## 3 Improved Stochastic Resonance Model with Output Saturation Limits
### Accelerated Stochastic Resonance Model  
It can be seen from the analysis of Figure 7 in Section 2.1 that the larger the slope of the inner measured potential well wall of the bistable system, i.e., the part of \(x<|x_{m}|\), the more it promotes the oscillation of the particles between the potential wells. The principle is as follows: in the part of \(x<|x_{m}|\), the slope of the potential well increases, so that the positive acceleration of the system increases and the change of the rate state of the system is obvious; in the part of \(|x_{m}|<x<|c|\), the average slope of the potential well increases, so that the negative acceleration of the particles increases and accelerates the convergence of the system. Combining the non-saturated stochastic resonance model proposed in Section 2.1, keeping the external potential linear, and setting the inner potential as exponential can enhance the slope of the internal potential and promote the system to leapfrog between the potential wells, while taking into account the advantages of the unsaturated output, to obtain the accelerated stochastic resonance model, which has the following potential function. \[U_{ASR}(x)=\left\{ \begin{array}{ll}-\frac{\alpha^{2}}{4b}\left(\frac{x+c}{c- \sqrt{a/b}}\right)&x<-c\\ \\ -\frac{\alpha^{2}}{4b}\left[\frac{\exp(x+c)-1}{\exp\left(c-\sqrt{a/b}\right)- 1}\right]&-c\leq x\leq-\sqrt{a/b}\\ \\ -\frac{\alpha^{2}}{4b}\left[\frac{\exp(-x)-1}{\exp\left(\sqrt{a/b}\right)- 1}\right]&-\sqrt{a/b}\,<x<0\\ \\ -\frac{\alpha^{2}}{4b}\left(\frac{x-c}{c-\sqrt{a/b}}\right)&x\geq c\\ \end{array} \right. \tag{42}\]  
A comparison of the potential function curves of the accelerated stochastic resonance system with those of the classical stochastic resonance system when the system parameters are accelerated is shown in Figure 8.  
As can be seen in Figure 8, the red dashed line is the original double potential well curve, and the blue curve is the accelerated stochastic resonance potential well curve. By comparison, we can conclude that the potential function curve of accelerated stochastic resonance possesses a steeper trap wall in the inner potential well part, which can promote the oscillatory leap of particles between the two potential wells because the stochastic resonance phenomenon mainly occurs in the range of \(x\leq|c|\), while the peripheral potential well  
Figure 8: Accelerated stochastic resonance vs. classical stochastic resonance potential curves. wall maintains the linear characteristics of the non-saturating stochastic resonance, so the amplitude of the output signal of the system no longer tends to a stable saturation value at time \(t\to\infty\), and instead, it increases linearly, avoiding the output saturation phenomenon. Disregarding the effect of noise, when the signal of Equation (13) is input to the system, the system potential function is modulated by the input signal, and the potential function curve is shown in Figure 9.  
As can be seen in Figure 9, the depth of the left and right potential wells of the accelerated stochastic resonance potential function undergoes alternating transformations, and the period of its dynamic change is consistent with the period of the signal to be detected. The slope of the internal potential function of the accelerated stochastic resonance detection system is always larger than that of the original stochastic resonance signal detection algorithm, indicating that it can promote the oscillation of the particles between the two potential wells, reduce the energy loss, and improve the conversion rate of the noise energy to the signal energy. Figure 9: Potential function curves under input signal modulation. (**a**) Potential function curves at quarter-cycle moments. (**b**) Potential function curve at the moment of three-quarter cycle.