## 1 Introduction
### Transient Steady-State Response Analysis of Underdamped Stochastic Resonance System  
#### 3.2.1 Transient Response Theory Derivation  
At the beginning of the second-order underdamped nonsaturated bistable stochastic resonance process, the particle does not immediately undergo transition, but randomly selects one of the two potential wells as the initial potential well for particle motion, and performs a local periodic oscillation motion, namely the intra-well motion, for a certain period of time within the initial potential well. This process is the transient response process of the system, and the corresponding time is the transient response time. In order to better study the transient response of the system, the Q-channel of the system is also selected for analysis. The output of the Q-channel of the system is \(x_{\text{Q}}(t)\), and the system output is only related to the motion of the particle within the initial well. Then, the nonsaturated bistable potential function can be equivalently regarded as two monostable potential functions, whose expressions are as follows, respectively:  
\[U_{+}^{*}(x_{\text{Q}})\approx U^{\prime}(x_{s1})(x_{\text{Q}}-x_{s1})^{2}/2 \tag{20}\]  
\[U_{-}^{*}(x_{\text{Q}})\approx U^{\prime}(x_{s2})(x_{\text{Q}}-x_{s2})^{2}/2 \tag{21}\]  
where the steady-state points \(x_{s1}\) and \(x_{s2}\) satisfy the following:  
\[\left\{ \begin{array}{l}U^{\prime}(x_{\text{Q}})=\frac{2\pi}{\tau^{2}}x_{ \text{Q}}\exp(-\frac{x_{\text{Q}}{\tau^{2}}}{\tau^{2}})-ax_{\text{Q}}-s_{ \text{Q}}(t)=0\\ U^{\prime\prime}(x_{\text{Q}})>0\end{array} \right. \tag{22}\]  
Consequently, the motion equations of the particle near the two stable points of the unsaturated potential function under the influence of a second-order underdamped system are obtained as follows:  
\[\frac{\text{d}^{2}x_{\text{Q}+}(t)}{\text{d}t^{2}}=-U^{\prime\prime}(x_{s1})( x_{\text{Q}+}(t)-x_{s1})-\gamma\frac{dx_{\text{Q}+}(t)}{dt}+n_{\text{Q}+}(t) \tag{23}\]  
\[\frac{\text{d}^{2}x_{\text{Q}-}(t)}{\text{d}t^{2}}=-U^{\prime\prime}(x_{s2})( x_{\text{Q}-}(t)-x_{s2})-\gamma\frac{dx_{\text{Q}-}(t)}{dt}+n_{\text{Q}-}(t) \tag{24}\]  
Based on Equations (23) and (24), the motion of the particle within the left and right wells can be represented by the following deterministic equations:  
\[\frac{\text{d}^{2}x_{\text{Q}+}(t)}{\text{d}t^{2}}=-bx_{\text{Q}+}(t)-\gamma \frac{dx_{\text{Q}+}(t)}{dt}+n_{\text{Q}+}(t) \tag{25}\]  
\[\frac{\text{d}^{2}x_{\text{Q}-}(t)}{\text{d}t^{2}}=-bx_{\text{Q}-}(t)-\gamma \frac{dx_{\text{Q}-}(t)}{dt}+n_{\text{Q}-}(t) \tag{26}\]  
Figure 2: The graph of the influence of noise intensity and damping coefficient on the transition probability. When a weak baseband OFDM signal \(s(t)\) is input into the system, the input signal is segmented according to the symbol period and subjected to orthogonal transformation to obtain the OFDM complex baseband signal of each symbol period. At this time, the system has Q input signals denoted as \(s_{\text{Q}}(t)\), and the output \(x_{\text{Q}\pm}(t)\) within the left and right potential wells can be expressed by the following equation:  
\[x_{\text{Q}+}(t)=\left\langle x_{\text{Q}}(t)\right\rangle_{0}+\int_{-\infty}^{ T_{d}}\left[\sum_{k=0}^{K-1}\beta_{k}\sin(2\pi f_{k}\tau+\theta_{k})\right].h(t- \tau)d\tau \tag{27}\]  
\[x_{\text{Q}-}(t)=\left\langle x_{\text{Q}}(t)\right\rangle_{0}+\int_{-\infty}^ {T_{d}}\left[\sum_{k=0}^{K-1}\beta_{k}\sin(2\pi f_{k}\tau+\theta_{k})\right].h(t- \tau)d\tau \tag{28}\]  
\[s_{\text{Q}}(t)=\sum_{k=0}^{K-1}\beta_{k}\sin(2\pi f_{k}\tau+\theta_{k}) \tag{29}\]  
where \(\left\langle x_{\text{Q}}(t)\right\rangle_{0}\) represents the average response value of the system output when there is no signal input, and \(T_{d}\) is the transient response time. \(h(t)\) is the SUUBSR system response function, and its equation is expressed as follows:  
\[h(t)=-\frac{u(t)}{D}\frac{\text{d}}{\text{d}t}K_{xx}^{0}(t) \tag{30}\]  
In the equation, \(u(t)=\left\{ \begin{array}{ll}0,t<0\\ 1,t>0\end{array}