† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 60551002) and the Natural Science Foundation of Hunan Province, China (Grant No. 2018JJ3680).
Based on adiabatic approximation theory, in this paper we study the asymmetric stochastic resonance system with time-delayed feedback driven by non-Gaussian colored noise. The analytical expressions of the mean first-passage time (MFPT) and output signal-to-noise ratio (SNR) are derived by using a path integral approach, unified colored-noise approximation (UCNA), and small delay approximation. The effects of time-delayed feedback and non-Gaussian colored noise on the output SNR are analyzed. Moreover, three types of asymmetric potential function characteristics are thoroughly discussed. And they are well-depth asymmetry (DASR), well-width asymmetry (WASR), and synchronous action of well-depth and well-width asymmetry (DWASR), respectively. The conclusion of this paper is that the time-delayed feedback can suppress SR, however, the non-Gaussian noise deviation parameter has the opposite effect. Moreover, the correlation time plays a significant role in improving SNR, and the SNR of asymmetric stochastic resonance is higher than that of symmetric stochastic resonance. Our experiments demonstrate that the appropriate parameters can make the asymmetric stochastic resonance perform better to detect weak signals than the symmetric stochastic resonance, in which no matter whether these signals have low frequency or high frequency, accompanied by strong or weak noise.
Weak signal detection is a cutting-edged methodology in which the modern electronics and information-processing methods are used to detect weak signals in strong background noise environment.[1–5] Since the 1940s, weak signal detection has been widely used in relevant theoretical researches and practical applications. As a new method different from the previous weak signal detection, the stochastic resonance (SR) has made great progress since the quaternary global meteorological glaciers was studied by Benzi et al. in 1981.[6] The SR studies mainly focus on bistable stochastic resonance but also including monostable stochastic resonance[7–9] and tristable stochastic resonance.[10–12] The SR of this paper is referred to as the bistable stochastic resonance.
The previous studies of SR were generally based on the symmetric potential function,[13–15] but there were few studies based on asymmetric potential functions, and most of them are asymmetric systems in real natural systems. So studying asymmetric systems is another key point in the development of SR theory. The traditional SR theory is mainly applied to the researches of the symmetric potential function. And then, Wio and Bouzat[16] extended the adiabatic approximation theory to the asymmetric potential function. In the adiabatic approximation theory, a transition between two wells is the reciprocal of the MFPT. Subsequently, some researchers studied the effect of asymmetry on the SR generated under multiplicative noise. Li[17] found that the asymmetry is negatively correlated with SR. Owing to the fact that different physical systems are affected by different noise sources and Gaussian white noise does not exist in nature, it is especially necessary to study the SR driven by different noises. Fuentes et al.[18] investigated the effect of non-Gaussian colored noise on SR phenomenon. The path integral approach[19–23] and the UCNA[24] are used to convert the non-Gaussian colored noise into Gaussian white noise, thus obtaining a Fokker–Planck equation and an output SNR analytical expression. Wang et al.[25] discussed in detail the effect of the correlation between two multiplicative noises and an additive noise on the MFPT and found that it possesses rich transition phenomena. Dybiech and Nowak[26] studied the generic double-well potential model perturbed by the alpha-stable Lévy-type noises. To freely adjust the width and depth of the barrier, Liu and Cao[27] designed a piecewise potential function to analyze the effect of the asymmetry of the well on the SNR in the background of non-Gaussian colored noise.
Since time lags are widespread in nature, the study of SR with time-delayed feedback conforms to the developing trend.[28–34] Shi et al.[28] derived the expression of the steady-state probability density function (SPD) and SNR of SR with time-delayed feedback, suggesting that the delay time and feedback strength of the time-delayed term can enhance the SNR. Introducing time-delayed terms into the potential function can reduce the SNR, but the degradation reaction has the opposite effect.[29] According to the time-delayed feedback and Gaussian white noise, Tang et al.[30] proposed an asymmetric SR detection method. This method overcomes the shortcomings of the traditional SR, which has a long convergence time and tends to easily fall into local optimization. Therefore, the detection of weak fault features is improved. Liu and Wang[31] discussed the effects of additive white Gaussian noise and time-delayed feedback on the SR under three asymmetric potential functions. Tan et al.[32] investigated the effects of additive colored noise and time-delayed feedback on two asymmetric systems. For asymmetric systems, they analyzed the effects of noise and time-delayed feedback on SR performance in detail, respectively. However, the effect of asymmetric SR under the combination of non-Gaussian colored noise and time-delayed feedback can be further studied.
In this paper, we study an SR with time-delayed feedback in the context of non-Gaussian colored noise. The rest of this paper is organized as follows. In Section
The classical bistable stochastic resonance (CSR) model can be represented by the Langevin equation
![](cpb_29_5_050501/cpb_29_5_050501_ieqn1.gif)
![](cpb_29_5_050501/cpb_29_5_050501_ieqn2.gif)
![](cpb_29_5_050501/cpb_29_5_050501_ieqn3.gif)
![](cpb_29_5_050501/cpb_29_5_050501_ieqn4.gif)
Based on the asymmetric bistable potential function of the previous section, adding the non-Gaussian colored noise and time-delayed terms, the Langevin equation can be written as follows:
The parameter τ denotes the correlation time of the non-Gaussian noise and q is a measure of deviation of the non-Gaussian noise Γ(t) from the Gaussian distribution. When parameter q ≠ 1, Γ(t) is the non-Gaussian noise. Otherwise, the Γ(t) becomes the Gaussian colored noise with the correlation time τ and the noise intensity D. If τ = 0, Γ (t) is reduced to white noise, η(t) is the Gaussian white noise (〈 η(t)〉 = 0,〈 η(t)η(t′)〉 = 2Dδ(t – t′)), and D is the noise intensity. The path integral approach[19,35] is used to convert the non-Gaussian colored noise Γ(t) into a Gaussian colored noise with effective noise correlation time τ0 and effective noise intensity D0 when | q – 1 | ≪ 1,
Applying the UCNA[24] to Eq. (
The non-Gaussian noise in Eq. (
After analysis, the MFPT[19] is obtained as follows:
Using the adiabatic approximation theory, we expand transition rates W± (t) up to the first order of A cos (ω t)
Therefore, the output SNR of the asymmetric SR with non-Gaussian colored noise and time-delayed feedback can be deduced as follows:
To analyze and compare the effects of non-Gaussian colored noise and time-delay terms on the SNR between the asymmetric SR system and symmetric SR system, the system parameters a = b = 1, and A = 0.5 are selected.
Figures
Figures
![]() | Fig. 5. SNRs changing with D under different q values for (a) DASR, (b) WASR, (c) DWASR where α = 0.8, τ = 0.1, k = 0.7, σ = 0.3. |
![]() | Fig. 6. SNRs changing with D under different τ values for (a) DASR, (b) WASR, (c) DWASR where α = 0.8, q = 0.2, k = 0.7, σ = 0.3. |
The relationship between the SNR and the non-Gaussian noise deviation parameter q is shown in Fig.
We draw the three-dimensional plots for the SNR which is induced by the correlation time τ in Fig.
Like the non-Gaussian colored noise, we explore the relationship between time-delayed terms and the SNR of DASR, WASR, and DWASR in Figs.
![]() | Fig. 7. SNRs changing with D under different k values for (a) DASR, (b) WASR, (c) DWASR where α = 0.8, q = 0.2, τ = 0.1, σ = 0.3. |
![]() | Fig. 8. SNRs changing with D under different σ values for (a) DASR, (b) WASR, (c) DWASR where α = 0.8, q = 0.2, τ = 0.1, k = 0.7. |
Figure
Figure
The inventors of particle swarm optimization algorithm are James Kennedy and Russell Eberhart.[19] This algorithm is developed to imitate the foraging behavior of a flock of birds. The main idea of the algorithm is to use the information shared by individuals in the swarm to obtain an optimal solution.
Particles are used to simulate birds. Each particle can be regarded as an individual searching bird in the search space. The current position of the particle is a candidate solution for the optimization problem. In the meantime, the flight path of the search agent is the individual’s search process. There are only two attributes of particles: speed and position. The former represents the speed of movement, and the latter represents the direction of movement. The optimal solution found by the particle is the locally best. The globally best is the optimal value among individual optimal solutions. The updating of speed and position is the main idea of particle swarm algorithm, which can be expressed as
![](cpb_29_5_050501/cpb_29_5_050501_ieqn5.gif)
The algorithm stops when evolution algebra reaches the number of iterations, then returns the final optimal value and the optimal position.
To analyze the performance of three asymmetric bistable SR systems and prove the superiority of asymmetric systems, the CSR is chosen for comparison. The steps of weak signal detection are given below.
(i) Signal pre-processing: Set the noise intensity and add noise into the weak signal.
(ii) Frequency scale transformation: Since SR is limited by a small frequency parameter, a frequency scale transformation method is required to achieve SR of high frequency periodic signals.
(iii) Use the particle swarm optimization: Determine the optimal system parameters to maximize SNR by using the particle swarm optimization algorithm.
(iv) Post-processing: Obtain the output signal by using the optimal system parameters and fourth-order Runge–Kutta method. Acquire its waveform and spectrum by Fourier transform and extract weak fault characteristics from the spectrum.
Figure
Figure
In order to verify the universality of the asymmetric system and the capability for detecting weak signals under strong noise background, a signal with a frequency of 400 Hz and a noise intensity D of 8 is taken as the input signal. The output signal obtained by processing the input signal through the SR system is shown in Fig.
![]() | Table 1. Performance of different methods in processing kinds of noise-added signals. . |
In nature, the time lag is widespread, and the noise is colored. Therefore, it is of great significance to study the asymmetric SR under the influence of the time-delayed feedback and the non-Gaussian colored noise. The output SNR of the model is calculated and the effects of the time-delayed feedback and the non-Gaussian colored noise on the asymmetric SR are analyzed. The results show that the asymmetric SR is superior to that of the symmetric SR. Under the action of the non-Gaussian colored noise, the best performance of SR is derived from the Gaussian-distributed noise while different values of the non-Gaussian noise deviation parameter have little effect on performance. At the same time, in comparison with the white noise, the colored noise is meaningful and plays a crucial role in motivating the SR phenomenon. On the contrary, the time-delayed term reduces the resonant peak for the SNR rapidly and weakens the SR effect. When the time lag exists, there is an optimal delay time σ approaching to 1 to maximize the performance of the SR. From the spectrum analyses of three asymmetric systems, we can find that the best SR is the DASR, the second best SR is the DWASR, and the worst SR is the WASR. At the same time, the asymmetric system performance is better than the symmetric systems. It is worth mentioning that under the background of the strong noise, the three asymmetric SR systems can effectively detect the weak signals, including high-frequency signals.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[39] |