-
二维平面中,两个矢量利用三角变换可以求得两者之间夹角的余弦值,如图1所示。以此类推,可将二维空间的两个矢量推广到n维空间,同理可得n维空间两个矢量的夹角。
基于矢量内积法进行正弦信号相位计算时,待测信号需满足实数序列[10]。将相位测距中出射、测量信号作为n维矢量,矢量维度取决于采集的数据量,将两路采集后的离散信号分别表示为矢量X、Y。两路信号数量积的值由矢量X、Y的长度及X、Y的夹角的余弦确定,可表示为:
$$ X \cdot Y = \left\| X \right\| \cdot \left\| Y \right\|\cos \varphi $$ (1) n维空间的内积可表示为:
$$ X \cdot Y = \left( {{x_1},{x_2}, \cdots {x_n}} \right) \cdot \left( {{y_1},{y_2}, \cdots {y_n}} \right) = \sum\limits_{i = 1}^n {{x_i}} {y_i} $$ (2) 矢量X、Y的长度可表示为:
$$ \left\| X \right\| = \sqrt {\sum\limits_{i = 1}^n {{x_i}^2} } \text{,}\left\| Y \right\| = \sqrt {\sum\limits_{i = 1}^n {{y_i}^2} } $$ (3) 则n维矢量X、Y的夹角即待求相位差可表示为:
$$ \varphi = \arccos \frac{{X \cdot Y}}{{\left\| X \right\| \cdot \left\| Y \right\|}} $$ (4) -
VIP法求解相位时,出射和反射两路信号均为实数序列,根据反余弦函数
$\arccos x$ 的取值范围推导出一路信号的鉴相范围为[0, π/2],因此VIP法的鉴相范围为[0, π],相较于VIP法,DFT法求解相位差时,信号幅度最大谱线对应的位置对应的实部虚部有正负之分,根据反正切函数$\arctan x$ 的取值范围可以推导出一路信号的相位范围为(−π/2,π/2),因此DFT法的鉴相范围为(−π,π)。当所求距离超过一个测尺长度时,VIP法和DFT法求解的相位值为真实相位差的“小数部分”,因此所求真实相位差值可表示为:
$$ \Delta {\varphi _{{\rm{VIP}}}} = N \cdot {\text{π}} + {\varphi _{{\rm{VIP}}}} $$ (5) $$ \Delta {\varphi _{{\rm{DFT}}}} = N \cdot 2{\text{π}} + {\varphi _{{\rm{DFT}}}} $$ (6) 其中,N的取值由测距系统的粗测确定。
-
在实际测量系统中,鉴相方法的计算速度影响系统的测量速度。连续信号经过采样后为有限长离散信号,对于带有相位信息的N点的有限测量数据,利用VIP法和DFT法进行相位求解的速度主要取决于两类算法的计算复杂度。上述鉴相方法硬件一般采用FPGA实现。针对N点的有限序列的DFT算法可以表示为:
$$ X(k) = \sum\limits_{n = 0}^{N - 1} {x(n)} W_N^{kn},k = 0,\;1, \cdots \;,N - 1 $$ (7) N点DFT的计算量为N2次复数乘法以及N×(N−1)次复数加法。基于旋转因子的周期性和对称性,将N点DFT分解为较短的DFT进行计算,可以大幅减少计算量,N点的DFT法的计算量可降低为:复数乘法共
$ N/2 \times {\text{log}_2}N $ 次,复数加法共$ N \times {\text{log}_2}N $ 次[16]。相较于DFT法,由公式(4)可得,基于FPGA实现VIP法所需的计算量为3N次乘法、3(N−1)次加法以及2次开方实数运算,则在同等数据量运算下,VIP法均为实数运算且运算量更小,所占用内存资源更小。
为直观地进行比较,以N=1024为例,计算两种方法的计算量。DFT法的计算量为:复数乘法共
$ 1024/2 \times {\text{log}_2}1\;024 = 5\;120 $ 次,复数加法共$ 1\;024 \times {\text{log}_2} $ $ 1\;024 = 10\;240 $ 次。VIP法的计算量为:实数乘法共3×1024=3072次,实数加法共3×(1024−1)=3069次,以及2次实数开方运算。从上述示例中可以看出,在同等数据量时,VIP算法的计算量约为DFT算法的1/3,在实际测量中能够实现更快的鉴相计算。
-
摘要: 测量速度和测量精度是相位式激光测距系统的两个重要指标。针对高速高精度测距需求,研究了基于矢量内积(Vector Inner Product, VIP)法的高速数字鉴相方法,从鉴相计算的点数、鉴相计算速度和鉴相精度等方面对VIP法的鉴相性能进行了仿真分析和实验测试,并与传统的数字频域鉴相法(Discrete Fourier Transform, DFT)进行了性能对比。仿真和实验结果表明,VIP法具有更高的鉴相速度和鉴相精度,当信号调制频率为50 MHz时,基于高速采样板卡实测得到的鉴相精度优于0.1°,测距精度0.2 mm以内,相同计算点数VIP法的鉴相处理速度是DFT法的3倍。研究结果表明,VIP法具有鉴相精度高、鉴相速度快的优点,适用于高速高精度激光测距系统。Abstract: Measurement speed and measurement accuracy are two important indicators of the phase laser ranging system. Aiming to the needs of high-speed and high-precision ranging system, a high-speed digital phase detection method based on the Vector Inner Product (VIP) method was studied. The phase detection performance of the VIP method was simulated and tested in terms of the number of phase detection calculation points, phase detection calculation speed, and phase detection accuracy. The simulation and experimental results were compared with the traditional digital frequency domain phase detection method (Discrete Fourier Transform (DFT) method), which show that the VIP method has higher phase detection speed and phase detection accuracy. When the signal modulation frequency is set as 50 MHz, based on high-speed sampling board, the measured phase detection accuracy is better than 0.1°, the ranging accuracy is within 0.2 mm, the phase detection processing speed of the VIP method is 3 times faster than that of the DFT method under the circumstance with the same number of calculation points. Referring to the high phase detection accuracy and fast phase detection speed, all the analysis data show that the VIP method has the significant advantages, which is suitable for the high-speed and high-precision laser ranging systems.
-
-
[1] Li Guicun, Fang Yahao, Zhang Hao, et al. Correction of power-to-phase conversion for distance measurement using synthetic wavelength method of a femtosecond laser [J]. Chinese Journal of Lasers, 2021, 48(1): 0104002. (in Chinese) [2] Tan Xiaorui, Zhang Pizhuang, Fan Yuanyuan. Design of new type phase laser ranging system circuit [J]. Laser & Infrared, 2016, 46(3): 279-283. (in Chinese) doi: 10.3969/j.issn.1001-5078.2016.03.007 [3] Xu Xianze, Weng Mingjie, Xu Fengqiu, et al. Phase laser ranger based on quadrature modem and frequency reduction [J]. Optics and Precision Engineering, 2017, 25(8): 1979-1986. (in Chinese) doi: 10.3788/OPE.20172508.1979 [4] Zhang Shilei, Cui Yu, Xing Muzeng, et al. Light field imaging target ranging technology [J]. Chinese Journal of Optics, 2020, 13(6): 1332-1342. (in Chinese) doi: 10.37188/CO.2020-0043 [5] Hu Qi, Wang Zhe, Liu Hongshun, et al. Step diffraction algorithm based on single fast Fourier transform algorithm [J]. Chinese Optics, 2018, 11(4): 568-575. (in Chinese) doi: 10.3788/co.20181104.0568 [6] Xu Yongyao, Zhang Tieli, Gao Xiaoqiang, et al. Review of digital processing techniques for phase signal in absolute distance measurement [J]. Journal of Astronautic Metrology and Measurement, 2020, 40(6): 1-6. (in Chinese) [7] 王心遥, 张珂殊. 基于欠采样的激光测距数字鉴相方法[J]. 红外与激光工程, 2013, (5): 1330-1337. Wang Xinyao, Zhang Keshu. Digital phase-shift measuring methods based on sub-sampling in laser range finder[J]. Infrared and Laser Engineering, 2013, 42(5): 1330-1337. (in Chinese) [8] 杨佳敏. 基于连续波相移的实时测量及触发技术研究[D]. 山西: 中北大学, 2019. Yang Jiamin. Research on real-time measurement and trigger technology based on continuous wave phase shift [D]. Taiyuan: North University of China, 2019. (in Chinese) [9] Jia Fangxiu, Ding Zhenliang, Yuan Feng. New phase difference measurement based on digital synchronous demodulation [J]. Instrument Technique and Sensor, 2009(4): 78-80. (in Chinese) doi: 10.3969/j.issn.1002-1841.2009.04.028 [10] Zhou Xiang, Lv Youxin. Research on estimation and compensation of I/Q imbalance in orthogonal channels [J]. Radar Science and Technology, 2017, 15(1): 8-12. (in Chinese) doi: 10.3969/j.issn.1672-2337.2017.01.002 [11] Wang Xuangang, Gou Ningyi, Zhang Keshu. Unbiased improvement of spectrum analysis phase discrimination for phase-shift laser range finder [J]. Optics and Precision Engineering, 2012, 20(4): 888-895. (in Chinese) doi: 10.3788/OPE.20122004.0888 [12] Guo Tianmao, Liu Ke, Miao Yinxiao, et al. Application of all phase FFT frequency measurement in FMCW laser ranging technology [J]. Journal of Astronautic Metrology and Measurement, 2019, 39(2): 45-50. (in Chinese) doi: 10.12060/j.issn.1000-7202.2019.02.10 [13] Huang Zhengying, Li Ji, Chen Jiexiang, et al. The phase difference detector by inner product applied in laser phase distance measurement system [J]. Optoelectronic Technology & Information, 2002, 15(6): 31-34. (in Chinese) [14] 田桂平, 万钧力, 陈滟涛. 基于矢量内积法的高精度数字相位差计[J]. 电测与仪表, 2004, 41(10): 16-18. Tian G P, Wan J L, Chen Y T. High precision digital phase detector based on inner product theory [J]. Electrical Measurement & Instrumentation, 2004, 41(10): 16-18. (in Chinese) [15] Zhu Jueying, Yang Jue, Fang Xiang, et al. Phase difference measurement based on vector inner product method and its application [J]. Measurement Technique, 2016(12): 35-37. (in Chinese) [16] Wang Linquan, Pi Yiming, Chen Xiaoning, et al. Realization of high-speed FFT by FPGA [J]. Journal of University of Electronic Science and Technology of China, 2005, 34(2): 152-155. (in Chinese) doi: 10.3969/j.issn.1001-0548.2005.02.003