## 3 Improved Stochastic Resonance Model with Output Saturation Limits  
### Unsaturated Stochastic Resonance Model  
After analyzing the potential function of the system, it is evident that the slope of the potential function significantly affects the performance of the stochastic resonance algorithm. The most influential part is the two major potential walls. Therefore, by reducing the slope of the potential wall--that is, lowering the slope of the part of the potential function curve that crosses over the stability--the output performance of the system can be improved. As shown by the above analysis, the output saturation of the system is mainly affected by the quadratic term \(x^{4}\) in the potential function; in order to avoid the existence of the quadratic term in the nonlinear potential function, the primary function function is used to construct the Unsaturated Bistable Stochastic Resonance (UBSR) signal detection algorithm, and its potential function is expressed as  
\[U_{UBSR}=\left\{ \begin{array}{ll}-\frac{a^{2}}{4b}\left(\frac{x+c}{c-\sqrt{a/ b}}\right)&x<-\sqrt{\frac{a}{b}}\\ -\frac{1}{2}ax^{2}+\frac{1}{4}bx^{4}&-\sqrt{\frac{a}{b}}\leq x\leq\sqrt{\frac {a}{b}}\\ \frac{a^{2}}{4b}\left(\frac{x-c}{c-\sqrt{a/b}}\right)&x>\sqrt{\frac{a}{b}} \end{array} \right. \tag{12}\]  
The Unsaturated Bistable Stochastic Resonance detection system has a linear and nonlinear two-part system structure, the linear part of the potential barrier wall is treated linearly, and the nonlinear part still retains the original potential function of the bistable system. Since the stochastic resonance phenomenon mainly occurs in the \([-x_{m},x_{m}]\) region [17], the slope of the potential function can be finely tuned to promote the formation of periodic oscillations of the particles between the two potential wells, and the curves of the potential function with different slopes are shown in Figure 6.  
Figure 6: Unsaturated stochastic resonance potential function curves. As can be seen from Figure 6, the red dashed line is the original dual-potential-well curve, and the rest of the curves are linear potential-well walls with different slopes. By comparison, it is concluded that the linearly improved bistable system can be avoided by changing the slope of the potential-well walls, so that the amplitude of the system's output signal at time \(t\rightarrow\infty\) no longer tends to a stable saturation value, but rather increases linearly, avoiding the phenomenon of output saturation. Disregarding the effect of noise, when the signal is input to the system, the system potential function is modulated by the input signal, at which time the potential function is  
\[U_{s}(x)=U(x)-x(A\sin(2\pi f_{0}t)) \tag{13}\]  
where A is the input periodic signal amplitude, \(f_{0}\) is the frequency of the input periodic signal, and the setting time is one-quarter cycle and three-quarter cycle, at which time the potential function curve under modulation is shown in Figure 7.  
Figure 7: Potential function curves under input signal modulation. (**a**) Potential function curves at quarter-cycle moments. (**b**) Potential function curve at the moment of the three-quarter cycle. As can be seen in Figure 7, the depth of the left and right potential wells of the potential function of the improved algorithm undergoes alternating transformations, and the period of its dynamic change is consistent with the period of the signal to be detected. When \(|x|>\sqrt{a/b}\), the potential function of the improved stochastic resonance signal detection algorithm is always wider than that of the original stochastic resonance signal detection algorithm, indicating that the output signal amplitude can be amplified to a greater extent. Meanwhile, when \(\sqrt{a/b}\leq|x|\leq c\), the slope of the outer potential wall increases relative to the average slope of the original potential well, which can promote the oscillation of particles between the two potential wells to some extent.