Higher-Order Difference Phase Unwrapping for Microwave-based Vibration Measurements

Microwave interferometry is a promising technique for non-contact vibration measurement. However, due to the inherent drawback of phase wrapping, the measurable vibration amplitude is limited to a quarter wavelength (γ). To pursue high resolution and compact size, high-frequency microwave radars, e.g., millimeter-wave radars, are increasingly adopted. Yet, a higher carrier frequency further reduces the effective sensing range, which makes accurate measurement of large-amplitude vibrations challenging. To overcome this limitation, we propose a higher-order difference-based phase unwrapping (HDPU) method that significantly expands the applicability of microwave radar to vibration measurement. This method originates from an interesting finding: for a multi-sinusoidal signal with a proper high sampling rate (larger than π-times the Nyquist rate), there always exists a sufficiently large order such that the higher-order difference of the signal samples is limited to [−γ, γ]. This finding implies that the higher-order difference of the signal samples can be obtained from the wrapped phase. Considering the intrinsic connection between the multi-sinusoidal signal sample and its higher-order difference in the frequency-domain, HDPU enables robust vibration reconstruction by utilizing the spectral sparsity. By combining the higher-order difference and sparse reconstruction, HDPU allows microwave radar to measure vibrations with amplitudes exceeding a quarter wavelength without distortion. More importantly, it overcomes Itoh’s limitation and shows higher robustness to noise than unlimited sampling.