## 5 Conclusions  
In this paper, a second-order underdamped system is applied to the enhancement and demodulation of OFDM signals. A method for enhancing and demodulating OFDM signals based on SUUBSR is proposed. The expressions for the first passage time, transient, and steady-state responses of the system driven by OFDM signals are derived, respectively. It is found that the existence of a transient response can lead to energy loss in OFDM. The output response analytical expressions of the FUBSR system and the SUUBSR system are analyzed, theoretically proving that the response speed of the SUUBSR system is superior to that of the FUBSR system. The relationship between system parameters, damping coefficient, input signal frequency, and system response speed and transient response is explored. It is found that the above parameters have an impact on both the magnitude of the system's transient response and its response speed. Larger values of parameters \(a\) and \(f_{k}\) result in shorter transient processes, while larger values of parameters \(v\) and \(r\) lead to longer transient processes. The damping coefficient is not linearly related to the length of the transient process, indicating that selecting appropriate parameters can minimize the impact of transient response on OFDM signals. During the steady-state response analysis, it is found that there is no DC offset component in the system that affects the steady-state output, and there exists an optimal damping that maximizes the steady-state output SNR. The simulation results show that the proposed system can eliminate the transient response of the system and enhance and demodulate OFDM signals with different subcarrier modulations such as QPSK, 16 QAM, and 64 QAM. Compared to the FUBSR system, the proposed system demonstrates better enhancement of weak OFDM signals, lower demodulation bit error rate, faster system response speed, and better noise resistance in low SNR environments.  
As the noise environment in current communication systems becomes increasingly complex, this paper only discusses and analyzes the system performance of the proposed system in Gaussian white noise environments. Therefore, for future research work on this paper, the first priority should be to comprehensively consider the system's ability to enhance and demodulate OFDM signals in different types of noise environments. Secondly, the objects processed by the proposed system should not be limited to communication signals. In the future, the system's good noise resistance can be utilized for image processing. Finally, increasing the system order means increased computational complexity, so future research should also focus on how to reduce computational complexity.  
Conceptualization, G.L. and D.W.; methodology, G.L.; software, D.W.; validation, G.L. and D.W.; formal analysis, G.L.; investigation, G.L.; resources, G.L. and D.W.; data curation, G.L. and D.W.; writing--original draft preparation, G.L.; writing--review and editing, G.L. and D.W.; visualization, G.L. and D.W.; supervision G.L. and D.W.; project administration, G.L.; funding acquisition G.L. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (Grant No. 61671375).  
Not applicable.  
The sharing of the data used in this study is limited due to the privacy restrictions of the project. If necessary, the data can be obtained by contacting the corresponding author.  
The authors declare no conflicts of interest.