## 1 Introduction
### Transient Steady-State Response Analysis of Underdamped Stochastic Resonance System
#### 3.2.1 Transient Response Theory Derivation
\right.\) represents the unit step function, and \(K_{xx}^{0}(\tau,D)\) is the autocorrelation function of the SUUBSR system without being disturbed by an external periodic signal, which is described as follows:  
\[K_{xx}^{0}(\tau,D)=x_{s1,2}^{2}\exp(-2R_{1,2}\tau) \tag{31}\]  
Without the addition of external noise, under the combined influence of a SUUBSR system and a weak OFDM signal, the equation of motion for the particle within the initial trap is as follows:  
\[\frac{\text{d}^{2}x_{\text{Q}}(t)}{\text{d}t^{2}}=-bx_{\text{Q}}(t)-\gamma \frac{dx_{\text{Q}}(t)}{dt}+\sum_{k=0}^{K-1}\beta_{k}\sin(2\pi f_{k}t+\theta_ {k}) \tag{32}\]  
The general solution to Equation (32) is as follows:  
\[x_{\text{Q}}(t)=e^{-\frac{\pi}{2}t}\left[C_{1}\sin(\frac{\sqrt{4b-\gamma^{2}} }{2}t)+\left[C_{2}\cos(\frac{\sqrt{4b-\gamma^{2}}}{2}t)\right]\right]-\frac{ \text{e}^{-2\kappa_{1}2^{t}\tau^{2}}}{D}\cdot\ln(\frac{2v}{k\tau^{2}})\cdot \int\sum_{k=0}^{K-1}\beta_{k}\sin(2\pi f_{k}t+\theta_{k})\text{e}^{2R_{1,2}t}dt \tag{33}\]  
Substitute the initial conditions \(t=0\), \(x_{\text{Q}}(0)=0\), and \(x_{\text{Q}}^{\prime}(0)=0\) into Formula (33) to obtain the values of the undetermined coefficients \(C_{1}\) and \(C_{2}\) in the general solution. \[C_{1}=\ln(\frac{2v}{ar^{2}})\sum_{k=0}^{K-1}\frac{2\pi f_{k}\beta_{k}\gamma( \sin\theta_{k}+R_{1,2}\cos\theta_{k})}{4 DR_{1,2}{}^{2}(1+\frac{\pi^{2}f_{k}^{2 }}{R_{1,2}})} \tag{34}\]  
\[C_{2}=-\ln(\frac{2v}{ar^{2}})\sum_{k=0}^{K-1}\frac{r^{2}\beta_{k}(R_{1,2}\sin \theta_{k}-\cos\theta_{k})}{2 DR_{1,2}{}^{2}(1+\frac{\pi^{2}f_{k}^{2}}{R_{1,2}})} \tag{35}\]By substituting \(C_{1}\) and \(C_{2}\) into Equation (33), the expression for the transient response of the particle's motion within one symbol period in the SUUBSR system is determined. \[ \begin{split} x_{\text{Q}}(t)&=\ln\left(\frac{2v}{ar^{2 }}\right)e^{-\frac{\gamma}{2}t\sum\limits_{k=0}^{K-1}\frac{2\pi f_{k}A_{k} \gamma(\sin\theta_{k}+R_{1,2}\cos\theta_{k})}{4 DR_{1,2}{}^{2}(1+\frac{\pi^{2} f_{k}^{2}}{R_{1,2}})\sqrt{(4b-\gamma^{2})}}\sin\left(\frac{\sqrt{4b-\gamma^{2}}}{2}t \right)\\ &+\ln\left(\frac{2v}{ar^{2}}\right)e^{-\frac{\gamma}{2}t\sum \limits_{k=0}^{K-1}\frac{r^{2}\beta_{k}(R_{1,2}\sin\theta_{k}-\cos\theta_{k}) }{2 DR_{1,2}{}^{2}(1+\frac{\pi^{2}f_{k}^{2}}{R_{1,2}})}\cos\left(\frac{\sqrt{4 b-\gamma^{2}}}{2}t\right)\\ &+\ln\left(\frac{2v}{ar^{2}}\right)\sum\limits_{k=0}^{K-1}\frac{r ^{2}\beta_{k}\cos(2\pi f_{k}t+\theta_{k})}{2 DR_{1,2}{}^{2}(1+\frac{\pi^{2}f_{ k}^{2}}{R_{1,2}})}-\ln\left(\frac{2v}{ar^{2}}\right)\sum\limits_{k=0}^{K-1} \frac{r^{2}A_{k}\sin(2\pi f_{k}t+\theta_{k})}{2 DR_{1,2}(1+\frac{\pi^{2}f_{k}^{ 2}}{R_{1,2}})}\end{split}  \tag{36}\]  
where \(0<t<T_{d}\). Similarly, the transient response expression of the OFDM signal enhancement and demodulation system based on FUBSR can be obtained. \[ \begin{split} z_{\text{Q}}(t)&=\sum\limits_{k=0}^{ N-1}\frac{\pi f_{k}A_{k}(r_{k}\sin\theta_{k}-\cos\theta_{k})}{r_{k}(1+\frac{4\pi^{2} f_{k}^{2}}{r_{k}^{2}})}e^{-bt}-\ln\left(\frac{2v}{ar^{2}}\right)\sum\limits_{k=0}^{ N-1}\frac{\beta_{k}r^{2}\sin(2\pi f_{k}t+\theta_{k})}{2 Dr_{k}(1+\frac{\pi^{2}f_{k} ^{2}}{r_{k}^{2}})}\\ &+\ln\left(\frac{2v}{ar^{2}}\right)\sum\limits_{k=0}^{N-1}\frac {\beta_{k}r^{2}\cos(2\pi f_{k}t+\theta_{k})}{2 Dr_{k}{}^{2}(1+\frac{\pi^{2}f_{ k}^{2}}{r_{k}^{2}})}\end{split}  \tag{37}\]  
In the equation, \(r_{k}\) is a function that represents the Kramers' escape rate for OFDM signal enhancement and demodulation system based on FUBSR [10]. \[r_{k}=\frac{ar^{2}\ln(2v/ar^{2})}{v\sqrt{2\pi}}\exp\left(-\frac{ar^{2}(\ln(2v/ ar^{2})+1)}{2D}\right) \tag{38}\]  
From Equations (36) and (37), it can be derived that the transient response component in the response of the SUUBSR system and the FUBSR system determines the speed of the system's response. Subsequently, an analysis of the transient processes of the two systems reveals that as time increases, the magnitude of the transient response will approach 0 indefinitely.