the angle of the current vector ahead of the \(q\) axis. Where the first operating point is located at the FEA sampling point, and the second operating point is far away from the FEA sampling point. To visually measure the accuracy of the model, this paper defines the Euclidean distance score (EDS):  
\[\text{EDS}_{\text{Comp}}=\sqrt{\sum_{k=1}^{m}\left(T_{\text{Comp}}^{(k)}-T_{ \text{FEA}}^{(k)}\right)^{2}} \tag{45}\]  
where \(T_{\text{FEA}}\) is the FEA torque sequence, \(T_{\text{Comp}}\) is the comparison torque sequence, and \(m\) is the length of the sequence. the smaller the EDS, the closer the comparison data are to the FEA data. Torque calculation under SE conditions.
## Appendix A  
Proportional allocation of the randomly sampled pixels as identified by Segmented VPR and Segmented RESTREND is shown in Table A1.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Ecoregions**} & \multicolumn{4}{c}{**Decreasing**} & \multicolumn{4}{c}{**Increasing**} & \multicolumn{4}{c}{**NSC**} & \multirow{2}{*}{**Total**} \\ \cline{2-10} \cline{3-10}  & **52** & **71** & **82** & **31** & **42** & **52** & **71** & **82** & **42** & **52** & **71** \\ \hline Arizona/New Mexico Mountains & 1 & 1 & 0 & 0 & 1 & 10 & 1 & 0 & 0 & 0 & 1 & 16 \\ Arizona/New Mexico Plateau & 13 & 4 & 0 & 0 & 0 & 16 & 7 & 0 & 1 & 2 & 0 & 43 \\ Chihuahua Desert & 21 & 1 & 0 & 1 & 0 & 73 & 6 & 0 & 0 & 5 & 0 & 106 \\ Colorado Plateaus & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 6 \\ High Plains & 4 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 2 & 11 \\ Madrean Archipelago & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 1 & 0 & 1 & 0 & 5 \\ Southwestern Tablelands & 10 & 16 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 31 \\ \hline Total & 53 & 24 & 1 & 1 & 1 & 106 & 18 & 1 & 1 & 9 & 4 & 219 \\ \hline \hline \end{tabular}
\end{table}
Table A1: Allocation of the randomly selected pixels as identified with Segmented VPR method over the different lower classes based on the National Land Cover Dataset of 2011 (NLCD 2011) along the corresponding ecoregions.  
Appendix B The percentages of the randomly selected pixels (i.e., 155 identified by Segmented-VPR method) over grasslands and shrublands that were evaluated for their significance of the difference in mean productivity between before and after the break years at the pixel level using the Rangeland Production Monitoring Service (RPMS) data.  
Appendix C A summary of the statistical significance test result of the difference in mean in productivity before and after the break years using the randomly selected pixels (i.e., 155 identified by Segmented-VPR method) based on the Rangeland Production Monitoring Service (RPMS) data at the ecoregion level.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Ecoregions**} & \multicolumn{2}{c}{**Insignificant Difference**} & \multicolumn{2}{c}{**Significant Difference**} & \multirow{2}{*}{**Total**} \\ \cline{2-2} \cline{5-6}  & **Decreasing** & **Increasing** & **Decreasing** & **Increasing** & \\ \hline Arizona/New Mexico Mountains & 1 & 0 & 0 & 0 & 1 \\ Arizona/New Mexico Plateau & 0 & 0 & 1 & 7 & 8 \\ Chihuahua Desert & 0 & 3 & 1 & 0 & 4 \\ Southwestern Talelands & 1 & 0 & 10 & 0 & 11 \\ \hline Total & 2 & 3 & 12 & 7 & 24 \\ \hline \end{tabular}
\end{table}
Table 3: The percentages of the randomly selected grassland pixels with respect to their significance in the difference in mean productivity before and after the break years based on the RPMS data along with their identified direction of change over the major ecoregions in New Mexico.
# Blade Crack Diagnosis Based on Blade Tip Timing and Convolution Neural Networks
## 3 Results
### Results of Lumped Parameter Model
#### 3.1.2 Prediction of Blade Crack Location and Depth  
This section shows the fault diagnosis results obtained by the CNN method in this study for four cases in terms of crack location and depth. The confusion matrix of the blade crack location detection results is depicted in Figure 9. Positions 0 to 3 indicate no crack, top cracks, mid cracks, and root cracks, respectively. In addition, the values in the matrix reflect the proportion of correct detections in the true value to the total number of results in that category. As shown in Figure 9, the classification of crack faults is weak for locations 2 and 3; locations 0 and 1 are better, and the classification results are more than 80% for each case. The results show that the CNN is very effective at this task. However, many crack faults are not diagnosed as cracks, since the crack fault characteristics are imparted upon the no-crack blades during the coupling action, leading to confusion in the diagnostic identification. It is evident from a comparison of Figure 9a-c that deviations have a stronger influence on crack fault prediction, particularly for the prediction of no cracks. Since noise has less impact on the identification method than systematic deviation does, system deviation directly affects the identification of blade vibration characteristics.  
Figure 10 shows the blade crack depth prediction results for four different cases. It can be seen that the detection results in Figure 10a,c are relatively stable throughout the whole change in crack severity, and the errors are mostly within 10%, while the detection results in Figure 10b,d have larger errors between the predicted and real values due to system deviation and noise. The prediction accuracies are calculated to be 95.4%, 90.7%, 94.6%, and 87.5% for the four cases, respectively. Comparing Figure 10a-c, the noise has less impact on the prediction of blade crack depth. It is clear that the system deviation causes the data to have a more discrete distribution, increasing the error between the predicted and true values. The reason is that the system deviation directly affects the results of the BTT signal recognition.  
Figure 8: Prediction accuracy of blade crack depth by different deep learning models (case 1: No system deviation and No noise; case 2: System deviation and No noise; case 3: Noise and No system deviation; case 4: System deviation and Noise).
## 5 Results and Discussion  
In the following subsections, the results of the ML models are presented and discussed using confusion matrix and various performance metrics such as overall accuracy, sensitivity (ability to detect positive instances), specificity (ability to detect negative instances), negative predicted value (NPV), positive predicted value (PPV), and balanced accuracy (the average of sensitivity and specificity). Due to the imbalanced dataset used for the training and testing purposes, additional informative performance metrics such as precision, recall, and F1_score are utilized to assess the efficacy of the ML models.  
\[Overall\ Accuracy=(TP+TN)\ /\ (TP+TN+FP+FN) \tag{6}\]  
where \(TP\) = True Positive (number of samples correctly predicted as positive), \(TN\) = True Negative (number of samples correctly predicted as negative), \(FP\) = False Positive (number of samples incorrectly predicted as positive), and \(FN\) = False Negative (number of samples incorrectly predicted as negative).  
\[Precision=TP\ /\ (TP+FP) \tag{7}\]  
\[Sensitivity=TP\ /\ (TP+FN) \tag{8}\]  
\[F1\_score=\frac{2\times precision\times Sensitivity}{precision+Sensitivity} \tag{9}\]
# Information Fusion in Autonomous Vehicle Using Artificial Neural Group Key Synchronization
## 3 Proposed Methodology
### Assessment of Security
#### 3.3.1 Authentication of the Information Source  
To avoid the problems caused by fraudulent and incorrect data from spoofing attacks, a verification system is employed to ensure that the information is transmitted from a genuine source. The most practical approach is to employ digital authentications for both the initiator and the recipient to sign and check. The ECC technique was chosen by the standards owing to the lower size of signs and keys, as per [39]. Signs are generated by using ECDSA using 256 or 224-bit keys [38].
# Robotic Coverage Path Planning for Ultrasonic Inspection
## 4 Using Curvature as Vector Field  
Any twice differentiable parametric surface \(S\) in \(R^{3}\) has two principal curvatures, \(\kappa_{1}\) and \(\kappa_{2}\), at each surface point. These principal curvatures correspond to two perpendicular vector \(\mathbf{t}_{1}\) and \(\mathbf{t}_{2}\) in the tangent plane. Given a parametric surface description, the respective tangent vectors of principal curvatures can be derived analytically. In the case of discretized surface representations, e.g., triangle meshes, principal curvature tangent vectors need to be approximated numerically.  
Different methods were proposed to calculate curvature for arbitrary surfaces. For example, one approach builds upon principal component analysis (PCA) of points in the local neighborhood [29]. We follow a similar approach. However, instead of an analysis of points, we estimate curvature based on surface normals. A singular value decomposition (SVD) of surface normals delivers curvatures \(\kappa_{i}\) and tangent vectors \(\mathbf{t}_{i}\).  
The goal of our CPP algorithm is to keep a suitable alignment of a line-shaped linear phased array ultrasound sensor relative to the surface of the inspected part, see Figure 2. We assume that suitable poses along the path can be derived from local surface properties, i.e., orientations of the principal curvatures. For ultrasonic inspection with a linear phased array, it is important that a large fraction of the complete energy of the emitted signal is also reflected back to the transducer head. Ideally, the surface normals of all points at which the reflection occurs point toward the phased array. Small variations (e.g., due to calibration errors or positioning inaccuracies) within the plane \(\epsilon\) spanned by the surface normal \(\mathbf{n}\) and \(\mathbf{t}_{1}\) lead to only moderate loss of the reflected signal intensity. If the variation is small enough, there is still a chance that the reflection hits an element from the phased array. Surface normals that are tilted out of this plane result in high loss of signal intensity. The reason for this is that reflections that exit the plane are not captured by any of the phased array transducer elements.  
Figure 1: Example of a streamline and its corresponding vector field in 2D. The selected seed point is shown in the center of the figure.
## 3 Literature Review
### UAV-Gesture  
[PERSON] et al. introduced the UAV-Gesture [17] dataset, which addresses the lack of research on gesture-based UAV control in outdoor settings. This dataset aims to fill the existing research gap, as most studies in this field are focused on indoor environments. The dataset consists of 119 high-definition video clips, totaling 37,151 frames, captured in an outdoor setting using a 3 DR Solo UAV and a GoPro Hero 4 Black camera. The dataset comprises annotations of 13 body joints and gesture classes for all frames, encompassing gestures appropriate for UAV navigation and command. The dataset was captured with variations in phase, orientation, and camera movement to augment realism. The authors employed an extended version of the VATIC [44] tool for annotation and utilized a Pose-based Convolutional Neural Network [45] (P-CNN) for gesture recognition. This approach resulted in a baseline accuracy of 91.9%. This dataset facilitates extensive research in gesture recognition, action recognition, human pose recognition, and UAV control, showcasing its efficacy and potential for real-world applications.
# A Systematic Revision of the NFIP Claims Hazard Data in Florida for Flood Risk Assessment
## 3 Results and Discussion
### NFIP Claims Hazard Revision
#### 3.1.3 Summary of Revised Claims
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Cause**} & \multicolumn{2}{c}{**Provided**} & \multicolumn{2}{c}{**Revised**} \\ \cline{2-5}  & **Count** & **\%** & **Count** & **\%** \\ \hline
0 & 11,069 & 7.2 & 215 & 0.1 \\
1 & **45,222** & **29.4** & **71,386** & **46.4** \\
2 & 11,303 & 7.4 & 7401 & 4.8 \\
3 & 1919 & 1.2 & 1522 & 1.0 \\
4 & **81,614** & **53.1** & **59,364** & **38.6** \\
5 & 0 & 0 & 9596 & 6.2 \\
6 & 2624 & 1.7 & 4267 & 2.8 \\ all & 153,751 & 100 & 153,751 & 100 \\ \hline \hline \end{tabular} \(0\): other causes; 1: tidal water overflow; 2: stream, river, or lake overflow; 3: alluvial fan overflow; 4: accumulation of rainfall; 5: indeterminate, cause 1 or 4; 6: all other causes.  
\end{table} 
Table 2: The NFIP claims count summary in accordance with the cause of loss. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Cause**} & \multicolumn{2}{c}{**Provided**} & \multicolumn{2}{c}{**Revised**} \\ \cline{2-5}  & \multicolumn{2}{c}{**Paid Loss**} & \multicolumn{2}{c}{**Paid Loss**} & \\  & **(Thousands of** & **\%** & **(Thousands of** & **\%** \\  & **USD)** & & **USD)** & \\ \hline
0 & 318,190 & 8.6 & 4046 & 0.1 \\
1 & **1,358,612** & **36.6** & **2,414,944** & **65.0** \\
2 & 260,414 & 7.0 & 132,077 & 3.6 \\
3 & 42,905 & 1.2 & 32,738 & 0.9 \\
4 & **1,729,295** & **46.6** & **810,876** & **21.8** \\
5 & 0 & 0 & 286,971 & _7.7_ \\
6 & 4770 & 0.1 & 32,534 & 0.9 \\ all & 3,714,186 & 100 & 3,714,186 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The NFIP claims paid loss amount summary in accordance with the cause of loss (see Table 2 for the cause number reference).
# User Association Performance Trade-Offs in Integrated RF/mmWave/THz Communications
## 3 System Model
### Beamforming Model
#### 3.2.1 Beam Management  
Beam training and tracking ensure proper alignment between the transmitter and receiver beams.  
* Training: At association/handover, beams are aligned through channel estimation and feedback over \(\tau\) seconds. To calculate the training overhead, let \(\tau\) be the beam training time, \(T\) be the data transmission time per beam, and the total overhead per beam = \(\tau+T\). The effective rate reduces to: \[R_{eff}=(1-\tau/T)R\] (10) where \(R_{eff}\) is the effective achievable rate accounting for the beam training overhead (bps), and \(R\) is the max achievable rate without the beam training overhead (bps).
* Tracking: The beam tracking mechanism accounts for the continuous adaptation of beams to changing channel conditions over time. Due to channel variations between BSs and UEs, the optimal beamforming weights need periodic adjustment to maintain high signal quality. The model captures this by estimating the channel state every \(T_{b}\) seconds to perform beam tracking. However, there is an inherent tracking error that causes an angular offset. Beams may not be perfectly aligned due to factors like mobility; this misalignment leads to reductions in the beamforming gain achieved. The beamforming gain equals: \[G=G_{max}\times e^{(-\frac{\theta}{5\Delta t})}\] (11)where \(\theta\) is the offset between the adapted beam and the true optimal beam orientation, which follows a Gaussian distribution, \(\theta_{3 dB}\) represents the half-power beamwidth, and \(G_{max}\) is the maximum possible beamforming gain attained with perfect beam alignment. The lower gain resulting from tracking errors then translates to an additional rate loss for each user. By incorporating beam tracking dynamics, the model realistically reflects the time-varying nature of wireless channels and associated impacts on beamformed transmissions.