esa-sceva/satcom-chunk-collection huggingface dataset:
# Artificial Intelligence Applications for MEMS-Based Sensors and Manufacturing Process Optimization
## 4 Artificial Intelligence Application in MEMS System: Current Trends
### AI Applications in the MEMS Manufacturing and Design Process
[88] proposed a methodical three-step approach to identify single and mixed defects using a deep ML-based method. In the first step, a spatial filter was used to remove random noise. Next, a splitter was used to separate the single and mixed patterns. Finally, a Deep Structured Convolutional Network (DSCN) model was used to find the composite pattern. A shallow Randomized General Regression Network (RGRN) was used to find the single pattern defects with an overall accuracy of 86.17%. [PERSON] and [PERSON] [89] proposed a theoretical framework for a MEMS oscillatory neurocomputer for pattern recognition using auto-correlative associative memory. In this paper, the authors' used MEMS as an analog information unit. They further claimed that this method could help build an information processing unit by eliminating micro-controllers in the process flow. [PERSON] et al. [90] proposed a PC-based Expert System, 'EASYMEMS', for MEMS design and manufacturing. EASYMEMS covers three main aspects of MEMS-based systems: materials, design, and manufacturing. It contains a knowledge-based engine to provide expert consultation and perform dynamic and static analysis. Although other software packages exist, such as ACS and IntelliSuite, the authors claimed that their proposed system could provide a more comprehensive and user-friendly framework based on the AI expert system. [PERSON] et al. [91] proposed an ML-based approach in place of the standard FEA to reduce the design time as FEA is computationally heavy. The authors claimed their system is almost 4000 times faster in detecting vibration modes for disk resonators. Although this is an exciting result, the research needs further comparative analysis using other AI approaches.
# Beam Grouping Based RS Resource Reuse and De-Contamination in Large Scale MIMO Systems
## 5 Numerical Results  
In a downlink LS-MIMO system with \(N_{t}\) TX antennas and \(K\) single antenna RXs, the \(K\times 1\) received signal vector at the UEs can be represented as follows:  
\[\mathbf{y}=\sqrt{\rho_{t}}\zeta\mathbf{H}\mathbf{D}\mathbf{W}\mathbf{x}+ \mathbf{z}, \tag{13}\]  
where \(\zeta\) is the normalized factor to make total TX power as \(\rho_{t}\), \(\mathbf{H}\) is the \(K\times N_{t}\) correlated channel matrix that was introduced in Section 2.2, and \(\mathbf{D}\) is the precoding matrix for transmit steering matrix, i.e., \(\mathbf{D}=[\mathbf{a}^{H}(\vec{\theta_{1}}),\ \cdots,\ \mathbf{a}^{H}(\vec{ \theta_{k}}),\ \cdots,\ \mathbf{a}^{H}(\vec{\theta_{K}})]^{H}\), \(\vec{\theta_{k}}\) is the average of angular spread, \(\mathbf{W}\) is the group precoding matrix, which can be zero-forcing (ZF) or matched filtering (MF) in each group. With the precoding matrix, \(\mathbf{H}\cdot\mathbf{D}\) can be represented as follows:  
Figure 7: A modified BGRR algorithm to guarantee the space separability. \[\mathbf{H}\cdot\mathbf{D} = \left( \begin{array}{c}\mathbf{g}_{1}\mathbf{A}_{1}\\ \mathbf{g}_{2}\mathbf{A}_{2}\\ \vdots\\ \mathbf{g}_{i}\mathbf{A}_{i}\\ \vdots\\ \mathbf{g}_{K}\mathbf{A}_{K}\end{array} \right)\cdot\left(\mathbf{a}^{H}( \vec{\theta}_{1})\;\mathbf{a}^{H}(\vec{\theta}_{2})\;\cdots\;\mathbf{a}^{H}( \vec{\theta}_{i})\;\cdots\;\mathbf{a}^{H}(\vec{\theta}_{K})\;\right), \tag{14}\] \[= \left( \begin{array}{ccccc}\mathbf{g}_{1}\mathbf{A}_{1}\mathbf{a }^{H}(\vec{\theta}_{1})&\mathbf{g}_{1}\mathbf{A}_{1}\mathbf{a}^{H}(\vec{ \theta}_{2})&\cdots&\mathbf{g}_{1}\mathbf{A}_{1}\mathbf{a}^{H}(\vec{\theta}_{ i})\;\cdots&\mathbf{g}_{1}\mathbf{A}_{1}\mathbf{a}^{H}(\vec{\theta}_{K})\\ \mathbf{g}_{2}\mathbf{A}_{2}\mathbf{a}^{H}(\vec{\theta}_{1})&\mathbf{g}_{1} \mathbf{A}_{2}\mathbf{a}^{H}(\vec{\theta}_{2})&\cdots&\mathbf{g}_{2}\mathbf{A }_{2}\mathbf{a}^{H}(\vec{\theta}_{i})\;\cdots&\mathbf{g}_{2}\mathbf{A}_{2} \mathbf{a}^{H}(\vec{\theta}_{K})\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{g}_{K}\mathbf{A}_{K}\mathbf{a}^{H}(\vec{\theta}_{1})&\mathbf{g}_{K} \mathbf{A}_{K}\mathbf{a}^{H}(\vec{\theta}_{2})&\cdots&\mathbf{g}_{K}\mathbf{A }_{K}\mathbf{a}^{H}(\vec{\theta}_{i})\;\cdots&\mathbf{g}_{K}\mathbf{A}_{K} \mathbf{a}^{H}(\vec{\theta}_{K})\end{array}
## 4 Methods
### Cluster-Head Election Phase
#### 4.2.2 Elect the Vice Cluster-Heads in DCK-LEACH
Therefore, the objective function of residual energy in this election is shown in Equation (6) as the same as that of the primary cluster-head selection. Considering the above two factors, the node with more residual energy and positioned closer to the base station is more likely to become the vice cluster-head. We have the formula as follows:  
\[Q(i)=\mu f_{1}(i)+\varphi f_{3}(i) \tag{12}\]  
where \(\mu\) and \(\varphi\) are weighting parameters. At the beginning, we set \(\mu=\varphi=1/2\). If the sum of residual energy of all nodes is less than half of the sum of the initial energy of the network (0.5 * n\(E_{0}\)), the update formula of \(\mu\) and \(\varphi\) are as follows:\[\mu(i)=\frac{E_{res}(i)}{E_{0}} \tag{13}\]  
\[\varphi(i)=1-\mu(i) \tag{14}\]  
where, \(E_{0}\) is the initial energy of node \(i\), and \(E_{res}(i)\) indicates the residual energy of node \(i\). Finally, for each node, the adaptive function value (\(Q\)) is dynamically recalculated based on the residual energy of the network in the current round, and a node with the largest \(Q\) is selected as a vice cluster-head within clusters. Finally, K vice cluster-heads are generated.
# Broadband Millimeter-Wave 5G Power Amplifier Design in 22 nm CMOS FD-SOI and 40 nm GaN HEMT
## 3 Results
### Differential Cascode CMOS PA Measurement Results
It achieves a maximum PAE \(\geq\) 15.1% and \(P_{OUT,SAT}\)\(\geq\) 12.9 dBm across 24-39 GHz, highlighting the broadband performance of this CMOS PA.  
As has been conducted with GaN design #1, this CMOS PA's linearity is tested with 50/100/400/9 \(\times\) 100 MHz 256-QAM NR input signals with the same PAPR = 8.0 dB and same NI PXI equipment at 24 GHz, and the PA output spectra are shown in Figure 15. With the same output power of \(P_{OUT,AVE}\) = \(\sim\)7 dBm, at the instantaneous signal BW = 50 MHz this PA achieves an ACLR -/+ of \(-\)28.0/\(-\)29.4 dB (\(PAE_{AVE}\) = 7.3%), with BW = 100 MHz, \(-\)27.6/\(-\)28.8 dB (\(PAE_{AVE}\) = 7.0%), with BW = 400 MHz, \(-\)27.3/\(-\)27.0 dBc (\(PAE_{AVE}\) = 7.7%), and with BW = 9 \(\times\) 100 MHz, \(-\)24.6/\(-\)25.9 dBc (\(PAE_{AVE}\) = 8.1%). Similarly, this PA's linearity is also tested at 50-900 MHz BW at several other frequencies in the 5G FR2 band. When varying the BW of a 256-QAM 5G NR signal (PAPR = 8 dB) at 24 GHz and 28 GHz, the \(P_{OUT,AVE}\) is about 7 dBm, and at 37 GHz and 39 GHz, \(P_{OUT,AVE}\) is approximately \(\sim\)6 dBm. Figure 16 also shows interesting ACLR measurement data vs. signal BW. When the BW is increased from 50 MHz to 100 MHz, the linearity is not affected much, but when increasing it from 100 MHz to 400 MHz, the PA's linearity is degraded by a small but noticeable amount of \(\sim\)1-3 dB. Then, increasing the BW from 400 MHz to 9 \(\times\) 100 MHz, the linearity becomes further degraded by an additional \(\sim\)2 dB at all frequencies measured. Although, this CMOS PA's linear output power only degrades by \(\sim\)1 dB when its operating frequencies change from 24/28 GHz to 37/39 GHz. Figure 14: Measured maximum PAE, \(S_{21}\), and \(P_{OUT,SAT}\) vs. frequency of the broadband CMOS FD-SOI PA.  
Figure 13: Large-signal post-layout PEX simulation vs. CW measurement of the broadband CMOS FD-SOI PA at (**a**) 24, (**b**) 28, and (**c**) 44 GHz.
# Horizontal Wind Effect on the Aerodynamic Performance of Coaxial Tri-Rotor MAV
## 2 Theoretical Analysis
### The Parameters of Aerodynamic Performance  
#### 2.4.1 Power Loading  
The total hover efficiency of a MAV can be quantified by means of effective power loading (_PL_). The _PL_ is defined as the ratio of the thrust to power required [27]:  
\[PL=\frac{T}{P}. \tag{4}\]  
The thrust coefficient \(C_{T}\) and power coefficient \(C_{P}\) are defined as [28]:  
\[C_{T}\,=\,\frac{T}{{\rho A\Omega^{2}R^{2}}}, \tag{5}\]  
\[C_{P}=\frac{P}{{\rho A\Omega^{3}R^{3}}}=\frac{Q\Omega}{{\rho A\Omega^{3}R^{3}} }=\frac{Q}{{\rho A\Omega^{2}R^{3}}}. \tag{6}\]  
Therefore, the power loading (_PL_) can be written as:  
\[PL=\frac{C_{T}}{{\Omega RC_{P}}}=\frac{T}{{Q\Omega}}, \tag{7}\]  
where \(T\) is the thrust, N; \(A\) is the area of the rotor, m\({}^{2}\); \(P\) is the power, W; \(\Omega\) is rotational speed of the rotor, r/min; \(R\) is the rotor radius, m; \(Q\) is the torque, Nm; \(\rho\) is the fluid density, kg/m3; \(C_{T}\) is the thrust coefficient; \(C_{P}\) is the power coefficient.  
To maximize the _PL_, that is, for a given thrust, the power demand is minimum. When designing a vehicle it is wanted to maximize the power loading such that energy requirements are minimized. This will give the vehicle the best endurance or payload capabilities possible.
# Calculation Method for Settlement Deformation of Existing Tunnel Induced by Underpass Construction
## 3 Calculation of Additional Stresses in Tunnel Undercutting  
The work performed by the increased load brought on by the new tunnel's excavation is a crucial component of the formula derivation in the energy calculation technique. Under the condition of underpassing construction, the shield machine has a complex interaction relationship with the surrounding soil, thus inducing additional stresses in the existing tunnel. The following four primary components [22] can be taken into account when analyzing the extra stresses created during the building process: (a) soil loss; (b) shield shell friction\(f\); (c) grouting pressure \(p\); and (d) shield cutter thrust \(q\). Therefore, the additional stresses due to these four factors need to be derived separately.
## 4 Modeling and Results  
### Datasets Used for Modeling  
To effectively train a model capable of recognizing handwritten Japanese kanji, we utilized two distinct datasets: CASIA-HWDB and Kuzushiji Kanji. Each dataset offers unique characteristics that contribute to different stages of the training process.
# On the Virtual Cell Transmission in Ultra Dense Networks
###### Abstract  
Ultra dense networks (UDN) are identified as one of the key enablers for 5G, since they can provide an ultra high spectral reuse factor exploiting proximal transmissions. By densifying the network infrastructure equipment, it is highly possible that each user will have one or more dedicated serving base station antennas, introducing the user-centric virtual cell paradigm. However, due to irregular deployment of a large amount of base station antennas, the interference environment becomes rather complex, thus introducing severe interferences among different virtual cells. This paper focuses on the downlink transmission scheme in UDN where a large number of users and base station antennas is uniformly spread over a certain area. An interference graph is first created based on the large-scale fadings to give a potential description of the interference relationship among the virtual cells. Then, base station antennas and users in the virtual cells within the same maximally-connected component are grouped together and merged into one new virtual cell cluster, where users are jointly served via zero-forcing (ZF) beamforming. A multi-virtual-cell minimum mean square error precoding scheme is further proposed to mitigate the inter-cluster interference. Additionally, the interference alignment framework is proposed based on the low complexity virtual cell merging to eliminate the strong interference between different virtual cells. Simulation results show that the proposed interference graph-based virtual cell merging approach can attain the average user spectral efficiency performance of the grouping scheme based on virtual cell overlapping with a smaller virtual cell size and reduced signal processing complexity. Besides, the proposed user-centric transmission scheme greatly outperforms the BS-centric transmission scheme (maximum ratio transmission (MRT)) in terms of both the average user spectral efficiency and edge user spectral efficiency. What is more, interference alignment based on the low complexity virtual cell merging can achieve much better performance than ZF and MRT precoding in terms of average user spectral efficiency.  
ultra dense networks; virtual cell; interference graph; zero-forcing precoding; interference alignment  
PACS:J0101
## 2 Model Creation and Data Processing
A general position vector formula of this \(\mathbf{p}_{i}\) structure will thus be:  
\[\mathbf{p}_{i+1}=R_{z1}(\varphi_{1})\times\left[\mathbf{r}_{1}+\sum_{k=1}^{i}\left[\prod _{J=1}^{k}\left(\mathbf{R}_{y(i+1)}\big{(}\theta_{j+1}\big{)}\times\mathbf{R}_{z(j+1)} \big{(}\varphi_{j+1}\big{)}\right)\right]\times\mathbf{r}_{k+1}\right] \tag{7}\]  
The resulting orientation of the \(\mathbf{r}_{i}\) vector defined by the Euler's angle can be seen in [27] according to the \(\mathbf{R}_{z}(\gamma)\mathbf{R}_{y}(\beta)\mathbf{R}_{x}(\alpha)\) option will be determined from the relation below based on the final arm rotation:  
\[\mathbf{R}_{ZYX(i+1)}=\mathbf{R}_{z1}(\varphi_{1})\times\prod_{j=1}^{i}\left(\mathbf{R}_{ y(j+1)}\big{(}\theta_{j+1}\big{)}\times\mathbf{R}_{z(j+1)}\big{(}\varphi_{j+1} \big{)}\right) \tag{8}\]  
Generally speaking, the final shape of the \(\mathbf{R}_{ZYX(i+1)}\) in terms of its \(i\)-segment will look as follows:  
\[\mathbf{R}_{ZYX(i+1)}=\left[ \begin{array}{ccc}\text{a}_{11}&\text{a}_{12}&\text {a}_{13}\\ \text{a}_{21}&\text{a}_{22}&\text{a}_{23}\\ \text{a}_{31}&\text{a}_{32}&\text{a}_{33}\end{array} \right] \tag{9}\]  
where \(\text{a}_{\text{ij}}\) for \(i=1\), \(2\), \(3\), \(j=1\), \(2\), \(3\) are the matrix elements. Considering the \(\mathbf{R}_{z}(\gamma)\mathbf{R}_{y}(\beta)\mathbf{R}_{x}(\alpha)\) option, the Euler's angles will then be as follows:  
\[\alpha=\arctan\left(\frac{\text{a}_{32}}{\text{a}_{33}}\right) \tag{10}\]  
\[\beta=\arcsin(-\text{a}_{31}) \tag{11}\]  
or also:  
\[\beta=\arctan\left(\frac{-\text{a}_{31}}{\sqrt{\text{a}_{32}^{2}+\text{a}_{3 3}^{2}}}\right) \tag{12}\]  
\[\gamma=\arctan\left(\frac{\text{a}_{21}}{\text{a}_{11}}\right) \tag{13}\]  
Thus, for example, in the case of an arm with 3 active degrees of freedom, i.e., for the \(\mathbf{r}_{4}\) vector, the final rotation according to Equation (8) will be as follows:  
\[\mathbf{R}_{ZYX4}=\mathbf{R}_{z1}(\varphi_{1})\times\mathbf{R}_{y2}(\theta_{2})\times\bm {R}_{z2}(\varphi_{2})\times\mathbf{R}_{y3}(\theta_{3})\times\mathbf{R}_{z3}(\varphi_{3 })\times\mathbf{R}_{y4}(\theta_{4}) \tag{14}\]  
Our case involves 6 degrees of freedom.
# An Autonomous Wireless Device for Real-Time Monitoring of Water Needs
## 3 Materials and Methods
### Hardware Description  
In this section, a complete description of the hardware that makes up our end-device is given. When choosing the hardware that forms our end-device, it should be noted that all of the components were selected with low consumption in mind, thus increasing battery life to the maximum.  
Each end-device comprised a radio transceiver, a micro-controller, sensors, and an antenna, along with other circuitry that would enable it to communicate with the gateway to transmit information collected by the sensors [34].
# Saturation-Based Airlight Color Restoration of Hazy Images
## 2 Dehazing for Color Restoration
### Gray World Hypothesis  
When only the aforementioned ASM is used, the biased haze color can degrade the color recognition of the image, and the dark haze color may also degrade visibility. Therefore, many dehazing methods for color-cast images restore image color based on the gray world hypothesis [15; 20; 21; 22].  
The gray world hypothesis indicates that in a typical RGB color image, the average of each color channel is identical. Given that the R, G, and B channel values of a color-cast image \(I(x)\) are \(I_{r}(x)\), \(I_{g}(x)\), and \(I_{b}(x)\), and the white balance result is \(\widehat{I}_{r}(x)\), \(\widehat{I}_{g}(x)\), and \(\widehat{I}_{b}(x)\), respectively, the gray world hypothesis is expressed as follows [25; 26]:  
\[\widehat{I}_{r}(x) =\frac{G_{avg}}{R_{avg}}I_{r}(x) \tag{4}\] \[\widehat{I}_{g}(x) =I_{g}(x)\] \[\widehat{I}_{b}(x) =\frac{G_{avg}}{R_{avg}}I_{b}(x)\]  
Given that the widths of the row and column of \(I(x)\) are \(M\) and \(N\), respectively, the average \(R_{avg}\), \(G_{avg}\), and \(B_{avg}\) of the R, G, and B channels can be expressed as follows:  
\[R_{avg} =\frac{1}{MN}\sum\limits_{X=1}^{MN}I_{r}(x) \tag{5}\] \[G_{avg} =\frac{1}{MN}\sum\limits_{X=1}^{MN}I_{g}(x)\] \[B_{avg} =\frac{1}{MN}\sum\limits_{X=1}^{MN}I_{b}(x)\]  
The resulting images of the gray world hypothesis from the aforementioned Equations (4) and (5) can be verified in Figure 1.  
In Figure 1, the typical hazy image produces a result very similar to the original image, as shown in Figure 1a,c. In contrast, the color-cast image shows a biased histogram,as shown in Figure 1b, and the gray world hypothesis restores color by moving and overlapping the corresponding histogram, as shown in Figure 1d.
# Hybrid Deep Learning Approach for Automated Sleep Cycle Analysis
## 3 Methodology
### Model Training
#### 3.4.3 Backpropagation  
To minimize errors, the weights and biases of the network are updated using the loss function gradients \(J(W,b)\) in a process called backpropagation [46]. In this process, the error flows from the output layer back toward the input layer, and the gradients are computed to update the model parameters via optimization algorithms such as gradient descent (GD):  
\[w^{(l)}=w^{(l)}-\eta\frac{\partial J(W,b)}{\partial w^{(l)}},\qquad b^{(l)}=b^ {(l)}-\eta\frac{\partial J(W,b)}{\partial b^{(l)}}, \tag{14}\]  
where \(\eta\) is the learning rate, and \(W\) represents the collection of all weight parameters in the network. The gradient computation follows the chain rule as follows:  
\[\frac{\partial J(W,b)}{\partial w^{(l)}}=\frac{\partial J}{\partial a^{(L)}} \frac{\partial a^{(L)}}{\partial z^{(L)}}\frac{\partial z^{(L)}}{\partial w^{ (l)}}, \tag{15}\]  
where each term is the derivative at a different layer. The error at layer \(l\) is given by  
\[\delta^{(l)}=\frac{\partial J(W,b)}{\partial z^{(l)}}, \tag{16}\]  
which propagates backward as  
\[\delta^{(l)}=\left(W^{(l+1)}\right)^{T}\delta^{(l+1)}\odot f^{\prime}(z^{(l)}), \tag{17}\]  
where \(\odot\) denotes the Hadamard product. Finally, the weight and bias gradients are  
\[\frac{\partial J(W,b)}{\partial w^{(l)}}=\delta^{(l)}(a^{(l-1)})^{T},\qquad \frac{\partial J(W,b)}{\partial b^{(l)}}=\sum_{i=1}^{m}\delta^{(l)}_{i}. \tag{18}\]  
The resulting gradients are applied in successive iterations to update the weights of the network, gradually improving the predictive performance of the model.
## 1 Introduction
an MBS WT model of the Vestas V52 and subjected it to simulated wind fields based on DLCs for the purpose of generating simulated data to develop a prototype of a virtual 6-DOF GB input loads sensor [12]. The investigation presented a methodology for generating a simulated time-series of the target variables, 6-DOF GB input loads, as well as the predictor variables, the simulated deformations, misalignments, and rotational speeds of various drivetrain components [12]. Since [PERSON] et al. aimed to develop the envisaged virtual sensor in the absence of real world data, it was necessary to model and simulate the predictor variables, or the ideal responses of the sensor signals to be expected in the real world. The current investigation, on the other hand, aims to design experiments for a purpose built test rig where it is only necessary to control the applied loads to a purposely instrumented V52 drivetrain in order to collect the true sensor signals. In addition, due to the need for identifying the range of loads that are expected to be experienced by the V52 drivetrain during its lifetime, it is necessary to perform simulations covering a large number of DLCs. Therefore, the current investigation utilizes a simplified variant of the modeling approach demonstrated by [PERSON] et al., which is more targeted to generate the independent variables, or factors, of the planned experiments, which are the 6-DOF rotor loads and rotational speed. This paper presents a methodology for further processing the resulting time-series data of the target factors from the various DLC simulations to generate a series of designed experiments designed to subject the V52 drivetrain to realistic factor combinations during the planned campaign. In brief, this investigation aims to provide a methodology to enable investigators to avoid the unmanageable, e.g., unrealistic load combinations or number of planned experiments, and manage the unavoidable, e.g., systemic bias and the need to identify and cover the realistic regions of the design space, when designing tests for a WT drivetrain that require covering realistic combinations of the 6-DOF rotor loads and rotational speed. This is achieved by first outlining a process for generating simulated time-series data of a WT experiencing wind fields covering the design load cases (DLC) according to the IEC 61400-1 [43] in a simulated environment and then presenting an approach for analyzing this data to generate designed experiments that cover the relevant regions of the design space at hand. The paper is organized as follows. Section 1 introduces and motivates the methods used in this investigation. Section 2 details the proposed methodology with Section 2.1 presenting the modeling and simulation approach and Section 2.2 providing a description of the resulting simulated data. Section 2.3 presents the data analysis. Section 2.4 elaborates on considerations and challenges in practical implementation of the proposed method. Section 3 presents the results of the investigation followed by a discussion of the results in Section 4. The conclusions are then outlined in Section 5.
## 3 Nature of Input Dimension and Overfitting
### Explanation for the Input Dimension Based on Vibration Signals
#### 3.2.2 The influence of Input Dimension of Sparse Filtering  
The method of [PERSON] et al. [28] was adopted to engage with the experiment signals described above. The selection of input dimension \(N_{in}\) of sparse filtering was investigated. Softmax regression was adopted as a classifier and the diagnosis results are shown in Figure 6. It can be seen that the testing accuracy decreased after an initial increase with the input dimension. Therefore, excessive human labor was required to select the input dimension for the high testing accuracy. To overcome this deficiency, we decided to provide a clear explanation of the nature of the input dimension.  
The bearing signals were employed to verify the above explanation of the input dimension on the basis of harmonic signals. As shown in Figure 7a, two weight vectors of \(N_{in}=50\) and 100 were selected respectively, and their spectra are shown in Figure 7b. It was determined that the weight vectors were striking similar to the one-dimensional Gabor filter which served as an excellent band-pass filter for signals. The Gabor function is shown as follows [34]:  
\[f(d)=A\exp(-\frac{(d-D)^{2}}{2\sigma^{2}})\cos(2\pi\omega(d-D)+\varphi)+B \tag{7}\]  
where \(A\), \(\omega\), and \(\phi\) are the amplitude, spatial frequency, and phase of the cosine term, respectively, \(\sigma\) is the standard deviation of the Gaussian, \(D\) denotes a position offset, and \(B\) is an offset parameter. The two different weight vectors exert the same bandwidth, which means that the features extracted by them theoretically have the same frequency information. However, when the input dimension diminishes, the frequency interval of weight matrix also shrinks. As a result, the energy of frequency is dispersed, learning to unclear learned features, which results in lower testing accuracy.  
Figure 5: Relationship between the frequencies of samples and the learned features: (**a**) \(N_{in}=200\); (**b**) \(N_{in}=100\).