esa-sceva/satcom-chunk-collection huggingface dataset:
In this sense, MD simulations give insight into the mechanism of action of new ligands and can provide a new direction for future designs of more effective inhibitors [26; 27; 28]. Furthermore, theoretical studies using the Density Functional Theory (DFT) have reported the antibacterial activity of the hydrozone and acarbose ligand [29] based on molecular structure, electronic properties, and spectral analysis of the acarbose and malonamide derivatives [30]. The aim of this work is to determine by DFT calculations, molecular docking, and molecular dynamics, all those molecular properties that allow us to predict the binding activity of the human intestinal maltase-glucoamylase N-terminal (NtMGAM) with the new ligands. One of the most important contributions of this work is the proposed sequence of theoretical calculations for future research with similar systems. This sequence involves the next steps:  
1. DFT calculations, which provide useful information about the chemical reactivity of the molecules and the most suitable areas for nucleophilic and electrophilic attacks;
2. Molinspiration Cheminformatics for the calculations of the polar surface area and the octanol-water partition coefficient (logP) values;
3. Molecular docking calculations, which determine the most favorable position of interaction between a ligand and a macromolecule by the binding energy;
4. Molecular dynamics calculations, for measuring the stability of the binding between these molecules, considering several contributions such as solvents, electrostatic energy, and temperature. The results obtained with this sequence allowed for defining the oxidative damage between the amino acids of the active site and the new alternatives of treatment (maysin and luteolin) and comparing these to acarbose. Another important property determined in this work was the molecular polar surface area (PSA) and logP values, which quantifies the permeability in cells, thus, it could be taken as a measure of intestinal absorption of the maysin and luteolin. Besides, the Lipinski rule was used for the determination of the highest absorption and permeability in the active site of maltase-glucose(MGA) in the presence of maysin and luteolin. This theoretical study could be a reference in the interpretation of the experimental results as those obtained by our research group in the work about the inhibitory effect of saccharides and phenolic compounds from maize silks on intestinal alpha-glucosidase [14], and a study comparative of different naturals compounds, such as quercetin, tetrocanosol, and acarbose. These molecules showed high binding ability towards \(\alpha\)-amylase and \(\alpha\)-glucosidase [31]. Finally, a comparative binding site analysis of two proteins (human neutral alpha-glucosidase and maltase-glucoamylase) indicated that the active site residues are more or less conserved, based on the docked ACA [32].
## Appendix A Algorithm Description of AP Clustering and Its Flow Chart
* **Reference value \(P\)** is the degree to which point \(v\) is considered a cluster center, and it corresponds to the diagonal values of the similarity matrix \(S\). The reference value directly affects the number of clusters obtained and the choice of cluster centers. A higher reference value indicates a higher likelihood that a data point will become a cluster center; conversely, a lower reference value reduces the likelihood of a data point being chosen as a cluster center. In the absence of prior knowledge, the median of similarity values is typically used as the reference value for each data point, generally ranging between [0, 1]. In this study, a reference value of 0.5 is used and set as a global shared preference. * **Damping factor \(\lambda\)** is introduced to prevent data oscillations during the information propagation process. A higher damping factor limits the magnitude of updates duringiterations, thus enhancing algorithm stability but slowing convergence. Conversely, a lower damping factor accelerates convergence but may lead to more oscillations. Based on expert experience, common settings for the damping factor are 0.5 or 0.9. In this study, a damping factor of 0.9 is used.
# Fault Diagnosis of Railway Point Machines Using the Locally Connected Autoencoder
## 4 Fault Data Analysis
### Analysis Results of Fault Diagnosis
Figure 10 reports the impact of the number of hidden units on the AUC scores when \(\alpha\) and \(L\) are fixed to 0.1 and 10, respectively, while \(M_{\text{loc}}\) and \(S_{\text{tr}}\) are set to 30. Then, for the number of hidden units equal to five, the AUC has a very small value, which can be explained by that a small number of hidden units implies relatively low dimensions of \(\mathbf{W}\), \(\mathbf{W}^{\prime}\), \(\mathbf{b}\) and \(\mathbf{b}^{\prime}\), which causes underfitting. When the number of units increases, the AUC scores stabilize at a relatively high value and so a high degree of accuracy is achieved. Further increasing the number of units results in decreasing AUC values, which is due to the problem of information redundancy. It shows that the excessive increase of the number of units negatively affects the fault detection. We conclude that through the performed fault analysis based on AUC scores the superiority of the locally connected autoencoder over the baseline method and other autoencoder-based methods has been verified. In the process of the analysis, we have devoted much effort to test Cu-3300. Note that it is difficult to obtain accurate labels to indicate healthy or unhealthy states. Moreover, finding fault signals of all possible types of anomalous behaviors is even more difficult, as new types of faults may arise due to the dynamic characteristic. Thus, we propose that Cu-3300 may lack completeness. We have made the dataset Cu-3300 and the code file freely accessible online at [[https://github.com/[PERSON]/SignalRepresentationAnalysis](https://github.com/[PERSON]/SignalRepresentationAnalysis)]([https://github.com/[PERSON]/SignalRepresentationAnalysis](https://github.com/[PERSON]/SignalRepresentationAnalysis)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**k** & **60** & **80** & **100** & **140** & **190** \\ \hline Featuresbased & 0.722 & 0.751 & 0.828 & 0.95 & 0.953 \\ SPDAE-80 & 0.708 & 0.783 & 0.854 & 0.923 & 0.936 \\ STDAE-150-50 & 0.719 & 0.703 & 0.689 & 0.905 & 0.935 \\ GRU-40 & 0.881 & 0.903 & 0.909 & 0.903 & 0.907 \\ GRU-70 & 0.916 & 0.930 & 0.926 & 0.924 & 0.923 \\ Average-I & 0.82 & 0.860 & 0.939 & 0.972 & **0.968** \\ Average-II & 0.912 & 0.973 & **0.979** & **0.976** & 0.97 \\ Weighted-I & 0.907 & 0.960 & 0.969 & 0.972 & **0.968** \\ Weighted-II & **0.949** & **0.981** & **0.979** & **0.976** & **0.968** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparative results of different schemes in terms of AUC values. Figure 9: Performance comparison with different \(\alpha\) and \(L\).
## 4 Development of ViPER+ for Construction Vehicles and Equipment Pose Estimation
### Data Collection Subsystem
#### 4.1.2 Tag/Anchor Communication  
As we mentioned, tags are entities whose location is tracked by the localization system. In TDOA systems, tags periodically transmit beacon messages using their UWB radio. When anchors receive this message, they send a report packet containing the timestamp of the received packet to the server. In addition to anchors, tags also plan their transmission time by utilizing time synchronization messages. The Time Division Multiple Access (TDMA) approach was used in our tags to send beacon messages to avoid collision between messages. For each tag, our system dedicates a transmission time slot based on the ID of the tag. Then each tag schedules its transmission time according to the time slot and the reception time of the time synchronization message.
## 5 Model-Based Predictive Control of the VSI  
The model in (6) and the inequality constraints in (8) and (10) are used to formulate the control algorithm as a quadratic programming (QP) problem, based on the following quadratic cost function [33]:  
\[V(\mathbf{\tilde{x}},\mathbf{\tilde{u}})=\mathbf{\tilde{x}}_{\mathrm{e},N}^{ \top}\mathbf{Q}_{N}\mathbf{\tilde{x}}_{\mathrm{e},N}+\sum_{k=0}^{N-1}\mathbf{ \tilde{x}}_{\mathrm{e},k}^{\top}\mathbf{Q}\mathbf{\tilde{x}}_{\mathrm{e},k}+ \mathbf{\tilde{u}}_{k}^{\top}\mathbf{R}\mathbf{\tilde{u}}_{k}, \tag{19}\]  
where \(\mathbf{Q}\) and \(\mathbf{R}\) are the same weighting matrices for the state and input defined of (15), and \(\mathbf{Q}_{N}\) is the solution of the discrete-time Riccati equation associated with the optimal control problem. This criterion ensures system stability while maintaining the same dynamic response as the LQI when the constraints are not active. The prediction horizon is \(N\), and \(\{\mathbf{\tilde{x}}_{\mathrm{e},k},\mathbf{\tilde{u}}_{k}\}\) represent deviation variables around de desired steady-state operating point,  
\[\mathbf{\tilde{x}}_{\mathrm{e},k} =\mathbf{x}_{\mathrm{e},f}-\begin{bmatrix}\mathbf{\tilde{x}}_{ \mathrm{e}}\\ \mathbf{\tilde{0}}_{2\times 1}\end{bmatrix}, \tag{20}\] \[\mathbf{\tilde{u}}_{k} =\mathbf{u}_{t}-\mathbf{\tilde{u}}_{\mathrm{s}}. \tag{21}\]  
The steady-state operating point \(\{\mathbf{\tilde{x}}_{\mathrm{s}},\mathbf{\tilde{u}}_{\mathrm{s}}\}\) is computed as follows:  
\[\begin{bmatrix}\mathbf{\tilde{x}}_{\mathrm{s}}\\ \mathbf{\tilde{u}}_{\mathrm{s}}\end{bmatrix}=\begin{bmatrix}\mathbb{I}_{4 \times 4}-\mathbf{A}_{\mathrm{d}}&-\mathbf{B}_{\mathrm{d}}\\ \mathbf{C}&\mathbf{0}_{2\times 2}\end{bmatrix}^{-1}\begin{bmatrix} \mathbf{B}_{\mathrm{pd}}\mathbf{i}_{\mathrm{o},f}\\ \mathbf{r}\end{bmatrix}. \tag{22}\]  
The state and input constraints are given by the following:  
\[\mathbf{\tilde{x}}_{\mathrm{e},k+1} =\mathbf{A}_{\mathrm{e}}\mathbf{\tilde{x}}_{\mathrm{e},k}+\mathbf{ B}_{\mathrm{e}}\mathbf{\tilde{u}}_{k}, \tag{23}\] \[\mathbf{\tilde{x}}_{\mathrm{min}} \leq\mathbf{J}_{\mathrm{e}}\mathbf{\tilde{x}}_{k}\leq\mathbf{ \tilde{x}}_{\mathrm{max}},\] (24) \[\mathbf{\tilde{u}}_{\mathrm{min}} \leq\mathbf{E}\mathbf{\tilde{u}}_{k}\leq\mathbf{\tilde{u}}_{ \mathrm{max}},\] (25) \[\mathbf{x}_{0} =\mathbf{x}_{t}, \tag{26}\]  
where \(\mathbf{J}_{\mathrm{e}}=\begin{bmatrix}\mathbf{J}&\mathbf{0}_{6\times 2} \end{bmatrix}\), and (24) and (25) represent the states and inputs constraints using the transformations (20) and (21). The control action is computed by solving the optimization problem and using a receding horizon scheme (RHC) at time \(t\). This approach accelerates the steady state convergence of the system by forwarding the steady state operating point, using the references and known disturbances, improving the system's time response. However, when current constraints are active, the integral of the error between the voltage and its references becomes an open integrator, leading to numerical problems in the MPC control. To address this challenge, this work adopts the incremental formulation proposed in [34].
## 3 SOH Estimation Methods  
### Preprocessing Methods  
#### 3.1.1 Continuous Wavelet Transform (CWT)  
Unlike the Fourier transform, which has a fixed resolution and cannot represent the characteristics of non-periodic signals well, wavelet transform methods have the advantage that signals can be analyzed efficiently using a wavelet function that has flexible resolution [31]. A time-series analysis using the wavelet transform utilizes \(\psi(t)\), the mother wavelet, which is defined in Equation (1).  
\[\psi_{s,t}(t)=\frac{1}{\sqrt{s}}\psi\bigg{(}\frac{t-\tau}{s}\bigg{)} \tag{1}\]  
Figure 3: The reference discharge profile of RW9: (**a**) voltage; (**b**) temperature.  
Figure 2: The RW steps measured for battery RW9: (**a**) first 50 RW steps; (**b**) last 50 RW steps.  
Equation (1) moves the wavelet function by \(\tau\) on the time axis and adjusts the scale by \(s\) to decompose the signal. Given the mother wavelet, the CWT of signal \(f(t)\) is defined in Equation (2).  
\[\text{CWT}(s,\tau)=\int_{-\infty}^{+\infty}f(t)\frac{1}{\sqrt{s}}\psi^{*}\Big{(} \frac{t-\tau}{s}\Big{)}dt \tag{2}\]  
The wavelet coefficients obtained from Equation (2) shows the similarity between \(f(t)\) and \(\psi(t)\). The wavelet transform decomposes the time domain of \(f(t)\) to produce signals only, however, the time-frequency domain can be obtained by pseudo-frequencies. \(\psi^{*}(t)\) is the inner product of the wavelet function. The wavelet function comprises a set of functions with different cycles. Based on this characteristic, it has the advantage that multi-resolution decomposition is possible. The relationship between the scale and pseudo-frequencies varies depending on the center of the wavelet function, scale performance, and sampling cycle. The spectrum of the wavelet function is maximized at a given scale. Equation (3) shows the relationship between these parameters.  
\[f_{s}=\frac{f_{c}}{s\cdot\Delta} \tag{3}\]  
In Equation (3), \(f_{c}\) denotes the center frequency, \(\Delta\) denotes the sampling cycle, and \(f_{s}\) denotes the result of the pseudo-frequency. All signals were synthesized by applying the inverse CWT defined by Equation (4) as follows:  
\[f(t)=\frac{1}{C_{\psi}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}W(s, \tau)\psi_{s,\tau}(t)\frac{dsd\tau}{s^{2}} \tag{4}\]  
\(C_{\psi}\) is an acceptance constant that can be obtained when the mother wavelet satisfies all the requirements. The results of CWT are shown in Figure 4.
# A Novel Approach for the Estimation of Doubly Spread Acoustic Channels Based on Wavelet Transform
## 1 Introduction
To overcome this difficulty, this paper proposes a novel wavelet transform (WT) based algorithm by employing the hyperbolic frequency modulation (HFM) signal as training signal. In UWA environment, Doppler effects cause frequency shifts of the transmitted signal. HFM signal is a preferable training signal in UWA communication as the frequency shifts caused by Doppler effects can be treated as a time shift [16; 17]. Thus the Doppler effects and time delay can be simultaneously estimated by analyzing the time-frequency characteristics of the signal. In addition, we choose WT in this paper as it is an effective measurable tool for time-frequency analysis and has been used for the signal analysis in engineering, biomedical and researcher application [18; 19]. WT methods consist of two categories: continuous wavelet transform (CWT) and discrete wavelet transform (DWT), DWT is nothing but discretization of CWT. The main contributions of this paper are as follows:  
* We propose to use a superimposed HFM signal as training signal and prove that the HFM up sweep and down sweep signals will not interfere each other for WT. * We propose an iterative algorithm to estimate MSML channel based on WT. * We use computer simulations based on BELLHOP to investigate the performance of the proposed algorithm, and make comparisons with OMP algorithm. * We present the complexity analysis of the proposed algorithm as well as OMP algorithm. The rest of this paper is organized as follows: Section 2 gives the MSML channel model. In Section 3, brief introductions of HFM signal and WT are included and we present the WT-based algorithm. Section 4 focuses on the simulation results and complexity analysis. Section 5 concludes the paper. Notation: we will use the following notations in this paper: Upper (lower) bold-face letters stand for matrices (vectors); Superscript * denotes conjugate.
## 2 Theoretical Basis
### Equivalent Strain-Softening Model for Jointed Rock Mass
#### 2.1.2 Strain-Softening Model of Equivalent Jointed Rock Mass
In this paper, the softening parameter \(\eta\) is defined as the plastic shear strain, which is obtained via the difference between the maximum and minimum main plastic strains:  
\[\gamma^{p}=\eta=\varepsilon_{1}{}^{p}-\varepsilon_{3}{}^{p} \tag{9}\]  
where \(\gamma^{p}\) is the plastic shear strain and \(\varepsilon_{1}{}^{p}\) and \(\varepsilon_{3}{}^{p}\) are the maximum and minimum plastic strains, respectively. The maximum host plastic strain of a pluton from Figure 3b can be expressed as:  
\[\varepsilon_{1}{}^{p}=\varepsilon_{1}{}^{pank,e}+\varepsilon_{1}{}^{drop}- \varepsilon_{1}{}^{e} \tag{10}\]  
where \(\varepsilon_{1}{}^{pank,e}\) is the maximum principal strain of elasticity before the peak, \(\varepsilon_{1}{}^{drop}\) is the strain softening after the peak, and \(\varepsilon_{1}{}^{e}\) is the maximum principal strain of elasticity. These can be further expressed as:  
\[\left. \begin{array}{c}\varepsilon_{1}^{pank,e}=\frac{\sigma_{1}^{p}( \sigma_{3})}{E}\\ \varepsilon_{1}^{drop}=\frac{\sigma_{1}^{p}(\sigma_{3})-\sigma_{1}^{e}( \sigma_{3})}{-M}\\ \varepsilon_{1}^{e}=\frac{\sigma_{1}^{e}(\sigma_{3})}{E}\end{array} \right\} \tag{11}\]  
where \(\sigma_{1}{}^{p}\) is the peak principal stress, \(\sigma_{1}{}^{r}\) is the residual principal stress, and \(E\) is the elastic modulus of the jointed plutons. In considering the shear swelling angle \(\psi\), \(\varepsilon_{3}{}^{p}\) can be represented as:  
\[\varepsilon_{3}{}^{p}=-\frac{1}{2}K_{\psi}\cdot\varepsilon_{1}{}^{p} \tag{12}\]  
where:  
\[K_{\psi}=\frac{1+\sin\psi}{1-\sin\psi} \tag{13}\]  
When \(\psi=0\), Equation (12) becomes:  
\[\varepsilon_{3}{}^{p}=-\frac{1}{2}\varepsilon_{1}{}^{p} \tag{14}\]  
The above analysis shows that the slope \(M\) of the softening curve is the key to controlling the softening process of jointed plutons. In this paper, the formulation proposed by [PERSON] et al. [26] is applied to consider the change of peripheral pressure and quality grade of the plutons:  
\[M=-\xi\cdot E \tag{15}\]  
when: \(\frac{\sigma_{3}}{\sqrt{S^{p}}\sigma_{ci}}\leq 0.1\)  
\[\xi=\left[0.0046e^{0.0768 GSH}\right]\left(\frac{\sigma_{3}}{\sqrt{S^{p}} \sigma_{ci}}\right)^{-1} \tag{16}\]  
when: \(\frac{\sigma_{3}}{\sqrt{S^{p}}\sigma_{ci}}\geq 0.1\)  
\[\xi=\left[0.0046e^{0.0768 GSH}\right]\left(\frac{\sigma_{3}}{2\sqrt{S^{p}} \sigma_{ci}}+0.05\right)^{-1} \tag{17}\]In summary, the analysis reveals that the post-peak mechanical behavior of plutons in deep fault fracture zones is affected by the pluton quality grade and the peri-pressure. The pluton softening parameter \(\eta^{*}\) varies during the post-peak unloading of the plutons and its size follows the peri-pressure changes experienced by the plutons. To verify the correctness of the analysis and reliability of the strain-softening model, the model was validated and analyzed using room triaxial compression tests.
# Effects of Introducing Rest Intervals in Functional Fitness Training
## 3 Results  
All participants completed the protocols fully. Our results indicated no significant differences in the time employed to complete the standard and adapted FFT workouts (FFTstandard = 12.45 \(\pm\) 2.97 min, FFTadapted = 12.89 \(\pm\) 2.45 min, \(p\)\(>\) 0.05), including three sets of 1 min rest time and exercise time in FFTadapted. Neither were significant differences detected between groups in HR (\(p\)\(>\) 0.05). Both mean HR values indicated high-intensity exercise (FFTstandard = 172 \(\pm\) 10 bpm, FFTadapted = 171 \(\pm\) 8 bpm), as both were 90% of age-predicted HR during the workouts and above 95% at the end (FFTstandard = 182 \(\pm\) 8 bpm, FFTadapted = 181 \(\pm\) 6 bpm) [32]. In the two-way repeated measures ANOVA (Table 1), significant differences in blood lactate were detected for the factor Time, i.e., between time points (F(1,24) = 363.75, \(p\) = 0.000, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.938, SP = 1.000); the factor Group, i.e., between the two FFT groups (F(1,24) = 5.314, \(p\) = 0.030, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.181, SP = 0.600); and for the interaction Time \(\times\) Group (F(1,24) = 7.769, \(p\) = 0.010, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.245, SP = 0.762). Using the Bonferroni adjustment to control for multiple post-hoc comparisons for the blood lactate data indicated a significantly lower lactate level (by more than 10%) post-exercise in FFTadapted (\(p\) = 0.015). Heart rate showed differences only for Time (F(1,24) = 4270.025, \(p\) = 0.000, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.994, SP = 1.000), and not for Group (F(1,24) = 0.032, \(p\) = 0.860, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.001, SP = 0.032) or Time \(\times\) Group (F(1,24) = 4.014, \(p\) = 0.057, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.143, SP = 0.485). RPE scores did differ significantly for Time (F(1,24) = 87.377, \(p\) = 0.000, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.785, SP = 1.000), Group (F(1,24) = 8.254, \(p\) = 0.008, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.256, SP = 0.787), and Time \(\times\) Group (F(1,24) = 4.240, \(p\) = 0.051, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.150, SP = 0.507). Multiple post-hoc comparisons detected significant differences between protocols 3 min after exercise (\(p\) = 0.003) (Table 1). When we compared jump ability between the FFT workouts (Table 2), a significant difference was observed in jump height for Time (F(1,24) = 22.010, \(p\) = 0.000, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.478, SP = 0.994), but not Group (F(1,24) = 0.858, \(p\) = 0.363, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.035, SP = 0.144) or Time \(\times\) Group (F(1,24) = 0.634, \(p\) = 0.434, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.026, SP = 0.119). Significant differences in maximum takeoff velocity were also noted for Time (F(1,24) = 15.010, \(p\) = 0.001, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.385, SP = 0.960), but not Group (F(1,24) = 0.951, \(p\) = 0.339, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.038, SP = 0.155) or Time \(\times\) Group (F(1,24) = 1.593, \(p\) = 0.219, \(\eta_{\mathrm{p}}\)\({}^{2}\) = 0.062, SP = 0.228). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
## 3 The Proposed Channel Estimation Methods
### Denoising Method 1  
Since a channel estimator outputs one by one once it achieves convergence, the filter could generate multiple estimated samples. Observing the received pilot symbols in the DD domain, which are described in Figure 2, we can find that \(N_{\text{total}}=(M_{\tau}-L)(N_{\
u}-2K)\) symbols can be used for channel estimation. By supposing the estimator is converged after \(N_{c}\) steps, it can output \(N_{h}=N_{\text{total}}-N_{c}\) effective samples.  
Figure 2: The received pilot symbols in DD domain.  
Due to it only choosing one from multiple samples to represent the channel, the traditional adaptive channel estimation method may deteriorate the estimation accuracy due to the impact of estimation noise, while in the proposed method, we define the \(N_{h}\) samples as the multiple observations associated with the same channel. Thus, the \(i\)-th estimated sample can be described as the sum of the channel's true value and the estimated error.  
\[\hat{\mathbf{H}}_{i}=\mathbf{H}+\mathbf{e}_{i},\ \ \ i=1,\cdots,N_{h} \tag{20}\]  
Similar to [PERSON]'s work, estimated error \(\mathbf{e}_{i}\) can be modeled as random noise. Based on the invariant characteristic of random noise, we develop two simple yet efficient denoising strategies to enhance the performance of the proposed method. The first one is to combine the \(N_{h}\) samples by the arithmetic average operation. Then, the final estimated channel of the filter is given by  
\[\hat{\mathbf{H}} =\frac{1}{N_{h}}\sum_{i=1}^{N_{h}}\hat{\mathbf{H}}_{i} \tag{21}\] \[=\mathbf{H}+\frac{1}{N_{h}}\sum_{i=1}^{N_{h}}\mathbf{e}_{i} \tag{22}\]  
Assume the noise \(\mathbf{e}_{i}\) is independent, and the variance is \(\sigma_{\varepsilon}^{2}\). Thus, the variance of the estimated error is reduced to \(\frac{1}{N_{h}}\sigma_{\varepsilon}^{2}\). It is noted that the noise variance is proportional to \(\frac{1}{N_{h}}\). As the value of \(N_{h}\) increases, the change in the variance becomes smaller and smaller. When \(N_{h}\) is large enough, the effect of increasing the number of \(N_{h}\) is very small.