Weak Periodic Signal Research Articles

Delay induced steady-state transition and stochastic resonance for an ecological vegetation growth system subjected to multiplicative and additive noises

In this paper, our aim is to investigate the steady state properties and stochastic resonance (SR) phenomenon for an ecological vegetation growth system with time delay induced by the multiplicative and additive noises. Numerical results show that the SR phenomenon caused by time delay, different noise terms and a weak periodic signal occurs in the vegetation growth model under different values of system parameters. With regard to the stationary state properties of the vegetation system, the results indicate that the terms of different noises and time delay can all accelerate the shift from the substantial state to the barren one of the ecological system, restrain the development of the vegetation system and weaken the stability of the ecological system. On the other hand, the additive noise strength always enhances the SNR and the SR phenomenon, while the intensity of multiplicative noise often reduces the effect of the SR. In particular, time delay can play different roles in exciting the SR phenomenon in different cases.

A new kind of metal detector based on chaotic oscillator

The sensitivity of a metal detector greatly depends on the identification ability to weak signals from the probe. In order to improve the sensitivity of metal detectors, this paper applies the Duffing chaotic oscillator to metal detectors based on its characteristic which is very sensitive to weak periodic signals. To make a suitable Duffing system for detectors, this paper computes two Lyapunov characteristics exponents of the Duffing oscillator, which help to obtain the threshold of the Duffing system in the critical state accurately and give quantitative criteria for chaos. Meanwhile, a corresponding simulation model of the chaotic oscillator is made by the Simulink tool box of Matlab. Simulation results shows that Duffing oscillator is very sensitive to sinusoidal signals in high frequency cases. And experimental results show that the measurable diameter of metal particles is about 1.5mm. It indicates that this new method can feasibly and effectively improve the metal detector sensitivity.

Stochastic resonance energy harvesting for a rotating shaft subject to random and periodic vibrations: influence of potential function asymmetry and frequency sweep

Stochastic resonance is referred to as a physical phenomenon that is manifest in nonlinear systems whereby a weak periodic signal can be significantly amplified with the aid of inherent noise or vice versa. In this paper, stochastic resonance is considered to harvest energy from two typical vibrations in rotating shafts: random whirl vibration and periodic stick-slip vibration. Stick-slip vibrations impose a constant offset in centrifugal force and distort the potential function of the harvester, leading to potential function asymmetry. A numerical analysis based on a finite element method was conducted to investigate stochastic resonance with potential function asymmetry. Simulation results revealed that a harvester with symmetric potential function generates seven times higher power than that with asymmetric potential function. Furthermore, a frequency-sweep analysis also showed that stochastic resonance has hysteretic behavior, resulting in frequency difference between up-sweep and down-sweep excitations. An electromagnetic energy harvesting system was constructed to experimentally verify the numerical analysis. In contrast to traditional stochastic resonance harvesters, the proposed harvester uses magnetic force to compensate the offset in the centrifugal force. System identification was performed to obtain the parameters needed in the numerical analysis. With the identified parameters, the numerical simulations showed good agreement with the experiment results with around 10% error, which verified the effect of potential function asymmetry and frequency sweep excitation condition on stochastic resonance. Finally, attributed to compensating the centrifugal force offset, the proposed harvester generated nearly three times more open-circuit output voltage than its traditional counterpart.

Stochastic resonance in an underdamped system with FitzHug-Nagumo potential for weak signal detection

Vibration signals are widely used for bearing fault detection and diagnosis. When signals are acquired in the field, usually, the faulty periodic signal is weak and is concealed by noise. Various de-noising methods have been developed to extract the target signal from the raw signal. Stochastic resonance (SR) is a technique that changed the traditional denoising process, in which the weak periodic fault signal can be identified by adding an expression, the potential, to the raw signal and solving a differential equation problem. However, current SR methods have some deficiencies such us limited filtering performance, low frequency input signal and sequential search for optimum parameters. Consequently, in this study, we explore the application of SR based on the FitzHug-Nagumo (FHN) potential in rolling bearing vibration signals. Besides, we improve the search of the SR optimum parameters by the use of particle swarm optimization (PSO). The effectiveness of the proposed method is verified by using both simulated and real bearing data sets.

Detecting impact signal in mechanical fault diagnosis under chaotic and Gaussian background noise

Abstract In actual fault diagnosis, useful information is often submerged in heavy noise, and the feature information is difficult to extract. Traditional methods, such like stochastic resonance (SR), which using noise to enhance weak signals instead of suppressing noise, failed in chaotic background. Neural network, which use reference sequence to estimate and reconstruct the background noise, failed in white Gaussian noise. To solve these problems, a novel weak signal detection method aimed at the problem of detecting impact signal buried under heavy chaotic and Gaussian background noise is proposed. First, the proposed method obtains the virtual reference sequence by constructing the Hankel data matrix. Then an M-order optimal FIR filter is designed, which can minimize the output power of background noise and pass the weak periodic signal undistorted. Finally, detection and reconstruction of the weak periodic signal are achieved from the output SBNR (signal to background noise ratio). The simulation shows, compared with the stochastic resonance (SR) method, the proposed method can detect the weak periodic signal in chaotic noise background while stochastic resonance (SR) method cannot. Compared with the neural network method, (a) the proposed method does not need a reference sequence while neural network method needs one; (b) the proposed method can detect the weak periodic signal in white Gaussian noise background while the neural network method fails, in chaotic noise background, the proposed method can detect the weak periodic signal under a lower SBNR (about 8–17 dB lower) than the neural network method; (c) the proposed method can reconstruct the weak periodic signal precisely.

Stochastic resonance with multiplicative heavy-tailed Lévy noise: Optimal tuning on an algebraic time scale

Stochastic resonance is an amplification and synchronization effect of weak periodic signals in nonlinear systems through a small noise perturbation. In this paper we study the dynamics of stochastic resonance in a bistable system driven by multiplicative Lévy noise with heavy tails, e.g., [Formula: see text]-stable Lévy noise. We determine the optimal tuning with respect to a probabilistic synchronization measure for both the jump-diffusion and the reduced two-state Markov chain. These results extend the theory of stochastic resonance to the case of heavy tail Lévy perturbations.

Stochastic resonance in a fractal dimensional bistable system

A fractal dimensional bistable system driven by multiplicative and additive noises and a periodic signal is investigated. We have derived analytically the fractal Fokker–Planck equation of the system, and obtained exact expression of its signal-to-noise ratio (SNR). Numerical results indicate that: (1) The curve of the SNR as a function of multiplicative noise intensity D or additive noise intensity Q exhibits a peak in the fractal dimensional system, i.e., a stochastic resonance phenomenon; (2) For the smaller values of D, the SNR first decreases then increases with increment of dimensionality \(\alpha \). At the integer dimension of \(\alpha =1\), response of the system to the weak periodic signal displays a minimum. Yet the SNR increases monotonically for the greater values of D. Our further investigation shows that the height of the potential barrier depends on the dimensionality, and influences on the SNR of the system.

Stochastic resonance of a tri-stable system with α stable noise

A tri-stable system excited by weak periodic signal is taken as a model and the stochastic resonance phenomenon is investigated by additive α stable noise in this paper. The laws for the resonance system parameters q, p, skewness parameter r and intensity amplification factor Q of α stable noise to act on the resonant output are explored under different stability indicies α and asymmetric skewness β of α stable noise. The results indicate that a weak signal can be realized by tuning the system parameters q, p and r under the joint action of additive α stable noise, and the interval of q and p which can induce stochastic resonance does not change with α or β. Moreover, a certain rule is found in which adjusting the intensity amplification factor Q of α stable noise can also realize a synergistic effect when studying the noise-induced stochastic resonance, and the interval of Q does not change with α or β; the best value of the characteristic index is α=1 under any system parameter, and the best value of the symmetry parameter is β=1 under any system parameter. So, the system performance is best when α=1 and β=1. The results will contribute to a reasonable selection of parameter-induced stochastic resonance system parameters and noise intensity of noise-induced stochastic resonance under α stable noise.

Detection the nonlinear ultrasonic signals based on modified Duffing equations

The nonlinear ultrasonic signals, like second harmonic generation (SHG) signals, could reflect the nonlinearity of material induced by fatigue damage in nonlinear ultrasonic technique which are weak nonlinear signals and usually submerged by strong background noise. In this paper the modified Duffing equations are applied to detect the SHG signals relating to the fatigue damage of material. Due to the Duffing equation could only detect the signal with specific frequency and initial phase, firstly the frequency transformation is carried on the Duffing equation which could detect the signal with any frequency. Then the influence of initial phases of to-be-detected signal and reference signal on the detection result is studied in detail, four modified Duffing equations are proposed to detect actual engineering signals with any initial phase. The relationship between the response amplitude and the total driving force is applied to estimate the amplitude of weak periodic signal. The detection results show the modified Duffing equations could effectively detect the second harmonic in SHG signals. When the SHG signals include strong background noise, the noise doesn’t change the motion state of Duffing equation and the second harmonic signal could be detected until the SNR of noisy SHG signals are −26.3, yet the frequency spectrum method could only identify when the SNR is greater than 0.5. When estimation the amplitude of second harmonic signal, the estimation error of Duffing equation is obviously less than the frequency spectrum analysis method under the same noise level, which illustrates the Duffing equation has the noise immune capacity. The presence of the second harmonic signal in nonlinear ultrasonic experiments could provide an insight about the early fatigue damage of engineering components.

Stochastic resonance in overdamped washboard potential system

Brownian motion in a washboard potential has practical significance in investigating a lot of physical problems such as the electrical conductivity of super-ionic conductor, the fluctuation of super-current in Josephson junction, and the ad-atom motion on crystal surface. In this paper, we study the overdamped motion of a Brownian particle in a washboard potential driven jointly by a periodic signal and an additive Gaussian white noise. Since the direct simulation about stochastic system is always time-consuming, the purpose of this paper is to introduce a simple and useful technique to study the linear and nonlinear responses of overdamped washboard potential systems. In the limit of a weak periodic signal, combining the linear response theory and the perturbation expansion method, we propose the method of moments to calculate the linear response of the system. On this basis, by the Floquet theory and the non-perturbation expansion method, the method of moments is extended to calculating the nonlinear response of the system. The long time ensemble average and the spectral amplification factor of the first harmonic calculated from direct numerical simulation and from the method of moments demonstrate that they are in good agreement, which shows the validity of the method we proposed. Furthermore, the dependence of the spectral amplification factor at the first three harmonics on the noise intensity is investigated. It is observed that for appropriate parameters, the curve of the spectral amplification factor versus the noise intensity exhibits a peaking behavior which is a signature of stochastic resonance. Then we discuss the influences of the bias parameter and the amplitude of the periodic signal on the stochastic resonance. The results show that with the increase of the bias parameter in a certain range, the peak value of the resonance curve increases and the noise intensity corresponding to the resonance peak decreases. With the increase of the driven amplitude, comparing the changes of the resonance curves, we can conclude that the effect of stochastic resonance becomes more prominent. At the same time, by using the mean square error as the quantitative indicator to compare the difference between the results obtained from the method of moments and from the stochastic simulation under different signal amplitudes, we find that the method of moments is applicable when the amplitude of the periodic signal is lesser than 0.25.

Novel synthetic index-based adaptive stochastic resonance method and its application in bearing fault diagnosis

Stochastic resonance (SR), which is characterized by the fact that proper noise can be utilized to enhance weak periodic signals, has been widely applied in weak signal detection. SR is a nonlinear parameterized filter, and the output signal relies on the system parameters for the deterministic input signal. The most commonly used index for parameter tuning in the SR procedure is the signal-to-noise ratio (SNR). However, using the SNR index to evaluate the denoising effect of SR quantitatively is insufficient when the target signal frequency cannot be estimated accurately. To address this issue, six different indexes, namely, power spectral kurtosis of the SR output signal, correlation coefficient between the SR output and the original signal, peak SNR, structural similarity, root mean square error, and smoothness, are constructed in this study to measure the SR output quantitatively. These six quantitative indexes are fused into a new synthetic quantitative index (SQI) via a back propagation neural network to guide the adaptive parameter selection of the SR procedure. The index fusion procedure reduces the instability of each index and thus improves the robustness of parameter tuning. In addition, genetic algorithm is utilized to quickly select the optimal SR parameters. The efficiency of bearing fault diagnosis is thus further improved. The effectiveness and efficiency of the proposed SQI-based adaptive SR method for bearing fault diagnosis are verified through numerical and experiment analyses.

Harvesting wind energy to detect weak signals using mechanical stochastic resonance

Wind is free and ubiquitous and can be harnessed in multiple ways. We demonstrate mechanical stochastic resonance in a tabletop experiment in which wind energy is harvested to amplify weak periodic signals detected via the movement of an inverted pendulum. Unlike earlier mechanical stochastic resonance experiments, where noise was added via electrically driven vibrations, our broad-spectrum noise source is a single flapping flag. The regime of the experiment is readily accessible, with wind speeds ∼20 m/s and signal frequencies ∼1 Hz. We readily obtain signal-to-noise ratios on the order of 10 dB.

Stochastic resonance and stability for an ecological vegetation growth system driven by colored noises and multiplicative signal

In this paper, we investigate the steady-state properties and the transition rate for an ecological vegetation growth system induced by the terms of the colored multiplicative and additive noises. Numerical results indicate that the multiplicative noise and the additive one can reduce the stability of the ecological system and slow down the development velocity of the vegetation, while two noise self-correlation times can increase the stability of the system and speed up the expansion process of the vegetation system. With respect to the stochastic resonance (SR) phenomenon caused by noise terms and a multiplicative weak periodic signal, the results show that the additive noise always enhances the SR effect, two noise self-correlation time terms can produce SR phenomenon, but play opposite roles in enhancing or inhibiting the SR effect under different parameter conditions. In particular, the two self-correlation times can keep up the maximum of the signal-to-noise ratio (SNR) invariant in specific situations. Analogously, the multiplicative noise can not only improve the SNR, but also restrain the SR phenomenon in different cases.

Periodic fault signal enhancement in rotating machine vibrations via stochastic resonance

This paper proposes a novel approach to periodic fault signal enhancement in rotating machine vibrations with a tristable mechanical vibration amplifier (TMVA) by exploiting stochastic resonance (SR). The TMVA is a nonlinear physical structure system that consists of a cantilever beam and a magnet system. Through the TMVA, the periodic weak signal can be amplified with the assistance of noise in the regime of SR. Benefitting from a wider interwell spacing and a smoother potential curve, the TMVA produces a more regular output waveform with lower noise in a wider operating bandwidth as compared to the monostable and bistable amplifiers. Different from the traditional signal enhancement approach which is based on digital signal processing (DSP) techniques, the designed physical structure can realize signal enhancement in a simple, intuitive, effective and adaptive way without too much complex operations. The effectiveness and efficiency of the proposed approach are validated by a simulated fault signal and the practical bearing and gearbox fault signals, in comparison with a traditional DSP-based SR method. The principle of the proposed approach shows potential applications on rotating machine fault diagnosis area and other areas related to weak periodic signal enhancement.

Detecting weak periodic signals in EEG time series

Recently many attempts had been devoted to the detection of weak periodic signals embedded in noisy chaotic signals using simulation methods and Duffing Oscillator System. In this paper, weak periodic signals in electroencephalogram (EEG) time series are investigated using a Duffing Equation based method. The EEG signals are collected from FR1-F3 and FR2-F4 channels in the scalp of an epileptic patient. In order to excite the Duffing equation by outer signals, the data is scaled down to a compatible range with the equation by multiplying it with a constant. After running the data with the Duffing Equation, weak periodic signals are obtained. As a result, the presence of weak periodic signals is observed in EEG time series. The amplitude and frequency values of the found weak periodic signals are calculated. Presence of weak periodic signals within the EEG signals can give us insight about the dynamic structure of the brain and functional brain activity.

Estimation of Combustion Engine Rotation Speed Based on Vibration Signal Analysis

Diagnostics of mechanisms and systems allows estimating their current state and taking measures if necessary to carry out preventive or other actions for elimination of possible malfunctions. Diagnostics of mechanical parts of combustion engines, as well as other mechanical devices, mechanisms and systems as a whole, is the relevant objective. The purpose of the paper is to research the possible use of time-frequency autocorrelated function to analyze the combustion engine vibration signals to determine possible defects in its work. The paper introduces the basic possible definition of engine operation processes characteristics, being accompanied by emergence of additional frequency components in a spectrum of engine vibrosignal. The time-frequency correlation analysis allows receiving out of spectrum of engine vibrosignal additional information on weak periodic signals characterizing existence of any malfunctions of its mechanical part.

Stochastic resonance based on frequency information exchange

In the past few decades, stochastic resonance (SR) has attracted considerable attention of researchers due to a curious phenomenon appearing in a nonlinear system:an input weak periodic signal can be amplified and optimized by the assistance of noise. It has been proved that the classical stochastic resonance (CSR) has the adiabatic limit, so the performance of CSR in high-frequency signal detection is restricted in practical engineering. To break the restriction, a number of methods have been suggested, such as re-scaling frequency stochastic resonance (RFSR), parameters normalized stochastic resonance, modulated stochastic resonance, etc. Although the high-frequency signal can be detected by the above methods in specific conditions, there are some problems that restrict their applications in different circumstances. In this paper, a new method, stochastic resonance based on frequency-information exchange (FIESR), is developed to deal with the adiabatic limit of CSR. The mechanism of FIESR is analyzed in detail by the theory of single-side band modulation (SSB) which is based on phase shift. The information in small-parameter frequency domain is swapped with the information of the high-frequency target signal. Then the amplitude and phase of the target signal are moved to the small-parameter frequency domain. Consequently the target signal can be enhanced and detected by CSR in small-parameter frequency domain. Besides, a necessary plan, narrow band spectrum exchange, is put forward to diminish the influence of the spectrum leakage of FIESR. It is well known that the RFSR is a method of detecting the practical signal with large-parameter frequency. Through rescaling the time interval of the signal and compressing its frequency according to the scale R, the large-parameter frequency is compressed into a small-parameter frequency. The RFSR has a good performance in mechanical incipient fault diagnosis. However, it has a high sampling ratio limitation. The ratio of sampling frequency to target signal frequency is more than 50. To overcome this weakness of RFSR, frequency-information exchange (FIE) is introduced into RFSR. A new signal detection method based on FIE and RFSR, named F-RFSR, is put forward simultaneously. The flow of F-RFSR consists of three steps. Firstly, the frequency of the original input signal is compressed linearly according to the estimated scale. Then, the frequency information is exchanged between the compressed target signal and the small-parameter signal in the frequency domain. Finally, the CSR is used to amplify and detect the weak target signal processed by re-scaling frequency and FIE. Performance analysis of signal detection and numerical simulation are carried out to demonstrate that F-RFSR has more efficient sampling ratio than RFSR for practical application.

Stochastic resonance of a linear harmonic oscillator with non-linear damping fluctuation

Although non-linear noise exists far more widely in actual systems than linear noise, the study on non-linear noise is far from meeting the needs of practical situations as yet. The phenomenon of stochastic resonance (SR) is a non-linear cooperative effect which is jointly produced by signal, noise, and system, obviously, it is closely related to the nature of the noise. As a result, the non-linear nature of the non-linear noise has an inevitable impact on the dynamic behavior of a system, so it is of great significance to study the non-linear noise's influence on the dynamic behavior of the system. The linear harmonic oscillator is the most basic model to describe different phenomena in nature, and the quadratic noise is the most basic non-linear noise. In this paper, we consider a linear harmonic oscillator driven by an external periodic force and a quadratic damping fluctuation. For the proposed model, we focus on the effect of non-linear nature of quadratic fluctuation on the system's resonant behavior. Firstly, by the use of the Shapiro-Loginov formula and the Laplace transform technique, the analytical expressions of the first moment and the steady response amplitude of the output signal are obtained. Secondly, by studying the impacts of noise parameters and system intrinsic frequency, the non-monotonic behaviors of the steady response amplitude are found. Finally, numerical simulations are presented to verify the effectiveness of the analytical result. According to the research, we have the following conclusions: (1) The steady response amplitude is a non-monotonic function of coefficients of the quadratic damping fluctuation. Furthermore, the non-linear damping fluctuation is easier to contribute the system's enhancing response to the external periodic signal than the linear fluctuation. (2) The evolution of the steady response amplitude versus noise intensity presents more resonant behaviors. One-peak SR phenomenon and double-peak SR phenomenon are observed at different values of coefficients of the quadratic noise, particularly, the SR phenomenon disappears at the positive quadratic coefficient of the quadratic noise. (3) The evolution of the steady response amplitude versus the system intrinsic frequency presents true resonance, i. e. the phenomenon of resonance appears when the external signal frequency is equal to the system intrinsic frequency. True resonance is not observed in the linear harmonic oscillator driven by a linear damping fluctuation as yet. In conclusion, all the researches show that the non-linear nature of non-linear noise plays a key role in system's resonant behavior, in addition, the non-linear damping fluctuation is conductive to the detection and frequency estimation of weak periodic signal.

Autaptic pacemaker mediated propagation of weak rhythmic activity across small-world neuronal networks

We study the effects of an autapse, which is mathematically described as a self-feedback loop, on the propagation of weak, localized pacemaker activity across a Newman–Watts small-world network consisting of stochastic Hodgkin–Huxley neurons. We consider that only the pacemaker neuron, which is stimulated by a subthreshold periodic signal, has an electrical autapse that is characterized by a coupling strength and a delay time. We focus on the impact of the coupling strength, the network structure, the properties of the weak periodic stimulus, and the properties of the autapse on the transmission of localized pacemaker activity. Obtained results indicate the existence of optimal channel noise intensity for the propagation of the localized rhythm. Under optimal conditions, the autapse can significantly improve the propagation of pacemaker activity, but only for a specific range of the autaptic coupling strength. Moreover, the autaptic delay time has to be equal to the intrinsic oscillation period of the Hodgkin–Huxley neuron or its integer multiples. We analyze the inter-spike interval histogram and show that the autapse enhances or suppresses the propagation of the localized rhythm by increasing or decreasing the phase locking between the spiking of the pacemaker neuron and the weak periodic signal. In particular, when the autaptic delay time is equal to the intrinsic period of oscillations an optimal phase locking takes place, resulting in a dominant time scale of the spiking activity. We also investigate the effects of the network structure and the coupling strength on the propagation of pacemaker activity. We find that there exist an optimal coupling strength and an optimal network structure that together warrant an optimal propagation of the localized rhythm.

Analysis of Chaotic Resonance in Izhikevich Neuron Model.

In stochastic resonance (SR), the presence of noise helps a nonlinear system amplify a weak (sub-threshold) signal. Chaotic resonance (CR) is a phenomenon similar to SR but without stochastic noise, which has been observed in neural systems. However, no study to date has investigated and compared the characteristics and performance of the signal responses of a spiking neural system in some chaotic states in CR. In this paper, we focus on the Izhikevich neuron model, which can reproduce major spike patterns that have been experimentally observed. We examine and classify the chaotic characteristics of this model by using Lyapunov exponents with a saltation matrix and Poincaré section methods in order to address the measurement challenge posed by the state-dependent jump in the resetting process. We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state. In order to assess the signal responses of CR in these classified states, we introduced an extended Izhikevich neuron model by considering weak periodic signals, and defined the cycle histogram of neuron spikes as well as the corresponding mutual correlation and information. Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals. Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.