## 1 Introduction
### First Passage Time Analysis of System Response  
The mean first passage time (MFPT) [14; 15; 16] is one of the indicators reflecting the response speed of the system, which is defined as the average time for a Brownian particle to make the first transition from an initial steady-state point to another steady-state point. Since the methods for solving the MFPT of the I- and Q-channels of the system are the same, the following analysis takes the Q-channel of the system as an example, and the MFPT of the I-channel can be obtained in the same way. Firstly, let \(dx_{\text{Q}}(t)/dt=y_{\text{Q}}(t)\), and reduce the Q-channel system's analytical expression in Equation (6) to a one-dimensional system of equations:  
\[\left\{ \begin{array}{ll}\frac{dx_{\text{Q}}(t)}{dt}=y_{\text{Q}}(t)\\ \frac{dy_{\text{Q}}(t)}{dt}=\frac{2\pi}{r^{2}}x_{\text{Q}}(t)\exp(-\frac{x_{ \text{Q}}^{2}(t)}{r^{2}})-ax_{\text{Q}}(t)-\gamma y_{\text{Q}}(t)+s_{\text{Q }}(t)+n_{\text{Q}}(t)\end{array} \right. \tag{9}\]  
Secondly, the steady-state probability density function \(\rho(x_{\text{Q}},y_{\text{Q}},t)\) of the system in state \((x_{\text{Q}},y_{\text{Q}})\) at time \(t\) is derived through the Fokker-Planck equation, which is expressed as follows:  
\[\frac{\partial\rho(x_{\text{Q}},y_{\text{Q}},t)}{\partial t}=D\frac{\partial^ {2}}{\partial y_{\text{Q}^{2}}}\rho(x_{\text{Q}},y_{\text{Q}},t)-\frac{ \partial}{\partial x_{\text{Q}}}[y_{\text{Q}}\rho(x_{\text{Q}},y_{\text{Q}}, t)]-\frac{\partial}{\partial y_{\text{Q}}}\{[f(x_{\text{Q}})-\gamma y_{\text{Q}}] \rho(x_{\text{Q}},y_{\text{Q}},t)\} \tag{10}\]  
Let \(\partial\rho(x_{\text{Q}},y_{\text{Q}},t)/\partial t=0\); according to the detailed balance condition [17; 18; 19], derive the conditions satisfied by the stationary probability density function \(\rho_{s}(x_{\text{Q}},y_{\text{Q}},t)\):  
\[\left\{ \begin{array}{ll}\frac{\partial}{\partial x_{\text{Q}}}[y_{\text{Q} }\rho_{s}(x_{\text{Q}},y_{\text{Q}},t)]-\frac{\partial}{\partial y_{\text{Q}}} \{[U(x_{\text{Q}})]^{\prime}\rho_{s}(x_{\text{Q}},y_{\text{Q}},t)\}=0\\ \gamma y_{\text{Q}}\rho_{s}(x_{\text{Q}},y_{\text{Q}},t)+D\frac{\partial y_{ \text{Q}}(x_{\text{Q}},y_{\text{Q}},t)}{\partial y_{\text{Q}}}=0\end{array} \right. \tag{11}\]  
The steady-state probability density function expression can be derived from Equation (11):  
\[\rho_{s}(x_{\text{Q}},y_{\text{Q}},t)=P\exp\left(-\frac{\widetilde{U}(x_{ \text{Q}},y_{\text{Q}},t)}{D}\right) \tag{12}\]  
Figure 1: OFDM signal enhancement and demodulation model based on SUUBSR system. where \(P\) is the normalization constant, and \(\widetilde{U}(x_{\rm Q},y_{\rm Q},t)\) is the generalized potential function, which is expressed as follows:  
\[\widetilde{U}(x_{\rm Q},y_{\rm Q},t)=\gamma(\frac{1}{2}y_{\rm Q}^{2}+U(x_{\rm Q })-x_{\rm Q}\sum_{k=0}^{K-1}\beta_{k}\sin(2\pi f_{k}t+\theta_{k})) \tag{13}\]  
Then, in Equation (9), let \(\frac{dy_{\rm Q}(t)}{dt}=0\), \(\frac{dx_{\rm Q}(t)}{dt}=0\), \(D=0\), and \(\beta_{k}=0\), then the values of the three singular points are determined, \(Q_{u1}(x_{s1},0)\), \(Q_{u2}(x_{s2},0)\), and \(Q_{uu}(x_{uu},0)\). And, the expression of the Hessian matrix of Equation:  
\[\mathbf{H}=\left[ \begin{array}{cc}0&1\\ M_{x}&-\gamma\end{array} \right] \tag{14}\]  
In the matrix \(\mathbf{H}\), the expression for \(M_{x}\) is as follows:  
\[M_{x}=-4x_{\rm Q}^{2}\exp(-x_{\rm Q}^{2})+2\exp(-x_{\rm Q}^{2})-a \tag{15}\]  
And then, the corresponding eigenvalues of matrix \(\mathbf{H}\) at the three singular points \(Q_{u1}\), \(Q_{u2}\), and \(Q_{uu}\) in Formula (14) are obtained as \(\lambda_{s1}^{\pm}\), \(\lambda_{s2}^{\pm}\), and \(\lambda_{uu}^{\pm}\), respectively. \[ \begin{array}{l}\lambda_{s1}^{\pm}=\frac{-\tau\pm\sqrt{\gamma^{2}-4U^{\prime }(x_{s1})}}{2}\\ \lambda_{s2}^{\pm}=\frac{-\gamma\pm\sqrt{\gamma^{2}-4U^{\prime}(x_{s2})}}{2} \\ \lambda_{uu}^{\pm}=\frac{-\gamma\pm\sqrt{\gamma^{2}-4U^{\prime}(x_{uu})}}{2} \end{array}