## 1 Introduction
### Steady-State Response Analysis  
After a transient response of duration \(T_{d}\), the system begins to enter a steady-state phase during which the particle moves within the initial trap. To solve for the steady-state output of the system, the generalized Langevin gradient equation is chosen to represent the steady-state output state equation of the system. \[\frac{d^{2}x_{\text{Q}}(t)}{dt^{2}}+\gamma\frac{dx_{\text{Q}}(t)}{dt}=f(x_{ \text{Q}}(t),t)+\varepsilon(x_{\text{Q}}(t),t) \tag{43}\]  
In the equation, \(f(x_{\text{Q}}(t),t)\) is the deterministic component, and \(\varepsilon(x_{\text{Q}}(t),t)\) is the external random force. Figure 4: The influence of system parameters on the response speed of the system: (**a**) the influence of damping coefficient on the response speed of the system, (**b**) the influence of parameter \(v\) on the system’s response speed, (**c**) the influence of parameter \(r\) on the system’s response speed, (**d**) the influence of parameter \(a\) on the system’s response speed, and (**e**) the influence of parameter \(f_{k}\) on the system’s response speed. For the convenience of solving, Equation (43) can be rewritten into the form of Equation (44) based on statistical theory:  
\[\frac{d^{2}x_{\text{Q}}(t)}{dt^{2}}+\gamma\frac{dx_{\text{Q}}(t)}{dt}=f(x_{ \text{Q}}(t),t)+\mathbf{G}\
abla_{x}\ln J(x_{\text{Q}}(t),t) \tag{44}\]  
where \(\mathbf{G}\) represents the diffusion matrix of the state vector in the state space, \(\
abla_{x}\) represents the gradient operator, and \(J(x_{\text{Q}}(t),t)\) represents the Jacobian determinant, \(\ln J(x_{\text{Q}}(t),t)=-\ln p(x_{\text{Q}}(t),t)\). First, the deterministic component in Equation (44) is analyzed. Assuming that when the system initially reaches \(x_{si}(i=1,2)\) steady-state condition, the particle is located near the steady-state point a of the SUUBSR system. The deterministic component is Taylor expanded at the two steady-state points, \(x_{s1}\) and \(x_{s2}\), of the nonsaturating potential function. Then, Taylor series expansions \(f_{1}(x_{\text{Q}}(t),t)\) and \(f_{2}(x_{\text{Q}}(t),t)\) are obtained, which can be expressed as follows:  
\[\begin{array}{c}f_{i}(x_{\text{Q}}(t),t)=\sum\limits_{j=1}^{n}a_{ij}(t)x_{j }(t)+\sum\  
where \(\kappa(t)\) can be described as follows:  
\[\kappa(t)=\sum\limits_{k=0}^{K-1}\frac{2R_{1,2}\beta_{k}x_{s_{i}}\sin(w_{k}t+ \theta_{k})}{D\sqrt{4{R_{1,2}}^{2}+{w_{k}}^{2}}} \tag{50}\]  
The time-domain average of the autocorrelation function of the steady-state output of the system excited by an OFDM signal with a single symbol period is obtained as follows:  
\[ \begin{split}&\langle\langle x_{\text{Q}}(t)x_{\text{Q}}(t+\tau) \rangle\rangle=\frac{1}{T_{s}}\int_{T_{d}}^{T_{s}}\langle x_{\text{Q}}(t)x_{ \text{Q}}(t+\tau)\rangle dt=\sum\limits_{k=0}^{K-1}x_{s_{i}}^{2}\exp(-2R_{1,2} |\tau|)\\ &\left[1-\frac{1}{2}\left(\frac{\beta_{k}x_{s_{i}}}{D}\right)^{2} \frac{4R_{1,2}^{2}}{4R_{1,2}^{2}+{w_{k}}^{2}}\right]+\sum\limits_{k=0}^{K-1} \frac{x_{s_{i}}^{2}}{2}\left(\frac{\beta_{k}x_{s_{i}}}{D}\right)^{2}\frac{4R_ {1,2}^{2}}{4R_{1,2}^{2}+{w_{k}}^{2}}\cos(w_{k}t+\theta_{k})\end{split}  \tag{51}\]  
Performing a Fourier transform on Equation (51) yields the analytical expression for the power spectrum of the steady-state output of the system:  
\[ \begin{split}& X(w)=X_{N}(w)+X_{S}(w)=\sum\limits_{k=0}^{K-1} \left[1-\frac{1}{2}\left(\frac{\beta_{k}x_{s_{i}}}{D}\right)^{2}\frac{4R_{1,2 }^{2}}{4R_{1,2}^{2}+{w_{k}}^{2}}\right]\frac{4R_{1,2}^{2}x_{s_{i}}^{2}}{4R_{1,2}^{2}+{w}^{2}}+\\ &\sum\limits_{k=0}^{K-1}\frac{\pi}{2}\left(\frac{\beta_{k}x_{s_ {i}}}{D}\right)^{2}\frac{4R_{1,2}^{2}}{4R_{1,2}^{2}+{w_{k}}^{2}}[\delta(w-w_{ k})+\delta(w+w_{k})]\end{split}  \tag{52}\]  
The steady-state output SNR is the ratio of the total signal power at each subcarrier frequency of the system output signal to the total noise power at each subcarrier frequency when the system reaches a steady state. Its expression is as follows:  
\[\left\{