Robust Vibration Displacement Extraction for FMCW Radar by Incorporating Optimized Beat Frequency and Maximum Likelihood Estimation

Frequency modulated continuous wave (FMCW) radar is an emerging non-contact measurement technology with strong antijamming capability, wide measurement range, and high accuracy. However, its performance can be severely degraded by environmental noise. In this paper, we propose a robust vibration displacement extraction method for FMCW radar by combining beat frequency optimization with maximum likelihood estimation (MLE). First, we use the Nelder-Mead algorithm to accurately optimize and estimate the beat frequency from noisy radar signals. Then, leveraging this optimized beat frequency, we apply the MLE approach across multiple radar sweep periods to obtain robust and precise phase time series, facilitating reliable vibration displacement extraction. Numerical results reveal that the proposed beat frequency optimization achieves a minimal estimation error of 0.0087 Hz. Additionally, by employing the optimized beat frequency with the MLE, our method significantly improves the robustness and accuracy of phase estimation under varying signal-to-noise ratio conditions. Specifically, improvements of at least 3.60% in root mean square error (RMSE) and 7.64% in mean variance of phase errors were observed compared to traditional methods lacking optimization. The proposed method thus demonstrates superior robustness and accuracy in vibration displacement extraction, providing substantial benefits for structural health monitoring, vibration analysis, and engineering control applications.