Frequency–phase coupled parameter estimation for vibration measurement with LFMCW radar

Linear frequency modulated continuous wave (LFMCW) radar is widely employed in vibration measurement. In the received intermediate frequency signal, the distance information is embedded in both the frequency and phase, i.e., frequency–phase coupling. Early studies rely primarily on classical phase unwrapping algorithms, which frequently fail during rapid or large-amplitude vibrations. Although recent advances have enhanced robustness by incorporating frequency-derived coarse distance estimates as side information for phase unwrapping, these sequential approaches still under-utilize the inherent frequency–phase coupling relationship. In this article, we propose a novel method that jointly leverages both frequency and phase information for direct distance estimation. To start with, we derive a simplified discrete-time baseband signal model, which clearly unveils the coupling. The maximum likelihood (ML) estimation is used to reap the promised performance, but its complexity grows exponentially with the number of observations. To address the complexity barrier, we propose a heuristic method to sequentially solve such a joint optimization problem, and the performance is close to the Cramér–Rao lower bound (CRLB). We analyze the behavior of the ML estimators and discuss the impact of various parameters on system performance. A real-world experiment validates the system model and the proposed algorithm, with results aligning with theoretical analysis.