Objective Distributed acoustic sensing systems based on phase- sensitive optical time- domain reflectometer (& Fcy;-OTDR) are commonly used for vibration signal detection. When external vibrations are applied to the sensing optical fiber, the fiber's refractive index changes, leading to variations in the phase of the backscattered Rayleigh light at the vibration location. Since the phase change of the backscattered Rayleigh light is linearly related to the vibration applied to the fiber, external vibration signals can be measured based on these phase changes. However, phase ambiguity often occurs during demodulation, resulting in distorted phase values, making it difficult to accurately reflect the vibration information. Common demodulation methods, such as the arctangent method, can extract the phase, but due to the phase's periodicity and variability, only the wrapped phase with a period of 2 pi and a principal value interval from -pi to pi can be obtained. As the vibration measurement range increases, if the phase change exceeds 2 pi, phase ambiguity occurs, leading to demodulated phase results that cannot accurately reflect the vibration amplitude and frequency.<br /> Methods To address the phase ambiguity issue, we propose a phase demodulation algorithm based on differential- compensation- accumulation, building on digital in- phase and quadrature (IQ) demodulation. This algorithm provides a reliable phase compensation scheme to solve phase ambiguity, making the differential phase no longer dependent on the jump threshold of pi. It effectively avoids phase misalignment caused by phase changes exceeding 2 pi, enabling accurate extraction of phase information from vibration signals. The & Fcy;- OTDR digital IQ demodulation process is divided into three processes: mixing, filtering, and phase demodulation. Since the phase information obtained by digital IQ demodulation is mainly distributed between -pi /2 and pi /2, arctangent unwrapping in a four- quadrant manner is required. A backward differential operation is then applied to the arctangent phase signal to eliminate accumulated phase noise and prevent its global propagation, minimizing errors. By selecting appropriate compensation coefficients in the differential domain to adjust the phase signal, the difference between adjacent elements is reduced to less than pi. The optimal compensated phase solution is then accumulated, yielding accurate phase values and effectively reducing errors caused by discontinuities.<br /> Results and Discussions In the & Fcy;- OTDR experimental system, vibration signals with frequencies of 20, 60, and 100 Hz are detected. When comparing the time- phase plots of the 60 Hz sinusoidal signal obtained using the proposed algorithm and the traditional unwrapping algorithm [Figs. 8(a) and 8(c)], it is clear that the proposed algorithm is less influenced by phase noise and frequency drift, with good continuity and high accuracy in the demodulation results. However, the results obtained by the unwrapping algorithm show errors when phase changes exceed the adjacent point's phase change limit [Fig. 8(c)]. A comparison of the phase power spectral density (PSD) of the 60 Hz vibration signal shows that the proposed algorithm experiences less noise interference, with a signal-to-noise ratio (SNR) of 24.7 dB. The PSD obtained by the unwrapping algorithm is more disturbed by noise, particularly in the <60 Hz frequency range, with an SNR of 17.4 dB. Therefore, the proposed demodulation algorithm improves the SNR by 7. 3 dB compared to the unwrapping algorithm [Figs. 9(a) and 9(b)]. Similarly, the 20 and 100 Hz sinusoidal signals are well constructed using the proposed demodulation algorithm [Figs. 11(a) and 11(b)]. The PSD analysis of the 20 Hz and 100 Hz signals reveals that the signal power is concentrated around their respective frequencies, with SNRs of 27.4 and 32.4 dB, respectively [Figs. 11(c) and 11(d)], demonstrating the accurate recovery of external vibration signals.<br /> Conclusions Phase ambiguity is a common limitation in many phase demodulation methods, restricting the vibration measurement range. We propose a differential- compensation- accumulation demodulation algorithm for recovering vibration signals in & Fcy;- OTDR distributed fiber sensing systems, accurately reconstructing sinusoidal signals loaded on the optical fiber. Unlike traditional demodulation algorithms, the proposed algorithm produces continuous demodulated phases over time, avoiding phase errors caused by changes exceeding 2 pi. Compared to unwrapping algorithms, the proposed algorithm significantly improves phase demodulation accuracy and reduces phase noise interference, enhancing the system's SNR.
