## 5 Conclusions  
Frequency leakage and the fence effect pose significant challenges in the estimation of FMCW signals. The commonly used Rife algorithm, while prevalent, is vulnerable to noise and has limitations concerning the frequency range. To address these issues, this paper proposes the SP-Rife algorithm, which leverages spectrum refinement and phase interpolation techniques. The algorithm enhances the frequency resolution through spectrum refinement, improving the signal's robustness against noise. Additionally, by employing amplitude-phase conversion followed by interpolation estimation, the SP-Rife algorithm achieves a greater accuracy and precision in frequency estimation.  
The simulation results indicate that the SP-Rife algorithm reduces the computational complexity by 30.1% compared to the Rife algorithm, demonstrating a significant advantage over the other two improved Rife algorithms. In comparison to the PAI-Rife algorithm, which shows a relatively strong estimation performance, the SP-Rife algorithm provides a 38.5% improvement in noise robustness and a 27.8% enhancement in frequency estimation accuracy. Overall, the SP-Rife algorithm achieves an effective balance between computational efficiency and accuracy, outperforming the Rife, ASIQ-Rife, and PAI-Rife algorithms in terms of overall estimation performance. Despite these advancements, the algorithm's effectiveness is contingent upon the appropriate setting of the spectrum refinement factor, which is sensitive to varying environmental conditions. In future work, adaptive methods for adjusting the spectral refinement factor will be explored to enable automatic calibration based on the characteristics of signals and environmental noise, thereby enhancing the generality and robustness of the algorithm. The current work has solved the estimation accuracy, which provides a foundation for future parameter adaptation.  
Figure 8: Comparison results of measurement errors.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**2378 mm**} & \multicolumn{2}{c}{**2420 mm**} & \multicolumn{2}{c}{**2561 mm**} & \multicolumn{2}{c}{**2653 mm**} \\ \cline{2-9}  & **Result** & **Error** & **Results** & **Error** & **Result** & **Error** & **Result** & **Error** \\ \hline Rife & 2324.3 & 53.7 & 2362.2 & 57.8 & 2603.4 & 42.4 & 2607.2 & 45.8 \\ ASIQ-Rife & 2328.4 & 49.6 & 2466.9 & 46.9 & 2586.5 & 25.5 & 2623.6 & 29.4 \\ PAI-Rife & 2363.9 & 14.1 & 2412.2 & 7.8 & 2567.6 & 6.6 & 2656.2 & 3.2 \\ SP–Rife & 2373.1 & 4.9 & 2416.3 & 3.7 & 2558.1 & 2.9 & 2654.7 & 1.7 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of distance values obtained by different algorithms.  
**Author Contributions:** Conceptualization, G.J. and M.C.; methodology, W.F.; software, M.C. and S.G.; validation, G.J. and M.C.; formal analysis, S.G.; investigation, G.J.; resources, W.F.; data curation, W.F. and M.C.; writing--original draft preparation, S.G.; writing--review and editing, M.C.; visualization, G.J.; supervision, M.C.; project administration, W.F.; funding acquisition, M.C. and G.J. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the Key R&D and Transformation Plan of Qinghai Province (Grant Number 2023-SF-148S-I).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available upon request from the authors.  
**Acknowledgments:** The authors would like to thank the editor-in-chief, editor, and anonymous reviewers for their valuable reviews.  
**Conflicts of Interest:** The authors declare no conflicts of interest.