## 2 Tri-Stable Stochastic Resonance Theory
### Parameter Compensation Stochastic Resonance  
With the in-depth study of SR theory by scholars, the problem that classical SR is limited by the adiabatic approximation theory is solved. At present, stochastic resonance can not only detect low frequency (\(f<<1\) Hz) signals but also detect high frequency (\(f>>1\) Hz) signals, and the signal frequency far exceeds the frequency range limited by traditional SR theory. Existing methods for detecting high-frequency signals include subsampling, modulation, normalization, and parameter compensation. The principle of the subsampling method is simple and easy to operate, but the method has requirements on the sampling frequency and is difficult to determine. The modulation method theory is simple, but the method is very restrictive due to the difficulty in determining the frequency of the modulation signal. The normalized transform can convert the high frequency frequency domain to the low frequency frequency domain, but the method will greatly reduce the amplitude of the measured signal during processing, so the amplitude of the output signal of the system is smaller than the amplitude of the input signal. The parameter compensation method is to compensate for the influence of the frequency of the high-frequency signal  
Figure 2: The curve of at \(a=3,b=5,c=0.038\). to be tested on the amplitude of the output signal of the system. Therefore, it is only necessary to know the approximate range of the frequency of the signal to be detected during the detection, which is not only simple in principle and convenient in operation but also subject to very few restrictions. The literature has carried out a preliminary study on the parameter compensation SR, which proves the effectiveness of the method. The specific principles of using the parameter compensation SR are as follows:  
The nonlinear multipotential function \(U(x)=\frac{a}{2}x^{2}-\frac{1+a}{4b}x^{4}+\frac{\xi}{6}x^{6}\) and the input signal \(s(t)\) Equation (23) are substituted by Equation (20) in the middle  
\[\dot{x}=-ax+\frac{1+a}{b}x^{3}-cx^{5}+\sum_{i=1}^{n}A_{i}\sin\left(2\pi f_{i}t \right)+D\xi(t), \tag{31}\]  
By integrating Equation (31), the output signal \(x\) of the SR system is  
\[x =\int\left[-ax+\frac{1+a}{b}x^{3}-cx^{5}+\sum_{i=1}^{n}A_{i}\sin \left(2\pi f_{i}t\right)+D\xi(t)\right]dt \tag{32}\] \[=\int\left[-ax+\frac{1+a}{b}x^{3}-cx^{5}\right]dt+\int\left[\sum _{i=1}^{n}A_{i}\sin\left(2\pi f_{i}t\right)\right]dt+\int\left[D\xi(t)\right]dt\] \[=-\frac{a}{2}x^{2}+\frac{1+a}{4b}x^{4}-\frac{c}{6}x^{6}-\left[ \sum_{i=1}^{n}\frac{A_{i}}{2\pi f_{i}}\cos\left(2\pi f_{i}t\right)\right]+ \int\left[D\xi(t)\right]dt\qquad i=1,2,\ldots,n\]  
As observed in Equation (32), the input signal \(s(t)\) is processed by the multi-stable SR system, and its amplitude \(A_{i}\) becomes \(\frac{1}{2\pi f_{i}}\) of the original signal. When \(f_{i}<<1\), then \(\frac{1}{2\pi f_{i}}>1\) the amplitude \(A_{i}\) of the input signal \(s(t)\) is amplified; when \(f_{i}>1\), then \(\frac{1}{2\pi f_{i}}<1\), the input signal \(s(t)\) value \(A_{i}\) is reduced, and it can be found that as the input signal \(s(t)\) frequency \(f_{i}\) increases, the amplitude \(A_{i}\) of the input signal \(x\) is greatly reduced. Therefore, high frequency signals cannot be seen in the output of the SR system excited by the high frequency signal. For this problem, the compensation parameter \(G\) can be added to cancel the signal amplitude \(A_{i}\) reduction caused by the high frequency signal frequency \(f_{i}\):  
\[\dot{x} =G\left[-ax+\frac{1+a}{b}x^{3}-cx^{5}+s(t)+D\xi(t)\right]\] \[=G\left[-ax+\frac{1+a}{b}x^{3}-cx^{5}+\sum_{i=1}^{n}A_{i}\sin \left(2\pi f_{i}t\right)+D\xi(t)\right], \tag{33}\]  
In Equation (33), \(G\) is the compensation parameter. When the value of \(G\) is equal to \(2\pi f_{i}\), the amplitude \(A_{i}\) of the input signal \(s(t)\) after passing through the SR system is unchanged. If the value of \(G\) is greater than \(2\pi f_{i}\), the input signal is \(s(t)\).