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图 1中右侧包含连续激光源;O1, O2, O3为3dB光耦合器;3根分别长为L1, L2, L3的铠装单模光纤;光电探测器D1, D2;环形器C1, C2。左侧是对光电探测器接收信号的处理流程。在没有振动时,经O1耦合的两路相干光同时到达D1, D2。而在长为L2的振动臂上某一点有入侵行为,将存在时间差Δt,且入侵位置P可表示为[4-9]:
$ P = \frac{{{L_1} + {L_3} - \Delta tc/{n_0}}}{2} $
(1) 设采样率为f, 单个采样点表示的定位距离精度r由相邻两个采样点之间的时长决定,r可以通过下式求得[8, 10]:
$ r = \frac{c}{{2f{n_0}}} $
(2) 式中, c为光速, n0为光纤折射率,约为1.46。
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RLS自适应滤波器是指能够根据环境的改变,采用累计误差平方和为代价函数的准则,自适应反馈调节滤波器系数使得代价函数最小[11]。图 2所示为M阶RLS横式滤波器结构。给定一个输入采样序列x(n), x(n-1)…x(n-M+1),并用其冲激响应w0, w1, …, wM-1来表征该滤波器。在某一离散时刻n,滤波器的输出为y(n),先验估计误差eerr用期望响应d(n)与y(n)之差表示,通过迭代反馈使得下一刻的输出值y(n+1)不断地逼近期望信号d(n+1),误差将迫近为0。图中,Z-1表示信号延迟1拍。
假定n时刻输入M×1维矢量X(n)=[x(n), x(n-1), …, x(n-M+1)]T,滤波器系数M×1维矢量W(n)=[w0, w1, …, wM-1]T,T表示矢量转置;M×M维迭代矩阵C(n)。振动信号为平稳信号,取遗忘因子λ=1;正则化参量δ为小正实数,与输入信号x(n)的信噪比有关[12-13]。对RLS采用前加窗法,算法流程见下[11-13]。
(1) 初始n=0时刻值:
$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{X}}\left( 0 \right) = \mathit{\boldsymbol{W}}\left( 0 \right) = {{\left[ {0,0, \cdots ,0} \right]}^{\rm{T}}}}\\ {\mathit{\boldsymbol{C}}\left( 0 \right) = {\delta ^{ - 1}}\mathit{\boldsymbol{I}}} \end{array}} \right. $
(3) 式中,I是M×M维单位矩阵;迭代过程对n=1, 2…,更新增益矢量g(n):
$ \mathit{\boldsymbol{g}}\left( n \right) = \frac{{\mathit{\boldsymbol{C}}\left( {n - 1} \right)\mathit{\boldsymbol{X}}\left( n \right)}}{{\lambda + {{\left[ {\mathit{\boldsymbol{X}}\left( n \right)} \right]}^{\rm{T}}}\mathit{\boldsymbol{C}}\left( {n - 1} \right)\mathit{\boldsymbol{X}}\left( n \right)}} $
(4) (2) 更新滤波器系数矢量W(n):
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{W}}\left( n \right) = \mathit{\boldsymbol{W}}\left( {n - 1} \right) + }\\ {\mathit{\boldsymbol{g}}\left( n \right)\left[ {d\left( n \right) - {{\left[ {\mathit{\boldsymbol{X}}\left( n \right)} \right]}^{\rm{T}}}\mathit{\boldsymbol{W}}\left( {n - 1} \right)} \right]} \end{array} $
(5) (3) 更新迭代矩阵C(n):
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{C}}\left( n \right) = {\lambda ^{ - 1}}\left[ {\mathit{\boldsymbol{C}}\left( {n - 1} \right) - } \right.}\\ {\left. {\mathit{\boldsymbol{g}}\left( n \right){{\left[ {\mathit{\boldsymbol{X}}\left( n \right)} \right]}^{\rm{T}}}\mathit{\boldsymbol{C}}\left( {n - 1} \right)} \right]} \end{array} $
(6) 由于滤波器的输入信号x(n)会遍历滤波器的0~M-1级延迟位置。由叠加定理知,相对于x(n)的延迟信号x(n-τ)可由滤波器各级冲激响应w0, w1, …, wM-1, 表示为:
$ \begin{array}{*{20}{c}} {x\left( {n - \tau } \right) = {w_0}x\left( n \right) + \cdots + }\\ {{w_{M - 1}}x\left( {n - M + 1} \right)} \end{array} $
(7) 式中,τ为延迟点数。又根据卷积定理:
$ x\left( {n - \tau } \right) = x\left( n \right) * \delta \left( {n - \tau } \right) $
(8) 式中, δ(n)为采样冲激响应函数。取:
$ {w_i} = \delta \left( {i - \tau } \right) = \left\{ {\begin{array}{*{20}{l}} {1,\left( {i = \tau ,0 \le i,\tau < M} \right)}\\ {0,\left( {{\rm{other}}} \right)} \end{array}} \right. $
(9) 式中,0≤i < M, i表示各级滤波器延迟的相应点数。在实际的采样中不存在理想的δ(n),而是以采样函数sinc近似代替。因此,延迟信号x(n-τ)可以由x(n)与sinc函数近似表示[14-15]:
$ x\left( {n - \tau } \right) \approx \sum\limits_{i = 0}^{M - 1} {x\left( {n - i} \right){\rm{sinc}}\left( {i - \tau } \right)} $
(10) 即有限长横向滤波器的系数wi与采样函数sinc(i-τ)重合时,滤波器输出近似为延迟信号x(n-τ), 且系数最大值wτ的位置为延迟点数τ。
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寻找系数峰值的位置i=τ直接判别在噪声存在的情况下易产生偏差[16]。假定不在峰值点对信号采样,在i=τ-φ位置早采样,在i=τ+φ位置迟采样,早、迟采样的绝对值都比峰值采样值的绝对值小, φ表示偏离系数峰值位置的点数。再以迟采样时刻值减去早采样时刻值,得到鉴相曲线。由于收敛的系数相对于最佳采样时刻i=τ是偶函数(sinc(i-τ)),早、迟采样值的绝对值应该相等。因此,采样峰值的位置就是在鉴相曲线为零的位置。
基于递推最小二乘算法的光纤振动定位系统
Fiber optic vibration positioning system based on recursive least-squares algorithm
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摘要: 为了解决光纤振动定位算法中互相关稳定性差、最小均方收敛速度慢的问题,采用递推最小二乘算法、滑动平均原理、早迟门原理,设计了一套稳定且响应迅速的光纤振动定位系统。光路部分基于双马赫-曾德尔干涉结构,硬件平台以高速采集板结合现场可编程门阵列实现定位。在一段160m的光路上进行重复敲击实验,对取得的振动数据分多段分别计算定位结果,去除最大值、最小值后剩余定位结果取平均值以提高定位精度。结果表明,相对于互相关,递推最小二乘随着迭代次数增加,单次定位结果稳定不会发生偏移,且其定位收敛速度是最小均方的3倍左右;在采样率为10MHz时,系统实际响应时间约0.3s,定位误差范围在±6m,且定位稳定可靠。该研究对于光纤定位系统中定位算法的改进、定位精度的提高是有积极意义的。Abstract: In order to solve the problems of poor cross-correlation stability and slow minimum mean square convergence in optical fiber vibration location algorithm, recursive least squares algorithm, moving average principle and early-late gate principle were adopted. A stable and fast response optical fiber vibration positioning system was designed. Optical path was based on double Mach-Zehnder interference structure. Hardware platform used high-speed acquisition board and field-programmable gate array to realize positioning. Repeated knocking experiments were carried out on 160m optical path. The obtained vibration data were computed separately in several segments. After removing the maximum and minimum values, the residual positioning results were averaged to improve the positioning accuracy. The results show that, as the number of iterations increases, recursive least squares method can stabilize the single positioning result without migration, compared with cross-correlation. Convergence rate is about 3 times of minimum mean square. At sampling rate of 10MHz, actual response time of the system is about 0.3s. The range of positioning error is ±6m. The positioning is stable and reliable. This research has positive significance for the improvement of location algorithm and positioning accuracy in optical fiber positioning system.
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[1] ZHANG H, LV H W, FENG J L, et al. Oil pipeline security system based on fiber-optic vibration sensor and FPGA[J]. Journal of Changchun University of Science and Technology (Natural Science Edition), 2016, 39(3): 116-119(in Chinese). [2] XU G, HE Ch Ch, ZHANG L N, et al. Research of position techno-logy of Mach-Zehnder interfermeter[J]. Laser Technology, 2019, 43(2): 195-200(in Chinese). [3] CUI G L, YI W S, ZHANG Y H, et al. Design of disturbance monitoring system based on dual Mach-Zehnder distributed optical fiber sensing[J]. Journal of Jilin University(Information Science Edition), 2017, 35(6): 590-596(in Chinese). [4] JIANG J Sh, JIANG Y, LIU D, et al. Fiber-optic perimeter security system based on dual Mach-Zehnder interferometer structure[J]. Optical Technique, 2015, 41(3): 193-196(in Chinese). doi: 10.3788/GXJS20154103.0193 [5] XUE Y B, LI H. Cross-correlation arithmetic algorithm in the application of micro sensors[J]. Electronic Design Engineering, 2015, 23(1): 93-95(in Chinese). [6] LIN W T, LIANG Sh, LOU Sh Q, et al. A novel fiber-optic distributed disturbance sensor system with low false alarm rate[J]. Infrared and Laser Engineering, 2015, 44(6): 1845-1848(in Chinese). [7] LIU R H, LUO Y J. Optical fiber disturbance location system based on LMS and linear interpolation[J]. Optical Communication Techno-logy, 2019, 43(4): 58-62 (in Chinese). [8] LUO Y J, FANG L. Design and implementation of fiber-optic vibration position system based on FPGA[J]. Embedded Technology, 2018, 44(10): 60-63 (in Chinese). [9] YANG Sh Zh, ZHANG Zh Y, SHAO L Y, et al. A quadratic correlation method for vibration location based on dual M-Z fiber sensor[J]. Acta Photonica Sinica, 2017, 46(7): 0706004(in Chinese). doi: 10.3788/gzxb20174607.0706004 [10] HU G Q. Studying distributed optical fiber perimeter and its location technology based on dual Mach-Zehnder[D]. Beijing: Beijing Jiaotong University, 2017: 17-25(in Chinese). [11] GONG Y H. Adaptive filtering and time domain adaptive filtering and smart antenna[M]. Beijing: Electronics Industry Press, 2003: 89-107(in Chinese). [12] SIMON H. Adaptive filter theory[M].5rd ed. Beijing: Electronics Industry Press, 2016:326-333(in Chinese). [13] DU Y. Digital filter implement based on MATLAB and FPGA[M]. Beijing: Electronics Industry Press, 2016: 220-252(in Chinese). [14] XIAN Y L. Digital filter design and high-speed data transmission based on FPGA[D]. Xi'an: Xidian University of Electronic Technology, 2009: 13-24(in Chinese). [15] QIAO Zh Y. Adaptive time delay estimation based on minimum mean square error[D]. Xi'an: Xidian University of Electronic Technology, 2014: 17-24(in Chinese). [16] WANG J, AN J P. An improving tracking speed method in the early-late gate synchronizer[J]. Journal of Circuits and System, 2005, 10(6): 111-114(in Chinese). [17] XU Y B. Design and inplementation of long-distance multi-node optical fiber transmission system based on FPGA[D].Hefei: Anhui University, 2017: 13-19(in Chinese). [18] WU G K. Research on FPGA high speed implementation of digital signal processing algorithm[D]. Chengdu: University of Electronic Science and Technology of China, 2011: 71-82(in Chinese). [19] UWE M B. Digital signal processing with field programmable gate arrays[M].4rd ed. Beijing: Tsinghua University Press, 2017: 513-523(in Chinese). [20] ZHOU Y, DING F.Comparison of least squares identification for moving average models[J]. Science Technology and Engineering, 2007, 7(18): 4570-4575(in Chinese).