Dual-counters-based real-time 1D phase unwrapping algorithm in laser measurement and sensing applications

doi:10.23919/JSEE.2026.000168

Abstract

Abstract:

Heterodyne laser interferometer has been widely used for displacement and velocity measurements in the fields of vibration calibration and acoustic metrology. When the displacement or velocity to be measured exceeds a certain limit, the phase of the heterodyne interference signal, corresponding to different displacement or velocity amplitudes, is wrapped within the range of (−π, +π], and various phase unwrapping methods have been proposed to obtain the corresponding true phase. However, these methods are computationally intensive and require large memory and expensive hardware resources, thus hindering the development of real-time, embedded solutions for wide bandwidth measurement applications. In this study, a one-dimensional dual-counters-based phase unwrapping (1D DCPU) algorithm is proposed to achieve real-time phase unwrapping while utilizing minimal resources, which can be easily implemented on a field programmable gate array (FPGA) in real-time. The performance of this 1D DCPU algorithm is simultaneously demonstrated by the simulated and experimental results on the FPGA in the frequency range of 10−500 kHz. The proposed 1D DCPU algorithm offers the advantages of small computation and high-efficiency operation on FPGA, which is especially suitable for the large bandwidth, embedded and real-time sensing applications.

Key words: Doppler signal, laser heterodyne interferometer, phase unwrapping, velocity measurement

Wei Ke, Longbiao He, Feng Niu, Ping Yang, Rong Fan, Yin Cao, Xiujuan Feng, Lijing Li. Dual-counters-based real-time 1D phase unwrapping algorithm in laser measurement and sensing applications[J]. Journal of Systems Engineering and Electronics, 2026, 37(5): 1429-1437.

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References 31

1	Yang M, Cai C G, Wang D G, et al. Symmetric differential demodulation-based heterodyne laser interferometry used for wide frequency-band vibration calibration[J]. IEEE Trans. on Industrial Electronics, 2023: 8129.
2	Ke W, Li L J, He L B, et al Accelerating the phase demodulation process based on a parallel hilbert transform with overlapping computation for a laser heterodyne doppler vibrometer. Optics Letters, 2025, 50 (2): 507. doi: 10.1364/OL.545367
3	Wang M, Koukoulas T, Xing G Z, et al. Measurement of underwater acoustic pressures in the frequency range 100 to 500 kHz using optical interferometry and discussion on associated uncertainties[C]// The 25th International Congress on Sound and Vibration, 2018: 8.
4	Liao F J, Wu K H, Liu S M, et al High-dynamical-range angular measurement based on heterodyne phase shifting interferometry. Optics Communications, 2024, 555, 130209. doi: 10.1016/j.optcom.2023.130209
5	Wang H, Hu J, Fu H Q, et al A novel quality-guided two-dimensional InSAR phase unwrapping method via GAUNet. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 2021, 14, 7840. doi: 10.1109/JSTARS.2021.3099485
6	Osmanoglu B, Dixon T H, Wdowinski S, et al On the importance of path for phase unwrapping in synthetic aperture radar interferometry. Applied Optics, 2011, 50 (19): 3205. doi: 10.1364/AO.50.003205
7	Yu H W, Lan Y, Yuan Z H, et al Phase unwrapping in InSAR: a review. IEEE Geoscience and Remote Sensing Magazine, 2019, 7 (1): 40. doi: 10.1109/MGRS.2018.2873644
8	Yu J, Da F P Absolute phase unwrapping for objects with large depth range. IEEE Trans. on Instrumentation and Measurement, 2023, 72, 1.
9	Zebker H A, Lu Y Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms. Journal of the Optical Society of America A, 1998, 15 (3): 586. doi: 10.1364/JOSAA.15.000586
10	Ehtesham A, Zabit U, Bernal O D, et al Analysis and implementation of a direct phase unwrapping method for displacement measurement using self-mixing interferometry. IEEE Sensors Journal, 2017, 17 (22): 7425. doi: 10.1109/JSEN.2017.2758440
11	Kong L, Cui K, Shi J B, et al 1D phase unwrapping based on the quasi-Gramian matrix and deep learning for interferometric optical fiber sensing applications. Journal of Lightwave Technology, 2022, 40 (1): 252. doi: 10.1109/JLT.2021.3118394
12	Li Y, Liu Z B, Liu Y, et al Interferometric vibration sensor using phase-generated carrier method. Applied Optics, 2013, 52 (25): 6359. doi: 10.1364/AO.52.006359
13	Chen Z L, Xie Y Y, Wang Y, et al. Design of high-precision optical heterodyne interferometer phase demodulation system[C]// IEEE International Conference on High Voltage Engineering and Applications, 2022: 1.
14	Zhou L F, Yu H W, Lan Y Deep convolutional neural network-based robust phase gradient estimation for two-dimensional phase unwrapping using SAR interferograms. IEEE Trans. on Geoscience and Remote Sensing, 2020, 58 (7): 4653. doi: 10.1109/TGRS.2020.2965918
15	Yang F S, Pham T A, Brandenberg N, et al Robust phase unwrapping via deep image prior for quantitative phase imaging. IEEE Trans. on Image Processing, 2021, 30, 7025. doi: 10.1109/TIP.2021.3099956
16	Huang W W, Mei X S, Wang Y G, et al Two-dimensional phase unwrapping by a high-resolution deep learning network. Measurement, 2022, 200, 111566. doi: 10.1016/j.measurement.2022.111566
17	Zhou L F, Yu H W, Lan Y, et al Deep learning-based branch-cut method for InSAR two-dimensional phase unwrapping. IEEE Trans. on Geoscience and Remote Sensing, 2022, 60, 5209615.
18	Wu Z P, Wang T, Wang Y J, et al Deep-learning-based phase discontinuity prediction for 2-D phase unwrapping of SAR interferograms. IEEE Trans. on Geoscience and Remote Sensing, 2022, 60, 5216516.
19	García L L, Arellano A G, Cruz-Santos W A parallel path-following phase unwrapping algorithm based on a top-down breadth-first search approach. Optics and Lasers in Engineering, 2020, 124, 105827. doi: 10.1016/j.optlaseng.2019.105827
20	Ghiglia D C, Romero L A Minimum Lp-norm two-dimensional phase unwrapping. Journal of the Optical Society of America A, 1996, 13 (10): 1999. doi: 10.1364/JOSAA.13.001999
21	Sukumar P R, Waghmare R G, Singh R K, et al. Phase unwrapping with Kalman filter based denoising in digital holographic interferometry[C]// 2015 International Conference on Advances in Computing, Communications and Informatics (ICACCI), 2015: 2256.
22	TRIBOLET J M A new phase unwrapping algorithm. IEEE Trans. on Acoustics, Speech, and Signal Processing, 1977, 25 (2): 170.
23	Itoh K Analysis of the phase unwrapping algorithm. Applied Optics, 1982, 21 (14): 2470. doi: 10.1364/AO.21.002470
24	Spoorthi G E, Gorthi R K S S, Gorthi S PhaseNet 2. 0: phase unwrapping of noisy data based on deep learning approach[J]. IEEE Trans. on Image Processing, 2020, 29, 4862.
25	Bauer M, Ritter F, Siegmund G. High-precision laser vibrometers based on digital Doppler signal processing[C]// 5th International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, 2002, 4827: 50.
26	Zhou L F, Yu H W, Lan Y, et al Artificial intelligence in interferometric synthetic aperture radar phase unwrapping: a review. IEEE Geoscience and Remote Sensing Magazine, 2021, 9 (2): 10. doi: 10.1109/MGRS.2021.3065811
27	Domínguez-Guzmán G, Castillo-Mixcóatl J, Beltrán-Pérez G, et al. Itoh algorithm to unwrap 2D phase[C]// 7th Symposium Optics in Industry, 2009: 105.
28	Yang P, Xing G Z, He L B Calibration of high-frequency hydrophone up to 40 MHz by heterodyne interferometer. Ultrasonics, 2014, 54 (1): 402. doi: 10.1016/j.ultras.2013.07.013
29	Castellini P, Paone N, Pietroni P. Supporting theory[M]// Tomasini E P, Castellini P. Laser Doppler vibrometry: a multimedia guide to its features and usage. Berlin: Springer, 2020: 1.
30	Winter M, Füser H, Bieler M, et al. The problem of calibrating laser-Doppler vibrometers at high frequencies[C]// AIP Conference Proceedings, 2012, 1457(1): 165.
31	Buehn U, Nicklich H. Calibration of laser vibrometer standards according to the standardization project ISO 16063-41[C]// 7th International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, 2006: 6345: 466.

Resource consumption and latency	Number of slice	Number of LUTs as Logic	Number of LUTs as Memory	Number of used DSP48E1	Latency (clock cycles)
24-bit	87	146	13	1	10
32-bit	132	178	17	2	11

[1]	Xianming XIE, Qingning ZENG. Multi-baseline extended particle filtering phase unwrapping algorithm based on amended matrix pencil model and quantized path-following strategy [J]. Journal of Systems Engineering and Electronics, 2019, 30(1): 78-84.
[2]	Xianming Xie. Enhanced multi-baseline unscented Kalman filtering phase unwrapping algorithm [J]. Journal of Systems Engineering and Electronics, 2016, 27(2): 343-351.
[3]	Xianming Xie. Multi-baseline phase unwrapping algorithm for INSAR [J]. Journal of Systems Engineering and Electronics, 2013, 24(3): 417-.
[4]	Xianming Xie and Yiming Pi. Phase noise filtering and phase unwrapping method based on unscented Kalman filter [J]. Journal of Systems Engineering and Electronics, 2011, 22(3): 365-372.
[5]	Junqi Duan1,* and Zishu He2. Doppler radar by using single multicarrier pulse based on optical fibre delay lines [J]. Journal of Systems Engineering and Electronics, 2010, 21(3): 404-407.
[6]	Wen Shuliang & Yuan Qi. Velocity measurement technique for high-speed targets based on digital fine spectral line tracking [J]. Journal of Systems Engineering and Electronics, 2006, 17(1): 6-12.