## 3 Improved Stochastic Resonance Model with Output Saturation Limits
### Unsaturated Stochastic Resonance Model
The Fokker-Planck equation [18] for the Langevin equation corresponding to the non-saturated bistable stochastic resonance algorithm is  
\[\frac{\partial\rho(x,t)}{\partial t}=-\frac{\partial}{\partial x}\left[U^{{} ^{\prime}}\rho(x,t)\right]+D\frac{\partial^{2}\rho(x,t)}{\partial x^{2}} \tag{14}\]  
where \(\rho\) denotes the probability distribution of the particle's location, which is related to the particle's position \(x\) and the time \(t\). \(D\) is the noise intensity. The initial probability distribution of the equivalent particle model is assumed to satisfy the adiabatic approximation theory [19]; i.e., it is assumed that the probability distribution at the moment \(t=0\) is concentrated in a certain potential well, i.e., the  
\[\rho(x,0)=\delta\bigg{(}x\pm\sqrt{\frac{a}{b}}\bigg{)} \tag{15}\]  
At the same time, it is assumed that the probability flow does not vary with time; i.e., when the  
\[\frac{\partial\rho(x,t)}{\partial t}=0 \tag{16}\]  
At the time of integration of Equation (14), the steady-state flow intensity is expressed in terms of \(\Theta\), and the stable solution of the Fokker-Planck equation is obtained as  
\[U^{{}^{\prime}}\rho(x,t)+D\frac{\partial\rho(x,t)}{\partial x}=\Theta \tag{17}\]  
Because of the previous assumption that the probability flow does not transform with time at this point, Equation (17) can be rewritten as  
\[\frac{d\rho(x)}{\rho(x)}=-\frac{U^{{}^{\prime}}(x)}{D}dx \tag{18}\]  
to solve Equation (18),  
Figure 6:0  
Figure 6:1  
where \(N\) denotes the constant term to be determined in the stabilized solution in the process of solving the differential equation. Since the object of study in this paper is signal processing and stochastic processes, it is more accurate to say that \(N\) denotes the constant process \(N(t)\) with respect to the time \(t\). Do not confuse it with \(N(f)\) in the simulation part. From the assumptions of Equation (15), assuming an initial positive potential well, we obtain  
Figure 6:2Integrating Equation (21) yields  
Figure 6:3  
According to the nature of the impulse function, Equation (22) can be simplified to obtain  
Figure 6:4  
Equation (20) is the chi-square solution of the Fokker-Planck equation, and the following solves for the non-chi-square solution of the Fokker-Planck equation, assuming that  
Figure 6:5  
follows  
Figure 6:6  
Substituting Equation (25) into Equation (17), we obtain  
Figure 6:7  
In turn, we can introduce  
Figure 6:8  
and set  
Figure 6:9  
Then, the steady-state flow intensity \(\Theta\) can be expressed as  
\[U_{s}(x)=U(x)-x(A\sin(2\pi f_{0}t)) \tag{13}\]0  
The total probability of being in the interval \((-\infty,A)\) at moment \(t\) is  
\[U_{s}(x)=U(x)-x(A\sin(2\pi f_{0}t)) \tag{13}\]1  
Since \(\Theta\) is the rate flow with probability out of this region, at a steady state, combining Equation (29) yields  
\[U_{s}(x)=U(x)-x(A\sin(2\pi f_{0}t)) \tag{13}\]2  
\[U_{s}(x)=U(x)-x(A\sin(2\pi f_{0}t)) \tag{13}\]3  
In Equation (32),  
\[U_{s}(x)=U(x)-x(A\sin(2\pi f_{0}t)) \tag{13}\]4\(R\) is the Kramers escape rate, the rate at which the reaction probability flows into the unstable region. Substituting Equation (12) into Equation (33) yields the probabilistic jump rate of the non-saturated stochastic resonance between the two potential wells:  
\[ \begin{array}{l}R_{UBSR-}^{-1}=\frac{1}{D}\left\{\int_{-c}^{-\sqrt{\frac{T}{ 6}}}\exp\left[-\frac{1}{D}\left(\frac{a^{2}}{4b}\left(\frac{x+c}{c-\sqrt{a/b}} \right)\right)\right]dx\right\}\\ \qquad\qquad\times\left\{\int_{-\sqrt{b}}^{0}\exp\left[\frac{1}{D}\left(-\frac {1}{2}ax^{2}+\frac{1}{4}bx^{4}\right)\right]dx\right\}\end{array}  \tag{34}\]  
Since the above equation contains the transcendental function, it is not possible to compute the exact analytical solution, so Equation (34) is solved approximately with the help of Taylor expansion by taking the constant term of the expansion equation while considering the adiabatic approximation condition constraints, i.e., \(D\ll 1\).