What Is the “Nyquist Criterion” for Microwave Vibrometry, and Can It Be Bypassed?

Microwave interferometry is a promising technique for noncontact vibration measurement. However, due to its inherent drawback of phase wrapping, the measurable vibration amplitude is limited to a quarter wavelength. To extend the detectable range of high-frequency microwave radar, vibration demodulation from wrapped samples is a critical postprocessing technique. In this article, we study two key questions: 1) what is the foundational condition that must be satisfied for phase interferometry-based vibration demodulation? 2) can this condition be bypassed? To answer the first question, we derive the unwrapping criterion in the phase domain, which depends on factors including carrier wavelength, sampling rate, and vibration characteristics. The derived phase-domain unwrapping criterion, which can be viewed as an Itoh-type condition specialized to microwave vibrometry, together with the classical time-domain Nyquist criterion, provides a practical measurability condition for phase-based microwave vibration measurement. Furthermore, we discover that the unwrapping criterion is not absolute and propose a dual-frequency microwave radar-based vibration demodulation algorithm to bypass it. Inspired by Chinese remainder theorem (CRT), the algorithm synthesizes an equivalent longer wavelength from two shorter wavelengths, enabling accurate demodulation and measurement of large displacements. Both simulation and experimental results show that the proposed algorithm achieves improved accuracy and broader applicability compared with representative conventional vibration demodulation methods under the tested conditions. In summary, this work provides theoretical guidance and practical demodulation algorithms for microwave vibrometry.