White Noise Research Articles

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

A numerical investigation on the vibration of a two-deck Euler–Bernoulli beam flooded by a potential flow

Abstract This work concerns the structural vibration of a bladeless wind turbine, modelled by a two-deck Euler–Bernoulli beam, due to a surrounding potential flow. The deflection is governed by the Euler–Bernoulli equation which is studied first by a linear theory and then computed numerically by a finite difference method in space with a collocation method over the arc length, and an implicit Euler method in time. The fluid motion in the presence of gravity is governed by the full Euler equations and solved by the time-dependent conformal mapping technique together with a pseudo-spectral method. Numerical experiments of excitation by a moving disturbance on the fluid surface with/without a stochastic noise are carried out. The random process involved in generating the noise on the water surface is driven by a Wiener Process. A Monte Carlo method is used for stochastic computations. The generated surface waves impinge on the beam causing structural vibration which is presented and discussed in detail. By elementary statistical analysis, the structural response subject to the stochastic hydrodynamic disturbance caused by white noise is found to be Gaussian.

Enhancement of single-lead dry-electrode ECG through wavelet denoising

Neonatal electrocardiogram (ECG) monitoring is an important diagnostic tool for identifying cardiac issues in infants at birth. Long-term remote neonatal dry-electrode ECG monitoring solutions can be an additional step for preventive healthcare measures. In these solutions, power and computationally efficient embedded signal processing techniques for denoising newborn ECGs can assist in increasing neonatal medical wearable time. Wavelet denoising is an appropriate denoising mechanism with low computational complexity that can be implemented on embedded microcontrollers for long-term remote ECG monitoring. Discrete wavelet transform (DWT) denoising for neonatal dry-electrode ECG using different wavelet families is investigated. The wavelet families and mother wavelets used include Daubechies (db1, db2, db3, db4, and db6), symlets (sym5), and coiflets (coif5). Different levels of added white Gaussian noise (AWGN) were added to 19 newborn ECG signals, and denoising was performed to select the appropriate wavelets for neonatal dry-electrode ECG. The selected wavelets then undergo real noise additions of baseline wander and electrode motion to determine their robustness and accuracy. Signal-to-noise ratio (SNR), mean squared error (MSE), and power spectral density (PSD) are used to examine denoising performance. db1, db2, and db3 wavelets are eliminated from analysis where the 30 dB AWGN led to negative SNR improvement for at least one newborn ECG, removing important ECG information. db4 and sym5 are eliminated from selection due to their different waveform morphology compared to the dry-electrode newborn ECG’s QRS complex. db6 and coif5 are selected due to their highest SNR improvement and lowest MSE of 6.26 × 10−6 and 1.65 × 10−7 compared to other wavelets, respectively. Their wavelet shapes are more like a newborn ECG’s QRS morphology, validating their selection. db6 and coif5 showed similar denoising performance, decreasing electrode motion and baseline wander noisy ECG signals by 10 dB and 14 dB, respectively. Further denoising of inherent dry-electrode noise is observed. DWT with coif5 or db6 wavelets is appropriate for denoising newborn dry-electrode ECGs for long-term neonatal dry-electrode ECG monitoring solutions under different noise types. Their similarity to newborn dry-electrode ECGs yields accurate and robust reconstructed denoised newborn dry-electrode ECG signals.

Experimental research on dynamic responses and hydrodynamic pressures of deep-water bridge piers under seismic and wave actions

Earthquakes and extreme waves pose significant threats to the safety of offshore bridges, particularly in deep-water environments. The temporal sequence of earthquakes and waves and the specific type of earthquake are critical factors that contribute to the potential risks involved. This study aimed to fill a key knowledge gap by conducting a comprehensive series of physical tests on a deep-water bridge pier at a scale factor of 1:30. White noise tests, shaking table tests, wavemaker tests, and combined shaking table and wavemaker tests were carried out. Near-field pulsed earthquakes, far-field long-period earthquakes and earthquake-wave phase differences were examined in the tests. The test results indicate that the presence of water has a noticeable impact on the natural frequency, particularly the 2nd-order frequency, of bridge piers, which may significantly influence the dynamic responses of high bridge piers. The effects of water–pier interactions are further intensified by seismic actions such as the pulsed Loma Prieta and long-period Chi-Chi earthquakes. Moreover, the earthquake-induced hydrodynamic pressures on the upstream and downstream faces of the pier are nearly identical, whereas there is a notable difference when the pressures are caused by incident waves. The magnitude of the combined earthquake and wave responses in relation to the individual responses is influenced by the earthquake–wave phase differences, and the combined responses are less than the sum of the individual responses.

Stochastic Roll Dynamics of Smooth and Impacting Vessels in Random Waves

Ship instability is frequently associated with extreme roll motion. In this paper smooth and vibro-impact dynamics of vessels rolling under random wave excitations including nonlinearities is investigated. The vessel is represented by a single degree of freedom roll model. The random excitation due to waves and ice is modeled by a sum of random pulses with Poisson arrival rate. Stochastic P-bifurcation analysis is carried out by examining the variation with parameter of the joint probability density functions (jpdf) of the response computed numerically by the solution of the corresponding Fokker–Planck–Kolmogorov (FPK) equation. The finite element (FE), finite difference (FD), path integral (PI) and radial basis neural (RBN) network formulation methods are used in the solution of the FPK equation. The D-bifurcation analysis is analyzed by computing the largest Lyapunov exponent. The mean upcrossing rate is computed using Rice’s formula. The random wave excitation is also modeled as the response of a second order linear filter to white noise and the random response of the ship is investigated by solution of the corresponding FPK equation of the four dimensional Markov system by the FD method. Non-smooth coordinate transformations like the Zhuravlev and Ivanov transformations are used converting the discontinuous systems to equivalent smooth systems in the impact dynamics analysis. The adaptive time stepping method (ATSM) approach is used to accurately locate the discontinuity point and a Brownian tree approach is used to direct the integration along the correct Brownian path. Different models for nonlinear damping and the restoring moment are considered and their effects on ship stochastic response such as chaotic motion, unbounded rotational motion, impact modulation motion and ship capsizing are investigated.

Stochastic response and stability analysis of nonlinear vehicle energy-regenerative suspension system with time delay

With the rapid development of new energy vehicles, a novel vehicle energy-regenerative suspension for simultaneous vibration suppression and energy harvesting has become one of the key development directions of the next generation of vehicle design. In order to improve the performance of suspension system, the time-delayed feedback control is applied to the novel suspension system subjected to different external excitation. At the same time, there are few studies related to the stochastic dynamics of nonlinear vehicle energy-regenerative suspension system with time delay. This paper studies stochastic response and stability of nonlinear vehicle energy-regenerative suspension with time-delayed feedback control under the excitation of Gaussian white noise. First of all, a stochastic dynamic model of the nonlinear vehicle energy-regenerative suspension with time delay system is established. Secondly, using the stochastic averaging method based on generalized harmonic function, the steady-state probability density function (PDF) of amplitude is obtained. The accuracy of the presented approach is verified through the comparison of analytical and numerical solutions. In addition, through combining the stochastic averaging method with the Khasminskii method, the top Lyapunov exponent (TLE) of the system is obtained, and then the influence of time delay and system parameters on the asymptotic Lyapunov stability is discussed. Finally, the effects of time delay and system parameters on system performance are also studied.

Casimir–Onsager matrix for weakly driven processes

Modeling of physical systems must be based on their suitability to unavoidable physical laws. In this work, in the context of classical, isothermal, finite-time, and weak drivings, I demonstrate that physical systems, driven simultaneously at the same rate in two or more external parameters, must have the Fourier transform of their relaxation functions composing a positive-definite matrix to satisfy the Second Law of Thermodynamics. By evaluating them in the limit of near-to-equilibrium processes, I identify that such coefficients are the Casimir–Onsager ones. The result is verified in paradigmatic models of the overdamped and underdamped white noise Brownian motions. Finally, an extension to thermally isolated systems is made by using the time-averaged Casimir–Onsager matrix, in which the example of the harmonic oscillator is presented.

A Baseline Drift-Elimination Algorithm for Strain Measurement-System Signals Based on the Transformer Model

Strain measurements are vital in engineering trials, testing, and scientific research. In the process of signal acquisition, baseline drift has a significant impact on the accuracy and validity of data. Traditional solutions, such as discrete wavelet transform and empirical mode decomposition, cannot be used in real-time systems. To solve this problem, this paper proposes a Transformer-based model to eliminate the drift in the signal. A self-attentive mechanism is utilized in the encoder of the model to learn the interrelationships between the components of the input signal, and captures the key features. Then, the decoder generates a corrected signal. Meanwhile, a high-precision strain acquisition system is constructed. The experiments tested the model’s ability to remove drift from simulated voltage signals with and without Gaussian noise. The results demonstrated that the transformer model excels at eliminating signal baseline drift. Additionally, the performance of the model was investigated under different temperature conditions and with different levels of force applied by the electronic universal testing machine to produce strain. The experimental results indicate that the Transformer model can largely eliminate drift in dynamic signals l and has great potential for practical applications.

Investigating electromagnetic acoustic emission signals denoising for alloy materials non-destructive detecting: a CRQA method

Aiming at the problem that signals collected from local electromagnetic loading operations are usually mixed with background noises (especially white noise), this paper proposed an electromagnetic acoustic emission signal denoising technology based on cross recurrence quantification analysis (CRQA). Firstly, the decomposition layer and penalty factor of variational mode decomposition (VMD) are set by experience or optimization algorithm, and then the original signal is decomposed. Secondly, the main components are selected by the CRQA algorithm, and the electromagnetic acoustic emission signal after denoising is obtained by superposition reconstruction. The simulation and experimental results show that when 5dB noise is added, CRQA can effectively remove the background noises in electromagnetic acoustic emission signals compared to the correlation coefficient algorithm, and it can assist in realizing the high-precision non-destructive testing of alloy materials.

Dynamic analysis and optimal control of stochastic information cross-dissemination and variation model with random parametric perturbations.

Information dissemination has a significant impact on social development. This paper considers that there are many stochastic factors in the social system, which will result in the phenomena of information cross-dissemination and variation. The dual-system stochastic susceptible-infectious-mutant-recovered model of information cross-dissemination and variation is derived from this problem. Afterward, the existence of the global positive solution is demonstrated, sufficient conditions for the disappearance of information and its stationary distribution are calculated, and the optimal control strategy for the stochastic model is proposed. The numerical simulation supports the results of the theoretical analysis and is compared to the parameter variation of the deterministic model. The results demonstrate that cross-dissemination of information can result in information variation and diffusion. Meanwhile, white noise has a positive effect on information dissemination, which can be improved by adjusting the perturbation parameters.

Amplification in parametrically-driven resonators near instability based on Floquet theory and Green’s functions

Here we use Floquet theory to calculate the response of parametrically-driven time-periodic systems near the onset of parametric instability to an added external ac signal or white noise. We provide new estimates, based on the Green’s function method, for the response of the system in the frequency domain. Furthermore, we present novel expressions for the power and noise spectral densities. We validate our theoretical results by comparing our predictions for the specific cases of a single degree of freedom parametric amplifier and of the parametric amplifier coupled to a harmonic resonator with the numerical integration results and with analytical approximate results obtained via the averaging method up to second order.

Bivariate tempered space-fractional Poisson process and shock models

Abstract We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$ -stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.

Stochastic approach to a delayed in-host model of DENV transmission

Abstract Motivation: The immune response to the viral transmission experiences delays due to diverse biological factors and is affected by several factors. These include infection rate, rate of apoptosis and impact of the antibody-virus complex which exhibit unpredictable nature in a random environment.
Objective: The main goal is to examine the impact of noise terms, introduced into every state variable, on a delayed in-host model of viral transmission.
Methodology: To determine the intensity fluctuations and variances of all concerned state variables in the stochastic delayed model, which includes a constant delay and Gaussian white noise, the Fourier transformation method is employed.
Results and Conclusions: The limiting value of the delay parameter is derived explicitly as well as numerically. The system experiences Hopf bifurcation, whenever the delay parameter crosses the limiting value which is shown graphically. The intensities and variances of different cells and virus populations are derived and the values are computed numerically. As the delay increases, the level of viremia decreases while other parameters maintain fixed values. The outcomes of data sensitivity, shown in graphical form, indicate that the transmission rate and supply rate of viruses are highly sensitive. Our findings suggest that the presence of noise causes fluctuations in the delayed model, leading to a noticeable impact on the transmission of the virus.

Texture-preserving low dose CT image denoising using Pearson divergence

Objective. The mean squared error (MSE), also known as L 2 loss, has been widely used as a loss function to optimize image denoising models due to its strong performance as a mean estimator of the Gaussian noise model. Recently, various low-dose computed tomography (LDCT) image denoising methods using deep learning combined with the MSE loss have been developed; however, this approach has been observed to suffer from the regression-to-the-mean problem, leading to over-smoothed edges and degradation of texture in the image. Approach. To overcome this issue, we propose a stochastic function in the loss function to improve the texture of the denoised CT images, rather than relying on complicated networks or feature space losses. The proposed loss function includes the MSE loss to learn the mean distribution and the Pearson divergence loss to learn feature textures. Specifically, the Pearson divergence loss is computed in an image space to measure the distance between two intensity measures of denoised low-dose and normal-dose CT images. The evaluation of the proposed model employs a novel approach of multi-metric quantitative analysis utilizing relative texture feature distance. Results. Our experimental results show that the proposed Pearson divergence loss leads to a significant improvement in texture compared to the conventional MSE loss and generative adversarial network (GAN), both qualitatively and quantitatively. Significance. Achieving consistent texture preservation in LDCT is a challenge in conventional GAN-type methods due to adversarial aspects aimed at minimizing noise while preserving texture. By incorporating the Pearson regularizer in the loss function, we can easily achieve a balance between two conflicting properties. Consistent high-quality CT images can significantly help clinicians in diagnoses and supporting researchers in the development of AI-diagnostic models.

Noise-based correction for electrical impedance tomography.

Noisy measurements frequently cause noisy and inaccurate images in impedance imaging. No post-processing technique exists to calculate the propagation of measurement noise and use this to suppress noise in the image. The objectives were (1) to develop a post-processing method for noise-based correction (NBC) in impedance tomography, (2) to test whether NBC improves image quality in electrical impedance tomography (EIT), (3) to determine whether it is preferable to use correlated or uncorrelated noise for NBC, (4) to test whether NBC improves image quality with in vivo data and (5) to test whether NBC is stable across model and perturbation geometries. EIT was performed in silico in a 2D homogeneous circular domain and a realistic, heterogeneous 3D human head domain for four perturbations and 25 noise levels. This was validated by performing EIT for four perturbations in a circular, saline tank in 2D and a human head-shaped saline tank with a realistic skull-like layer in 3D. Images were assessed on the weighted spatial variance (WSV) with respect to the true, target image. The effect of NBC was also tested for in vivo EIT data of lung ventilation in a human thorax and cortical activity in a rat brain. On visual inspection, NBC maintained or increased image quality for all perturbations and noise levels in 2D and 3D, both experimentally and in silico. Analysis of the WSV showed that NBC significantly improved the WSV in nearly all cases. When the WSV was inferior with NBC, this was either visually imperceptible or a transformation between noisy reconstructions. For in vivo data, NBC improved image quality in all cases and preserved the expected shape of the reconstructed perturbation. In practice, uncorrelated NBC performed better than correlated NBC and is recommended as a general-use post-processing technique in EIT.

Use of the overexpectation effect to reduce conditioned seeking behavior controlled by nicotine.

Nicotine produces robust stimulus effects that can be conditioned to form associations with reinforcing nondrug stimuli. We examine how established associations to the nicotine stimulus may be weakened via the overexpectation effect. In two experiments, we separately conditioned sucrose associations to the interoceptive nicotine stimulus (0.4 mg/kg, SC) and to a "noisy" exteroceptive contextual stimulus (oscillating houselight and white noise) via the discriminated goal-tracking task. Thereafter, we presented additional sucrose pairings with the nicotine and noisy stimuli, now in compound. Testing of the conditioned goal-tracking evoked by the nicotine and noisy stimuli in isolation-before versus after compound conditioning (Experiment 1) or between treatment and control groups (Experiment 2)-demonstrated an attenuation of conditioned responding via the overexpectation effect. We suggest that applications of the overexpectation effect may provide some promise for treatments seeking to attenuate drug-evoked conditioned responses in situations where extinction-based interventions are not suitable.

Terahertz Nondestructive Measurement of Heat Radiation Performance of Thermal Barrier Coatings Based on Hybrid Artificial Neural Network

Effective control of the micro- and nanostructure of thermal barrier coatings is essential to enhance the thermal radiation performance of the coating, which helps to determine the remaining service life of the coating. This paper proposed a method to measure the radiation properties of thermal barrier coatings by terahertz nondestructive testing technique, using APS-prepared thermal barrier coatings as the object of study. Radiative properties were a comprehensive set of properties characterized by the diffuse reflectance, transmittance, and absorptance of the thermal barrier coating. The coating data in actual service were obtained by scanning electron microscopy and metallographic experiments, and the data were used as the simulation model critical value. The terahertz time-domain simulation data of coatings with different microstructural features were obtained using the finite-different time-domain (FDTD) method. In simulating the real test signals, white noise with a signal-to-noise ratio of 20 dB was added, and fast Fourier transform (FFT), short-time Fourier transform (STFT), and wavelet transform (WT) were used to reduce the noise and compare their noise reduction effects. Different machine learning methods were used to build the model, including support vector machine algorithm (SVM) and k-nearest neighbor algorithm (KNN). The principal component algorithm (PCA) was used to reduce the dimensionality of terahertz time-domain data, and the SVM algorithm and KNN algorithm were optimized using the particle swarm optimization algorithm (PSO) and the ant colony optimization algorithm (ACO), respectively, to improve the robustness of the system. The K-fold cross-validation method was used to construct the model to improve the adaptability of the model. It could be clearly seen that the novel hybrid PCA-ACO-SVM model had superior prediction performance. Finally, this work proposed a novel, convenient, nondestructive, online, safe and highly accurate method for measuring the radiation performance of thermal barrier coatings, which could be used for the judgment of the service life of thermal barrier coatings.

Multifractal Behavior of Cryptocurrencies During Periods of Economic Uncertainty

Background: In recent years, investors' interest in cryptocurrencies has increased due to their notable price volatility and rapid price increases. These investors view cryptocurrencies as suitable financial assets for portfolio rebalancing strategies. Purpose: The main objective of this study is to examine the multifractality of the cryptocurrencies Bitcoin (BTC), Lisk (LSK), Quantum (QUA), Litecoin (LTC), Ripple (XRP), Augur (REP), Darkcoin (DASH), EOS, IOTA (MIOTA). Methods: The Detrended Fluctuation Analysis (DFA) econophysics model supports the methodology. Results: The results suggest that during the 2020 pandemic period, the digital currencies LSK, QUA, MIOTA, XRP, REP, BTC, ETH, LTC and DASH showed very significant persistence, indicating that price formation is not random. However, validating that cryptocurrency prices are predictable based on historical time series was impossible. On the other hand, the digital currency EOS proved to be in equilibrium; in other words, price formation follows the random walk pattern, suggesting that prices are not autocorrelated over time. During the 2022 geopolitical conflict, long-term memory patterns shifted significantly towards short-term memories, i.e. anti-persistence. The digital currencies ETH, MIOTA, EOS, LTC, REP, LSK and DASH showed anti-persistence slopes, indicating that prices were less influenced by past events and more by recent events. On the other hand, the cryptocurrencies BTC (0.50), QUA (0.50), and XRP (0.50) demonstrate that prices contain a significant random component and that the residuals are independent and identically distributed (i.i.d.), supporting the idea that white noise might be present. Conclusion: From a risk management perspective, these findings are highly relevant to investors, traders and market participants.

The effect of the AIM2 inflammasome in noise-induced cognitive dysfunction in rats

Objective: To explore the effect of the absent in melanoma 2 (AIM2) -mediated neuroinflammation in noise-induced cognitive dysfunction in rats. Methods: In April 2023, sixteen male Wistar rats were randomly divided into control group and noise group, with 8 rats in each group. The rats in the noise group were placed in 50 cm×50 cm×40 cm transparent boxes and exposed to 100 dB (A) white noise with a sound pressure level of 100 dB (A) (4 h/d for 30 d) . At the same time, rats in the control group were kept in similar boxes with environmental noise less than 60 dB (A) . After 30 days of noise exposure, the Morris water maze experiment was applied to test the learning and memory abilities of the rats; the pathological morphology of hippocampal tissues was observed by Hematoxylin-Eosin (HE) staining. Western blot was used to detect the protein expression levels of AIM2, cysteinyl aspartate specific proteinase-1 (caspase-1) , apoptosis-associated speck-like protein (ASC) , interleukin-1β (IL-1β) , IL-18, ionic calcium-binding articulation molecule-1 (Iba-1) , and glial fibrillary acidic protein (GFAP) . The expression of both Iba-1 and GFAP in hippocampal tissue was assessed by immunohistochemical staining. The co-localization of AIM2 with Iba-1 or GFAP was determined by immunofluorescence double staining. Results: Compared with the control group, the escape latency of rats in the noise group was increased by 16.29 s, 17.71 s, and 20.26 s on days 3, 4, and 5, respectively. On day 6, the noise-exposed rats spent shorter time in the target quadrant and had fewer times in crossing the platform[ (7.25±2.27) s and (1.13±0.64) times] than the control group[ (15.64±3.99) s and (4.25±2.12) times] (P<0.05) . After noise exposure, hippocampal neurons of rats displayed marked nuclear hyperchromatic and pyknosis phenomenon. The noise-exposed rats had higher numbers of both microglia and astrocytes (27.00±2.65 and 43.33±5.51) in the DG area of the hippocampus relative to the control group (14.67±3.06 and 20.00±4.58) (P<0.05) . Moreover, the glial cells in the noise group had larger cell cytosol with more and thicker branches. The protein expression levels of inflammatory cytokines Cleaved-IL-1β and Cleaved-IL-18 in the hippocampus of rats in the noise group (1.55±0.19 and 1.74±0.12) were significantly higher than the control group (1.00±0.11 and 1.00±0.13) (P<0.05) . After noise exposure, the protein expression levels of AIM2, Cleaved-Caspase-1 and ASC (1.19±0.09, 1.34±0.07 and 1.14±0.01) were higher than the control group (1.00±0.07, 1.00±0.14 and 1.00±0.06) and differences between the two groups were statistically significant (P<0.05) . A significant increase in the number of cells co-localizing AIM2 with Iba-1 or GFAP in the noise group (28.67±4.04 and 40.67±5.13) compared with the control group (15.67±4.04 and 17.67±3.79) , and statistically significant differences were observed between the two groups (P<0.05) . Conclusion: Noise exposure may activate the AIM2 inflammasome in hippocampal glial cells of rats, releasing excessive inflammatory cytokines and causing neuroinflammation that damages neurons.

PSD and Cross-PSD of Responses of Seven Classes of Fractional Vibrations Driven by fGn, fBm, Fractional OU Process, and von Kármán Process

This paper gives its contributions in four stages. First, we propose the analytical expressions of power spectrum density (PSD) responses and cross-PSD responses to seven classes of fractional vibrators driven by fractional Gaussian noise (fGn). Second, we put forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by fractional Brownian motion (fBm). Third, we present the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators driven by the fractional Ornstein–Uhlenbeck (OU) process. Fourth, we bring forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by the von Kármán process. We show that the statistical dependences of the responses to seven classes of fractional vibrators follow those of the excitation of fGn, fBm, the OU process, or the von Kármán process. We also demonstrate the obvious effects of fractional orders on the responses to seven classes of fractional vibrations. In addition, we newly introduce class VII fractional vibrators, their frequency transfer function, and their impulse response in this research.